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,o. 128 
 
 °f- -3 UNIVERSITY OF ILLINOIS 
 
 GRADUATE COLLEGE 
 DIGITAL COMPUTER LABORATORY 
 
 REPORT NO. 128 
 
 A STUDY OF ARITHMETIC RECODING WITH APPLICATIONS 
 IN MULTIPLICATION AND DIVISION 
 
 J . . Penhollov 
 
 September 10, 1962 
 
 (This is being submitted in partial fulfillment of the 
 requirements for the Degree of Doctor of Philosophy 
 in Electrical Engineering, September 1962.) 
 
UNIVERSITY OF ILLINOIS 
 GRADUATE COLLEGE 
 DIGITAL COMPUTER LABORATORY 
 
 REPORT NO. 128 
 
 A STUDY OF ARITHMETIC RECODING WITH APPLICATIONS 
 IN MULTIPLICATION AND DIVISION 
 
 by 
 J . . Penhollow 
 
 September 10, 1962 
 
 (This is being submitted in partial fulfillment of the 
 requirements for the Degree of Doctor of Philosophy 
 in Electrical Engineering, September 1962.) 
 
_J , l_, . l_> / 
 
 ACKNOWLEDGMENT 
 
 The guidance, encouragement and many helpful suggestions of my 
 advisor, Professor James E. Robertson, are gratefully acknowledged. 
 Professor Gemot Metze also contributed valuable advice and criticism. 
 
 Thanks are extended to Mrs. Phyllis Olson and Mrs. Anna Rita 
 Ferris who typed the thesis in its final form. 
 
TABLE OF CONTENTS 
 
 Page 
 
 1. INTRODUCTION ............... . . 1 
 
 2. ARITHMETIC RECODING OF POSITIVE INTEGRAL RADIX REPRESENTATIONS . . 3 
 
 2.1 Basic Concepts of Arithmetic Recoding ............ 3 
 
 2.2 A Special Class of Arithmetic Recodings . . . 4 
 
 2.3 The One-to-Many Mapping 6 
 
 2.3-1 The Digital Mapping Function ............. 6 
 
 2.3.2 The Set of Mapping Functions 10 
 
 2.3.3 The Number of Possible Mappings 12 
 
 2.4 Mode Functions o ............ 17 
 
 2.4.1 Left and Right Directed Mode Functions ........ 19 
 
 2.4.2 General Mode Functions ................ 23 
 
 2.5 Recoding Functions. ..................... 24 
 
 3. LEFT AND RIGHT DIRECTED RECODING IN THE BINARY CASE 27 
 
 3.1 Left and Right Directed Mode Functions in Boolean Form. ... 27 
 
 3.2 General Mode Functions in Boolean Form . 29 
 
 3-3 Equivalent Left and Right Directed Recoding Functions .... 33 
 
 4. MINIMAL RECODING OF BINARY REPRESENTATIONS 37 
 
 4.1 Analysis of the Minimal Representation. 39 
 
 4.1.1 Decomposition of the Binary Representation ...... 40 
 
 4.1.2 The Gain Function 42 
 
 4.1.2.1 Analysis of the Zero Sequence 44 
 
 4.1.2.2 Analysis of the Unit Sequence 46 
 
 4,1.2 = 3 Analysis of the Alternating Sequence 48 
 
 4.1.3 A Summary of Maximum Gains . 53 
 
 4.1.4 Choice of the Next Mode Digit. 56 
 
 4.1.4.1 An Example. ........ 56 
 
 4.1.4.2 Special Restrictions on the Set of 
 
 Independent Variables 58 
 
 4.1.4-3 Criteria for Choosing the Next Mode Digit . . 60 
 
 4.2 The Subclass of Restricted, Minimal Left Directed 
 
 Mode Functions. 69 
 
 4.2.1 Case Analysis of the Partial Left Directed Sequence. . 69 
 
 4.2.2 Derivation of the Restricted, Minimal Left Directed 
 
 Mode Functions 75 
 
 4.3 The Subclass of Restricted, Minimal Right Directed 
 
 Mode Functions. 87 
 
 4-3<l Case Analysis of the Partial Right Directed Sequence . 87 
 4.3*2 Derivation of the Restricted, Minimal Right Directed 
 
 Mode Functions . 94 
 
 4.4 The Classes of Minimal Mode Functions ............ 105 
 
 4.5 Summary Il6 
 
 -IV- 
 
TABLE OF CONTENTS (CONTINUED) 
 
 Page 
 
 5. APPLICATIONS OF MINIMAL RECODING TO MULTIPLICATION AND DIVISION 
 ALGORITHMS 118 
 
 5-1 Minimal Recoding Functions for Infinite Representations . . . 119 
 
 5.2 An Arithmetic Interpretation of Minimal Mode Functions. . . . 122 
 
 5.2.1 The Left Directed Case 123 
 
 5.2.2 The Right Directed Case 127 
 
 5.3 Minimal Multiplication Algorithms 130 
 
 5.^ Minimal Division Algorithms 135 
 
 5.5 Interrelation of Minimal Multiplication and Division 
 
 Algorithms IU9 
 
 6. CONCLUSION ...... 152 
 
 6.1 Summary 152 
 
 6.2 Results and Conclusions I5U 
 
 6.3 Areas for Further Study 155 
 
 BIBLIOGRAPHY I56 
 
 APPENDIX 158 
 
 -v- 
 
1 . INTRODUCTION 
 
 Arithmetic procedures based on redundant number representations have 
 played a major role in the evolution of high speed digital computers. Beginning 
 with the introduction of the stored carry concept by Burks, Goldstine and 
 von Neumann [3] , various redundant representations have been employed to reduce 
 the operating time for addition, multiplication and division. The use of stored 
 carries or signed-digit representations permits a significant speed improvement 
 in all three of these operations. Additional speed may be achieved in multi- 
 plication and division if the multiplier and quotient are recoded in a redundant 
 representation having a minimum or near minimum of nonzero digits. This tech- 
 nique reduces the number of times the adder must be used. If the time required 
 to add and shift is large compared with the time to simply shift, a considerable 
 reduction in multiplication and division time can be realized when an appropriate 
 recoding is employed. 
 
 The objective of this dissertation is the study of recoding functions 
 in general and minimal recoding functions in particular as applied to multipli- 
 cation and division. A recoding function is defined as a transformation which 
 maps every element of a set of unique, integral radix representations into a 
 unique element of a larger set of redundant, integral radix representations 
 with algebraic value preserved. A left directed recoding transforms the digits 
 of the unique representation from right to left. A right directed recoding is 
 applied in the opposite direction. If a left and right directed recoding both 
 produce identical redundant representations for all unique representations 
 they are equivalent. After reducing the unique representation to the binary 
 case, the relationship between equivalent left and right directed recodings is 
 
 * 
 
 Numbers in brackets refer to references listed in the bibliography. 
 
-2- 
 
 established* By adding the requirement that the redundant representation 
 must contain a minimum of nonzero digits a minimal recoding is obtained. 
 Multiplication algorithms which utilize minimally recoded multipliers are 
 defined by minimal left directed recodings while division algorithms which 
 generate minimally recoded quotients are defined by minimal right directed 
 recodings « An inverse relationship between minimal multiplication and 
 division algorithms is established via the equivalence of minimal left and 
 right directed recodings . 
 
2. ARITHMETIC RECODING OF POSITIVE INTEGRAL RADIX REPRESENTATIONS 
 
 This chapter provides a mathematical foundation for the study of a 
 certain class of arithmetic recodings as applied to positive integral radix 
 representations. To emphasize the character and scope of a general arithmetic 
 recoding, a broad definition is given initially and later narrowed to include 
 only the case of interest. The definitions and theorems presented in the 
 remainder of the chapter are based on the restricted definition. 
 
 2 .1 Basic Concepts of Arithmetic Recoding 
 
 As used in this dissertation, an arithmetic recoding is the trans- 
 formation of a given positive integral radix representation into another with 
 algebraic value preserved. A more precise definition is given below. 
 
 Definition 2-1 : General Arithmetic Recoding 
 
 Let C be a finite set of N distinct algebraic values. Let each 
 c 6 C have T (c) > 1 distinct positive integral representations of character- 
 istic a and T R (c) > 1 distinct positive integral representations of charac- 
 teristic p. Let T (c) and T (c) be finite for all c e C. 
 
 Let A be the set of distinct representations of characteristic a for 
 
 all c G C. Let N. = Z T (c) > N_ be the number of elements in A. Likewise, 
 A cr ' — C ' 
 
 let B be the set of distinct representations of characteristic f3 for all c e C, 
 
 Let N_. = Z T Q (c) > N^ be the number of elements in B. 
 B c 3 - C 
 
 Any transformation of A into B or B into A which preserves algebraic 
 value is an arithmetic recoding. 
 
Under this definition a conversion from one radix representation to 
 another, such as from decimal to binary or binary to ternary, is classified 
 as an arithmetic recoding. The conversion from radix complement to carry-borrow 
 representation [8] and conversely are also arithmetic recodings . Another 
 example is the transformation of a unique representation, such as radix comple- 
 ment, diminished radix complement or sign absolute value, into a signed-digit 
 representation [l] utilizing the same radix. The converse of this is also a 
 recoding „ 
 
 The conversion of a unique representation of a given integral radix 
 to a signed-digit representation of the same radix has been employed effectively 
 in the design of fast multiplication and division processes for digital com- 
 puters [12], [6]. In the work which follows, a well-defined class of such 
 arithmetic recodings is studied in detail. This preliminary study provides a 
 basis for the later investigation of arithmetic recodings employed in multipli- 
 cation and division algorithms and their interrelation. 
 
 2.2 A Special Class of Arithmetic Recodings 
 
 The special class of arithmetic recodings to be investigated is 
 characterized by the transformation of a finite radix complement representation 
 onto a finite redundant radix representation both having an integer radix r > 2, 
 The redundant representation chosen is a natural extension of the modified 
 signed-digit representation defined by Avizienis [l] and has the desirable 
 property that the representation of is unique. Each digit, y! , may assume 
 any integer value in the range -r + 1 < y! < r - 1. 
 
 The range of values, y, to be represented in radix complement form is 
 -1 < y < 1. Because of certain simplifications in the analysis, it is convenient 
 to modify the usual radix complement representation such that the second position 
 to the left of the radix point rather than the first has negative weight. 
 
 
-5- 
 
 Definition 2-2: Modified Radix Complement Representation 
 
 n 
 y = -y r + E y.r ; Integer r > 2 
 i=0 x 
 
 < y_! < !; y Q = y_ x ( r - x ); o < y t < r - 1 
 
 Definition 2-3: Extended Signed-Digit Representation 
 
 n 
 
 -l 
 
 y' = E y! r ; Integer r > 2 
 1=0 ± 
 
 1 - r < y! < r - 1 
 
 From this point on, the symbols y and y' will be used to designate 
 numbers in modified radix complement and extended signed-digit representations 
 respectively. 
 
 Definition 2-k : Specialized Arithmetic Recoding 
 
 Let Y be the set of all y as defined above and let Y" be the set of 
 all y' as defined above. Let Y' be the proper subset of Y" which contains all 
 y' that are algebraically equal to some y £ Y. 
 
 Let F be a transformation of Y onto Y' such that for every y £ Y 
 there exists a unique y 1 £ Y' with y' = y. The transformation, F, is an 
 arithmetic recoding of Y onto Y' . 
 
-6- 
 
 Throughout the remainder of the dissertation an arithmetic recoding 
 
 as defined here will be referred to as a recoding. 
 
 It is apparent from the definition of Y 1 that there are y for which 
 
 there is more than one y' satisfying the condition y' = y. For example, 
 
 n . n-1 
 
 y'=0+ E (r - l)r ~~ and y ' = 1 + E • r ~" - r are valid elements of Y' 
 
 1 i-1 2 i-1 n 
 
 and both are algebraically equal to y = + E (r - l)r . In precise termi- 
 
 i=l 
 nology the relation y' = y partitions Y' into 2r disjoint subsets Y' . 
 
 It follows that the specification of a recoding of Y onto Y' can be 
 achieved in two distinct steps. First, a one-to-many mapping between every 
 y G Y and its corresponding subset, Y' , is established. This mapping 
 guarantees that the recoding of any y e Y will always result in a y' £ Y' 
 which satisfies the relation y' = y. The second step provides constraints 
 which reduce the one-to-many mapping to a one-to-one mapping. Sections 2.3 and 
 2.4 contain the development of the one-to-many mapping and the necessary con- 
 straints to establish a recoding. 
 
 2.3 The One-to-Many Mapping 
 
 2.3-1 The Digital Mapping Function 
 
 The development of a one-to-many mapping which preserves algebraic 
 value is best approached by considering the restrictions imposed on the mapping 
 of a typical digit, y., into some y! . It is clear that the addition of a unit 
 in the i position must be compensated by the subtraction of r units in the 
 (i - l) position. Either or both of these operations on y. will in general 
 yield a y! that falls within the prescribed limits, 1 - r < y! < r - 1, of the 
 extended signed-digit representation. It is easily shown that the addition of 
 two or more units in the i position followed by a subtraction or r times that 
 
-7- 
 
 number of units in the (i - l) position does not in general yield a y! within 
 these limits. These intuitive observations lead to the following theorem con- 
 cerning the digital mapping function. 
 
 Theorem 2-1 : Given that y € Y and y' £ Y', then y' = y if and only if the digital 
 mapping function assumes the following form. 
 
 y i = y i + m i " rm i-l ( 2l±S) 
 
 The mode digits, m., are subject to the restrictions: m . = y ., , m =0 and 
 ' i -1 -1 n 
 
 m. = or 1 for < i < n - 1 such that the range restriction, 1 - r < y! < r - 1, 
 is always satisfied. 
 
 Proof : Part I 
 
 Since y' = y } it follows that 
 
 n 
 
 •J _ 
 
 y' - y = y -.r + Z (y' - y )r 
 j=0 J J 
 
 Let m.r represent the difference between y' and y at the i 
 
 position, 
 
 i . n 
 
 m.r"" = y r + Z (y! - y )r" J = Z (y - y! )r" J 
 j=0 J J j=i-l J J 
 
 . -. i . . n . . 
 
 m = y r 1+ + Z (y! - y )r 1_J = Z (y - y ' )r 1_J 
 j=0 J J j=i+l J J 
 
 It is clear by inspection of the most significant half of the above 
 equation that m. is always an integer for -1 < i < n. Furthermore, 
 
m = y r = r 
 
 ° _ ~^f Y f„ _ „<\~J 
 
 ( E (y, - y:)r" J ) = y 
 
 j-o J 
 
 m = (y r + E (y* - y )r" J )r n = 
 j = J J 
 
 We next examine the expression m. - rm. 
 
 l l-l 
 
 . , i . . . , i-1 . . 
 
 m - rm = y r + L (y! - y )r - y r - L (y - y )r 
 
 j=0 J J j=0 J J 
 
 I (y, - y:)r i_j - I (y. - y:)^^ 
 
 J=i+1 J J j=i J J 
 
 Therefore, y! - y. = m. - rm. , for < i < n. 
 ' i J i i i-1 - - 
 
 Thus far no restrictions have been placed on m. (0 < i < n - l) 
 except that it must be an integer. However, the limiting values of yj and y. 
 restrict the set of integer values that m. may assume. Let this set be 
 designated by u. From the limits on y! and y., it follows that 
 
 ^ y i^Min " ^ y i^Max - y i " y i - ' y i^Max " ^ y i^Min 
 
 2-2r<y! -y. < r - 1 
 
 2 - 2r < m. - rm. _ < r - 1 
 — l l-l — 
 
 This inequality must be satisfied by all m. and m. (0 < i < n) 
 chosen independently from the set u. The procedure for determining the elements 
 of u may be described as follows. Let m. assume a particular value of u say m 
 and allow m. = m to range over the set u. The above inequality places limits 
 on m relative to m as given by 
 
. m - 2 . 
 2 + > m > 1 
 
 M 
 
 In a like manner let m. . = m and allow m. = m. The limits on m relative to 
 
 l-l i 
 
 m in this case are 
 
 -1 + r(m + l) > m > 2 + r(m - 2) 
 
 The most restrictive set of limits must apply for all m, and m must always be 
 contained within these limits. 
 
 Assume m = 0. The corresponding limits are 
 
 2 1 
 
 2-->m>-l+- 
 r — — r 
 
 -l + r>m>2-2r 
 
 It follows that 1 > m > are the most restrictive limits on m for finite 
 r > 2. The value is included so we next choose m = 1. The corresponding 
 limits are 
 
 2 -->m>-l+- 
 r — — r 
 
 -1 + 2r > m > 2- r 
 
 Again 1 > m > are the most restrictive limits on m for finite r > 2. The 
 values and 1 are included in "both the first and second limits and no new values 
 of m have appeared. Therefore the set u contains only the two integer values 
 and 1. 
 
 We conclude that if y 1 = y, then the digital mapping function must 
 assume the form given in the hypothesis. 
 
-10- 
 Part II 
 
 Given the digital mapping function as defined in the hypothesis, we 
 
 show that y 1 = y. This part of the proof was originally demonstrated by 
 
 J. E. Robertson [11] . 
 
 n n n n 
 
 r = ^ y[r' x = l^ y ± r- J + Y m i r_1 - 1 ra i-i r_1+1 
 
 i=0 i=0 i=0 i=0 
 
 The upper limit in the center summation on the right may be replaced by n - 1 
 
 since m =0. In this same summation let i = ,i - 1. By collecting terms and 
 n n 
 
 simplifying we have y' = -m ,r + ) y.r . Since m = y , it follows that 
 
 ■i 
 
 y = y_ x r + / y\r 
 i=0 
 
 Corollary ; The limits on y' are -1 < y 1 < 1 for any r > 2. 
 
 Proof: By Eck 2:1 we have y' = y +m -rm n . Since y = y _ (r - l) and 
 J o J o o -1 o -l v ' 
 
 m=y ; y'=yr-y+m - ry = m - y . Taking the limits on m 
 and y -, into account, it follows that -1 < y l < 1. 
 
 2.3-2 The Set of Mapping Functions 
 
 The one-to-many mapping which takes each y e Y into its corresponding 
 Y' is the set of digital mapping functions for < i < n with the associated 
 mode digit restrictions. Let E represent this set of functions. Since E can 
 be easily expressed in matrix form it is convenient to define the sequence of 
 mode digits as a vector. 
 
 Definition 2-5 : Mode Vector 
 
 Let m= -^m ., , m , rm , ...,m ., , m f represent a mode vector. 
 (_ -1 o 1 n-1 nj 
 
-11- 
 
 Because of the range restriction, 1 - r < y. f < r - 1, on the digital 
 mapping function, m. can not always he chosen arbitrarily as or 1„ For 
 example, if y. = r - 1 and m. = 1, satisfaction of the range restriction requires 
 
 that m. n = 1. As another example, if y. =0 and m. _, =1, then it is necessary 
 
 l-l l l-l ' J 
 
 that m. = 1. It is apparent that the choice of m. is not always independent of 
 y., and likewise the choice of m is not always independent of y. 
 
 Definition 2-6 : Allowable Mode Vectors 
 
 A mode vector, m, is an allowable mode vector with respect to a 
 
 given y e Y if and only if m n = y n , m =0 and l-r<y. + m. -rm. ., < r - 1 
 
 -1 -1 n —li l-l — 
 
 for < i < n. Let V denote the set of all allowable mode vectors associated 
 
 - y 
 
 with a given y e Y. 
 
 Definition 2-7 ; The Set of Mapping Functions 
 
 Let y' £ Y', y 6 Y and m e V be vectors having n + 2 components 
 and let JO be a nonsingular square matrix of dimension n + 2. The set of 
 
 J 
 
 mapping functions, E(y,m), is then defined by the following matrix equation, 
 
 y' = E(y,m) = y + 0m 
 
 (2:2) 
 
 y 
 
 i 
 
 n-l 
 
 r 
 
 n 
 
 — ~" 
 
 
 - y -l 
 
 
 y o 
 
 
 y i 
 
 + 
 
 y n-l 
 
 
 y n 
 
 
 L_ —J 
 
 
 1 
 ■r 1 
 
 
 
 -r 1 
 
 J 
 
 m 
 
 m 
 
 i-1 
 
 m. 
 
 i 
 
 m 
 
 n-l 
 
-12- 
 
 Theorem 2-2 : For a given y £ Y, E(y,m) is a one-to-one mapping of V into Y' . 
 
 Proof : By Theorem 2-1 and the definition of V every y' £ Y' is the image of 
 at least one m £ V . 
 
 y 
 
 Assume that m* and m** are two distinct elements of V which map y 
 
 y 
 
 into the- same y' under E, 
 
 y' = y + 0m* = y + 0m** 
 
 5m* = 0m** 
 
 Since <p is a nonsingular square matrix of rank n + 2, it possesses 
 
 an inverse, 
 
 f 
 
 n+1 n 
 r r 
 
 Premultiplication of on both sides of the above equation yields 
 m* = m** which contradicts the assumption that they were different. 
 
 2.3-3 The Number of Possible Mappings 
 
 By Theorem 2-2 each m £ V corresponds to a unique y' £ Y' and 
 conversely. The number of possible mappings of y on Y' is therefore equal to 
 the number of elements in V or Y' . 
 
 y y 
 
-13- 
 
 Definition 2-8 : Special Subsets of Y 
 
 Let Y be a subset of Y containing those y which satisfy the con- 
 a 
 
 dition 1 < v. < r - 2 for 1 < i < n-i and 1 < y < r - 1. Let Y, be the 
 
 - J i - -- - J n- b 
 
 complement of Y with respect to Y. 
 
 cL 
 
 Theorem 2-3 : Let N(y) denote the number of elements in V „ N(y) =2 if and 
 only if y e Y . 
 
 Proof : Part I 
 
 Assume that N(y) = 2 . Since there are only n mode digits m. 
 (0<i<n-l) which may be specified for any m, the number of allowable mode 
 vectors for a given y can be expressed as 
 
 n-1 
 N(y) = TTN.(y) 
 i=0 
 
 where N. (y) is the number of different values m. may assume and still satisfy 
 
 1 - r < y. + m. - rm. , < r - 1 for < i < n. Since m. can only assume two 
 
 — i i l-l — — — i 
 
 different values at most, the maximum value of N(y) is 2 . 
 
 If N(y) = 2 f each m. must be allowed to assume a value of or 1 
 independent of the value of any other mode digit and still satisfy 
 
 l-r<y. + m. -rm. , <r-l for < i < n - 1. A typical m. is involved 
 
 — l i l-l — — — l 
 
 in two digital mapping functions. Therefore y. and y. must be restricted 
 
 such that the following inequalities are always satisifed for any of the eight 
 
 possible combinations of m. _, , m. and m. , . 
 
 l-l i l+l 
 
■Ik- 
 
 1 - r < y. + m. - rm. . <r-l 
 - 1 1 l-l - 
 
 1 - r < y. , + m. . -rm. <r-l 
 - J i+1 i+I i - 
 
 Substitution of mode digit values and simplification yields eight 
 inequalities . 
 
 (a) 1 - r < y. < r - 1 
 
 (b) -r < y. < r - 2 
 
 (c) 1 < y. < 2r - 1 
 
 (d) < y. < 2r - 2 
 
 (e) 1 - r < y < r - 1 
 
 (f ) -r < y. +1 < r - 2 
 
 (g) 1 < y. +1 < 2r - 1 
 
 (h) < y. _ < 2r - 2 
 - J i+1 - 
 
 It is immediately evident that inequalities (a), (d), (e) and (h) 
 are always satisfied for < i < n - 1. However, inequalities (b), (c), (f ) 
 and (g) can only be satisfied if 1 < y. <r-2 for l<i<n-2„ If i = 0, 
 satisfaction of 1 - r < y ' < r - 1 is guaranteed by the Corollary to 
 Theorem 2-1 so y is not restricted. To satisfy (f ) and (g) it is necessary 
 that 1 < y < r - 2, If i = n - 1, satisfaction of (b) and (c) requires 
 1 < y i < r - 2. Inequalities (f ) and (h) do not apply since m = in all 
 cases o Under this condition (e) and (g) are satisfied if 1 < y < r - 1. 
 
 It follows that if N(y) = 2 n , then 1 < y. < r - 2 for 1 < i < n - 1 
 
 and 1 < y < r - 1. Any y which satisfies these conditions is an element of 
 
 Y by Definition 2-8. 
 a 
 
 Part II 
 
 If y e Y , the values that m. may assume are restricted by the 
 
 a l 
 
 following pair of inequalities for 1 < i < n - 2. 
 
■15- 
 
 -r < m. - rm. , < 1 
 — 1 l-l - 
 
 ■r < m. -, - rm. < 1 
 
 - l+l i - 
 
 In these inequalities y. and y. , have "been allowed to assume either their 
 
 l l+l 
 
 maximum or minimum value in order to achieve the most restrictive limits on 
 
 m. - rm. , and m. , - rm. . The mode digits m. _, and m. . are now allowed to 
 l l-l l+l l l-l l+l 
 
 assume values of or 1 so as to produce the most restrictive limits on m. . 
 
 < m. < 1 
 -r < -rm. < 
 
 Both of these inequalities are satisfied if m. is arbitrarily 
 or 1 for 1 < i < n - 2. 
 
 If i = 0, then j q = y_ 1 (r - l), m = y , 1 < y < r - 2 and m 
 may be or 1. In all cases the following inequalities must be satisfied . 
 
 1 - r < m - y . < r - 1 
 
 — o -1 — 
 
 ■r < m n - rm < 1 
 — 1 o — 
 
 The second inequality provides the most restrictive limits by 
 choosing m appropriately. Since r > rm > 0, m may also be or 1 
 
 arbitrarily. 
 
 If i = n - 1, the most restrictive limits on m _, - rm _ and 
 
 n-1 n-2 
 
 m - rm are 
 
 n n-1 
 
 ■r<m n -rm _, < 1 
 — n-1 n-2 — 
 
 ■r < m - rm 
 
 - n n-1 < 
 
-16- 
 
 Since m = and m „ may be arbitrarily or 1, the most restrictive limits 
 
 n n-2 ' 
 
 on m , are 
 n-1 
 
 < m ., < 1 
 — n-1 - 
 
 ■r < rm , < 
 
 — n-1 — 
 
 Thus m ., may also be or 1 arbitrarily. 
 n-1 
 
 It follows that m. may be or 1 arbitrarily for < i < n - 1 and 
 
 hence the number of allowable mode vectors associated with any y £ Y is 
 
 a 
 
 N(y) = 2 n . 
 
 Corollary : For a given ye Y, 1 < N(y) < 2 if and only if y e Y „ 
 
 Proof : Part I 
 
 If 1 < N(y) < 2 , then y is not an element of Y . Therefore y e Y . 
 
 cl U 
 
 Part II 
 If y e Y , then y is not an element of Y . Therefore N(y) J 1 2 . 
 
 D cL 
 
 Since 2 is the maximum value N(y) may assume for any y e Y, N(y) < 2 for 
 
 y £ Y „ By the trivial mapping which leaves y unchanged except for y ! = -y , 
 
 it is clear that there is at least one allowable mode vector^ 
 
 m 
 
 ■\y -,, 0, ... Oh for every y e Y. Thus for y e Y , 1 < N(y) < 2 n , 
 
 Theorem 2-h: If y = or y = -1, then N(y) = 1, 
 
-17- 
 
 Proof : By Definition 2-8,, y = and y = -1 are both elements of Y . Thus., 
 
 N(y) < 2 . By the Corollary to Theorem 2-1, 1 < y' < 1 is satisfied for m 
 
 equal to or 1 for any y e Y and thus for y = or y = -1. 
 
 Assume m = 1. Since y = 0, it is necessary that m = 1 to satisfy 
 
 l-r<m-r<r-l. An inductive argument follows. Assume hl = 1. Since 
 
 y = 0, it is necessary that im = 1 in order to satisfy 1 - r < im - r 
 
 < r - 1. Therefore, m = 1 which violates the requirement that m =0. 
 — n n 
 
 It is thus necessary to choose m - and by the same argument all m. = for 
 
 C i < n. The only allowable mode vector for either y = or y = -1 is 
 
 .1 
 
 m =iy .,, 0, ... . 
 
 ■1 
 
 »;■ 
 
 No attempt is made to determine the form of N(y) when y e Y, . 
 
 n-1 
 From the preceeding theorems it is clear that (2r - 2)(r - 2) of the sub- 
 
 n 
 sets, Y' where y 6 Y , contain 2 elements. The remaining 
 y a 
 
 2r - (2r - 2)(r - 2) subsets, Y 1 where y € Y , contain at least one 
 element but always less than 2 elements. The subsets Y 1 = and Y' = -1 
 
 y y 
 
 contain only one element by Theorem 2-U. It can be shown that these are the 
 only y e Y for which there is only one allowable mode vector. 
 
 2.h Mode Functions 
 
 The problem of specifying a recoding of Y onto Y' has now been 
 reduced to the problem of specifying an allowable mode vector for every y e Y. 
 Since a wide choice of allowable mode vectors is available in almost all 
 cases, it is possible to control certain characteristics of the recoded repre- 
 sentation. For instance, the appropriate choice of an allowable mode vector 
 for each y e Y can guarantee the frequency of occurrence of certain digits or 
 
-18- 
 
 digit combinations. It is exactly this feature which is effectively utilized 
 in high speed multiplication and division processes . The fact that certain y 
 can only be mapped into one or a few y' is obviously not important provided 
 at least one of the possible representations in Y' embodies the desired 
 
 y 
 
 characteristics . 
 
 A unique m e V for every y € Y is specified by a mode function set, 
 M(y), which is defined as follows. 
 
 Definition 2-9 * Set of General Mode Functions 
 
 Let M(y) denote a set of general mode functions which for every 
 
 y £ Y specifies a unique m £ V . M(y) contains n general mode functions, M., 
 
 plus the two boundary conditions m _, = y n and m = 0„ Each M. (Y*) defines 
 
 -1 -1 n 11 
 
 m. such that the range restriction is satisfied for < i < n - 1 with 
 
 m n = y , and m = 0, 
 -1-1 n 
 
 The set of digits, Y*. includes y. or y. , or both and may include 
 
 l l l+l 
 
 an arbitrary number of digits to the left and right of y. or y. . , 
 
 Definition 2-10 : Equivalent Sets of Mode Functions 
 
 Two mode function sets M* and M** are equivalent if and only if 
 M*(y) = M**(y) for all y e Y. 
 
 It will be shown in section 2,4.2 that a set of general mode functions, 
 as defined here, can be derived from a set of left or right directed mode 
 functions. These special sets of mode functions are discussed in the next 
 section. 
 
-19- 
 
 2.^.1 Left and Right Directed Mode Functions 
 
 It is easier to incorporate the range restriction in the formation 
 
 of an allowable mode vector if the mode digits are generated in an ordered 
 
 manner from right to left or left to right,, Under these conditions each m. 
 
 is a function of the preceding mode digit m. . or m. . as well as Y*. If 
 
 * & to l+l l-l i 
 
 the mode digits are generated from right to left, the corresponding mode 
 
 functions are said to be left directed. Conversely, right directed mode 
 
 functions generate mode digits from left to right. 
 
 If y. . and m. , are specified, then by Theorem 2-1 there is an 
 l+l l+l 
 
 m. € u such that 1 - r < y! . < r - 1 is satisfied. Likewise if y. and m. , 
 l - J i+l - l i-l 
 
 are specified, then there is an m. £ u such that 1 - r < y.' < r - 1 is satis- 
 fied. It is apparent that in either case there are conditions under which the 
 range restriction can not be satisfied by an arbitrary choice of m. , For 
 
 example, if y. , = r - 1 and m. , = 1, then it is necessary that m. = 1 to 
 ' l+l l+l l 
 
 satisfy l-r<y'<r-l. As a second example, if y, =0 and m. = 1, 
 
 then m. =1 necessarily. The exact effects of the range restriction on the 
 
 choice of m. are stated in the next two theorems, 
 l 
 
 Theorem 2-5: If y. -, and m. . are given, the range restriction forces 
 l+l l+l • to 
 
 m. =m. . ify. , + m. n =0 (Mod r) but leaves m. arbitrary otherwise, 
 l l+l l+l l+l v i J 
 
 Proof: If y. _ + m. , =0 (Mod r), then y. , - -m. n (Mod r). Since 
 J i+1 l+l v n J i+1 i+1 v ' 
 
 m. = or 1, the corresponding values of y. are and r - 1. The range 
 restriction requires that 
 
 1 - r < -rm. < r - 1: m. ., = 
 - l — ? l+l 
 
 1 - r < r - rm. <r-l;m. , =1 
 - i — ? l+l 
 
-20- 
 
 The first inequality can only be satisfied by m. =0 and the second by m. = 1, 
 
 Thus in both cases m. = m. 
 
 i l+l 
 
 'if y. , I m. , 4- (Mod r), then y. , 4 -m. , (Mod r). Since 
 l+l l+l ' " i+l ' l+l v 
 
 m. . = or 1, y. , 4 and y. - 4 r - 1. If y. n is selected to produce the 
 i+l ' J i+1 ' J i+1 ' J i+1 
 
 most restrictive conditions on ra. for m. . - and m. _ = 1, the following 
 
 l i+l i+l ' ° 
 
 inequalities are obtained „ 
 
 •r < -rm. < 1; m. .. = 
 — i — 7 i+l 
 
 -r - 1 < -rm. < 0; m. , = 1 
 - l — ' i+l 
 
 Both inequalities can be satisfied with m. = or 1, 
 
 i 
 
 Theorem 2-6: If y. and m. _, are given, the range restriction forces m. = m. , 
 
 l l-l ' l l-l 
 
 if m. . - y. =1 (Mod r) but leaves m. arbitrary otherwise . 
 l-l l l 
 
 Proof: If m. _, - y. =1 (Mod r), then m. n = y.+l (Mod r). Since m. n = 
 
 l-l J ± v n i-1 J i v ' l-l 
 
 or 1 the corresponding values of y. are r - 1 and 0. The range restriction 
 requires that 
 
 l-r<r-l + m. < r - 1; m. , =0 
 — i — l-l 
 
 1 - r < m. -r<r-l= m. _ = 1 
 
 — i — * i-I 
 
 These inequalities can only be satisfied by making m. = in the first and 
 
 m. = 1 in the second. In both cases m. = m. 
 i l i-I 
 
 If m. , - y. / 1 (Mod r), then m. . 4 y.+l (Mod r). Since m. = 
 i-I l ' ' i-I J J x 1 
 
 or 1^ y. ^ r - 1 and y. / 0. The allowed value of y. is selected to yield 
 
 the most restrictive limits on m. . 
 
 l 
 
-21- 
 
 ■r < m. < 1; m. , = 
 — i — l-l 
 
 0<m. < r + 1; m. , =1 
 — i — y i-I 
 
 Both inequalities can be satisfied with m. = or 1. 
 
 The left and right directed mode functions must incorporate the 
 restrictions stated in Theorems 2-5 and 2-6 respectively. In those cases where 
 the mode digit is arbitrary with respect to the range restriction^ the left or 
 right directed mode function must define it in such a manner as to guarantee 
 the desired characteristics of the recoded representation. 
 
 Definition 2-11 : Set of Left Directed Mode Functions 
 
 Let L(y) denote a set of left directed mode functions which for every 
 y € Y specifies a unique allowable mode vector X £ V where 
 
 X = jy , X , . .. X } Or. L(y) contains n left directed mode functions^ L., 
 
 plus the two boundary conditions X . = y _ and X = 0. Each 
 
 -1 -1 n 
 
 L.(a,. , » y. , j Y7 , ) defines X. such that the restriction of Theorem 2-5 is 
 l i+l' i+l^ i+I i 
 
 satisfied for < i < n - 1 with X -, = y . and X = 0. The set, Y. . . excludes 
 - - -1 -1 n ' i+I 
 
 y. but may include an arbitrary number of digits to the left and right of 
 
 y i+r 
 
 Definition 2-12 : Set of Right Directed Mode Functions 
 
 Let R(y) denote a set of right directed mode functions which for 
 every y e Y specifies a unique allowable mode vector P e V where 
 
 P = iy-,jP> ...P , , Or. R(y) contains n right directed mode functions. R.. 
 -1 o^ n-1 ' i' 
 
-22- 
 
 plus the two boundary conditions P n = y . and p = 0. Each R.(p. ,, v., YV) 
 
 -1 -1 n i x i-l' J i' i' 
 
 defines P. such that the restriction of Theorem 2-6 is satisfied for 
 l 
 
 < i < n - 1 with P = y and p =0. The set, YV, excludes y. but may 
 include an arbitrary number of digits to the right or left of y. . 
 
 From this point on ; m. will be used to designate the i mode digit 
 in general, whereas X. and P. will be used to distinguish the i mode digit 
 as generated by left and right directed mode functions respectively. Likewise, 
 X and p will be used to distinguish left and right directed mode vectors. 
 
 It is important to note that the boundary condition \ = y affects 
 
 none of the L. while the condition P =0 may affect an arbitrary number of R. . 
 l n i 
 
 The Corollary of Theorem 2-1 guarantees -1 < y' < 1 for all r > 2 and 
 
 X , = y ., with X = or 1„ Thus the range restriction on y' is always satis- 
 -1 -1 o ° o 
 
 fied regardless of how X is defined by L (\, , y n , Y7). If y =0 and 
 
 R , (p ^>y -, ? ^ -, ) = P , = lj then Theorem 2-6 demands that p = 1. It 
 n-l v n-2' J n-V n-l y n-1 n 
 
 is clear that in this case R , must insure that P n = 0. The condition under 
 
 n-1 n-1 
 
 which an arbitrary R. is restricted by P =0 can be stated as follows, 
 
 l n 
 
 Theorem 2-7: If y =0 for n-k+l<j<n, then R , = P n = where 
 i — — ' n-k n-k 
 
 1 < k < n. 
 
 Proof: The proof here is similar to the one used in Theorem 2-k. If y = 0, 
 
 it is then necessary that R , - P = to insure that p =0 and still 
 
 n-1 n-1 n 
 
 satisfy the range restriction. If y . = also, then R n = P _= to 
 
 n-1 ' n-2 n-2 
 
 guarantee p = and satisfy the range restriction. 
 
-23- 
 
 Assume it is necessary that R . , = P , , = 0, if y. = for 
 
 n-k+1 n-k+1 * j 
 
 n-k + 2<j<n„ If y , , = 0. then it is required that R , = p . = 
 - - n-k+1 ^ n-k n-k 
 
 to satisfy the range restriction and the condition P . _ = 0, 
 J to n-k+1 
 
 If k = n, the Corollary of Theorem 2-1 guarantees that R = p = 
 does not conflict with the requirement that P = y . 
 
 2.^,2 General Mode Functions 
 
 A left or right directed mode function is distinguished from a 
 general mode function by the presence of a mode digit in the set of independent 
 variables. The mode digit can be eliminated by writing the next mode function 
 in terms of the present mode function beginning at the right or left boundary. 
 This technique reduces a left or right directed mode function to a general 
 mode function and thus forms a basis for the proof of the following theorems. 
 
 Theorem 2-8 : Given a set of left directed mode functions, L(y), there exists 
 an equivalent set of general mode functions, M(y). 
 
 Proof : If M(y) is equivalent to L(y), then m = X for all y £ Y by 
 
 Definition 2-10, It is clear that X -, - m n = y _, and \ = m = for all 
 
 --L -1-1 n n 
 
 y e Y, 
 
 Note that X , = L , (\ . y , i ) is also a general digital mode 
 n-1 n-1 n' n' n 
 
 function since X = 0, and Y* _, can represent y U i . 
 n n-1 n w n 
 
 1 . (Y* . ) = L . (0, y , Y L ) = X . 
 
 n-V n-l y n-l v ' J n' n y n-1 
 
 m _ = M 
 n-1 
 
 Now 
 
 consider X _ = L ^ (X ., y , , I ,). Substitute M n for 
 n-2 n-2 n-1' J n-1 ; n-1 n-1 
 
 \ . and define Y* = Y* U y ,1]^. Then m „ = M „(Y* ) = X . 
 n-1 n-2 n-1 J n-1 ^ n-1 n-2 n-2 ' n-2 y n-2 
 
-2J+- 
 
 Assume that m , = M . (Y* ) = A, , . Substitute M , for \ , in 
 n-k n-k n-k n-k n-k n-k 
 
 L n-k-l° 
 
 *• , n=L . n (M , , y n , Y 1 , ) 
 n-k-1 n-k-1 v n-k' J n-k' n-k' 
 
 If we define Y* . = Y* , U y , U^u then 
 n-k-1 n-k n-k n-k J 
 
 m . = M . ,(Y* .) = \ . .. 
 
 n-k-1 n-k-1 n-k-1 n-k-1 
 
 Theorem 2-9 ' Given a set of right directed mode functions, R(y), there exists 
 an equivalent set of general mode functions, M(y). 
 
 Proof: It is clear that P , = m , = y , and p = m = for all y £ Y. The 
 -1-1-1 n n 
 
 proof that there exists an M (Y*) = R (p , y , YT) for < k < n - 1 is 
 similar to the one for Theorem 2-8, 
 
 2. 5 Recoding Functions 
 
 The set of mapping functions as represented by Eq. 2:2 becomes a set 
 of recoding functions when a unique mode vector is specified for every y £ Y 
 by a set of mode functions. 
 
 Definition 2-13 : The Set of Recoding Functions 
 
 In the set of mapping functions, E(y,m), let m be defined by a set 
 of mode functions, M(y), for all y £ Y. Let M(y) be represented as a vector 
 
 of mode functions with m , = y _ and m =0. The set of recoding functions, 
 
 -1 J -1 n to ' 
 
 F(y), is then defined by the following matrix equation. 
 
 
-25- 
 
 y' = F(y) = E[y, M(y)] = y + 0M(y) 
 
 (2:3) 
 
 y 
 
 n 
 
 y i 
 
 1 
 •r 1 
 
 
 -r 1 
 
 •r 1 
 -r 1 
 
 M (Y*) 
 o o 
 
 M. , (Y* . 
 l-l l-l 
 
 M. (Y*) 
 l l 
 
 M (Y* ) 
 
 Definition 2-lk- : Sets of Left or Right Directed Recoding Functions 
 
 Let F (y) and F (y) denote sets of left and right directed recoding 
 L R 
 
 functions respectively,, F and F are defined by the following equations. 
 
 L K 
 
 y' = F L (y) = y + 0L(y) 
 
 (2*) 
 
 » = F R (y) = y + 0R(y) 
 
 (2:5) 
 
 Definition 2-l4 : Equivalent Sets of Recoding Functions 
 
 Two sets of recoding functions F* and F** are equivalent if and only 
 if F*(y) = F**(y) for all y £ Y. 
 
 Theorem 2-10 : F*(y) and F**(y) are equivalent if and only if the corresponding 
 M*(y) and M**(y) are equivalent. 
 
-26- 
 
 Proof : Since m = M(y) and y' = F(y) = E[y, M(y)J for all y e Y, the proof is 
 immediate by application of Theorem 2-2 to every y e Y. 
 
 Theorem 2-10 applies in case F* = F and F** = F . Therefore, a set 
 
 L R 
 
 of left directed recoding functions is equivalent to a set of right directed 
 
 recoding functions if and only if L(y) is equivalent to R(y)< In Chapter 3> the 
 
 problem of determining an R(y) which is equivalent to a given L(y), and con- 
 versely, is investigated for the special case of r = 2 . 
 
3, LEFT AND RIGHT DIRECTED RECODING IN THE BINARY CASE 
 
 Left and right directed recodings of a binary radix complement 
 representation are investigated in this chapter. A formulation of L(y) and 
 R(y) is derived which separates the restrictions of Theorems 2-5 and 2-6 from 
 those which control the characteristics of the recoding „ These new forms of 
 L(y) and R(y) are then reduced to a set of general mode functions . Using these 
 results a method is developed for obtaining a set of right directed recoding 
 functions which is equivalent to a given set of left directed recoding functions 
 and vice versa, 
 
 3=1 Left and Right Directed Mode Functions in Boolean Form 
 
 Since v., X. and p. are now either or 1, the mode functions L. and 
 l 11 i 
 
 R. can be expressed as Boolean functions . If L. and R. are expanded about 
 
 X. f y. and p. , y. respectively^ the restrictions imposed by Theorems 2-5 
 
 and 2-6 can be easily incorporated. 
 
 Theorem 3-1 : In the binary case the Boolean form of L. is given by 
 
 \. = L.(X. ., y. ., Y L J = \. _y. . V (*•• ■, V Y- JK. (y. ., Y 1 .) (3:l) 
 
 i i v i+i' ■'i+i' 1+1 i+i J i+i v v 1+1 v J i+i y i w i+y i+i ' w J 
 
 where K. is independent of the range restriction. 
 
 Proof: By expanding L. {X. _,, y. „ . Y7 _, ) about X. , and y . _, we obtain 
 J r to i i+i' J i+1' i+l i+l J i+l 
 
 X. = \. v. _L. (1, 1, ^JVV ,y. ,L. (1. 0, Y 1 , ) 
 
 l i+l J i+l i v ' } i+l y i+l J i+l i v ' ' i+l ' 
 
 \t\. n y. ,L.(0, 1, Y7 , ) V\. n y. ,L. (0, 0, ^1 _ ). 
 
 i+l J i+l i v } ' i+l i+l J i+l i v 3 } i+l ' 
 
 -27- 
 
-28- 
 
 According to Theorem 2-5, \. = \. , if y. , + \. , = (Mod 2). In Boolean 
 
 ' i l+l l+l l+l v J 
 
 form, X. = X. _ if y. . e X. . = 0. Therefore if \ v. , = 1. then \. = 1 
 ' i i+l l+l l+l i+1 l+l ' l 
 
 and if X. n y. . = 1. then X. = necessarily. The choice of X. is arbitrary 
 i+l i+l l l J 
 
 otherwise. It follows that L. (l, 1, Y7 , ) s 1 and L. (0, 0, Y^ ) = 0. By 
 
 l v ' ' i+l ' i v ' ' i+l ' J 
 
 making these substitutions and joining the identically zero terms 
 
 X. .y. n y. ,L.(0, 1, YT" . ) and X. n y. n y. n L (l. 0, YT' , ) we obtain 
 i+l J i+l J i+l i v ' ' i+l i+l i+l J i+l x } ' i+l 
 
 \. = X. _y. . V (\. . e y, n )K.(y. _ , Y 1 . ) (3:2) 
 
 l i+l J i+l T v i+l J i+1' i w i+l' i+l y w ' 
 
 where 
 
 K.(y. n , Y L J = y. n L.(0, 1, ^JVy. ,L.(l, 0, Y 1 . ) (3:3) 
 
 i VJ i+l' i+l y J i+1 i v ' ' i+l ' J i+1 i v ' ' i+l ' x ' 
 
 It is clear that K. is not affected by the range restriction as imposed by 
 Theorem 2-5 since y / X . Simplification of Eq. 3:2 leads to Eq. 3:1- 
 
 Theorem 3-2 : In the binary case the Boolean form of R. is given by 
 
 p i ■ V p i-i' v $ - P i-A v (p i-i * Wv *?> (3:4) 
 
 where S. is independent of the range 'restriction. 
 
 Proof: Expansion of R. (p. , . y., Y?) about P. . and y. yields 
 l v i-l' i' l l-l l 
 
 p t - ViW 1 * L > ^» Vp i.^i R i (1 - °< # 
 
 Vp._ iy .R.(0, 1, Y^JV P^y^O, 0, Y*). 
 
-29- 
 
 By Theorem 2-6, P. =p. . if p. . = 1 + y. (Mod 2). In Boolean form, P. = p. . 
 J l l-l l-l l ' l l-l 
 
 if P. , = y. . It follows that P. = 1 if P. j. = 1 and p. = if p. .y, = 1. 
 l-l ''i l i-I i i l-l i 
 
 Otherwise the choice of P. is arbitrary. Therefore_, in the expansion of R. it 
 is necessary that R. (l, 0, Y?) - 1 and R. (0, 1, Y?) - 0. Making these sub- 
 stitutions and joining the identically zero terms P. y.y.R. (l,, 1, T.) and 
 M.^y.y.R. (0, 0, Y?) yields 
 
 P i = P i-l^i V (p i-l e y i )S i (y i' Y i ) (3:5) 
 
 where 
 
 S.(y., Y^) = y.R. (1, 1, Y?) V y.R^O, 0, Y*) (3:6) 
 
 S. is not affected by the range restriction since it is defined under condition 
 that p. j- y. . Simplification of Eq„ 3^5 yields Eq. 3^- 
 
 As defined in section 2,^.1, L(y) and R(y) represent the set of 
 
 Boolean functions L. and R., as defined by Eqs . 3:1 and 'i'.h, for 
 
 < i < n - 1 with the boundary conditions X n = P , = y , and X - p = 0. 
 - - J -1 -1 J -1 n n 
 
 3.2 General Mode Functions in Boolean Form 
 
 Theorems 2-8 and 2-9 state that sets of left and right directed mode 
 functions can be reduced to equivalent sets of general mode functions. The 
 next two theorems are special cases of Theorems 2-8 and 2-9- The method of 
 proof will be by a process of resubstitution. 
 
-30- 
 
 Theorem 3~3 ° Given a set of left directed mode functions, L(y), with 
 
 X. = L.(\. _, y. ., Y^) = \. _y. . v (*.. -, V y. , )K. (y. ., Y 1 _) 
 
 1 1 i+l' i+l' i l+l l+l l+l J i+1 i w i+l' l+l 
 
 for < i < n - 1, there exists an equivalent set of general mode functions, 
 J(y), with 
 
 n-i-1 p-1 
 
 m. = J.(Y*) = y. ,K. w "V" J- . K - TT (j ■ ■> ■ V K. .) (3:7) 
 
 i i v i y J i+1 l v Y n J i+l+p l+p .'I w i+l+ j i+j y 
 
 ■p=l .1=0 
 
 for < i < n - 1, 
 
 Proof; Note first of all that m n = \ , = y , and m = X = for all y e Y. 
 -1-1-1 n n 
 
 An inductive argument follows . Substitute L. n for \. .. in the 
 
 to l+l l+l 
 
 expression for L. . 
 
 \ ■ [ \ +2 ^i +2 * K i + i» v yi +2 K i + i ] ^i + i v V v Wk 
 
 Assume that after the (k - l) substitution we have 
 
 k-1 
 
 \. =\. , 7T(y. n .VK. .) \/y. -.K. 
 i l+k .'' i+1+j i+j i+I i 
 
 k-1 p-1 
 
 v * J l+i+p i+p .*• w i+i+j i+j 
 
 p=l j=o 
 
 By substituting L. , for X. , we obtain 
 l+k l+k 
 
 k k-1 
 
 h - \ +fc fi ]!^ (y i + i + j v "W Vy i + i K i v 'Wi+k T (y i + i +J v K i +j } 
 
 k-1 p-1 
 
 V Yy. , K. 1T(y. , • V K. .) 
 v * J i+l+p i+p .'' w i+l+j i+j 
 
 p=l j=0 
 
-31- 
 
 By collecting terms we have 
 
 k k p-1 
 
 K = \. , , T(y. , . V K. .) V y- n K. V/ Vy. . K. "TT (y. _ . V K. .), 
 
 1 i+k+1 .»' i+1+j i+j l+l l v , i+1+p l+p .»' i+1+j i+j 
 j=0 p=l j=0 
 
 This proves that the general expression for \. is valid for all y £ Y provided 
 the terms involved are defined. 
 
 In the last expression above let k = n - 1 - 1. Since \ = 0,, we have 
 
 n-i-1 p-1 
 
 l l+l l V J i+l+p i+p .1' W i+1+J l+J 
 r>=l ^ .1=0 
 
 If we define the right hand side of this equation as J. (Y* ) for < i < n - 1, 
 the theorem follows . 
 
 Theorem 3-^ - Given a set of right directed mode functions, R(y), with 
 
 P. = R.(p. ., y., Y?) = P. _y. >/ (p. . V y.)S.(y., Y?) 
 
 i l l-l i i l-l i l-l i 11 i 
 
 there exists an equivalent set of general mode functions, T(y), with 
 
 i-1 p-1 i-1 
 
 m. = T.(Y*) = y.S. V ~\T y. S. Tf (y. .VS. .) V S "TT (y- • ^ s - •) 
 
 i i v i' J i l ^ J i-p i-p ""i-j i-j y o " VJ i-J i-j' 
 
 (3:8) 
 
 for < i < n - 1. 
 
 Proof: Note that m . = P -, = y , and m = p = for all y e Y. 
 -1-1-1 n n 
 
 An inductive argument follows. Replace P. "by P.. in the 
 expression for R. . 
 
-32- 
 
 Assume that after the (k - l) substitution we have 
 
 k-1 k-1 p-1 
 
 p. = p. , TT (y- -V s. .) v y.s. yYy. s. TT (y- • V s. .) 
 
 1 1_k jL'o 1_J 1_J x x p =i 1_p 1-p j=o 1_J 1_J 
 
 By substituting R. , for p. , we obtain 
 l-k l-k 
 
 'i ■ p i-i-k TTfri.j * s._j) v 7^ v ^.A.i T(y ± .j v s.^) 
 
 k-1 p-1 
 
 V ~V y- s. Tf (y. .vs. .) 
 
 By collecting terms we have 
 
 k k _ p-1 
 
 p - =p - i i TT(y- -vs. Jvy.s. v V" y- s - If (y- • v s - •). 
 
 i i-1-k " w i-j v i-j 7 J i i V J i-p i-p JJ w i-J v i-J 
 
 This completes the induction proof and shows that the general expression for 
 P. is valid for all y e Y if the terms involved are defined. 
 
 In the last expression for p. let k = i. Since P = y = y for 
 r = 2, this expression becomes 
 
 'i " 'A * % ^i-p s i- P T^.j v s ± _ 3 ) y s o JG^ v Sl ..). 
 
 If we define the right hand side of this equation as T. (Y*)> the theorem 
 follows . 
 
-33- 
 
 In summary, these two theorems provide more explicit relationships 
 between a given set of left or right directed mode functions and the cor- 
 responding set of general mode functions. These relationships will be used 
 to obtain an equivalent set of right directed mode functions from a given set 
 of left directed mode functions and conversely, 
 
 3. 3 Equivalent Left and Right Directed Recoding Functions 
 
 By Theorem 2-10, F is equivalent to F if and only if L(y) is 
 
 L R 
 
 equivalent to R(y). Assume that L(y) is given. Each functional element of 
 
 this set, L., generates a corresponding mode digit, X. . We wish to establish 
 
 a set R(y) such that each corresponding functional element, R., generates a 
 
 mode digit, P., that is equal to X. for all y € Y. 
 
 The Boolean interpretation of Theorem 2-6 was developed in the proof 
 
 of Theorem 3-2. It states that P. = 1 necessarily if p. y. = 1. If P. = y., 
 
 then P © y. = 1 and p. may be or 1 with respect to the range restriction. 
 
 If P . n y. = 1, then P. = necessarily. 
 l-l 1 1 
 
 By Eq. 3:5; it is clear that S. determines the value of P. when 
 
 P. . = y. . In order for R. to be equivalent to L. , P. should equal X. for all 
 1-1 1 i * i' i * i 
 
 y e Y whenever P. . © y. = 1. Therefore, p. = P. n y. v(p. , © y. )\. . The 
 l-l 1 ' 1 l-l 1 1-1 J 1 1 
 
 X. as used here must be independent of all mode digits except possible X. 
 Since \ = L (\ , y. , YT -,) = J. (Y*) by Eq. 3:7, substitution and simpli- 
 fication yields P. = p. j. V (P. -, V y. )J. (Y-*). 
 1 l-l 1 l-l 1 1 1 
 
 Theorem 3- 5 : If p. is defined by 
 
 R i (p i-i> y i> Y i ) = P i-i^i v(p i-i v y±) J ±( Y V (3:9) 
 
-3k- 
 
 for < i < n - 1, then R(y) is equivalent to a given L(y) in which 
 
 X. = L.(X. _, y. n , Y L _ ) = J.(Y*) where J. (Y*) is defined by Eq. 3:7. 
 l l l+l i+l' l+l i v i i v l J ^ J 
 
 Proof: R(y) is equivalent to L(y) if and only if p = X for all y e Y. An 
 
 inductive argument follows., By the definition of R(y) and L(y), it is clear 
 
 that p n - X n = y -, and p = X = for all y 6 Y„ Assume P. . = X. n for 
 -1 -1 -1 n n J l-l i-I 
 
 all y £ Y and substitute into the equation for P. as stated in the hypothesis 
 
 P. = *». ,y. V (X , V y. )J.(Y*) 
 
 l i-l J i v i-I Y J i i l 
 
 Since X. = J. and X,. . = X.. y. V (X. V y. )K. - , by substitution in the above 
 l l i-I 1 l i J i i-I' ^ 
 
 equation we obtain P. = X.y. V X.y. V X . K. = X. . Thus P. = X. for 
 
 l 111111 l l l 
 
 <; i < n - 1 and all y 6 Y. 
 
 Now assume R(y) is given and that we wish to determine the equivalent 
 
 L(y). In this case the Boolean form of Theorem 2-5 is examined. If 
 
 X. _,y. _, = 1, then X. = 1 necessarily. If X. , 4 y. ,, then X. , ® y. , = 1 
 i+l J i+l ' l J l+l ' J i+l' l+l J i+1 
 
 and X. is arbitrary. If X. , y. , =1, then X. = necessarily, 
 l l+l l+l l 
 
 By Eq. 3<2 it is apparent that K. determines X. only when 
 
 X. _ e y. _ = 1. In manufacturing a L. which is equivalent to the corres- 
 l+l l+l i 
 
 ponding R., it is necessary that X. = P. for all y e Y whenever X e y = 1, 
 
 Therefore, X. - X. n y n V (X ., © y. _ )p.. As used here, P. can not be a 
 ' i l+l i+I i+I J i+1 i ' i 
 
 function of any mode digits except possibly p. . This requirement is easily 
 
 satisfied since P. = T. (Y*) by Eq. 3:8- Substitution in the above equation 
 
 and simplification yields X. = X. j. _ V (X. _ \/y. , )T.(Y*)- 
 
 i i+I i+I v i+I i+I i l 
 
■35- 
 
 Theorem 3-6: If X. is defined by 
 — 1 
 
 for < i < n - 1, then L(y) is equivalent to a given R(y) in which 
 
 P = R (p. n , v., Y R ) = T.(Y*) where T. (Y*) is defined by Eq. 3:8. 
 l i v i-l' i' l l l i i 
 
 Proof : By the definition of L(y) and R(y) we have X = p = y and 
 
 \ = P =0 for all y e Y. An inductive argument follows. Assume that 
 n n 
 
 X , = p. . for all y e Y. But substitution in the expression for X. as given 
 l+l l+l J ■ i 
 
 in the hypothesis we obtain 
 
 X. = p. _y. . y (p- i V y. -, )t. (y*). 
 1 1+1 1+1 v 1+1 1+1 1 1 
 
 Since P. , = p.y. n \/ (p V y. -, )S. n and P. = T. (Y*), substitution and 
 l+l i J i+l i r "'l+l l+l i i v l" 
 
 simplification yields 
 
 X. = P.y. ,Vp.S. n V P.y. , = P. . 
 i i J i+l i l+l li+l i 
 
 It follows that X. = p. for < i < n - 1 and all y eY. 
 11— — 
 
 By comparison of Eqs . 3*9 an( i 3«10 with Eqs . 3:+ and 3-1; it is 
 
 clear that J.(Y*) represents S. (y., YV) and T. (Y*) represents K. (y. .. Y7 1 n ) . 
 i v l ' * i w i' i' i v i / ^ i w i+l' i+l 7 
 
 It is apparent from this observation that the development of a set of left or 
 right directed mode functions equivalent to a given set of right or left 
 directed mode functions is accomplished in two steps. An equivalent set of 
 general mode functions is derived and then used to manufacture a standard set 
 of left or right directed mode functions. It follows from Theorem 2-10 that 
 
-36- 
 
 the corresponding sets of recoding functions F and F will also be equivalent 
 
 L K 
 
 Several examples of this technique are given at the end of Chapter k. 
 
k. MINIMAL RECODING OF BINARY REPRESENTATIONS 
 
 The results of Chapters 2 and 3 indicate that the properties of a set 
 
 of left or right directed recoding functions F (y) and F (y) are determined by 
 
 L R 
 
 the properties of the corresponding sets of left or right directed mode 
 
 functions L(y) and R(y). Furthermore, given L(y) an equivalent R(y) can "be 
 
 determined and conversely. It follows that for every F (y) there exists an 
 
 J-j 
 
 equivalent F (y) and conversely. The class of F (y) is therefore equivalent 
 R L 
 
 to the class of F (y). 
 K 
 
 In this chapter we confine our attention to the subclass of minimal 
 
 F T (y) and the equivalent subclass of minimal F (y) where y is restricted to 
 L R 
 
 binary representations. The sets of minimal recoding functions included in 
 these classes map Y onto Y' with the added constraint that y' is a minimal 
 representation of y. 
 
 Definition k-1 : Minimal Representation 
 
 A representation, y ? , is a minimal representation of y if and only 
 if y' = y and y' contains a minimum of nonzero digits. 
 
 A well-known element of these minimal subclasses is the set of 
 
 recoding functions, F (y) and its equivalent F (y), which produce a canonical, 
 
 L R 
 
 minimal representation for all y £ Y. This special representation has been 
 defined by Reitwiesner [9] and studied by Lehman [5], Robertson [11], 
 Tocher [13], Weinberger and Smith [Ik] and others. 
 
 ■37- 
 
-38- 
 
 Definition k-2 : Canonical, Minimal Representation 
 
 A representation, y', is a canonical, minimal representation of y 
 if and only if y' = y and y' contains a minimum number of nonzero digits 
 separated by at least one zero digit. 
 
 The canonical, minimal F (y) has been effectively employed to recode 
 
 the multiplier in fast multiplication schemes. Reitwiesner [9] has developed 
 
 a division algorithm which, in effect, uses the equivalent F (y) to generate 
 
 a recoded quotient. Other multiplication and division algorithms based on 
 
 minimal F (y) and F (y) are discussed in Chapter 5« 
 L R 
 
 For every set of minimal recoding functions, F T (y) or F (y), there 
 
 L R 
 
 is a corresponding set of minimal mode functions, L(y) or R(y). The subclasses 
 
 of minimal F (y) and F (y) are therefore defined by the subclasses of minimal 
 L R 
 
 L(y) and R(y) which are also equivalent. All possible minimal L(y) or 
 
 minimal R(y) are known if all possible minimal L. or minimal R. are known. 
 
 1 1 
 
 For this reason the prime emphasis in this chapter is on the development of 
 
 the classes of minimal L. and R. . 
 
 1 1 
 
 Definition ^-3 ' Minimal Left Directed Mode Functions 
 
 L. (\. ,i y. ,j Y7 J is a minimal left directed mode function if and 
 i v l+l' •'i+I' 1+1 
 
 only if it defines \. such that all partial sequences (...y. , y., y. ) 
 
 can be recoded with a minimum of nonzero digits. 
 
 Let the class of all such L. be denoted by 
 
 £. 
 
-39- 
 
 Definition k-k : Minimal Right Directed Mode Functions 
 
 ■p 
 
 R- (p- -, > y.j Y.) is a minimal right directed mode function if 
 1 l-l l l 
 
 and only if it defines p. such that all partial sequences (y. , y. } y. ? ...) 
 
 can he recoded with a minimum of nonzero digits. 
 
 Let the class of all such P.. be denoted by 
 
 (R. 
 
 The development of oC and (J\ begins with an analysis of the 
 
 minimal representation in section Ud. This analysis yields criteria for the 
 
 optimum choice of ^. and p.. These criteria are used to partition 
 
 (... y i _ 1 . y ± , y i+1 , *- 1+1 ) and (p i _ 1 , y ± , y ±fl , y i+2 . -0 into three mutually 
 
 exclusive subsets such that \. and p. are either or 1 or arbitrary. In 
 
 sections k,2 and k.~5 these subsets are used to establish the Boolean form of 
 
 all minimal L. and R. in which YT -, is restricted to y. to the left of y, n 
 1 l 1+1 J j J i+1 
 
 ID 
 
 and Y. is restricted to y , to the right of y. . These restricted, minimal L. 
 l J j B °i ' i 
 
 and R. represent subclasses of cT and CX . A method of obtaining the 
 unrestricted elements of cXL and (K is presented in section k-.k. It is based 
 
 on knowledge of the restricted minimal L. and R. and the equivalence relations 
 
 li 
 
 established by Theorems 5-5 and 5-6. Restricted, minimal L. and R. are 
 
 7 11 
 
 generally equivalent to unrestricted, minimal R. and L. . The particular L. 
 and R. which produce the canonical, minimal representation are examples of an 
 exception to this . 
 
 k-.l Analysis of the Minimal Representation 
 
 To obtain a minimal recoded representation of y it is necessary to 
 maximize the number of zeros occurring in y' . The approach used to solve 
 this problem is outlined below. 
 
-llO- 
 
 (a) The whole or partial representation of y is broken into a 
 unique series of special sequences. These sequences are of three types: the 
 zero sequence, the unit sequence, and the alternating sequence. 
 
 (b) The gain in zeros achieved by recoding a binary sequence is 
 defined as a function of the sequence type, its length, and the boundary mode 
 digits. By maximizing this gain, the optimum mode digit pattern is determined 
 for each of the three sequence types and for all possible combinations of 
 boundary mode digit values. 
 
 (c) The optimum value for the next mode digit is not always 
 determined by the special sequence with which it is associated. Therefore, 
 criteria are established for choosing this digit such that the entire series 
 of recoded sequences contains a maximum number of zeros. Fortunately, in most 
 cases only two or three special sequences, including the one which contains 
 the digit being recoded, need to be examined. 
 
 ^.1.1 Decomposition of the Binary Representation 
 
 Any segment of y £ Y can be decomposed into a series of special 
 sequences defined as follows. 
 
 Definition h-5 i The Zero Sequence 
 
 Let Z denote a sequence of one or more zeros. 
 
 Definition k-6 : The Unit Sequence 
 
 Let U denote a sequence of one or more units. 
 
-1*1- 
 
 Definition 4-7 : The Alternating Sequence 
 
 Let A denote an alternating sequence of zeros and units. The A 
 sequence contains an even number of digits having a zero on the left and a 
 unit on the right. 
 
 Definition h-8 i Order of Decomposition 
 
 Any segment of y £ Y is decomposed into a series of A, U and Z 
 sequences in the following order. 
 
 (a) All A type sequences are removed, 
 
 (b) The remaining sequences are then classified as type U or Z, 
 
 The definitions of A and the order of decomposition were chosen such 
 that the resulting series of sequences would be unique for a given segment of 
 y 6 Yo This is proven as a theorem below. There are others sets of defini- 
 tions [lk] which also satisfy the uniqueness requirement and no special claim 
 is made for the particular set chosen. 
 
 Theorem 4-1 : The decomposition of a given segment of y G Y in accordance with 
 Definition 4-8 yields a unique series of A, U and Z sequences. 
 
 Proof : Any segment of y £ Y may or may not include one or more A sequences. 
 If it contains one or more A sequences, each of these sequences must be bounded 
 by a single zero or two adjacent units on the left and by a single unit or two 
 adjacent zeros on the right. This also applies to the left and right boundaries 
 of y if we define y = y = y (a natural left extension of y in the binary 
 
-In- 
 case) and y , = y _ = 0. In all cases the initial zero and the terminal 
 n+1 n+2 
 
 unit of the A sequence is well defined. After the A sequences have been 
 deleted, the remaining sequences must he U or Z sequences containing one or 
 more elements . 
 
 If the segment of y contains no A sequences, then it must consist 
 of a single Z sequence, a single U sequence, or a single U sequence followed 
 by a single Z sequence. 
 
 Theorem h-2 : The ZU sequence can not occur. 
 
 Proof: If ZU occurs, the representation was not properly decomposed. The 
 terminal zero of Z followed by the initial unit of U should have been con- 
 sidered a two element A sequence. Therefore ZAU occurs in place of ZU. 
 
 In summary, a segment of y £ Y may be decomposed into a unique 
 series of special sequences. Under the given definitions only the following 
 pairs of special sequences may occur in this series. 
 
 UZ, UA, AU, ZA, AZ 
 
 4.1.2 The Gain Function 
 
 Since our goal is to develop minimal recoded representations, it is 
 useful to define a function which provides a quantitative measure of the dif- 
 ference in the number of zeros between the recoded and the original representa- 
 tions. Consider a segment of y starting with y. and ending with y. . 
 Suppose this sequence of p bits is recoded using Eq. 2:1 with given values 
 
•4> 
 
 for the boundary mode digits m . . and m. , . The number of nonzero digits 
 
 0-1 J+P-1 
 
 in the original sequence can be expressed as follows. 
 
 J+P-1 
 
 N = E y. (4:1) 
 
 Since y' = y. + m. - 2m. , and -1 < y! < 1, the number of nonzero digits in 
 i l l l-l — l — 
 
 the recoded representation is given by Eq. 4:2, 
 
 J+P-1 P 
 
 N' = E (y. + m. - 2m. . ) (4:2) 
 
 . . i i l-l 
 i+J 
 
 Definition 4-9 : The Gain Function 
 
 The gain function, G, represents the number of zeros gained by 
 
 recoding a binary sequence, P = (y., y. , , ... f j. ) } into a base 2 
 
 signed-digit sequence, P' = (y!, y! , «.., y' ), with a given pair of 
 
 boundary mode digits (m. , , m. , ). 
 
 J-l J+P-1 
 
 G(m. _, P, p, m. _ ) = N - N* 
 
 j-1' ' *' J+P-1 
 
 In this equation N and N' are defined by Eqs . 4:1 and 4:2 respectively. 
 
 2 
 Since y. = y., it is written as 
 
 J+P-1 
 
 G(m. , P, p, m. . ) = - E (2y. + m. - 2m. _ )(m. - 2m. . ) (4:3) 
 •]_> > n i+p-1 . . J i l i-l /v i l-l' v ' 
 
 J+p - i-j 
 
-44- 
 
 Definition 4-10: Optimum Mode Digit Sequence 
 
 An optimum mode digit sequence is any sequence (m , m . . . m ) 
 
 j-l' J j+P-1 
 
 which recodes P = (y., y. , > . . . y . ,) with a given pair of boundary mode 
 
 digits (m. ,, m. ., ) such that G(m. .. P. p. m. n ) assumes its maximum 
 J-l J+P-1 J-l J+P-1 
 
 value G. 
 
 We now determine the optimum mode digit sequence or sequences and 
 the corresponding G for each of the special sequences with all allowable 
 boundary mode digit pairs. 
 
 4.1.2.1 Analysis of the Zero Sequence 
 
 Since y. = for j<i<j+p-l, Eq. 4:3 becomes 
 
 j+P-1 p 
 
 G(m. . , Z, p, m. . ) = - E (m. - 2m. . ) (4:4) 
 
 j-l j+p-1 ._. i l-l 
 
 - 1 - J 
 
 Theorem 4-3 : Given a Z sequence, the allowable boundary mode digit pairs are 
 (0, 0), (0, l) and (1, 1). 
 
 Proof : Given that m. = 0, m. may be or 1 and still satisfy the range 
 
 res 
 
 triction. However if m. =1, m. . must be 1 to satisfy -1 < m. , - 2 < 1. 
 
 .1 ' .1+1 - .1+1 
 
 By induction it is clear that m. . must be 1. Since it is always possible 
 
 J+P-1 
 
 to choose m. . =0 for < k < p - 1, the choice of m. , =0 does not deter- 
 J+k ' j-l 
 
 mine the choice of m. , . If m. n - 1, the range restriction forces m. = 1 
 
 j+p-1 j-l J 
 
 and by induction m. _ = 1. The boundary pair (l, 0) is therefore not 
 
 j+p-1 
 
 allowed . 
 
-10- 
 
 Theorem h—k: Given a Z sequence with m. _, = m . , = 0, the optimum mode 
 * J-l J+P-l 
 
 digit sequence is m. =0 for j-l<i<j+p-l with G(0, Z, p, 0) = 0. 
 Proof: In terms of the given boundary conditions the gain function becomes 
 
 J+p-2 
 G(0, Z, p, 0) = -m - ^m - Z (m. - 2m ± )' 
 
 i=j+l 
 
 Since m , = the remaining m. n =0, so the optimum mode digit sequence is 
 j+p-1 j+k 
 
 m. = for j -l<i<j+p-l with G(0, Z, p, 0) = 0. 
 
 Theorem h-5: Given a Z sequence with m , = and m. , = 1, the optimum 
 j-l J+P-l 
 
 mode digit sequence is m. =0 for j-l<i<j+p-2 with m. , = 1. The 
 
 l "__^^ i+p-1 
 
 corresponding G(0, Z, p, l) = -1. 
 
 Proof: For the given boundary conditions Eq. k:k becomes 
 
 J+P-2 p 
 
 G(0, Z, p, 1) = -m - 1 - Z (m - 2m ) , 
 
 Again it is apparent that the optimum mode digit sequence is m. =0 for 
 
 j - 1 < i < j +p-2 with m. , = 1. Evaluation of Eq. h:6 yields 
 - - J+P-l 
 
 G(0, Z, p, 1) = -1. 
 
 Theorem K-6: Given a Z sequence with m. n = m. , = 1. the optimum mode 
 J-l J+P-l 
 
 digit sequence is m. =1 for j - 1 < i < j + p - 1 where G(l, Z, p, l) = -p. 
 
■46- 
 
 Proof: As demonstrated in the proof of Theorem 4-3? the boundary condition 
 
 m. ., = 1 forces all m. to be 1 to satisfy the range restriction. Therefore, 
 j-1 1 j & > 
 
 the optimum mode digit sequence must be m. =1 for j-l<i<j+p-l. 
 Evaluating Eq. 4:4 we have 
 
 J+P-l p 
 G(l, Z, p, 1) = - E (1-2) = -p. 
 
 4.1.2.2 Analysis of the Unit Sequence 
 
 In this case y. = 1 for j ~ 1 < i < J + p - 1 so Eq. 4:3 becomes 
 
 J+P-l 
 
 G(m. ., U, p, m. . ) = - E (2 - m. - 2m. _ )(m. - 2m. . ) (4:5) 
 
 v j-1' ' ' j+p-1 ._. i i-I i i-I v 
 
 Theorem 4-7 : Given a U sequence the allowable boundary mode digit pairs are: 
 (0, 0), (1, 0) and (1, 1). 
 
 Proof: Given m. _, =0, m. must be to satisfy -1 < 1 + m. < 1. By induction 
 J-1 J - J - 
 
 it follows that m. must also be which rules out the boundary pair (0, l), 
 If m. = 1, then m. may be or 1 and still satisfy the range restriction. 
 By induction m. may be or 1 provided all m. = 1 for -1 < k < p - 2. 
 
 Theorem 4-8: Given a U sequence with m. , = m. , = 0. the optimum mode 
 H J-1 J+P-l 
 
 digit sequence is m. =0 for j - 1 < i < J + p - 1 with G(0, U, p, 0) = 0. 
 
■47- 
 
 Proof: As proven in Theorem 4-7, the choice of m. , =0 forces the remaining 
 
 J-l 
 
 m , to be 0, The optimum mode digit sequence is therefore m. = for 
 J+k * i 
 
 j-l<i<j+p-lo Evaluating Eq. ^:5 we obtain G(0, U, p, 0) 
 
 j+p-1 
 = E (2)(0) = 0. 
 
 Theorem 4-9: Given a U sequence with m. _ = 1 and m. , = 0, the optimum 
 J-l J+P-l 
 
 mode digit sequence is m. =1 for j-l<i<j+p-2 with m. _, = 0. The 
 ^ i - - J+P-l 
 
 corresponding G(l, U, p, 0) = p - 1. 
 
 Proof: After substituting the boundary conditions, Eq. 4:5 becomes 
 
 J+P-2 
 
 G(l, U, p, 0) = m. - E (2 + m. - 2m. _ )(m. - 2m. , ). 
 v ' ' r7 j . . ., i l-l l l-l 
 
 i=J+l 
 
 The 
 
 term (2 + m. - 2m , )(m. - 2m. -, ) has a minimum value of -1 when 
 i i-I l l-l 
 
 m. = m. , = 1. The optimum mode digit sequence is therefore m. = 1 for 
 l i-I i 
 
 j -l<i<j+p-2 with m. , = 0. The corresponding G(l, U, p, 0) 
 — — J+P-l 
 
 = 1 + (j + p - 2) - (j + 1) + 1 = p - 1. 
 
 Theorem 4-10: Given a U sequence with m. , = m. , = 1, the optimum mode 
 
 J-l J+P-l 
 
 digit sequence is m. =1 for j-l<i<j+p-l with *G(l, U, p, l) = p. 
 
 Proof: The boundary condition m. , = 1 forces all other mode digits to 1. 
 
 J+P-l 
 
 The optimum mode digit sequence is therefore m. = 1 for j-l<i<j+p-l, 
 
 J+P-l 
 
 Evaluating Eq, 4:5 we have G(l, U, p, l) = - E (-l) = p. 
 
-1+8- 
 
 i+.1.2„3 Analysis of the Alternating Sequence 
 
 p-2 
 Let y = and y. = 1 where o < k < £ -—z — and p is an even 
 
 integer. The general gain function is expressed as follows. 
 
 P-2 
 2 2 
 
 G(m A, p, m ) = - I (a - 2a ± ) 
 
 (h:6) 
 
 P-2 
 2 
 
 Z (2 + m. _. .. - 2m. _. )(m. .. . - 2m . ^ n ) 
 j+2k+l j+2k v j+2k+l j+2k' 
 
 Theorem U-ll : Given an A sequence, the allowable boundary mode digit pairs 
 are: (0, 0), (0, l), (l, 0) and (l, l). 
 
 Proof: If m. =0, then m. may be or 1 to satisfy -1 < a. < 1. If 
 J — ■*- J J 
 
 m. . = 1. then m. must be 1 to satisfy -1 < m. - 2 < 1. When m. =1, however, 
 
 J-l J - J J 
 
 m. . may be either or 1 to satisfy -1 < -1 - a. . < 1. Since p > 2, the 
 J+l J+l - - ' 
 
 choice of m. . = or 1 does not dictate the choice of m. , . Likewise the 
 J-l J+P-l 
 
 choice of m. , does not dictate the choice of m. 
 
 J+P-l J-l 
 
 Theorem U-12: Given an A sequence with m. , = m. , = 0. the optimum mode 
 J-l J+P-l 
 
 digit sequence is m. =0 for j-l<i<j+p-l with G(0, A, p, 0) = 0. 
 
-k9- 
 Proof : For the given boundary conditions Eq. h:6 becomes 
 
 P-g 
 
 2 2 
 
 G(0, A, p, 0) - - m. - k ^(m. +2k - Sm.^) 
 
 ,-4 
 
 2 
 J (2 + m j + 2k + l - 2m j + 2k )(m j + 2k + l " 2m j + 2k } 
 
 To maximize this gain function^ it is necessary to minimize the summands such 
 
 that the resulting mode digit sequence is allowable. Since the summands in 
 
 the first sum are all squared, the best we can do is to choose m. , n . = 
 
 m = 0o In the second sum, the summands have a minimum value of -1 when 
 j+2k-l ' 
 
 m. = m. = 1. The corresponding mode digit sequences are allowable but 
 J+2k+l j+2k 
 
 not compatible. If we choose m. = for j - 1 < i < j + p - 1, G(0, A, p, 0) = 0. 
 
 If we choose m. = 1 for j < i < j + p - 2 with m. _ = m. , = 0, 
 i - j-1 J+p-1 
 
 G(0, A, p, 0) = -1 - ^ p " 2 ' + ±2-^ - + 1 = -1. It follows that the optimum 
 
 mode digit sequence is m. =0 for j-l<i<j+p-l with "G*(0, A, p, 0) = 0. 
 
 Theorem *4--13: Given an A sequence with m. , = and m. , = 1. there are f- 
 J-1 J+P-1 2 
 
 optimum mode digit sequences of the form m. = for J -l<i<j+2k-l 
 
 P - 2 
 and m. = 1 for j+2k<i<j+p-l where < k < *-r — . The corresponding 
 
 maximum gain is G(0, A, p, l) = 0, 
 
 Proof: Since m. , = 1, it is necessary that m. _ = 1 to satisfy 
 J+P-1 J+P-2 
 
 -1 < 2 - 2m. _ < 1. Therefore Eq, k:6 becomes 
 
 J+P-2 - * 
 
■50- 
 
 2 
 
 0(0, A, p, 1) = -m. - ^ (» j4fik - ^j^k-i) 2 
 
 2 
 
 - Z (2 + m. _. . - 2m. _. )(m. _. . - 2m. _, ), 
 k=Q v j+2k+l j+2k /v j+2k+l j+2k y ' 
 
 The policy used to maximize this gain function was described in the proof of 
 Theorem 4-12. 
 
 If p = 2, m. =1 necessarily. The only, and therefore optimum, 
 mode digit sequence is Oil. Substitution in Eq. 4:12 yields G(0, A, 2, l) = 
 which is the maximum gain for p = 2. 
 
 n h 
 
 If p > k, we consider k = and 1 < k < — — — separately. For 
 
 k = 0, we have -m. - (2 + m. n - 2m.)(m. n - 2m.). Note that this term is 
 J J+l J J+l J 
 
 zero for m. , = m. = 1 or m. ., =m. =0 but is negative for any other allowable 
 J+l J J+l J 
 
 o 
 combinations of m. , and m.. If k > 0, we have -(m. „., - 2m. ^, , ) 
 J+l J J+2k j+2k-l 
 
 - (2 + m. - 2m. _. )(m. 01 . - 2m. __ ). If m. _. . = 1, then m. must 
 j+2k+l j+2k j+2k+l J+2k j+2k+l ' J+2k 
 
 be 1 to satisfy -1 < 2 - 2m. _, < 1. The above expression is zero when 
 
 - J+2k — 
 
 m. . = m. __ =1 and m. _. n = or 1 or when m. 01 . = m. _. = m. ,_, , = 0. 
 j+2k+l j+2k j+2k-l j+2k+l j+2k j+2k-l 
 
 It is negative for any other allowable combination of these mode digits. 
 
 This shows that an optimum mode digit sequence must consist of a 
 
 series of zeros on the left followed by a series of units on the right. Since 
 
 m. , = m. _ = 1. the series of units always contains two elements. If it 
 j+p-1 J+P-2 
 
 contains three units it must contain four to satisfy the range restriction. 
 Thusj there are ^ different optimum mode digit sequences, including the case 
 p = 2, for which G(0, A, p, l) = 0. 
 
-51- 
 
 Theorem ^-14: Given an A sequence with m. _, = 1 and m. , = 0, there are ^ 
 J-l J+P-l 2 
 
 optimum mode digit sequences of the form m. = 1 for j ~ 1 < i < J + 2k and 
 
 p - 2 
 m. = for j + 2k + l<i<j + p - 1 where < k < *— ■ - — . The corresponding 
 
 maximum gain is G(l, A, p^ 0) = -1. 
 
 Proof: Since m. , =1, m. must be 1 to satisfy -1 < m. - 2 < 1. Therefore 
 Eq. h:6 is written as 
 
 P-2 
 2 2 
 
 G(l, A, p, 0) = -1 + m - Z ^m - 2m ^ 
 
 p^ 
 2 
 
 " k ^ (2 + m j + 2k + l " 2m j + 2k )(m j + 2k + l " 2m j + 2^' 
 
 If p = 2, m. ., =m. ., =0. The optimum mode digit sequence is 
 ' J + p-1 J+1 
 
 110 and G(l, A, 2j 0) = -1. 
 
 If p = k, G(l, A, 4, 0) = -1 + m. , - (m. „ - 2m. , ) . In case 
 
 J+1 J+2 j+1 
 
 m. , = 1. m. _ must be 1 to satisfy -1 < -2 + m. _ < 1 and G(l, A, k. 0) = -1, 
 j+1 j +2 - J+2 - 
 
 The same maximum gain is obtained if m. , = m. „ = 0. For all other allowable 
 
 J+1 J+2 
 
 combinations of m. , and m. _. G(l, A, h, 0) < -1. The two optimum mode 
 j+1 j+2' ' ' ' 
 
 digit sequences are therefore 11110 and 11000. 
 
 If p > G } we consider the sum of the (k + l) term in the first 
 summation and the k term in the second summation. The combined term for 
 
 1 ^ k ^ ^ 1S - (m j+2k+2 " 2m j+2k+l )2 - (2 + m j+2k+l- 2m j+2k )(m j+2k+l " 2m j+2k : 
 
 If m. _. , = 1. then m. _, must be 1 to satisfy -1 < 2 - 2m. _, < 1 and 
 j+2k+l ' j+2k J - j+2k - 
 
 m. must be 1 to satisfy -1 < m. _. _ - 2 < 1. Substitution in the above 
 j+2k+2 J - j+2k+2 - 
 
 term yields a zero result. This term is also zero if m. .. = or 1 and 
 
 J+2k 
 
 m = m. ^ n _ = 0. It is negative for any other allowable combination of 
 j+2k+l j+2k+2 & J 
 
 UNIVERSI: 
 ILUNQIS LIBRARY 
 
■52- 
 
 these mode digits. Since the term corresponding to k = 1 in the first sum 
 
 is associated with -1 + m. . as in the case p = K, the maximum gain when 
 
 J+ 1 
 
 p > 6 is also G(l, A, p, 0) = -1. 
 
 The optimum mode digit sequence therefore consists of a series of 
 
 units on the left followed "by a series of zeros on the right. There are always 
 
 two elements in the series of units since m. , = m. = 1. If there are three 
 
 J-l J 
 
 units in the series, there must be four to satisfy the range restriction. It 
 follows that there are ^ different optimum mode digit sequences for p > 2 
 with G(l, A,, p, 0) = -1 in all cases. 
 
 Theorem 4-15: Given an A sequence with m. _ = 1 and m. , .= 1> the optimum mode 
 J-l J+P-1 
 
 digit sequence is m. - 1 for j-l<i<j+p-l with G(l, A, p, l) = 0. 
 
 Proof: The boundary conditions require that m. . = m. =1 and m. 
 
 ■ J * j-l j j+p-2 
 
 = m. = 1 to satisfy the range restriction. Under these conditions Eq. 4:6 
 J+P-1 
 
 becomes 
 
 p^ 
 2 2 
 
 G(l, A, p, 1) = -1 + m - Z ^(m - 2m ^ 
 
 p^ 
 2 
 
 - E / 2 + m j + 2k + l - 2m j + 2k )(m j + 2k + l " 2m j + 2^' 
 
 K. — _L 
 
 If p = 2, m. , = m. , = 1 and G(l, A, 2, l) = where the optimum 
 ' J+P-1 J+l 
 
 mode digit sequence is 111. 
 
 If p = 4, m , = m. _ = 1 and m. _ = m. _ = 1. The range 
 j+p-1 J+3 J+P-2 J+2 
 
 restriction on y! , and y! „ can be satisfied by choosing m. . as or 1. 
 J j+1 J j+2 J & j+l 
 
 
-53- 
 
 G(l, A, k- , l) = -1 + m. . and the maximum value occurs for m. . = 1. The 
 v ' ' ' ' J+l J+l 
 
 optimum mode digit sequence is 11111 with G(l, A, k, l) = 0. 
 
 If p > 6, we consider the sum of the k terms in the first and 
 
 p - k 
 second summations „ The combined term for 1 < k < — - — is therefore 
 
 ■ (m j + 2k " 2m j + 2k-l )2 " (2 + m j + 2k + l ' 2m j + 2k )(m j + 2k + l " 2m j + 2k } ° If 
 
 m. _. . = 1. then m. ^ must be 1 to satisfy -1 < 2 - m. _. < 1. Under this 
 j+2k+l ' J+2k J - J+2k - 
 
 condition the above expression is zero for m. ^ n , = or 1. Note that if 
 
 j+2k-l 
 
 m. , = 0, the only way to make the next lower term (replacing k by k - l) 
 
 J+2K-1 
 
 equal to its maximum value of zero is to choose m. „, _ = m. „, _ = 0. By 
 
 j+2k-2 j+2k-3 
 
 induction this same choice must be made when k = 1 which would demand that 
 
 m. = m. =0 contrary to the hypothesis . On the other hand, if m. = 1 
 J J-1 j+2k-l 
 
 m. „, _ must be 1 and the next lower term is zero. If k = 1, m. =m. , =1 
 J+2k-2 j j-1 
 
 which satisfies the given boundary conditions. 
 
 It follows that the only optimum mode digit sequence is m. =1 for 
 j-l«:i<j+p-l with G(l, A, p, l) = for p > 2. 
 
 ^-,1.3 A Summary of Maximum Gains 
 
 In later sections the value of m. ^ or m . associated with the 
 
 J+P-2 J 
 
 maximum gain over a special sequence with given boundary mode digits will be 
 
 of interest. Whether the value of m. _ or m. is desired will depend on 
 
 J+P-2 J 
 
 whether a left or right directed recoding is being analyzed. To distinguish 
 
 these two cases G(m. _, P, p, m. n ) will be listed as 
 
 J-1 J+P-l 
 
 G(m. _, Po p. m. _ i m. _) and G(m.: m. ., P. p, m. , ) in Tables ^-1 and 
 j-1' *' J+P-1' J+P-2' v j' j-1' ' *' J+P-l 
 
 k-2 respectively. 
 
 The optimum mode digit sequence associated with a given set of 
 allowable boundary mode digits usually does not allow both values of 
 
-5fc- 
 
 OJ 
 I 
 
 ft 
 
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 •i~3 
 
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 ts) 
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 S3 
 
 ft 
 
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 OJ 
 
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 OJ 
 
 -=f 
 
 OJ 
 
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■55- 
 
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 ri 
 
 T"3 
 
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 tf 
 
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 ■•"a 
 
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 S 
 
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 '0\ 
 
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 CVJ 
 H 
 
 pq 
 
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 ft 
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 ft 
 
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 ft 
 
 IS1 
 
 ft 
 
 S3 
 
 ft 
 
 [Xl 
 
 1=3 
 
 ft 
 
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 ft 
 
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 OJ 
 
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 ii 
 
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■56- 
 
 m and m.o In many cases the value which is not included can be specified 
 J+P-2 J 
 
 as an additional restriction on the maximum gain. The method of evaluating 
 
 the maximum gain under this restriction is similar to the one used in the 
 
 preceding sections. The results of these calculations are included in 
 
 Tables 4-1 and 4-2. A dotted line indicates that the value of m . _ or m. 
 
 J+P-2 J 
 
 is not compatible with the given boundary mode digits. The maximum gain with 
 
 m unspecified is always the larger of G(m. n , P, p, m. ; 0) and 
 
 G(m ,, P, p, m. ., ; l). The maximum gain with m. unspecified is similarly 
 j-1' * *' J+P-l J 
 
 defined. 
 
 4.1.4 Choice of the Next Mode Digit 
 
 In left or right directed mode functions the choice of m. is 
 determined by K. or S. when m. is arbitrary with respect to the range restric- 
 tion. In this analysis K. and S., as defined by Eqs . 3-3 an( i 3-6, are minimal 
 characteristic functions. They must determine m. such that the recoded 
 representation has a maximum number of zero digits for all y £ Y. 
 
 4.1.4.1 An Example 
 
 It can be shown that the minimal requirement is not sufficient to 
 uniquely define m. in all cases. For example, consider a left directed minimal 
 recoding in which m. -0 ; y. =y. = 1 and y. = for < j < i - 1. 
 Decomposition of this partial representation in accordance with Definition 4-8 
 yields a ZAU series of special sequences, where Z contains (i - l) elements, 
 
 A contains two elements and U contains one. The choice of m. in this case is 
 
 l 
 
 clearly independent of the range restriction. It is equally clear by the 
 results of the preceding section that the maximum gain in zeros over the 
 three sequences combined is a function of the mode digits at their boundaries. 
 
 
-57- 
 
 
 
 
 z 
 
 
 A 
 
 
 U 
 
 
 m 
 
 
 
 ra o 
 
 m l *•• 
 
 m i-2 
 
 m i-l 
 
 m. 
 l 
 
 Vi = 
 
 
 
 X 
 
 
 
 
 ... 
 
 o 
 
 o 
 
 1 
 
 i 
 
 
 The boundary mode digits are 0, m. , and m. . Since m. =0, 
 Theorems 4-8 and 4-9 apply. Because p = 1 in the U sequence, G(m., U, 1, 0) = 
 for m. = or 1. Our goal is to maximize the zero gain over all three 
 sequences, so it is necessary to examine the effect of m. on A. By Theorem 4-12, 
 G(m. , A, 2, 0) = when m = 0. By Theorems 4-13 and 4-15, 
 G(m. , A, 2, l) = for m. = or 1. Thus G(m. , A, 2, m., U, 1, 0) = 
 G(m. , A, 2, m. ) + G(m., U, 1, 0) = when m. = or 1. 
 
 We still can not say that an arbitrary choice of m. will yield a 
 
 minimal recoding of the given representation. Note that m. = demands that 
 
 m. _ = to achieve a maximum gain of zero over AU. If the maximum gain over 
 1-2 
 
 the special sequence to the left of A is always less than m. = than when 
 m. = 1, the optimum choice for m. is 1. In this case the Z sequence appears 
 on the left of A and has a maximum gain of zero when m. = since m =0. 
 Since m. = necessarily or by choice when m. = or 1, a minimal recoding 
 of ZAU can be achieved with m. = or 1. The maximum gain in either case is 
 zero. The two possible minimal recodings are given below. 
 
 
 
 
 z 
 
 
 A 
 
 u 
 
 
 
 z 
 
 
 A 
 
 u 
 
 m 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 1 
 
 
 
 
 1 1 
 
 
 
 X | 
 
 1 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 1 
 
 
 
 
 1 
 
 1 
 
 X 
 
 
 
 
 
 
 
 1 
 
 1 
 1 
 
 1 
 
 1 
 
 
 
 
 1 
 
 I 
 
-58- 
 
 It is easy to construct other examples in which the minimal char- 
 acteristic uniquely determines m. . If the partial representation is a U 
 sequence with m. = or 1 and p > 3^ then m. = 1. If the partial represen- 
 tation is a Z sequence with m. n = or 1, then m. =0. 
 
 l+l ' l 
 
 Although these examples are stated in terms of a left directed 
 recoding_, their counterparts can be constructed for right directed recodingSo 
 In either case^, the choice of m. to achieve a minimal representation may or 
 may not be arbitrary . This suggests a class of minimal characteristic 
 functions corresponding to the class of minimal mode functions . 
 
 4.1, ^.2 Special Restrictions on the Set of Independent Variables 
 
 The theorems of section 4,1.2 and the preceding example make it 
 apparent that the minimal characteristic function, K. , must depend on y.,-, and 
 some number of digits to its left. To simplify the development of the class 
 of MiniMal K. and, therefore, the class of m ini m al L.,<£, * , is initially 
 
 1 ' » 1/ ' 1+1 
 
 restricted to those digits which lie to the left of y. n . It will be shown 
 
 & J i+1 
 
 later that these classes include K. and L. in which i. n contains digits on 
 
 l i l+l 
 
 both sides of y. 
 
 J i+1 
 
 A minimal L(y) may consist of n identical minimal L. . Even with 
 
 Y. , restricted. L _ may be a function of n + 2 bits, i.e., \ 
 l+l ' n-1 n-1 
 
 = L _ (X 3 y , i ) where Y = (y.; < i < n - l). To allow L. to have the 
 n-1 n' n n n x — — i 
 
 same Boolean form for < i < n - 1, we let YT 1 , include at least 'n digits 
 
 - - ' l+l 
 
 to the left of y. even though (n - i - l) of these are always duplicates 
 of y . Because of this convention and the fact that we are dealing with 
 
 minimal I., it is also possible to define y n = \ ■, = 1> n (\ . y . i ). 
 
 l' r J -1 -1 -l x o' o^ o 
 
 Y 1 
 
 It is easy to visualize a recoder in which Y7 . is restricted as 
 J l+l 
 
 described above. Imagine that the recoding occurs at the n position of a 
 
-59- 
 
 shift register containing y. During each recoding step, y is shifted right 
 
 th 
 one position with y duplicated in the sign position. The n position and 
 
 any number of positions to the left of the n up to and including the 
 position are inputs to the recoder. 
 
 A similar set of comments apply to right directed recodings. Here 
 again the theorems of section U.1.2 make it apparent that the minimal char- 
 acteristic function, S., must depend on y. and some number of digits to its 
 right. The development of the class of minimal S. and^ therefore^ the class of 
 minimal R.,l/l, is simplified by restricting YV to those digits which lie to 
 the right of y. . As in the left directed case,, it will be shown later that 
 
 these classes include S. and R. in which YV contains digits on both sides of 
 
 11 l 
 
 y i' 
 
 A minimal R(y) may consist of n identical minimal R. . We note that 
 
 R may be a function of n + 2 bits even though YV is restricted, i.e., 
 
 P = R (p ... y , i ) where i = (y. : 1 < i < n). By allowing TV to include 
 
 o o -1' o o o i' — — 1 
 
 at least n digits to the right of y. even though i of these are always zero, 
 
 R. may have the same Boolean form for 1 < i < n. Because of this convention 
 
 and the fact that we are dealing with minimal R., it is also possible to define 
 
 = p = R (p ., y , Y^). 
 n n n-1 n' n 
 
 A recoder in which YV is restricted as described above is easily 
 visualized, The recoding in this case occurs at the position of a shift 
 register containing y. During each recoding step, y is shifted left one 
 position with a zero inserted in the n position. The position and any 
 number of positions to its right up to and including the n are inputs to the 
 recoder. 
 
-60- 
 
 4.1.4.3 Criteria for Choosing the Next Mode Digit 
 
 The method of selecting the next mode digit in either a left or 
 right directed recoding may be described as a generalization of the technique 
 used in section 4.1.4.1 In the left directed case the partial representation 
 is treated as a unique series of special sequences with X. representing the 
 right hand boundary condition for the right most sequence, In the right 
 directed case the partial representation is treated as a unique series of 
 special sequences where p is the left hand boundary condition for the left 
 most sequence. In either case the analysis begins by considering the given 
 boundary mode digit and the special sequence associated with it. In certain 
 cases several special sequences beyond the first must be considered before 
 the choice of m. is determined. The following definitions and theorems 
 establish the criteria upon which this choice is based. 
 
 Definition 4-11 : Partial Sequences 
 
 Let X(w) define a left or right directed partial sequence. 
 
 X T (w) = (y. ., , ... v., y. .,) 
 L \ / VJ i+l-w' J i' J i+l y 
 
 X R (w) - (y., y. +1 ... y. +w ) 
 
 where < w < n + A with A - if n is even and A = 1 if n is odd. 
 
 X and X can each be represented by a unique series of special 
 L n 
 
 sequences. Let these sequences be denoted by P where P is associated with 
 
 iC _L 
 
 the given boundary mode digit X. or p . . 
 
 x L (w) = p t ... p k ... P p l 
 
 x r («) = p i; p 2 ... P k ... P 
 
-6i- 
 
 The variable w is related to t by 
 
 t 
 
 w(t) = Ep (4:7) 
 
 where p n is the number of digits in P, . If w - n + A, the total number of 
 k k 
 
 partial sequences will be designated by T. Thus,, w(t) = n + A. 
 
 Definition 4-12 : Allowable Sequence of Boundary Mode Digits 
 
 A sequence of boundary mode digits is allowable with respect to a 
 
 given m. , m. and X(w) if and only if adjacent boundary mode digits satisfy 
 
 Theorems 4-3, 4-7 and 4-11 . 
 
 An allowable boundary mode digit sequence with respect to a given 
 
 X(w) will be denoted as follows, 
 
 B(m._ ,,\. m._ / 1 1 m. ) = (m._ / , \ ... m._ , n \. m._ / \; m. ) 
 
 v i+u(t)' i+u(o.r i' v iTu(t) iTu(l)' iTu(o)^ i' 
 
 The function u(t) is defined as 
 
 t 
 u(t) = -1 + Ep, (4:8) 
 
 k=l k 
 
 where u(o) = -1 
 
 Definition 4-13: Composite Gain Function 
 
 The composite gain over X(w) with m. and m. given is defined as 
 
 follows 
 
-62- 
 G[X(w), B(m. Tu(t); m. ±1 ; m.)] = G^^y P ±t P] _, m. ±1 ; m. ) 
 
 + J Q G(m i-u(k)> V V %(k-l) ] 
 
 .K. — c. 
 
 Definition ^--l^ : The Set of Optimum Boundary Mode Digit Sequences 
 
 Let Q(m._ / , \= m. n ; m.) denote the set of all B(m._ /, >, . m. _, ; m. ) 
 v i+u(t)' 1+1/ l i+u(tj' i+l' l 
 
 which for a given X(w), m. and m. cause G[X, B] to assume its maximum 
 value, G[X(v), Q(m._ ,,\. m. , ; m.)J„ 
 
 The following theorem provides a method of generating 
 
 Q( m - — t^M m - -, y m - ) given X(w), m. _ and m. . More importantly, it yields the 
 i+u(tr i±l i i±l i 
 
 maximum composite gain,, G[X, Q] . 
 
 Theorem U-l6: Given X(w), m. , and m. , let 
 l+l i' 
 
 G[X(w - p t ), Q(m._ u(t _ ir m. ±1 ; m.)] = gO*^^)). 
 Let G Q (t) - G(m._ u(t) , P^ p t , 0) + gO*^^ = 0) 
 
 and Gl (t) = G(m._ u(t) , P^ p t , l) + gO^^) = D 
 
 where m._ / \ = or 1, 
 lTu ( t ) 
 
 If G (t) > G^t) (^:10) 
 
 then m._ / ,\ = is the (t - l) boundary mode digit in any element of 
 
 Q(m._ /, v. m. .; m. ) and g(m._ /, %) = G (t). 
 i+u(t)' i+l' i toV iTu(t) o 
 
-63- 
 
 If G Q (t) < G 1 (t) (4:ll) 
 
 \th 
 m. /, \ = 1 is the (t - l) boundary mode digit in any element of 
 
 Q(m i+u(t)> II W m i } and g(m i-u(t) } = G l (t) - 
 
 If G Q (t) = G 1 (t) (4:12) 
 
 th 
 m / \ = or 1 may be the (t - l) boundary mode digit in any element of 
 
 Q(m._ /.>, m. .. ; m. ) and g(m._ ,,% = G (t) = G, (t). 
 i+u(tj i±l i iTu(tJ o 1 
 
 In these equations G(m._ , , \, P, , p, , m /, ., \ is assigned an 
 
 ^ v i+u(t)' V t' i+u(t-l) 
 
 arbitrarily large negative value if m /, % and m.^. - _, s are not allowable 
 
 J to to i+u(t) lTu(t-l) 
 
 boundary mode digits with respect to P 
 
 t 
 
 Proof: For a given value of m._ / ,\. G(m._ / \, P , , p, , m._ /, n \) must have 
 to i+u(t)' i+u(tj' t' t' i+u(t-l) 
 
 a finite value for either m._ /_, n \ = or 1 or both since at least three sets 
 
 i+u(t-l ) 
 
 of boundary mode digits are allowable for any special sequence. 
 
 Since P. . m. , and m. are given, g(m._ /, \ = 0) = G(0, P n , p, , m. . ; m. ) 
 1' i±l i ' i+u(lj ' V 1 i±l i 
 
 and g(m._ ,. \ = 1 ) = G(l, P, , p, , m. n ; m. ) . If m /n \ and m. . are not 
 
 iTu(l) v ' 1 } *l/ i±l' i i+u(l) i±l 
 
 st 
 
 allowable mode digits with respect to P n , then m /. \ can not be the 1 
 
 1 i+u(l; 
 
 digit of any optimum boundary mode digit sequence. Taking the other value of 
 
 ■ -r /-, \) g( m -^ f n \ ~ 0) an ^ g(m._ /_x = l) are formed as follows. 
 i+u(l) i+u(2) i+u(2j 
 
 m 
 
 g(m i+u(2) = 0) = G( °' P 2' P 2' m i+u(l) } + g(m iTu(l) ) 
 
 g(m i+u(2) = X) = °" (l > P 2' P 2> m i+u(l) } + g(m iTu(l) ) 
 
 If either of these are negatively infinite, the maximum composite gain over 
 
 P n P^ and the choice of m._ fn \ are clearly defined. The process continues 
 1 2 l+u ( 2 ) 
 
 when g(m._^ /.% = 0) and g(m._ / , \ = l) have finite values. 
 iTu(t) iTu(t) 
 
-6k- 
 
 Assume that g(m._ /, , \ = O) and g(m._ , . N = l) are finite for 
 toV lTu(t-l) &v i+u(t-l) 
 
 some t > 1. The value m / s must assume for maximum composite gain over 
 X(n) is not obvious since the special sequences beyond the (t - l) have not 
 been examined. It is possible to avoid simultaneous consideration of all 
 these sequences by computing the maximum composite gain corresponding to the 
 
 sets Q(m /.\ = 0, m. n ; m.) and Q(m._ ,,\ = 1, m. _,: m.). Depending on 
 
 i+u(t) ' i±l' l i+u(t) 1+1 1 
 
 the special sequences which follow the (t) sequence, the elements of either 
 or both of these sets represent partial optimum boundary mode digit sequences . 
 This follows from the fact that the optimum boundary mode digit sequence must 
 
 include m._ ,, % = or 1. 
 
 i +u ( t ) 
 
 If m._ /, N = 0, then by trichotomy G (t) must be greater than, less 
 i+u(t) ' o ■ ' 
 
 than or equal to G (t). If G (t) > G, (t), then G (t) is clearly the maximum 
 composite gain over X(w) given m. _, and m.. with m._ ,, \ = 0. Therefore, 
 
 * & v / b 1±1 x i+u(t; ' 
 
 g(m.- ,, % = 0) = G (t ) . It follows that m._ /, n \ = must be the (t - l) 
 i+u(t) o v i+u(t-l) 
 
 boundary mode digit of Q(m,_ / , » = 0, m. , ; m. ). A similar argument follows 
 
 i+u(t) ' 1±1' 1 
 
 if G (t ) < G., (t ) . In this case g(m._^ ,, % = 0) = G.ft) and m._ ,, . \ = 1 must 
 o v l x ov iTu(t) 1 i+u(t-lj 
 
 be the (t - l) boundary mode digit of Q(m / \ = 0, m. ; m. ). Likewise if 
 
 G (t) = G n (t) = g(m.__ . v = 0), then m._ ,, _ v = or 1 may be the (t - l) 
 o 1 toV i+u(t) ' i+u(t-l) 
 
 boundary mode digit of Q(m._ / \ = 0, m. ; m.). 
 
 If m._ /, ^ = 1» the same arguments apply but note that G (t) may 
 lTu ( t ) ' o v 
 
 not bear the same relationship with G, (t) as in the case m._ ,.\ = 0. It 
 
 r l v i+u(tj 
 
 follows that the (t - l) boundary mode digit in Q(m / \ = 0, m. ; m. ) 
 
 and Q(m._ / . \ = 1, m. ,; m. ) may not be the same. Note also that 
 v i+u(t) ' i±l' i' J 
 
 g(m / , \ = 0) is not necessarily equal to g(m ,,\ = l) even if G (t) bears 
 
 i+u(t) j i e\ 1+u (t) o v 
 
 the same relationship to G (t) in both cases. 
 
-6 5 - 
 
 Since P is the only special sequence examined initially, it is 
 necessary to consider the choice of m. when m.— ,_, \ = and 1 assuming that 
 
 J i i+u(i; to 
 
 both of these values are allowed. The possible choices of m /n \ and m. can 
 
 i+u(l) 1 
 
 be divided into three basic combinations. Each of the following three 
 
 theorems concerning the optimum choice of m. deals with one of these 
 
 l 
 
 combinations . 
 
 Theorem h-YJ i If P., and m. , are such that only one value of m._ /., \ is 
 1 i±l iTu(lJ 
 
 allowed, then m. is never arbitrary and is always determined by the optimum 
 
 mode digit sequences corresponding to the allowed value of m._ /_, \. 
 
 i+u(l; 
 
 Proof : Tables ^-1 and k-2 show that the following cases satisfy the hypothesis 
 
 Left Directed 
 
 Right Directed 
 
 G(0, Z, p > 1, 0; 0) = 
 
 G(l; 1, Z, p > 1, 1) = -p. 
 
 G(l, U, p > 1, 1; 1) = p 
 
 U 
 
 G(0; 0, U, p > 1, 0) = 
 
 In all cases m. _, P., and the allowed choice of m._ /. N establish 
 i±l' 1' i+u(l) 
 
 a single optimum mode digit sequence over P in which the value of m. is fixed, 
 By Theorems h-k } h-G } k-Q and J+-10 any alteration to permit a different value 
 for m. will produce a disallowed mode digit sequence. 
 
 Theorem U-l8: If P., and m. . are such that m._ /n N 
 1 l+l iTu(l) 
 
 may be or 1 and the m. 
 
 corresponding to the maximum gain over P.. has the same fixed value in either 
 case, the optimum choice for m. is this value. 
 
■66- 
 
 Proof : Tables 4-1 and 4-2 show that the following cases satisfy the hypothesis 
 
 Left Directed 
 
 Right Directed 
 
 G(0; 0, Z, p > 2, 0) = 
 G(0; 0, Z, p > 2, l) = -1 
 
 G(0, A, p > 2, 1; l) =• 
 G(l, A, p > 2, 1; l) = 
 
 G(l; 1, U, p > 2, 0) = Pu -1 
 
 G(l; 1, U, p > 2, 1) = 
 
 J \J 
 
 G(l; 1, A, p > 2; 0) - -1 
 G(l; 1, .A, p > 2; l) = 
 
 Since the optimum mode digit sequences associated with m.__ , \ - 
 and 1 all include the same fixed value for m., any change in m. will produce 
 either a disallowed mode digit sequence or a lower gain over P . Therefore, 
 regardless of the special sequences beyond P , the optimum value of m. is 
 determined. 
 
 Theorem k-lQi If P n and m. n are such that m.— ,. \ may be or 1 and the m. 
 1 l+l i+u(lj 1 
 
 corresponding to the maximum gain over P has a different value in each case, 
 
 the optimum choice of m. is determined by the following procedure. 
 
 Starting with t = 1. Q(m. /,\ = 0, m. n ; m. ) and 
 
 ' v i+u(tj ' 1+1 1 
 
 Q(m / \ = 1, m. ; m. ) are developed according to Theorem k-±6 until one 
 of the following conditions occurs. 
 
 m. = if 
 
 i 
 
 G[ 
 
 X > Q(m i-u(t)' m i±l' ° )] > G[X ' Q(m iTu(t)' m i±l' 1)] (k:13) 
 
 for m /, \ = and 1. 
 
 i+u(t ) 
 
-67- 
 
 m. = 1 if 
 
 1 
 
 G[X, Q(m 
 
 m 
 
 iTu(t)' i±l 
 
 i 0)] < G[X, Q(m iTu ( t ), ^ i±1 ^ 1 )J (^:1^) 
 
 for m /, v = and 1. 
 
 i+u(t ) 
 
 m. = or 1 if 
 
 i 
 
 G ' X > «»iTu(t)' °W 0)1 = G[X ' Q(m iTu(t)' "W 1)J (k:13) 
 
 for m . _ / ^ \ = and 1 , 
 i+u(t ) 
 
 Proof ; Tables ^--1 and k-2 show that the following cases satisfy the hypothesis 
 
 Left Directed 
 
 G(0, Z, p > 1, 1; 0) = -1 
 
 G(l, Z, p > 1, 1} 1) = -p z 
 
 G(0, U, p > 1, 0; 0) - 
 
 G(l, U, p > 1, 0; 1) = Pu -1 
 
 G(0, A, p > 2, 0; 0) = 
 
 G(l, A, p > 2, 0; 1) = -1 
 
 Right Directed 
 
 G(0; 0, Z, p = 1, 0) = 
 
 G(l; 0, Z, p = 1, 1) = -1 
 
 G(0; 1, U, p = 1, 0) = 
 
 G(l; 1, U, p = 1, 1) = 1 
 
 G(0; 0, A, p > 2, 0) = 
 
 G(l; 0, A, p > 2, l) = 
 
 In these cases the conditions described by Eqs . k\Y^, ^:1^- and U:15 
 are never satisfied for t = 1 since Tables ^— 1 and k-2 show that 
 
 G[X, Q(m._ u(l) = 0, m. ±1 ; 0)] > o[X, Q(m„ u(l) = 0, m.^; l)] 
 G[X, «(m l:pu(l) = 1, mi±1 ; 1)] > 3[X, <J(« 1Tu(l) ■ 1, * 1±i J 0)] 
 
 The maximum gain appearing on the right side of these inequalities does not 
 exist in all cases. 
 
-68- 
 
 As indicated in the proof of Theorem 4-l6, G[X, Q(m , , <., m. ,: 0)1 
 
 i+u(t; i±l 
 
 must be >, <, or equal to G[X, Q(m._ ,,\. m. n ; l)] for either value of 
 
 i+u(t) i±l' 
 
 m._ / , \ "by the law of trichotomy. If one of the conditions described by 
 iTu(t) 
 
 Eqs. 4:13, 4:l4 and 4:15 is satisfied after examining t special sequences, 
 
 it is not necessary to consider additional sequences. This follows from the 
 
 fact that the maximum composite gain over the special sequences beyond P must 
 
 occur for m._ , « = or 1, 
 i+u(t ) 
 
 If the Eq. 4:13 is satisfied for example, then 
 
 ° [X > «K*u(t)< °W ° )] > ~ GlX ' Q(m Wu(t)' m i±l' 1)] 
 
 for either value of m /, \. Thus m. = is the only value of m. that can be 
 
 i+u(t) 1 l 
 
 associated with the maximum composite gain over the entire partial sequence 
 regardless of what the special sequences beyond P happen to be„ Similar 
 
 statements apply if Eqs. k:lk or 4:15 are satisfied for m , \ = and 1. 
 
 If none of the conditions given by Eqs. 4:13, 4:l4 and 4:15 occur 
 for t < T, then one must occur when t = T. Since m / \ = y . in a left 
 
 ' l+u(Tj -1 
 
 directed recoding and in a right directed recoding, one of the conditions 
 
 must be satisfied for either m , \ = or m ,\ = 1. 
 
 i+u(T) i+u(t) 
 
 The criteria established by Theorems 4-17, 4-l8 and 4-19 will be 
 used to develop the subclasses of restricted, minimal L. and R. in sections 
 4.2 and 4.3 respectively. 
 
•69- 
 
 k.2 The Subclass of Restricted, Minimal Left Directed Mode Functions 
 
 This section is divided into two subsections. In the first sub- 
 section the set consisting of all left directed partial sequences, X (n + A), 
 
 J_i 
 
 and X. -, is partitioned into three subsets by an exhaustive case analysis, 
 l+l 
 
 These subsets correspond to the three possible ways of choosing X. to achieve 
 a minimal recoded representation. In the second subsection these subsets are 
 used to establish a necessary and sufficient restricted, minimal L. with an 
 auxiliary set of min terms. The subclass of restricted, minimal L. contains 
 all L. that can be formed by joining any combination of min terms to the 
 necessary and sufficient L. . The particular L. which yields the canonical, 
 minimal representation is investigated. 
 
 ^+.2.1 Case Analysis of the Partial Left Directed Sequence 
 
 Theorems ^--17^ ^-18 and U-19 are used to partition the set 
 
 (X (n + A), X. ) into three subsets corresponding to X. = 0, X. = 1 and 
 
 X. - or 1. For convenience, the notation for the maximum composite gain 
 
 is modified so that the boundary mode digits separate the special sequences. 
 
 For example, if UA has boundary mode digits X. ,^\ = 1, X, /n N = 1 and 
 
 i-u(2J i-u(l) 
 
 X - 0, then the maximum composite gain appears as G(1U1A0; X. ) = p - 1. 
 
 For identification, the maximum gain over each sequence is summed in order of 
 occurrence. No attempt is made to show all possible optimum boundary mode 
 digit sequences . 
 
 Case I: X , = 1 
 
 l+l 
 
 Let P., = U so that y. . =1. Theorem +-17 applies so X. - 1 
 1 l+l l 
 
 Let P n ^ A so that y. n =1. Theorem U-l8 applies so X. = 1 
 1 J i+1 ** l 
 
-TO- 
 
 Case II: 
 
 \. - = 1 
 
 1+1 
 
 Let P = Z so that y = 0. Theorem 4-19 applies, so the opt 
 
 lmum 
 
 choice of X. depends on sequences to the left of Z 
 
 UZ; Pu > 1, 1 < p z < 2 
 
 Table k-1 and Theorem 4-l6 are used to obtain the following composite gains 
 
 G(0U0Z1; 0) = - 1 
 
 G(0U0Z1; l) = + q 
 
 G(1U0Z1; 0) = ( Pu -1) - 1 
 
 G(1U1Z1; 1) = Pu - p z 
 
 If p z = 1, q = -co; if p = 2, q = -2, 
 satisfied, so AUZ must be analyzed. 
 
 None of Eqs . 4:13, k:lk or 4:15 are 
 
 AUZ; p A > 2, Pu > 1, 1 < p z < 2 
 
 G(0A1U0Z1; 0) = + (p„-l) - 1 
 G(1A1U0Z1; 0) = + (Py-l) - 1 
 
 G(0A1U1Z1; l) = + Py - p z 
 G(1A1U1Z1; l) = + p T . - p 7 
 
 U Li 
 
 If p = 1, Eq. 4:l4 is satisfied so X. = 1. If p = 2, Eq. 4:15 is satisfied 
 
 so X. =0 or 1. 
 
 l 
 
 UZ; p > 1, p > 3 
 
 G(0U0Z1; 0) = - 1 
 
 G(0U0Z1; l) = - 2 
 
 G(1U0Z1; 0) = ( Pu -1) - 1 
 
 G(1U0Z1; 1) = (Py-1) - 2 
 
 Eq. 4:13 is satisfied, so X. - 0, 
 
 
-71- 
 
 AZ; P A > 2, 1 < p z < 2 
 
 G(0A0Z1; 0) = - 1 
 G(1A0Z1; 0) = -1 - 1 
 
 G(0A1Z1; l) = - p 
 G(1A1Z1; l) = - p r 
 
 Since none of Eqs . 4:13, h:lk or 4:15 are satisfied, UAZ and ZAZ are analyzed 
 next. 
 
 UAZ ; p n > 1, p > 2, 1 < p < 2 
 
 U 
 
 A 
 
 G(0U0A0Z1; 0) = + - 1 
 
 G(0U0A1Z1; l) = + - p r 
 
 G(1U1A0Z1; 0) = p - 1 - 1 
 
 G(1U1A1Z1; 1) = ?u + - p z 
 
 AUAZ ; P A > 2, P y > 1, P A > 2, 1 < p z < 2 
 
 G(0A1U1A0Z1; 0)=0+p-l-l 
 G(1A1U1A0Z1; 0)=0+p-l-l 
 
 G(0A1U1A1Z1; 1 ) = + p TT + - p 7 
 
 U Zi 
 
 G(1A1U1A1Z1; 1 ) = + p TT + - p^ 
 
 U Zi 
 
 If p = 1, Eq. 4:l4 is satisfied so X. = 1. If p = 2, Eq. 4:15 is satisfied 
 
 Li 1 Zj 
 
 so X. =0 or 1. 
 
 l 
 
 ZAZ ; p z > 1, P A > 2, 1 < p z < 2 
 
 G(0Z0A0Z1; 0) = + 
 
 G(0Z0A1Z1; l) = + - p r 
 
 G(1Z1A0Z1; 0) = -p - 1 - 1 
 
 G(1Z1A1Z1; 1) = -p z + - p z 
 
 UZAZ; p > 1, p > 1, p > 2, 1 < p < 2 
 
 U 
 
 A 
 
 G(0U0Z0A0Z1; 0)=0+0+0-l 
 
 G(0U0Z0A1Z1; l)=0+0+0-p r 
 
 G(1U0Z0A0Z1; 0) = (p n -l) +0+0-1 G(1U0Z0A1Z1; l) 
 
 ( Pu -1) +0+0 
 
 If p„ = 1, Eq. 4:15 is satisfied so \. =0 or 1. 
 Zi i 
 
 If p„ = 2, Eq. 4:13 is satisfied so X. = 0. 
 z i 
 
-72- 
 
 AZAZ; P A > 2, p z > 1, p A > 2, 1 < ?z < 2 
 
 G(0A0Z0A0Z1; 0) = + + 0-l 
 G(1A0Z0A0Z1; 0) = -l+O+O-l 
 
 G(0A0Z0A1Z1; l) 
 G(1A0Z0A1Z1; l) 
 
 + + - p r 
 
 = -1 + + 
 
 If p = 1, Eq. 4:15 is satisfied so X. =0 or 1, 
 
 CJ 1 
 
 If p = 2, Eq. 4:13 is satisfied so X. = 0. 
 
 Li 1 
 
 AZ; p A > 2, p z > 3 
 
 G(0A0Z1; 0) = - 1 
 G(1A0Z1; 0) = -1 - 1 
 
 G(0A0Z1; l) = - 2 
 G(1A0Z1; l) = -1 - 2 
 
 Eq. 4:13 is satisfied so X. = 0, 
 
 Case III: 
 
 X. _ = 
 l+l 
 
 Let P = U so that y. =1. Theorem 4-19 applies so the optimum 
 choice of X. depends on sequences beyond U. The order of decomposition 
 insures that only an A sequence may appear on the left of U. 
 
 AU; P A > 2, P y = 1 
 
 Table 4-1 and Theorem 4-19 a ^ e used to obtain the following composite 
 gains . 
 
 G(0A0U0; 0) = + 
 G(1A0U0; 0) = -1 + 
 
 G(0A1U0; l) = + 
 G(1A1U0; l) = + 
 
■73- 
 
 UAU; Pu > 1, p A > 2, Pu = 1 
 
 G(OUOAOUO; 0) = + + 
 
 G(0U0A1U0; l) = + + 
 
 G(1U1A0U0; 0) = p - 1 + 
 
 G(1U1A1U0; l) = p + + 
 
 AUAU; p A > 2, Pu > 1, P A > 2, ^ = 1 
 
 G(0A1U1A0U0; 0) = + p-l + 
 
 G(1A1U1A0U0; 0) = + p-l + 
 
 G(0A1U1A1U0; l) = + p+0 + 
 G(1A1U1A1U0; l) = + p+0 + 
 
 Eq c ktlk is satisfied so X. = 1 
 
 l 
 
 ZAU; p > 1, p > 2, p = 1 
 
 A 
 
 U 
 
 G(OZOAOUO; 0) = + + 
 
 G(0Z0A1U0; l) = + + 
 
 G(1Z1A0U0; 0) - -p - 1 + 
 
 G(1Z1A1U0; l) = -p_ + + 
 
 Lx 
 
 AZAU; p A >2, p z > 1, p A > 2, V]J = 1 
 
 G(OAOZOAOUO; 0)=0+0+0+0 
 G(1A0Z0A0U0; 0)= -1+0+0+0 
 
 G(1A0Z0A1U0; l)=0+0+0+0 
 G(1A0Z0A1U0; l)= -1+0+0+0 
 
 Eq. 4:15 is satisfied so X. = or 1, 
 
 UZAU; Pu >1, V„> 1, P A > 2, p TT = 
 
 Z - ' ^A - 
 
 U 
 
 G(OUOZOAOUO; 0)=0+0+0+0 
 
 G(0U0Z0A1U0; l)=0+0+0+0 
 
 G(1U0Z0A0U0; 0) = (Py-l) +0+0+0 G(lUOZOAlUO; l) = (p y -l) +0+0+0 
 
 Eq. i+ : 15 is satisfied so A.. =0 or 1, 
 
■lh- 
 
 AU; P A > 2, Pu > 2 
 
 G(0A1U0; 0) = + (Py-2) 
 G(1A1U0; 0) = + (Py-2) 
 
 G(0A1U0; l) = + (Py-l) 
 G(1A1U0; l) = + ( Pu -1) 
 
 Eq. k'.lk is satisfied so X. - 1. 
 
 l 
 
 Let P., = A so that y. . 
 1 i+l 
 
 "beyond A must be considered. 
 
 1. Theorem U- 19 applies, so sequences 
 
 UA; Pu > 1, p A > 2 
 
 G(0U0A0; 0) = + 
 
 G(0U0AG; l) = - 1 
 
 G(1U0A0; 0) = (Prj-l) + 
 
 G(1U1A0; l) = p - 1 
 
 AUA ; p A >2, Pu > 1, p A > 2 
 
 G(0A1U0A0; 0) = + (Py-l) + 
 G(1A1U0A0^ 0) = + (Py-l) + 
 
 G(0A1U1A0; l) = + p - 1 
 G(1A1U1A0; l) = + p - 1 
 
 Eq. U : 15 is satisfied so \. =0 or 1. 
 
 ZA; p z > 1, p z > 2 
 
 G(0Z0A0; 0) = + 
 
 G(0Z0A0; l) = - 1 
 
 G(1Z1A0; 0) - -p + q 
 
 G(1Z1A0; l) = -p - 1 
 
 If p = 2, q = -oo; if p > 4 ; q = -1. None of Eqs . ^:13, htlk or U:15 are 
 satisfied, so UZA and AZA are analyzed next. 
 
 
■75- 
 
 UZA; p n > 1, p > 1, p > 2 
 
 U 
 
 A 
 
 G(OUOZOAO; 0) = + + 
 G(lUOZOAO; 0) = (Py-l) +0+0 
 
 Eq, 4:13 is satisfied so X. = 0. 
 
 G(OUOZOAO; l) = + - 1 
 G(1U0Z0A0; l) = (Py-l) +0-1 
 
 AZA ; p > 2, p 7 > 1, p > 2 
 
 A 
 
 A 
 
 G(OAOZOAO; 0) = + + 
 G(1A0Z0A0; 0) - -1+0+0 
 
 G(OAOZOAO; l) = + - 1 
 G(1A0Z0A0; l) = -1 + - 1 
 
 Eq. 4:13 is satisfied so X. = 0, 
 
 Case IV: 
 
 \. , = 
 
 i+I 
 
 Let P., = Z so that y. . =0. Theorem 4-17 applies so X. = 0. 
 1 i+I l 
 
 This completes the partitioning of the set (X (n + A), X. ). 
 Table 4-3 contains a summary of the results. In this table X is used to 
 represent the portion of the partial sequence that is not given explicitly 
 in terms of special sequences. The special sequences and their corresponding 
 lengths are listed in order of occurrence. If the length is fixed, the number 
 of digits is given, otherwise p , p or p represents an arbitrary length. 
 
 4.2.2 Derivation of the Restricted, Minimal Left Directed Mode Functions 
 
 The minimal L. derived in this section have X. , , y. n and 
 l i+I' i+I 
 
 YT J =(y. .;0<j<n + A-l)as independent variables. According to 
 
 Table 4-3, L. must have a value of 1 whenever X. = 1 and may have a value of 
 l l 
 
 1 whenever X. = or 1. 
 
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■76- 
 
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-78- 
 
 Definition 4-15: X. = NL. (\. n , y. ,, Y 1 _ ) 
 
 1 i i+l' i+l' l+l 
 
 Let NL. be a minimal left directed mode function in which 
 
 l 
 
 Y i + 1= (y i-j' 0<j<n + A-l). 
 
 NL. =1 whenever X. must be 1 according to Table 4-3. 
 11 & -j 
 
 NL. =0 whenever the choice of \. is arbitrary or according to 
 Table 4-3. 
 
 Theorem 4-20 : NL. is the necessary and sufficient restricted, minimal left 
 directed mode function. 
 
 Proof: The proof is immediate by appeal to Table 4-3. 
 
 Inspection of Table 4-3 shows that 
 
 X. = L.(\. ., y. _, Y^ .) 
 1 1 l+l' l+l' l+l 
 
 (4:16) 
 
 = X. j. _ V X. n y. n L. (1, 0, Y7 , ) V X. ,y. n L.(0, 1, Y 1 n ) 
 i+r'i+l i+l J i+l i y * ' l+l ' i+l^i+l i v } } l+l ' 
 
 Eq. 4:l6 is the same as Eq. 3:1 when Eq. 3:3 is substituted for K. (y. ., 1. ) 
 
 In order to illustrate the symmetry between L. (l, 0, Y7 ) and L. (0, 1, Y7 , ), 
 
 these two functions are developed first. The K. functions are then generated 
 
 by means of Eq. 3:3° 
 
 The family of L.(l, 0, YT .. ) functions is developed by considering 
 
 the partial sequences under Case II of Table 4-3. 
 
 L.(l, 0. Y7 _ ) must be 1 for all of the following left directed 
 1 ' } l+l 
 
 partial sequences beginning with y. . 
 
-79- 
 
 y. . . .. y. j. = (... 1101 . .. 01; 
 
 . .. 1101; . .. 11) 
 
 It follows that the necessary and sufficient L. (l, 0, Y7 ) is defined Toy 
 
 n* 
 
 J-J- 
 
 NL 
 
 ± (i, 0, Y^ +1 ) = Yl. ± y J y i-2k-i y i-2k (i+a7) 
 
 . =0 .-2J-1 i-2j k=Q 
 
 where n* = — - — with A* = if n is even and A - 1 if n is odd. On the first 
 
 step of a left directed recoding y. , = y so i = n - 1. When i = n - 1 } 
 
 the definition of n* insures that y. _ „ = y _ when n is odd and y. _ „ = y . 
 
 i-2n* -2 i-2n* -1 
 
 when n is even. Note that when n is even the n* term in Eq. kil'J is since 
 by definition y_ 2 = y = y . 
 
 L. (l, Y7 ) may be 1 for any of the following partial sequences 
 beginning with y. . 
 
 y i-j •'• y i-i y i = ( °°° 00101... 01); ... ; (... 0010l);(... 001) 
 
 y i-j °°° y i-i y i = ( °°° 1101 •" ° 10); '•• > ( °°° HOlO); (... 110) 
 
 Therefore, two distinct types of Boolean functions may be joined to 
 NL. (l, 0, Y7 , ). The general form of each is expressed below. 
 
 AL. . (1, 0, Y 1, . ) = y. „, 
 ij v ' ' i+1' i-2j 
 
 ir - 
 
 5k-i y i-2k 
 
 a 10 (k:l&) 
 
 HL„(l, 0, Y^) ==y._ 2 . 
 
 1C— U 
 
 ij 
 
 ■i+i' 
 
 rd-i 
 
 b 10 
 
 (^■19) 
 
 In these equations, a. . and b. . represent arbitrary Boolean functions of the 
 
 / \ o(n + A - 2j ) 
 
 (n + A - 2j ) digits to the left of y. . Any one of the 2^ 
 
 -i- *"^- j 
 
 different a. . and b. . functions may be associated with each AL. . (l, 0, Y7 -, ) 
 ij ij J ij ' ' i+l 
 
-80- 
 
 and BL. (l, 0, Y7 , ), and any combination of these functions for 1 < j < n* 
 may be joined to NL. (l, 0, Y7 , ) to form a particular L. (l, 0, Y7 ) . 
 
 The partial sequences associated with Case III of Table 4-3 are used 
 to develop the family of L. (0, 1, Y7 ) functions . 
 
 L. (0, 1, Y7 ) must be 1 for all of the following partial sequences 
 beginning with y. „ 
 
 ... y ± _j ... y^i = (••• 1101 ••• 01; ... ; ... 1101; ... 11) 
 
 Since these are the same sequences as those listed for L. (l, 0, YT" n ), the 
 
 l ' ' l+l 3 
 
 necessary and sufficient L. (0, 1, Yl ) is also defined by Eq. 4:17. 
 
 NL.(0, 1, Y 1 _ ) = NL.(1, 0, Y 1 n ) (4:20 ) 
 
 i\- l+l i v ' ' l+l v 
 
 L. (0, 1, Yf - ) may be 1 for any of the following partial sequences 
 
 beginning with y. . 
 
 ... y **« y±-i? ± = (••• ° 0101 ■•■ 01 ^ ••■ > (••• 00101 )i (••• ° 01 ) 
 
 ... y i . J ... y ± . 1 y ± = («.. noi ... 010); ... ; (... 11010); (... 110) 
 
 This set of partial sequences is identical to the set obtained for 
 L. (l, 0, YT 1 , ). Therefore, the two general types of Boolean functions that 
 may be joined to NL. (0, 1, YT 1 ) to form a particular L. (0, 1, Y7 , ) are 
 defined by Eqs h:lQ and k:lQ with a. . and b. . replacing a. . and b. . . 
 
 The subclass of restricted, minimal K. functions is formed by using 
 Eq. 3-3« Let NK. represent the necessary and sufficient K. . 
 
 NK. (Y 1 _) - NL.(0, 1, Y^ _ ) = NL.(1, 0, Y^ . ) (>:2l) 
 l i+l' l ' } l+l i v } 3 l+l 
 
■8l- 
 
 Note that NK. does not depend on y. , . The general forms of AK. . and BK. . 
 
 1 i+l ij ij 
 
 are given below. 
 
 AK. .(y. n , Y L n ) = y. n AL. . (0, 1, Y 1 _ )V y. .AL, .(l, 0, ^ . ) 
 lj^i+l' i+l y J i+1 ij i+l' J i+1 ij v ' ' i+l 
 
 (4:22) 
 
 BK. .(y. ., Y 1 . ) = y. n BL. .(0, 1, Y 1 . ) V y. ,BL. .(l, 0, Y L , ) 
 ij w i+l' i+l i+l ij i+l J i+1 ij i+l 
 
 (4:23) 
 
 Any combination of AK. . and BK. . joined to NK. forms a particular K. . 
 
 ij ij i i 
 
 These functions are used to define NL., AL. . and BL. .. 
 
 NL.(\. ., y. _, Y L _) = X. _y. . V (\. n V y. JNK. (4:24) 
 i i+l i+l' i+l i+l i+l i+l i+l i 
 
 AL. .(X. _, y. _, Y 1 . ) = (X. _ V y. , )AK. . (4:25) 
 
 IJ V 1+1' J l+1 ; 1+1 V 1+1 J l+1 IJ \ Si 
 
 BL. .{X. ., y. ., Y 1 _ ) = (\. . V y. , )BK. . (4:26) 
 
 l j v i+l' J i+1' i+l ' v i+l J i+l y ij v ' 
 
 Any minimal L. (X. ,, y. _, Y7,)in which FT" n includes only those digits to 
 J i v i+l' J i+1' i+l ' i+l J & 
 
 the left of y. can be constructed by joining an appropriate combination of 
 AL. . and BL. . to NL. . 
 
 To reduce the number of L. which must be considered, the following 
 
 i 
 
 restrictions are imposed. Let L. (0, 1. r J = L. (l, 0, Y7 1 , ) with 
 
 i ' i+l i v ' ' i+l 
 
 a.. =b.. = a. . =b.. =1 for all j. Under this restriction Eqs. 3:22 and 3:33 
 
 ij id id ij 
 
 become 
 
 AK. .(Y L .) = AL. .(0, 1, Y 1 J = AL. ,(l, 0, Y 1, J (4:27) 
 ij i+l ij i+l id 1+ 1 
 
 BK 
 -J 
 
 . .(Y 1 _) = BL. .(0, 1, Y? .) = BL. ,(1, 0, Y 1 _) (4:28) 
 ij i+l i,i i+l ij i+l 
 
■82- 
 
 Since NK. is not a function of y. , . any K. formed under these restrictions 
 1 i+l' l 
 
 Y 1 
 
 is a function of Y7 only. 
 
 The minimal K. constructed under these restrictions are called 
 
 l 
 
 symmetric, minimal left directed characteristic functions. Likewise all L. 
 formed with symmetric, minimal K. are called symmetric, minimal left directed 
 mode functions. Symmetric, minimal mode functions are also restricted. 
 
 The functions defined below are used to illustrate certain symmetries 
 in the subclass of symmetric, minimal L. „ 
 
 Definition k-l6: AK. , BK., AL. and BL. 
 l' l' l i 
 
 n* 
 AK. =~V AK. . (4:29) 
 
 j=l J 
 
 n* 
 BK = V BK (4:30) 
 
 j=l J 
 
 AL i = ^i + l^i + l^i (U:31) 
 
 BL. = (X. +1 Vy i+1 )BK. (4:32) 
 
 In the developments which follow, it is convenient to simplify NK. , 
 
 AK. and BK. and obtain their complements and duals. The lemmas in the 
 11 
 
 Appendix are introduced for this purpose. 
 
 Using Lemmas A-l, A-2 and A-3, NK. , AK. and BK. may be written as 
 follows . 
 
•83- 
 
 n* j-1 
 
 ™ ± - Vyi.2j.iTyi.2k-2 (4:33) 
 
 j=0 J k=-l 
 
 n* j-l _ n 
 
 M i = "V Vi- 2J T yi. 2k -i ■ BK i <^> 
 
 J— 1 K-U 
 
 i-fe^S^i-v (u:35) 
 
 j-i - 2J k=o ' i - 
 
 A simplified expression for NK. is obtained starting with Lemma A-k , 
 
 l 
 
 n*-l j-l _ _ n*-l 
 
 M i = V y i-2j % y i-2k-l V (y i-2n^ ^n*-^ % y i-2j-l 
 j=0 ° k=0 j=0 
 
 In the worst case i = n - 1, so y^.^V V^^-l = y -l-A^ y -2-A" By 
 definition y_ ± _ A = y_ 2 _ A for A = or 1. Thus y ± _ 2n * = y^n*-! for a11 i ' 
 
 n* j-l 
 
 "i = y y i- 2 j tt y i- 2 k-i (u:36) 
 
 j=0 J k=0 
 
 Lemma A-6 is used to derive AK. . 
 
 i 
 
 _ n*-l j-l n*-l 
 
 M i "" y i ^ .V y i-2j+l ,•' y i-2k * ^ y i-2n*+l^ y i-2n*^ .'' y i-2j 
 J=l k=l j=l 
 
 In the worst case, y n _ 1 _ 2 n* + l V Vl-2n* = y -A V y -l-A* ForA=0orl 
 y -A = y -l-A SO y i-2n*+l = y i-2n* f ° r a11 ± ' 
 
 K - y i v V yi . 2J+1 f y ± . 2k = K (^37) 
 
 j=l ° k=l 
 
-84- 
 
 The simplified expression for BK. is derived from Lemma A- 5 by usin£ 
 the same argument to eliminate the final product term. 
 
 n* ,1-1 
 
 "i-^ywiFi-a-"? (U:38) 
 
 BK. 
 
 The dual of NK. is obtained by taking the dual of the complement of 
 Eq. 4:36. 
 
 j=0 d k=0 
 
 The following identities can now be easily verified, 
 
 AK. = y.NK. (4:4o) 
 
 ill 
 
 BK. = y.NK (4:4l) 
 
 ill 
 
 NK. V NK° = NK (4:te) 
 
 ill 
 
 AK. V AK = AK (4:43) 
 
 ill 
 
 BK. N/ BK - BK (4:44) 
 
 ill 
 
 Of the many restricted, minimal L. that can be formed, there are 
 three of particular interest. One is NL. as defined by Eq. 4:24. The other 
 two are defined below. 
 
-85- 
 
 Definition 4-17: X. = SL. (X. _. y. n , Y7 , ) 
 L i l - 1+1/ "'i+l' i+l 
 
 SL. is called the simplest, minimal left directed mode function. 
 
 SL.(\. n , y. n , Y 1 n ) = \. v , / (\. n V y. Jy. (4:45) 
 i v i+l' ^i+I' i+l' i+I^i+I T v i+l r J i+1 /J i v 
 
 Definition 4-l8: \. = EL. (\. ., y. ,, Y 1 . ) 
 
 l l i+l ; J i+1^ i+l 
 
 EL. is called the entire, minimal left directed mode function with 
 i 
 
 *L ■ (J ± .y < j < n + A - 1). 
 
 EL. = 1 whenever X. has a value of 1 either by necessity or choice 
 11 
 
 according to Table 4-3. 
 
 EL. =0 whenever X. must be according to Table 4-3. 
 li B 
 
 .. D 
 
 Theorem 4-21: SL. = NL. V AL. = SL 
 
 1111 
 
 Proof: NL. V AL. = X. _y, , V (\. , V y. n )(NK. V AK. ). Using Eq. 4 : kO and 
 
 l l i+l J i+l ' v i+l J i+l' v 1 i' & *1 
 
 observing that y.NK. = NK.. it follows that 
 li i' 
 
 NL. V AL. = X. _y. . V (\. , V y. , )(y.NK. V y.NK.) = SL. . 
 l i i+l J i+l v i+l v J i+l' w i i J i V i 
 
 SL D =(V.Vy.. )(\. v. . V y. ) = SL. . 
 
 l i+l i+l v l+l^i+l J i i 
 
 Theorem 4-22 : SL. yields the canonical , minimal representation as defined by 
 Definition 4-2. 
 
 Proof : Since SL. yields a minimal representation, it is only necessary to 
 show that nonzero recoded digits are always separated by at least one zero 
 
-86- 
 
 recoded digit. To do this, we assume that y! . = y. , + \. _ - 2X. is nonzero 
 
 ' "'i+l J i+1 i+l 1 
 
 and show that y! must he zero when X. = SL. . 
 l 11 
 
 The only way y! can be nonzero and still satisfy the range 
 
 restriction is for (y. _, , X. _, , X. ) to equal one of the following combinations 
 
 i+l' i+l l 
 
 (010), (Oil), (100), (101 ). In each of these y. f \. Therefore, 
 
 substitution in Eq. 4:45 yields X. = y. . Using this result in SL. yields 
 
 X. , = X.y. = X. . It follows that y! = y. + X. - 2X. , = 0. 
 l-l li i ill l-l 
 
 Theorem +-23: EL. = NL. V AL. V BL. = SL. V BL. = NL D (4:47) 
 i i ' l i 11 i v ' 
 
 Proof : As defined by Eqs . 4:31 and 4:32, AL. V BL. has a value of 1 whenever 
 
 X. = or 1 according to Table 4-3. Since NL. has a value of 1 whenever X. 
 i & J i l 
 
 must be 1 according to Table +-3, EL. = NL. V AL. V BL. . 
 
 B ' l l l l 
 
 Substituting Eqs. 4:40 and 4:4l into Eqs. 4:31 and 4:32, the 
 expression for EL. becomes 
 
 EL. - X. _y. _ V (X. _ V y. . )(NK. V y.NJT V y.NK ) 
 l i+l^i+l i+l v i+l i i li l 
 
 EL. = \. _y. .V (X.^Vy.,. )NK D 
 l i+l i+l ' i+l v i+l l 
 
 EL D = X. v. . V (X n V y. , )NK. = NL. 
 
 l i+l i+l J i+l i+l l l 
 
 An interesting series of symmetric, minimal L. can be formed as 
 
 follows. Successively larger AL. . min terms are joined to NL. without using 
 
 any BL. .. When all of the AL. . have been joined to NL., SL. is obtained. 
 J ij iJ l' l 
 
-87- 
 
 Successively smaller BL. . min terms are then joined to SL. . EL. is obtained 
 
 ij i i 
 
 when all BL. . are included, SL. represents the center of this series with 
 L. of' progressively greater complexity lying on either side. These L. and 
 the many others which can be formed using combinations of AL. . and BL. . are 
 not investigated in this dissertation. 
 
 h. 3 The Subclass of Restricted, Minimal Right Directed Mode Functions 
 
 The organization of this section is identical to that of section 4.2, 
 
 The set consisting of all right directed partial sequences, X (n + A), and 
 
 K 
 
 P. , is partitioned into three subsets in the first subsection. These subsets 
 l-l 
 
 correspond to choosing p. =0, p. =1 and p. = or 1 to achieve a minimal 
 recoded representation. In the second subsection a necessary and sufficient 
 restricted, minimal R. function is established with an auxiliary set of min 
 
 terms, The subclass of restricted, minimal R. is derived from these. This 
 
 ' l 
 
 subclass contains all minimal R. that can be formed by joining any combination 
 of min terms to the necessary and sufficient R. , 
 
 h-.^.l Case Analysis of the Partial Right Directed Sequence 
 
 Theorems k-l'J, U-l8 and 4-19 are used to partition (p. , X (n + A) ) 
 
 l-l K 
 
 into three subsets corresponding to p . - 0, p. =1 and p. = or 1. The 
 
 notational conventions adopted in section 4.2.1 will also be used in this 
 
 section with appropriate modifications for the right directed case. For 
 
 example, if AU has boundary mode digits p. _, =0, P. ,., N = 1 and p. ,_\ = 1 
 
 l-l ' i+u(lj i+u(2J 
 
 then the maximum composite gain appears as G(p.; OAlUl) = - p . 
 
-88- 
 
 Case I: p. , = 1 
 
 l-l 
 
 Let P, = U. Note that U is the only special sequence containing 
 y. = 1. Theorem 4-19 applies if p = 1. Theorem k- 18 applies if p > 2. 
 
 UA; Py > 1, P A > 2 
 
 Table 4-2 and Theorem 4-l6 are used to obtain the following composite 
 
 gains. 
 
 G(0; 1U0A0) =0+0 G(l; 1U1A0) = p - 1 
 
 G(0; 1U0A1) = + G(l; 1U1A1 ) = p + 
 
 If p = 1, none of Eqs . 4:13> h:lk or 4:15 are satisfied; therefore UAU and 
 
 UAZ must be analyzed. If p > 2., Theorem 4-l8 applies so p. =1. 
 
 G(0; 1U0A1U0) = + + (Py-l) G(l; 1U1A1U0) = 1 + + (Py-l) 
 
 G(0; 1U0A1U1) = + + p G(l; 1U1A1U1) = 1 + + p 
 
 Eq. k:lk is satisfied so p. =1* 
 
 i 
 
 UAZ ; p = 1, p > 2, p z = 1 
 
 G(0; 1U0A0Z0) =0+0+0 G(l; 1U1A0Z0) =1-1+0 
 
 G(0; 1U0A0Z1) =0+0-1 G(l; 1U1A1Z1) =1+0-1 
 
 UAZA; Pu = 1, p A > 2, p z = 1, p A > 2 
 
 G(0; 1U0A0Z0A0) =0+0+0+0 G(l; 1U1A0Z0A0) =1-1+0+0 
 G(0; lUOAOZOAl) =0+0+0+0 G(l; 1U1A1Z1A1 ) =1+0-1+0 
 
 Eq. 4:15 is satisfied so p. = or 1. 
 
-89- 
 UAZ; Pu = 1, p A > 2, P z > 2 
 
 G(0; 1U0A0Z0) =0+0+0 G(l; 1U1A0Z0) =1-1+0 
 
 G(0; 1U0A0Z1) =0+0-1 G(l; 1U1A0Z1 ) =1-1-1 
 
 Eq. 4:15 is satisfied so p. = or 1 
 
 i 
 
 UZ; P y > 1, 1 < P z < 2 
 
 G(0; 1U0Z0) =0+0 G(l; 1U0Z0) = q + 
 
 G(0; 1U0Z1) = 0-1 G(l; 1U1Z1 ) = p.. - p 7 
 
 U Li 
 
 If Pu = 1, q = -co; if p y > 2, q = Pu -1. If p y = 1, none of Eqs . 4:13, 4:l4 
 
 or 4:15 are satisfied so UZA is analyzed below. If p > 2 } Eq. 4:l4 is 
 
 satisfied so P. = 1. 
 
 i 
 
 UZA ; p n = 1, 1 < p z < 2, P A > 2 
 
 G(0; 1U0Z0A0) =0+0+0 G(l; 1U1Z1A0) = 1 - p„ - 1 
 
 Li 
 
 G(0; 1U0Z0A1) =0+0+0 G(l; 1U1Z1A1) = 1 - p - 
 
 If p = 1, UZAU and UZAZ are analyzed below. If p = 2, Eq. 4:13 is satisfied 
 
 so p. = 0. 
 
 l 
 
 UZAU; ?u = 1, p z = 1, p A > 2 } P y > 1 
 
 G(0; 1U0Z0A1U0) = + + + (Py-l) G(l; 1U1Z1A1U0) = 1 - 1 + + (Py-l) 
 G(0; 1U0Z0A1U1) = + + + p G(l; 1U1Z1A1U1) = 1 - 1 + + 
 
 P U 
 
 Eq. 4:15 is satisfied so p . = or 1. 
 
-90- 
 
 UZAZ; ?u = 1, p z = 1, p A > 2, P z > 1 
 
 The last composite gain shown below is conditional on the length of the right 
 most Z sequence. 
 
 G(0; 1U0Z0A0Z0) =0+0+0+0 
 
 G(0; 1U0Z0A0Z1) =0+0+0-1 
 
 G(l; 1U1Z1A0Z0) =1-1-1+0 
 
 G(l; 1U1Z1A1Z1) =1-1+0-1 (Valid if p„ = l) 
 
 G(l; 1U1A1A0Z1) =1-1-1-1 (Valid if p„ > 2) 
 
 If p = 1, none of Eqs . ^:13, 4:1^ or k:15 are satisfied so UAZAZ must be 
 
 Zj 
 
 analyzed. If p > 2, Eq. k:13 is satisfied so P. =0. 
 
 Zj 1 
 
 UZAZA ; Pu = 1, p z = 1, p A > 2, p z = 1, p A > 2 
 
 G(0; 1U0Z0A0Z0A0) =0+0+0+0+0 G(l; 1U1Z1A0Z0A0) =1-1-1+0+0 
 G(0; 1U0Z0A0Z0A1) =0+0+0+0+0 G(l; 1U1Z1A1Z1A1) =1-1 
 
 Eq. U:13 is satisfied so P. = 0. 
 
 + 0-1 + 
 
 UZ; Pu = 1, p z > 3 
 
 G(0; 1U0Z0) =0+0 
 G(0; 1U0Z1) =0-1 
 
 G(l; 1U0Z0) = -co + 
 G(l; 1U1Z1) = 1 - p_ 
 
 Eq. ^:13 is satisfied so p. = 0, 
 
 UZ; Pu > 2, P z > 3 
 
 G(0; 1U0Z0) =0+0 
 G(0; 1U0Z1) =0-1 
 
 Eq. k:lk is satisfied so p. = 1, 
 
 G(l; 1U0Z0) = (Py-l) + 
 G(l; 1U0Z1) = (Py-l) - 1 
 
■91- 
 
 Case II: P. , = 1 
 
 l-l 
 
 Let P., = A so that y. = 0. Theorem if-l8 applies so P. = 1. 
 1 i i 
 
 Let P = Z so that y. =0. Theorem ^--17 applies so p. = 1, 
 
 Case III: p. = 
 
 l-l 
 
 Let P = U so that y. = 1. Theorem k-17 applies so p. = 0, 
 
 Case IV: p. . = 
 
 l-l 
 
 Let P = A so that y. = 0, Theorem if-19 applies so the choice of 
 P. depends on sequences to the right of A. 
 
 AU; p A = 2, p n > 1 
 
 G(0; 0A0U0) = + G(l; OAIUO) = + (lU-l) 
 
 G(0; OAOUl) = - oo G(l; OAlUl) = + p 
 
 If p„ = 1, Eqs. k:13, ij-:lU and if: 15 are not satisfied so ADA and AUZ must he 
 
 analyzed. If p > 2, Eq. k:lh is satisfied so p. = 1. 
 
 AUA ; p A = 2, Pu = 1, p A > 2 
 
 G(0; 0A0U0A0) =0+0+0 G(l; OAIUIAO) =0+1-1 
 
 G(0; 0A0U0A1) =0+0+0 G(l; OAlUlAl) =0+1+0 
 
 AUAUi p. = 2, p TT = 1, p. > 2, p TT > 1 
 
 G(0; 0A0U0A1U0) = + + + (Py-l) G(l; 0A1U1A1U0) = + 1 + + (p^-l) 
 
 G(0; 0A0U0A1U1) =0+0+0+ Pu G(l; 0A1U1A1U1 ) =0+1+0+ ^ 
 
 Eq. k:lk is satisfied so P. = 1. 
 
 i 
 
■92- 
 
 AUAZ; p A = 2, Pu = 1, p A > 2, P z = 1 
 
 G(0; OAOUOAOZO) =0+0+0+0 G(l; OAIUIAOZO) =0+1-1+0 
 G(0; OAOUOAOZl) =0+0+0-1 G(l; OAlUlAlZl) =0+1+0-1 
 
 AUAZA; p A = 2, Pu = 1, ?A > 2, ?z = 1, p A > 2 
 
 G(0; OAOUOAOZOAO) =0+0+0+0+0 G(l; 0A1U1A0Z0A0) =0+1-1+0+0 
 G(0; OAOUOAOZOAl) =0+0+0+0+0 G(l; OAlUlAlZOAl) =0+1-1+0+0 
 
 Eq. 4:15 is satisfied so P. =0 or 1. 
 
 AUAZ; p A = 2, Pu = 1, p A > 2, P z > 2 
 
 G(0; OAOUOAOZO) =0+0+0+0 G(l; OAIUIAOZO) =0+1-1+0 
 G(0; OAOUOAOZl) =0+0+0-1 G(l; OAlUlAOZl) =0+1-1-1 
 
 Eq. 4:15 is satisfied so p. = or 1. 
 
 AUZ; p A = 2, ?u = 1, p z = 1 
 
 G(0; OAOUOZO) =0+0+0 G(l; 0A1U0Z0) =0+0+0 
 
 G(0; OAOUOZl) =0+0-1 G(l; OAlUlZl) =0+1-1 
 
 AUZA ; p A = 2, Pu = 1, p z = 1, p A > 2 
 
 G(0; OAOUOZOAO) =0+0+0+0 G(l; 0A1U0Z0A0) =0+0+0+0 
 
 G(0; OAOUOZOAl) =0+0+0+0 G(l; OAlUlZlAl) =0+1-1+0 
 
 Eq. 4:15 is satisfied so P. =0 or 1. 
 
-93- 
 
 AUZ; p = 2, p n = 1, p_ > 2 
 
 A 
 
 U 
 
 G(0; OAOUOZO) =0+0+0 
 G(0; OAOUOZl) =0+0-1 
 
 Eq. 4:15 is satisfied so p . = or 1. 
 
 G(l; 0A1U0Z0) =0+0+0 
 G(l; OAlUOZl) =0+0-1 
 
 AU; P A > h, Pu > 1 
 
 G(0; 0A1U0) = + (Py-l) 
 G(0; 0A1U1) = + p 
 
 G(l; 0A1U0) = + (Prj-l) 
 G(l; 0A1U1) = + p y 
 
 Eq. 4:15 is satisfied 
 
 so p . 
 
 = or 1. 
 
 AZ; p A > 2, p z = 1 
 
 G(0; OAOZO) =0+0 
 G(0; OAOZl) =0-1 
 
 G(l; OAOZO) = -1 + 
 G(l; 0A1Z1) =0-1 
 
 AZA; p A > 2, p z = 1, p A > 2 
 
 G(0; OAOZOAO) =0+0+0 
 G(0; OAOZOAl) =0+0+0 
 
 Eq. 4:13 is satisfied so p. = 0, 
 
 G(l; OAOZOAO) = -1 + + 
 G(l; 0A1Z1A1) = 0-1 + 
 
 AZ; p A > 2, p z > 2 
 
 G(0; OAOZO) =0+0 
 G(0; OAOZl) =0-1 
 
 G(l; OAOZO) = -1 + 
 G(l; OAOZl) = -1 - 1 
 
 Eq. 4:13 is satisfied so p = 0, 
 
 l 
 
-94- 
 
 Let P = Z so that y. =0. Since p =0, Theorem 4-19 applies if 
 p = 1. If p > 2, Theorem 4-18 applies. 
 
 Zi Li 
 
 ZA; p z = 1, p A > 2 
 
 G(.0; 0Z0A0) = + 
 G(0; OZOAl) =0+0 
 
 Eq. 4:13 is satisfied so p. = 0, 
 
 G(l; 0Z1A0) = -1 - 1 
 G(l; 0Z1A1) = -1 + 
 
 ZA; p z > 2, p A > 2 
 
 Theorem 4-l8 applies so p. = 0„ 
 
 This completes the partitioning of the set (p. , X (n + A) ) . 
 
 i-1 R 
 
 Table 4-4 contains a summary of the results. As in Table 4-3, X is used to 
 designate the portion of the partial sequence that is not given explicitly in 
 terms of special sequences. The special sequences and their corresponding 
 lengths are listed in order of occurrence . 
 
 4.3.2 Derivation of the Restricted, Minimal Right Directed Mode Functions 
 
 The minimal R. derived in this section have p. , , y. and YV 
 1 i-l' 1 1 
 
 = (y. .? 1 < j < n + A) as independent variables. Table 4-4 forms the basis 
 for the following definitions and analysis. 
 
 Definition 4-19: p. = NR.(p. , , v., YV) 
 1 i v l-l 1' 1 
 
 Let NR. be a minimal right directed mode function with 
 
 y* = (y i+1 ; 1 < j < n + A). 
 
 NR. =1 whenever P. must be 1 according to Table 4-4. 
 11 
 
 NR. = whenever the choice of P. is or arbitrary according to 
 
 Table 4-4, 
 
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-97- 
 
 Theorem 4-24 : NR. is the necessary and sufficient restricted, minimal right 
 directed mode function. 
 
 Proof : The proof follows directly from Definition 4-19 and Table 4-4, 
 
 The following general expression for p. can "be obtained by inspection 
 of Table 4-4. 
 
 P. = R^P^, J ± , ij) = P,.^ V Pi.^ACl, 1, Y*)Y P^ACO, 0, Y*) 
 
 (4:48) 
 
 Note that Eq. 4:48 is the same as Eq. 3-4 when Eq. 3:6 is substituted for 
 S.(y., Y*). 
 
 Table 4-4 is used to develop the R. (l, 1, ^) and R. (0, 0, Y R ) 
 functions. Case I applies to the former and Case IV to the latter. 
 
 R. (l, 1, Y?) must be 1 for all of the following right directed 
 partial sequences beginning with y. . 
 
 y i+i y i+2 ••• y i+j ••• = (1 ••• ' 011 ••• ' ° 1011 ■••;■•■; 01 ••• 011 •••) 
 
 It is convenient to define NR. (l, 1. YV) in terms of NR. (l, 1, 0, YV ., ) . 
 
 NR.(l, 1, Y*) = y. +1 V NR.(l, 1, 0, Y** +1 ) (4:49) 
 
 nr. (i, i, o, ij +1 ) = 'f y i+2j+3 y i+2j+2 T yi+ 2k + 3 y i+2k+2 ^S50) 
 
-98- 
 
 where n* = — - — with A - if n is even and A = 1 if n is odd. On the first 
 
 step of a right directed recoding y. = y . Thus when i = the definition 
 
 of n* guarantees that y. ~ „ ~ = y A ^ = with n odd or even. 
 to i+2n*+2 n+A+2 
 
 R. (l, 1, YV) may be 1 for any of the following partial sequences 
 
 beginning with y. 
 
 l+l 
 
 y i+i y i+2 °°° y i+j "°° = (010 ° "•• ); (° 10100 ...);...; (oi ... oioo ... ) 
 y i+i y i+2 °°° y i+j °°° = (0011 ••• ^ (ooioii ...);...; (ooioi ... oil ... ) 
 
 The Boolean functions given below represent an arbitrary sequence in each of 
 these sets. 
 
 CR..(1, 1, Y^) = y. _CR. .(1, 1, 0, Y R - _ ) 
 
 ij ' i l+l ij i+l 
 
 (+:51) 
 
 CR. . (1, 1, 0, Y^ n ) = y. n . n 
 
 ij v ' ' ' i+l ' J i+2j+2 
 
 TT 
 
 k=0 
 
 _TT y i+2k+3 y i+ 2 k+ 2j c ij 
 
 ii 
 
 DR. .(1, 1, Y R ) = y. n DR. . (l, 1, 0, ^ . ) 
 
 ij v ' ' 1' J i+1 ij i+l 
 
 (4:53) 
 
 DR. .(1, 1, 0, ^ _ ) = y. . _ 
 ij v ' ' ' i+l / J i+2j+2 
 
 j-l 
 
 LkK y i +2k+ 3 y i+2k+2_ 
 
 d 1 ^ (k:5k) 
 
 As in the left directed case, c. . and d. . represent arbitrary Boolean functions 
 
 of the (n + A - 2j ) digits to the right of y. . Any one of these functions 
 
 may be associated with each CR . (l, 1, Y?) and DR. . (l, 1, Y?). Any combination 
 
 ij i ij i 
 
 of these functions for 1 < j < n* may be joined to NR.(l, 1, Y?) to form a 
 
 particular R. (l, 1, Y?). 
 
 R. (0, 0, YV) must be 1 for all of the following partial sequences 
 
 beginning with y. , . 
 D J i+l 
 
-99- 
 
 y. . y. _ ... y, . ... = (ill ...; 11011 . ..; . ..; 110101 ... Oil ...) 
 °i+l J i+2 J i+j v ' ' ' ' 
 
 R R 
 
 Here it is convenient to define NR. (0, 0, Y.) in terms of NR. (0, 0, 1, Y. ,) 
 
 NR.(0, 0, Y R ) = y. . NR.(0, 0, 1, Y R _ ) 
 
 NR. (0, 0, 1, Y R +1 ) = NR. (1, 1, 0, Y R +1 ) 
 
 (+:55) 
 (+:56) 
 
 R.(0, 0, Y.) may "be 1 for any of the following partial sequences 
 
 "beginning with y 
 
 i+1' 
 
 y i+l y i+2 
 
 y, . ... = (1100 ...); (110100 . ..); . ..; (1101. . . 0100 . ..) 
 
 i+J 
 
 y i+l y i+2 
 
 y ... = (1011 ...); (101011 ...); ...; (101 ... 1011 ...) 
 
 The Boolean functions which represent an arbitrary sequence in each 
 of these sets are given below. 
 
 CR. .(0, 0, Y R ) = y. . CR. .(0, 0, 1, Y R _ ) 
 
 (+•57) 
 
 CR. .(0, 0, 1, Y R _) = y. _. _ 
 lj ' ' i+l y 1+2J+2 
 
 ,*L y i+2k+3 y i+2k+2 
 - k=0 
 
 00 
 
 c . . 
 
 ij 
 
 (+:58) 
 
 DR. ,(0, 0, Y R ) = y. . DR. .(0, 0, 1, Y R _ ) 
 ij l J i+1 ij i+l 
 
 (+:59) 
 
 DR..(0, 0, 1, Y i+1 ) =y. +2 . +2 
 
 j-l 
 
 TJ" y i+2k+3 y i+2k+2 
 L k=0 
 
 d 00 
 
 ij 
 
 (k:6o) 
 
 As in Eq. k:52 and h'.^k, c. . and d. . are arbitrary Boolean functions of the 
 
 ij ij 
 
 ,R- 
 
 (n + A - 2j) digits to the right of y. _. n . Any combination of CR. . (0, 0, Y.) 
 
 R R 
 
 and DR. .(0, 0, Y.) with 1 < j < n* may be joined to NR.(0, 0, Y.) to form a 
 
 particular R.(0, 0, Y R ) . 
 
 The subclass of restricted, minimal S. functions is constructed by 
 
 substituting the above equations into Eq. 3:6. Let NS. represent the necessary 
 
 and sufficient S. 
 
-100- 
 NS.(y., Y R ) = y.y. +1 V (y. © y. +1 ) NR.(l ; 1, 0, Y* +1 ) (h:6l) 
 
 CS . . and DS . . are defined as follows . 
 
 • DS i.^i' Y l) =^i DR ij^ ^ Y i> Y ^i D V°' °> Y i) (^ 6 5) 
 
 Any combination of CS. . and DS . . joined to NS. forms a particular S.. 
 
 ij ij i i 
 
 These functions are used to define NR. , CR. . and DR. ,. 
 NR^p,.,, y ± , I») = Pul y. V (p.^ V J ± ) NS. (U:6U) 
 
 D Rl .( Pl . r y ± , Y E ) . (p.^ V y.) CS. . (*:66) 
 
 Any R.(p. , , y., Y.) in which Y. includes only those digits to the right of y. 
 
 can "be constructed by joining an appropriate combination of CR. . and DR. . 
 
 to NR. . 
 i 
 
 The number of minimal R. that may be defined in this manner is very 
 large. In the work that follows we limit our attention to those R. for which 
 R.(l, 1, 0, Y? n ) = R.(0, 0, 1, Y R J with c 1 ] = d 11 = c°° = d°° = 1 for all j. 
 Equation Ik 56 shows that the necessary and sufficient portions of these partial 
 functions are always equal* 
 
 Since c. . = d. . = 1, comparison of Eqs . ki^2 and ki^k with Eqs . ki^Q 
 and *4-;60 shows that 
 
 0R ± (1, 1, 0, Y R +1 ) = CR (0, 0, 1, Y R +1 ) (ki6l) 
 
 DR.^1, 1, 0, Y R +1 ) = DR ±J (0, 0, 1, Y R +1 ) (4:68) 
 
 To guarantee that R ± (l, 1, 0, Y R +1 ) = R.(0, 0, 1, Y R +1 ), CR (l, 1, 0, Y^ +1 ) 
 
-101- 
 
 must "be joined to NR.(l, 1, 0, Y. .) if CR..(0, 0, 1, Y7 , ) is joined to 
 NR.(0, 0, 1, Y. ). A similar statement applies to DR..(l, 1, 0, Y. ) and 
 DR.^0, 0, 1, Y^ +1 ). 
 
 Under these restrictions Eqs . 4:62 and 4:63 "become 
 
 DS ij ■ <*i • '1*1) "V 1 ' 1; °' ^ ih '- 70) 
 
 As in the left directed case, the minimal S. formed under these 
 restrictions are called symmetric, minimal right directed characteristic 
 functions. All R. formed with these S. are called symmetric, minimal right 
 directed mode functions. 
 
 The functions defined below are used to illustrate certain symmetries 
 in the subclass of symmetric, minimal R. . 
 
 Definition 4-20: CS., DS., CR. and DR.. 
 
 ■ ill i 
 
 n 
 
 CS 
 
 . = "V cs - • (+:7i) 
 
 1 j=l 1J 
 
 n 
 
 DS. = }/ DS. . (4:72) 
 
 1 j=l 1J 
 
 CR i = (p i-l V ^1 } CS i (4:73) 
 
 DR. = (p i _ 1 V y.) DS. (4:7+) 
 
 NS., CS. and DS. are now simplified by using Lemmas A-l, A-2 
 
 and A-3. 
 
 
NS, = y 
 
 n 
 
 -102- 
 
 y. 
 
 if 
 
 i+2j+3 k 'J _ x y i+2k+4 
 
 * 
 
 CS. 
 
 i 
 
 DS, 
 
 - <*i • y 1+ i) r y 
 
 TV ~ 
 
 T y i+2 y i+2j+2 jJJq y i+2k+3j 
 
 = DS. 
 
 l 
 
 D 
 
 (y . © y. . ) 
 w i ** J i+l' 
 
 V 
 
 0=1 
 
 y-? j_o y-,- . o ■? 
 
 7T 
 
 y.- 
 
 i+2 J i+2j+2 '_' J i+2k+3_ 
 
 CS. 
 l 
 
 (^75) 
 
 (hilG) 
 
 (.km) 
 
 Lemmas A-k, A-5 and A-6 are used to obtain NS., CS. and DS. 
 
 i' l l 
 
 NS i " *i y i+ i v ^i 9 y 1+1 ) 
 
 V 
 
 o-i 
 
 lT y : 
 
 j=0 i+2J+2 k'='o Ji+2k+ 3j 
 V (Y ± Q y 1+1 )(y i+2n ^2 y y i + 2n* +3 ) 
 
 n*-l 
 
 TT 
 
 0=0 
 
 y i+2j+3 
 
 In the worst case i = 0, so 
 
 y 2n*+2 y 2n*+3 y n+A+2 * y n+A+3° Sin ce y n+A+2 := y n+Af 3 
 
 for A = or 1. y. ,- „ _= Y. n » , for all i. 
 
 ; J i+2n*+2 J i+2n*+3 
 
 = 1 
 
 #■ 
 
 »S = y y V (jr © y ) 
 
 V yi+2j+2 Jl y i+2k+3 
 j=0 k=0 
 
 (^78) 
 
 Lemma A-6 is used to obtain CS. „ 
 
 l 
 
 n*-l 
 
 CS, - (y. 
 
 y. 
 
 ) v h + S Y 
 
 y. 
 
 j-i 
 
 y.-. 
 
 i " i " " i+1' ' J i+2" .V- J i+2j+l .'" J i+2k+2 
 
 .1-1 J k=l 
 
 n*-l 
 
 * ^ y i+2n*+l V y i.+2n*+2^ /' 
 
 0=1 
 
 i+2j+2 
 
 In the worst case i = 0, so 
 
 v yp n *, p y n .i-/V4-i V y r 
 
 Since y^ . , = y„^.o = ° 
 
 for A = or 1, y. _, „ . =y. _ ,, _ for all i. 
 ' °i+2n*+l J i+2n*+2 
 
-103- 
 
 _ _ n* 0-1 
 
 CS = (y. © y. +1 ) V y i+2 V y y 1+2i+1 TT y 1+2k+2 = ^ (4.79) 
 
 0=1 k=l 
 
 DS. is derived from Lemma A-5 "by using the same argument to eliminate the last 
 
 product term. 
 
 Yl,,,, it 
 
 ,D 
 
 ds, . ( 7l e y 1+1 ) v j i+2 v y y 1+2J+1 II y 1+21[+2 =<^ (4:8o) 
 
 J =1 k=l 
 
 The dual of the complement of Eq. 4:88 yields NS.. 
 
 NS ? ■ Vi+i * ^ © r i+ i> 
 
 
 ^V y i+2 j+2 IT y i+2k+3j 
 
 (4:81) 
 
 These equations can now "be used to verify the following identities. 
 
 CS. = (y. y ) y. +2 NS~ (4:82) 
 
 DS. = (y. y. +1 ) y i+2 NS° (4:8 5 ) 
 
 NS_. V NS? = NS? (4:84) 
 
 (4:8 5 ) 
 
 (4:86) 
 
 Three symmetric, minimal R. are of particular interest „ One is NR. 
 as defined by Eq. 4:64. The other two are defined below. 
 
 Definition 4-21: p. = SR. (p. , . v., Y7) 
 
 SR. is called the simplest, minimal right directed mode function. 
 
 1 1 
 
 l 
 
 cs. V cs D = 
 
 i i 
 
 . cs D 
 
 1 
 
 DS. V DS D = 
 
 l l 
 
 . DS D 
 
 1 
 
 SR 
 
 i = p i-i y i v Pi.i(y 1+ i v y i+2 ) y j&wPm ^ :8 ?) 
 
 Definition 4-22 : p. =ER.(p. ., y., Y?) 
 
 ER. is called the entire, minimal right directed mode function with 
 
 Y i = (y i+j ; 1 - J - n+A) * 
 
-104- 
 
 ER. = 1 whenever p. has a value of 1 either by necessity or choice 
 
 iccording to Table 4-4< 
 
 ER. = "whenever p. must be according to Table 4-4, 
 
 Theorem 4-23 : SR. = NR. V CR. = SR D (4:88) 
 
 — — ■ <*• 1111 v ' 
 
 Proof: NR. V CR. = p,^ y. \/ (o ± _ 1 V y 1 )(NS.V CS.) 
 
 Note that y i y 1+1 V (y ± © y ±+1 ) y i+2 ns. = ns.„ (4:89) 
 
 By using this equation and Eq. 4:82, we obtain 
 
 NS. V CS. =y.(y. ,V y . J \/ y . n y„ . Therefore, 
 1 1 J i J i+l J i+2 / v J i+l J i+2 > 
 
 SE i = Pi-i^i V Pi-i(y i+ i V y 1+2 ) V y> 1+1 y 1+2 
 
 SR i - ^i-i v *i><pi-iV Wi +2 >^iV y 1+1 V y i+2 ) - sh,. 
 
 Theorem 4-26: ER. = NR. V CR. V DR. = SR. V DR. = NR D (4:90) 
 
 — — - — — ■ liiiii l 
 
 Proof: CR. V DR. as defined by Eqs . 4:73 an( i 4:74 has a value of 1 whenever 
 
 n. = or 1 according to Table 4-4= NR. has a value of 1 whenever p. must 
 ■ l & l l 
 
 be 1 according to Table 4-4. It follows that ER. = NR. V CR. V DR. . 
 
 ° 1111 
 
 By substituting Eqs. 4:82 and 4:83 into Eqs. 4:73 an d 4:74 and 
 expressing NS . as in Eq. 4:89, ER. becomes 
 
 y 1 y 1+1 V(y 1 © y 1+1 ><y 1+2 V ns?) 
 
 ER i " Pi-iTi v (Pi-lV jr ± ) 
 
 Since the dual of Eq. 4:89 is equal to the expression in square 
 brackets, 
 
-105- 
 
 D 
 
 ER, = p, ,y, V (p. ,V y.) NS 
 
 ER D - p. .y. S/ (p. . V y.) NS. = NR. 
 1 l-l l T "i-l li l 
 
 As in the left directed case, an interesting series of symmetric, 
 
 minimal R. can be constructed as follows. Successively larger CR. . min terms 
 
 l ij 
 
 are joined to NR. without using any DR. .. When all CR. . terms are joined to 
 i ij ij 
 
 NR., SR. is obtained. Successively smaller DR. . min terms are then joined to 
 ii ij 
 
 SR. until ER. is obtained. SR. lies at the center of this series with progres- 
 111 to 
 
 sively more complex R. lying on either side. There are, of course, many other 
 symmetric, minimal R. besides these, but only NR., SR. and ER. are investigated 
 in this dissertation. 
 
 k.k The Classes of Minimal Mode Functions 
 
 This section contains an outline of a suggested procedure by which 
 
 the classes of minimal left and right directed mode functions, <^ and CA. , 
 
 can be theoretically obtained. This procedure is based on knowledge of the 
 
 subclasses of restricted, minimal L. and R. and the equations that relate 
 
 7 ii 
 
 equivalent sets of left and right directed mode functions as established by 
 
 Theorems 3-5 an d 3-6 • 
 
 Equations 3^7 an d 3-8 can be modified slightly if we postulate a 
 
 natural extension of the y representation. Assume that y=....y=y = y 
 
 and that y ... = Y . ~ = . . . . y _ = where a and 6 are arbitrarily large 
 ^n+1 n+2 n+p 
 
 positive integers. Any series of minimal R. defined for -n < j < -1 with 
 
 J 
 
 p , = y -i must yield p = y . Likewise, any series of minimal L. 
 
 defined for n < j < 2n - 1 with \„ =0 must yield \ = 0. Under these condi- 
 — — 2n n 
 
 tions Eqs . 3-7 an( i 3^8 may toe written as follows. 
 
■io6- 
 
 n q-1 
 
 J. = \/ y K. TT (y. . V K. ) 
 
 l Y J i+l+q i+q II w i+l+r v i+r y 
 
 q=0 r=0 
 
 (^:9D 
 
 < i < n - 1 
 
 n-1 q-1 
 
 T, = ~YT y. S. TT (y. VS. ) V 
 
 i Y^ i-q i-q ' ' w i-r T i-r' 
 
 q=0 r=0 
 
 n-1 
 
 "1 
 
 > 
 
 TT (y- v s. ) 
 
 -n II l-r i-r 
 
 r=0 
 
 < i < n - 1 
 
 J 
 
 Ok 92) 
 
 If «?I were known, all possible minimal K. could be determined by 
 writing each L. e c£ in the form of Eq. 3:1. Likewise, if \J\ were known, all 
 possible minimal S. could be determined by writing each R. e(j\ in the form 
 of Eq. 3°^' Knowledge of cC is sufficient to determine Q\ by use of Eqs . 4:91 
 and 3°9» Similarly, knowledge of (A is sufficient to determine cs- by use of 
 Eqs. 4:92 and 3:10. 
 
 At present, only the subclasses of restricted, minimal L. and R. 
 are known. A typical restricted, minimal L. has a corresponding K. = NK. V 
 
 y AK. . V V BK. . where the functions on the right are defined by Eqs. 4:21, 
 
 J J 
 4:22 and Ik 23. A typical restricted, minimal R. has a corresponding S. = 
 
 Vcs . . y V e 
 
 Ik 62 and 4:63. 
 
 NS. V VCS. . V YDS. . where the functions on the right are defined by Eqs. ki6l, 
 
 J J 
 
 Equations 4:91 and 4:92 involve (n + 1) K. and S. functions respectively, 
 These may be chosen independently of each other. Equation 4:91 yields a new 
 class of J. by substituting fn + li independent choices of K. from the subclass 
 
 of restricted, minimal K. . Likewise, Eq. 4:92 yields a new class of T. by 
 
-107- 
 
 substituting (n + 1) independent choices of S. from the subclass of restricted, 
 
 minimal S.. These new classes must contain minimal J. and T. since the K. 
 1 ill 
 
 and S. used to define them are minimal. Furthermore, J. and T. are, in general, 
 l l l ' " 
 
 functions of y. to the left and right of y. and y. . Consequently, by sub- 
 
 v •*• X~r_L 
 
 stituting J. and T. in Eqs. 5*9 an( i 3:10 minimal R. and L. are obtained which, 
 ° 1 l 11 
 
 in general, are not restricted. 
 
 The classes of minimal J. and T. are used to generate new classes of 
 
 11 
 
 minimal R. and L. respectively. After all identities have been taken into 
 
 account, the new and larger classes of minimal S. and K. are used to obtain 
 ' D 11 
 
 even larger classes of minimal J. and T.. The process continues until the new 
 
 o 1 1 
 
 classes of J. and T. contain the same elements as their predecessors. 
 11 
 
 ■Two examples are given below to illustrate the process by which J. 
 
 and T. are obtained. In the first example the R. which is equivalent to SL. 
 
 is obtained assuming that L. = SL. for i < ,i < i + n. In the second example 
 
 J i - - 
 
 the L. which is equivalent to SR. is obtained assuming that SR. = R. for 
 1 l l j 
 
 i - n < j < i. 
 
 The canonical, minimal L(y) is defined by SL. for < i < n - 1. 
 
 The R. which defines the equivalent R(y) is obtained by use of Eqs. 4:91 and 
 
 3:9. We let L. = SL. for n < i < 2n - 1 so that K. is defined in Eq. 4:91 
 ^ ' j l - d - l+q ^ 
 
 for < i < n - 1. Equation 4:45 shows that SK. = y. . Substituting this 
 in Eq. 4:91 yields ST. which is then simplified by the following theorem. 
 
 Theorem 4-27: Under the assumption that 0=y „ = y „=...: 
 
 ■- * °n+l J n+2 
 
 n q-1 
 
 SJ i = Vy i+ ( q+ l) y i + q TT (y i+(r+l) V y i+r } 
 q=0 x ' r=0 
 
 (4:93) 
 
 = "V" y i + 2j + l V T 1 y i + 2k + 2 V X y i+2j l" y i+ 2k + l 
 .1=0 ° k = -1 .1=1 k=0 
 
-108- 
 where < i < n-1, and n* = — - — -with A = if n is even and A = 1 if n is 
 odd. 
 
 Proof : Only the relative portion of the indices is retained in the proof. Let 
 
 n q-1 
 
 VX + i y oL 7T 
 
 q=t r=t 
 
 It is clear that H(0) represents the left side of the equation given in the 
 hypothesis. 
 
 Consider H(n-l) and H(n-2). 
 
 **> V Vi \ T (y r+i v r r ) . 
 
 Assume that 
 
 H(n-i) =y n y n . 1 Vy n+1 y n 
 
 H(n- 2 ) = y n y n _ ± V y n _iy n _ 2 Vy n+1 y n y n . 2 
 
 H(n-2s-l) = Y n y } JT y n -2h V 
 
 g=0 & h=g 
 
 s-1 s-1 
 
 V" y n -2g JT y n-(2h+l) 
 g=0 & h=g 
 
 H(n-2s) . V^-CSg-l) "TT y„-2h V 
 
 g=0 & h=g 
 
 s-1 -1 
 
 VI y n-2g Jl y n-(2h+l) 
 g=0 & h=g 
 
 We use H(t = n-2s+l) to verify that the expression for H(t-1 = n-2s) 
 is correct. 
 
 H(n-2s) = (y _ ^y _ ) H(n-2s+l) \/ y n n y 
 x ' w n-2s+l v J n-2s' x ' y J n-2s+l ,, n-2s 
 
 s-2 s-1 
 
 = y n-2s+2 y n-2s+l V V^ y n-2g Jl y n-(2h+l) V 
 
 g=0 fa h=g 
 
 s —1 s 
 
 V'n-te.U 1fV2hVV2s + l y n-2s " H(n " 2s) 
 
 g=0 & h=g 
 
-109- 
 
 By using H(t = n-2s+2) we verify that the expression for H(t-1 = n-2s+l) is 
 correct. 
 
 H(n- 2s+ l) = (y n . 2s+2 V y n . 2s+1 ) H(n-2s + 2) V V 2 s + ^n-2s + l 
 
 ~ y n-2s+3 y n-2s+2 V 
 
 s-2 s-1 s-2 s-1 
 
 V y n -(2g-l) T y n -2hV Y y n-2g JT y n -(2h + l) V 
 
 g=0 to h=g g=0 & h=g 
 
 y n-2s + 2 y n-2s + l = H(n-2s + l) 
 
 Set s = n* and let y = -j ■+• n* and n = -k +> n* - 1. 
 If n is odd, 
 
 n* j-1 n* -1 
 
 H(o) - V y 2 j X y 2k+ i v V" y 2yl IT y 2k . 
 
 j=l d k=0 j=l d k=0 
 
 If n is even, 
 
 n * J" 1 - n - J -1 
 
 h(o) ■ x y 2j + i ,T , ^ +2 v y 
 
 J* -— k-="-l — j=l ^ iJo 2k+1 ' 
 
 The two expressions for H(o) are now made identical. Substitute 
 j - j '-fl and k = k'+l in the second join of H(o) when n is odd. The resulting 
 join is equal to the first join in H(o) when n is even except that the upper 
 limit is n*-l instead of n*. Let the upper limit he n* and note that 
 y. o_.jfc.n = f° r n even or odd and any < i < n-1. The expression for H(o) 
 with n even or odd is therefore the right side of Eq. 4:93 as given in the 
 hypothesis . 
 
 Substituting SJ. in Eq. 3°9 an( i simplifying yields 
 
■110- 
 
 R 
 
 i (p i-l' y i> ^ = p i-i y i V p i-l I V y i + 2j + l IT y i + 2k +2 J V 
 
 j-l 
 
 
 - j=0 
 
 r n* 
 
 y i 
 
 V- 
 
 j-1 
 
 S yi+2k+i 
 
 (4:94) 
 
 This is the minimal R. which is equivalent to SL.. It is clearly restricted 
 
 •p 
 since Y. includes only those y. to the right of y.. 
 
 As used in Eq» 3°9> SJ. assumes the role of S. . It is shown below 
 
 that SJ. does not represent a symmetric, minimal S. as defined in section 4.3-2. 
 
 A comparison of the right-hand side of Eq» 4:93 with Eqs. 4:75 , 4:76 and 4:77 
 
 reveals that J. can not be constructed by joining CS. and/or DS. to NS.. 
 
 The minimal S. which equals SJ. can be constructed as follows 
 l ^ l 
 
 R.(l, 1, Y R ) = NR.(l, 1, Y R ) 
 
 i x ' } l i v l 
 
 (4:95) 
 
 n* n* 
 
 .(0, 0, Y*) = HE.(0, 0, Y*) V)f C (0, 0, Y*) V ">/ D (0, 0, Y*) 
 
 j=i - J x A 1J 
 
 In these equations c. . = d. . = 1. It follows that 
 
 S. = y. R.(l, 1, Y R ) V y. R.(0, 0, Y R ) 
 
 l i i x ' ' i * ^ i i ' ' l 
 
 (4:96) 
 
 (4:97) 
 
 By using Lemmas A-l, A-2 and A-3, R.(l, 1, Yv) and R.(0, 0, Y.) may 
 
 be written as 
 
 R 
 
 n* 
 
 r ± <i, i, ^=y i+1 v V 
 
 7T y 
 
 g =0 i+2g+5 h'L _i"i+2h^- 
 
 R.(0, 0, Y R ) = y. . 
 
 x \ > > x / j 1+± 
 
 r n* 
 
 Vy< 
 
 ■1 
 
 >J+3 h T]\i y i+2h + 4 
 
 y 
 
 y 
 
 i+l 
 
 n* 
 
 "V" 
 
 i+2 ,/ i+2g+2 H , 'i+2h+3 
 
 =1 h=0 
 
 -1 
 
 n* 
 
 i+l Y y i+2 y i+2g+2 
 
 c==l 
 
 It y± 
 
 h=0 
 
 +2h+3 
 
-Ill- 
 
 Lemma A-7 is now applied to the first two joins in R.(0, 0, Y.) 
 
 _R 
 
 r.(o, o, ij) = y . +iy . +2 V G 
 
 n* 
 
 g=0 
 
 V y ) 
 
 i+2g+2 °i+2g+3 _ 
 
 y i + l 
 
 r n* _ g-1 
 
 V. y i+2 y i+2g+2 TL y i+2h+3 
 - g=l h=0 
 
 Since y. . ~ = 1 for any i > with A = or 1, this expression becomes 
 
 r \ -\ n r g_1 
 
 r ± (o, o, y.) = y i+1 y i+2 V y i+1 V y i+2 g + 2 JT y ±+2h+3 • 
 
 '- e=l h=0 J 
 
 R 
 
 We now let g = j - 1 and h = k - 1 in both R.(l, 1, Y.) and 
 
 ■p 
 R.(0, 0, Y.). Since y. = for any i > with A = or 1, the upper 
 
 A 
 
 limit on the two joins can remain n*. With these modifications S. can be 
 
 l 
 
 written as 
 A 
 
 S i = y i y i+l V 7 ± 
 
 n* 3-1 
 
 V 1 y i+2j+l "IT y i+2k+2_ 
 
 v y i+ iy i+2 v 
 
 r n* 
 
 i+1 
 
 V y. 
 
 t 
 
 y. 
 
 i + 2j ^ *i + 2k + l 
 
 After collecting terms this expression equals SJ. as given by Eq. 4: 93- 
 
 As a second example, we consider the restricted, minimal R(y) that 
 is defined by SR. for < i < n - 1. The L. which defines the equivalent L(y) 
 is obtained by means of Eqs . 4:92 and 3:10. We let R. = SR. for - n < j < - 1 
 
 so that S. is defined for < i < n - 1 in Eq. ki92. The SS. which corre- 
 
 l-q — — ^ y l 
 
 sponds to SR. was obtained in the proof of Theorem k-25- It may be written as 
 
 SS. = NS. V CS. = y.(y. . V y. J V y. J. . 
 
 i 11 J i VJ i+l Y J i+2' v J ±+r i+2 ' 
 
 Substitution in Eq. h:92 yields ST. which is then simplified by the following 
 
 theorem. 
 
-112- 
 
 Theorem 4-28 : Under the assumption that ... y =y . = y : 
 
 ST i " "Jf Wi-(l-l) y i-(q-2) J[ G i-r V y l-(r-l) V y ± . (r . 2) > V 
 
 n-1 
 yi-n (y i-(n-l) V y i-(n-2) } TJ (y i-r V y i-(r-l) V ^.(^a) ) 
 
 n* n* j-1 
 
 " ")£ y i-2j + l Jf- 1 yi - 2kV X yi " 2J kTTx y i-2^-l 
 
 where < i < n-1, and n# = — - — with A = if n is even and A = 1 if n is odd- 
 
 Proof; Only the relative portion of the indices is retained in the proof. Let 
 
 H(t) = V y y y Jf (y r V Y T _ ± V Y T _ 2 ) V 
 q=t ^ ^ r=t 
 
 n-1 
 
 y n (y n-l V y n . 2 ) TT ^ y r V y r-l * y r-2 } ' 
 
 r=t 
 
 H(o) represents the left side of the equation given in the hypothesis. 
 Consider H(n-l) and H(n-2). 
 
 H(n-i) = y n y n _ 2 v ^.^.g V v^^.j V y n . 2 y n -3 
 
 H(n-2) = y n J n _ 2 y n _ k tf ^.1^-2^-4 V y n - 5 y n-4 V 
 y n y n-l y n-3 * y n-2 y n-3 
 
 Assume that 
 
 s —1 s —1 s 
 
 H(n- 2s+ i) = V y„.(2eti) TT y n . (2h+2 ) V T v 2g V 
 
 g=0 & h=g g=0 
 
 s s 
 
 V. y n-2g Jl y n-(2h+l) • 
 g =0 to h=g 
 
 Using H(t = n-2s+l) we verify that the expression for H(t-1 = n-2s) 
 
 is correct. 
 
 
-113- 
 
 H(n- 2s ) = (y n _ 2s v 3 r n . 2s y n . 2s . 1 v y n . 2s y n . 2s . 2 ) X«**D V 
 
 y n-2s y n-2s -l y n-2s -2 
 
 s-1 s 
 
 y n-2s y n-2s-l V \A y n-2g Jl y n-(2h+l) V 
 g =0 & h=g 
 
 s-1 s s+1 
 
 V y n-(2 g+ l) IT y n-(2h + 2) * IT y n -2g = H(n " 2s) 
 g=0 x to h=g N ' g=0 
 
 We now use H(t = n-2s+2) to verify that the expression for H(t-1 = n-2s+l) is 
 correct. 
 
 H(n-2s+l) = (y „ . \/ y „ v n V y n J „ J H(n-2s+2) V 
 K ' VJ n-2s+l y J n-2s+l J n-2s Y ,y n-2s+l ,y n-2s-l y v ' 
 
 y n-2s+l y n-2s y n-2s -1 
 
 y n-2s+i y n-2s 
 
 r s-2 s-2 
 
 "V" y n-(2q+l) 7^" « / n-(2h+2) V 
 - g=0 h=g J 
 
 -1 
 
 y n-2s+i y n-2s IL y n-2g V y n-2s+i y n-2s 
 g=0 
 
 y n-2s+i y n-2s-l 
 
 s-1 s-1 s 
 
 "TT y n-2g^ V" y n-2g JT y n -( 2 h+l) V 
 g=0 & g=0 & h=g 
 
 y n-2s + l y n-2s y n-2s-l = H ( n - 2s +D 
 
 Set s = n* and let g = -j+n* and h = -k + n* -1. If n is odd, 
 
 n* j-1 n* n* j-1 
 
 h(o) - V r 2i _ 2 T ^ v J Q ^ V -£ y^ JT mi y 2k 
 
 If n is even, 
 
 n* -1 n* n* j-1 
 
 H(o) ■ )£ ^j-i k T-i 7a v jT-i y2 J y X y£J icT-i y2M 
 
 Since < i < n - 1, the two product terms can be absorbed by the 
 corresponding joins even when i = n-1. If the indices in the two expressions 
 
-111*- 
 
 for H(o) are temporarily written in the form i - x, we note the following when 
 i = n-1. 
 
 y n-l-2n^l = y -l = y o = y n-l-2n* + 2 (n ° dd) 
 
 y n-l-2n y ~l ~ y o " y n-l-2n*+l (n even) 
 
 Having absorbed the product terms, the two expressions for H(o) are 
 now made identical. Let j = j " +■ 1 and k = k ! + 1 in the first join of H(o) 
 when n is odd. The transformed join looks like the second join of H(o) when 
 n is even except that the upper limit is n* - 1 instead of n*. This dis- 
 crepancy is corrected by allowing the upper limit to be n* and. using the fact 
 that y p = y -, = y « Observe that when i = n-1 and n is odd, 
 
 n*-l n*- 2 
 
 y i-2n* Tl . y i-2k-l = y i-2(n*-l) . |l . y i-2k-l * 
 k = -1 ' k = -1 
 
 The resulting H(o) with n odd or even is the right side of Eq. k: 98 given in 
 the hypothesis. 
 
 By substituting ST. in Eq. 3^10 and simplifying we obtain 
 
 V x i+i' y i + r Y i + i } = \ + i y i + i v y ±+1 y ±+2 v 
 
 (^99) 
 
 n* j-1 
 
 n* -1 
 
 \ + i X y i-2j+i V T n y i-2kJ v ~\f y ± _ 2 j JT , y±. 
 
 - J-l K - =1 
 
 j=o - 2 J k 1 !./ 1 * 1 
 
 Since y. is included in the set of independent variables, the L. defined by 
 
 this equation is not restricted. 
 
 ST. represents K. as used in Eq. 3;10. Inspection of Eq. ^98 shows 
 
 tbat ST. involves y. ^. Therefore ST. is not restricted and furthermore can not 
 1 i+2 1 
 
-115- 
 
 be constructed by simply joining an appropriate combination of AK. . and 
 
 -^ J 
 
 BK. . to NK. . 
 
 The minimal K. that equals ST. is constructed as follows. 
 
 n* n* 
 
 L.(0, 1, YT _) = ML.(0, 1, YT,,) V "V AK. .(0, 1, YT , ) V'"V~BK. .(0, 1, YT _) 
 
 1 v ; > 1+1 / x \ j > 1+1 / V 1J ' 1+1 ."_ 1«] 1+1 
 
 0=1 ° j=i 
 
 L.(l, 0, Y L . ) = NL.(1, 0, Y L _) 
 In these equations a. . = b. . =1. It follows that 
 
 K - y i+ i L i (0 ' x » *£n> v y i +2 "i + i L i (1 ' °' ^ v y i+ ^i + i 
 
 By using Lemmas A-l, A-2 and A-J we obtain 
 
 a* 
 
 j=o 
 
 j-1 
 
 y 
 
 k = -1 
 
 i-(2k+2) V* 
 
 n* 
 
 y y i"i-2.i 
 
 j-i 
 
 . =1 _-2j ^ y i-(2k+l) V 
 
 n* _ j-1 
 Y y i y i-2,i JT yi-(2k + l) . 
 
 3=1 
 
 n* 
 
 k=0 
 
 1-1 
 
 , ± (1, 0, YT +1 ) = "\f y ± _ 2 
 
 g=U h = -1 
 
 T y ^ 
 
 h = -1 
 Applying Lemma A-7 to the first two joins in L.(0, 1, YT , ) yields 
 
 (4:100) 
 
 (4:101) 
 
 (4:102) 
 
 1+1' 
 
 L.(0, 1, YT" n ) = y. 
 
 i v ' ' i+l' °i 
 
 n* 
 
 .y ^i- 2J v 7^^)j v 
 
 n* _ j-1 
 
 V7 y i y i-2j TC y i-2 
 J=l 
 
 k=0 
 
 Since y. A = y . . . for all < i < n-1 with A = or 1, 
 J l-n-A "'l-n-A-l — — ' 
 
 l. ( o, i, t5 +1 ) = ?1 vY y± . 2j if y^^ 
 
 k=0 
 
-116- 
 
 In L.(l, 0, YT .) let g = j - 1 and h = k - 1. Since 
 i l+l ° 
 
 y i-2(n*+l)+i y i-2n* = y i-2n*+l f ° r ° - i - n_1 with n even or odd ^ the u PP er 
 limit on the join can "be left as n*. Substituting these modified forms of 
 
 L.(0, 1, Y. ) and L.(l, 0, Y. ) in the expression for K. yields 
 
 *i = y i-H y i V y i + l 
 
 v y i_ 2 j TT y i_j 
 
 j=l L ^ J k=0 
 
 y i+2 
 
 n* j-1 -. 
 
 V y i-2Ul Tf y i.-2k * y i+2 y i+l 
 j=l k=0 J 
 
 After collecting terms, this expression equals ST. as given by Eq. ki^8, 
 
 k. 5 Summary 
 
 Sets of recoding functions which transform a binary representation 
 
 into an extended signed-digit representation having a minimum of nonzero digits 
 
 were investigated in this chapter. Both left and right directed sets of 
 
 minimal recoding functions, F (y) and F (y), were considered with primary 
 
 li K 
 
 emphasis on the corresponding sets of minimal mode functions, L(y) and R(y) • 
 
 It was observed that all possible minimal L(y) and R(y) correspond to all 
 
 possible sets of minimal L. and R.o 
 
 li 
 
 In section I4..I the minimal recoded representation was analyzed and 
 a procedure for determining the optimum choice of the next mode digit, m., 
 was established. A unique decomposition of an arbitrary binary sequence was 
 defined and the concept of a maximum gain of zeros in the recoded representa- 
 tion was introduced. In both the left and right directed cases, the maximum 
 gain over each of the three special sequences, Z, U and A was determined for 
 all possible boundary mode digit conditions. The results are listed in 
 Tables ^-1 and k-2 according to the choice of m. . A maximum composite gain 
 in zeros was defined as the sum of maximum gains over a series of special 
 
-117- 
 
 sequences. A theorem was introduced concerning the calculation of this gain 
 over a series of special sequences with "both and 1 as the terminating 
 "boundary mode digit. Three theorems followed which "based the optimum choice 
 of m. on a comparison of maximum composite gains. It was shown that in certain 
 cases the choice of m. is arbitrary. 
 
 In section k.2 and 4.3 "the theorems established in section 4.1 were 
 used to partition the sets of left and right directed partial sequences 
 (X T (n+A), \. ) and (p. , X (n+A)), into three mutually exclusive subsets. 
 In two of these subsets the optimum choice of \. or p, is or 1 while in the 
 third it is arbitrary. These results are summarized in Tables 4-3 and 4-4. 
 Using these tables, subclasses of restricted, minimal mode functions were 
 developed in terms of necessary and sufficient minimal mode functions and sets 
 of arbitrary min terms. In both sections subclasses of symmetric, minimal 
 mode functions were defined and some of the basic symmetries were investigated. 
 
 In section 4.4 a theoretical procedure for obtaining all minimal L. 
 
 and R. belonging to cJT and Ul was suggested. This procedure is based on 
 
 knowledge of the subclasses of restricted, minimal L. and R. and the equivalent 
 
 relations developed in section 3-3- Two special cases involving the simplest, 
 
 minimal mode functions, SL. and SR„, were considered as examples. 
 
 ' l x 
 
5. APPLICATIONS OF MINIMAL RECODING TO MULTIPLICATION 
 AND DIVISION ALGORITHMS 
 
 In the past various minimal recoding techniques have been used or 
 suggested as the basis of fast multiplication and division algorithms for 
 digital computers. In this chapter minimal left and right directed recoding 
 functions are used to define multiplication and division algorithms which 
 minimally recode multipliers and generate minimally recoded quotients. 
 
 The canonical, minimal recoding of multipliers has been widely used. 
 This is undoubtedly due to the fact that it involves the simplest, minimal 
 left directed recoding and possesses the unique property that at least one 
 zero must separate every nonzero digit in the recoded representation. As 
 indicated in the introduction to Chapter h, Reitwiesner [9] has developed a 
 division algorithm which generates a quotient in canonical, minimal form. 
 This algorithm can also be derived from the right directed recoding which is 
 equivalent to the canonical, minimal left directed recoding. Freiman [h] has 
 shown that the S-R-T division algorithm, proposed independently by Sweeney 
 and described by MacSorley [6], Robertson [10] and Tocher [13] » produces 
 a minimally recoded quotient for normalized divisors, D, in the range 3/5 < 
 |d| < J>jk. Wilson and Ledley [15] have presented a division algorithm which 
 yields a minimally recoded quotient for all normalized divisors. Metze [7] 
 has recently shown that the S-R-T method also yields a quotient in minimal 
 form for all normalized divisors if the shifted partial remainders are compared 
 with one of four constants each associated with a particular range of |D|. 
 
 In the first section of this chapter we extend certain results of 
 the preceding chapters to permit infinite representations. This is done so 
 that Y includes the binary radix complement representation of all y in the 
 interval - 1. < y < 1. Having established these extensions, the representation 
 of y may be considered, finite or infinite as the situation demands. 
 
 -118- 
 
-119- 
 
 An arithmetic interpretation of the minimal recocting functions is 
 developed in section 5«2. This permits the determination of the next mode 
 digit by knowledge of the value or approximate value of a weighted representa- 
 tion which includes the preceding mode digit and pertinent y. digits. It also 
 provides an important link between the minimal right directed recoding functions 
 and the minimal division algorithms. 
 
 Minimal multiplication and division algorithms are discussed in the 
 next two sections. These algorithms are defined in terms of minimal left and 
 right directed mode functions respectively. Their use guarantees minimally 
 recoded multipliers and quotients. Examples are given for n = 6 to illustrate 
 the multiplication and division algorithms defined by SL. and SR. . These 
 examples also provide an illustration of the interrelation of minimal multi- 
 plication and division algorithms. The classes of semi-universal, minimal 
 multiplication and division algorithms are defined. The latter class is shown 
 to include the Wilson-Ledley algorithm and the Metze modification of the S-R-T 
 algorithm. 
 
 In the final section the interrelation of minimal multiplication 
 and division algorithms is discussed. 
 
 5.1 Minimal Recoding Functions for Infinite Representations 
 
 The definition of minimal multiplication and division algorithms is 
 
 accomplished with less difficulty if infinite representations of y are allowed. 
 
 We rule out all binary expansions which end in an infinite sequence of l's so 
 
 that every real number, y, in the interval - 1 < y < 1 has a unique binary 
 
 representation [2]. 
 
 Having established uniqueness for the representation of y, most of 
 
 the definitions and theorems of the preceding chapters may be extended to the 
 
-120- 
 
 infinite case without difficulty. In this chapter we will be particularly- 
 interested in the interpretation of Tables 4-3 and 4-4 when n is infinite. 
 
 Inspection of Table 4-3 shows that Cases I and IV present no problem. 
 Only the boundary sequences for X. = 0, 1 need to be examined in Cases II and 
 
 00 
 
 III. If A denotes an infinite alternating sequence, these boundary 
 sequences are: 
 
 y i-l y i y i+l 
 
 J 
 
 A 
 
 A 
 
 A 
 
 
 
 
 1 
 
 Case II 
 
 \. ., = 1 
 
 l+l 
 
 Case III 
 
 \. n = 
 
 l+l 
 
 In Table 4-4 we note that Cases II and III are not affected by 
 allowing an infinite n. In Cases I and IV. only the boundary sequences for 
 
 p. =0, 1 need to be examined. These are? 
 
 1 A 
 
 y i y i+l y i+2 
 
 < 
 
 1 A 
 
 1 1 A 
 
 Case I 
 Pi-1 = ± 
 
 Case IV 
 
 p i-l 
 
 
 
 It is necessary to determine the values of X. and p. which guarantee 
 a maximum gain in zeros over these sequences. To accomplish this, we define 
 
 00 
 
 A as an A sequence of infinite length. The inductive arguments used to prove 
 Theorems 4-11, 4-12, 4-13, 4-14 and 4-15 are valid for p infinite. It follows 
 that the maximum gains for A sequences listed in Tables 4-1 and 4-2 also apply 
 
 00 
 
 to A sequences. Theorem 4-16 is also valid and is used to determine the 
 maximum composite gains over the sequences in question. 
 
■121- 
 
 In the left directed sequences, we allow a single unit to appear on 
 
 00 
 
 the left of A and set X = 1 as the left boundary condition if y < 0. If 
 
 y > 0, the unit does not appear and \ , = 0. By using Theorem k—l6 and Table k-1 
 
 the following results. 
 
 y > 
 AVZ: p A = -, p z = 2; X. +1 = 1 => X.=0 
 
 — , 00 . — . 00 . 
 
 G(0A 0Z1;0) = - 1 > G(0A 0Zl;l) =0-2 
 
 Al_Z: p A . », p z . i; X±+i . 1 =-, X. = 0, 1 
 G(0A°° 0Z1;0) =0-1 = G(0A°° 1Z1;1) =0-1 
 
 A°° U : P A = », P g = 1; \ +1 = z-$. \ ± = 0, 1 
 G(0A°° 0U0;0) =0 + = G(0A°° 1U0;1) =0 + 
 
 : P A = °°; X. n = ^^ \. = 
 
 A l+l ^^ l 
 
 G(0A 0;0) =0 > G(0A 0;l) = - 1 
 
 y < 
 
 00 
 
 UA Z: Pu = 1, p a = oo, p z = 2; x ±+1 = 1 =^ \. = 0,1 
 
 G(1U0A°° 0Z1;0) =0 + 0-1 = G(1U1A°° 1Z1;1) =1 + 0-2 
 
 m_^: Pu = 1, p A = oo, p z = 1; x. +i = i -p, x± = i 
 
 G(1U1A°° 0Z1;0) =1-1-1 < G(lUlA°° lZl;l) =1 + 0-1 
 
 UA_U: Pu = 1, p A = oo, Pu = !- x. +1 = => X. = 1 
 G(1U1A°° 0U0;0) =1-1 + < G(lUlA°° lU0;l) =1 + + 
 
 UA°° : p TT = 1, p ft = oo; \ - o — -^ \. = 0, 1 
 *U A ' l+l 9 l ' 
 
 — / 0° \ i 00 . 
 
 G(1U1A 0;0) =1-1 ss G(1U1A 0;l) =1-1 
 
-122- 
 
 00 00 
 
 It is convenient to assign \. a value of for the A 00, and A sequences 
 
 00 00 
 
 and a value of 1 for the A and A 1 sequences . This eliminates the need 
 to differentiate on the basis of the sign of y. 
 
 In the analysis of the four right directed sequences, we always 
 choose the right boundary condition p = as n — > °° . The following results 
 are obtained with the aid of Theorem k-l6 and Table k—2. 
 
 UZA i P TT = 1, -P z = 1, p a = °°; p i _ 1 = 1 -=^- o i = 
 G(0;1U0Z0A°° 0) = + + > G(1.;1U1Z1A°°0) =1-1-1 
 
 ua°° < p y = i, p a = °°; p ± _ ± = i ==^- p ± = o, i 
 
 G(0_;1U0A°° 0) = + = G(l;lUlA°° 0) = 1 - 1 
 
 AUA : p A = 2, Pu = 1, p A = °°; p i _ i = o :=£► p. = 0, 1 
 G(0;0A0U0A°° 0) = + + = G(l;0AlU0A°° 0) = + + 
 
 oo 
 
 A__: P A = «. p i _ i = =^ p. = 
 
 G(0;OA 0) = > G(l;OA 0) = - 1 
 
 00 00 00 00 
 
 Therefore, p. =0, 1 for 1A and OllA , and p. = for 10A and A . 
 
 5 .2 An Arithmetic Int erpretatio n of Minimal Mode Functions 
 
 It is convenient to assign an arithmetic value to the partial 
 sequences listed in Tables h-J> and k-k when discussing the application of 
 minimal recoding functions to multiplication and division processes. Both 
 the left and right directed partial sequences for n finite or infinite are 
 interpreted as unique binary representations, y*, lying in the range 
 -2 < y* < 2. Therefore, every minimal recoding function has a value of 1 over 
 a unique set of values, y*, in the range -2 < y* < 2. 
 
-123- 
 
 5.2.1 The Left Directed Case 
 
 Definition 5~1 « The Arithmetic Value of a Partial Left Directed Sequence 
 
 Let every partial sequence, 3C. (n + A), X. , for n finite or 
 infinite, be assigned a unique value, y , as defined below 
 
 Li 
 
 n+A-1 
 
 * 
 
 y L (l+l) = y i + l 
 
 2X. . + 
 l+l 
 
 y i-j 2 
 
 -j-l 
 
 (5:D 
 
 j=0 
 
 By considering all possible partial sequences, it is clear that 
 -2 < y T < 2. Table 4-3 as extended in section 5-1 yields the following results 
 
 Case I: 
 
 Case II: 
 
 - 1 < y L (i+i 
 
 Case III: 
 
 - V3 5 y L ( i+1 
 
 - 3/2 < y^ 14 " 1 
 
 - 5/3 < y£(i+i 
 
 - 4 
 
 - 2 y T *U+i 
 
 Case IV: 
 
 5/3 < yj(i+i 
 
 3/2 < y*(i+l 
 
 V3 < y*U+i 
 
 l < y£(l+l 
 
 o < y*(i+l 
 
 < o 
 
 < - l 
 
 < - V3 
 
 < - 3/2 
 
 < - 5/3 
 
 < 2 
 
 < 5/3 
 
 < 3/2 
 S V3 
 
 < l 
 
 X. = 
 
 1 
 
 1 
 
 
 X. = 
 
 1 
 
 1 
 
 
 X. = 
 
 1 
 
 o, 
 
 1 
 
 X. = 
 
 1 
 
 o, 
 
 1 
 
 X. = 
 
 1 
 
 
 
 
 X. a 
 
 1 
 
 1 
 
 
 X. = 
 
 1 
 
 o, 
 
 1 
 
 X. = 
 
 1 
 
 o, 
 
 1 
 
 \. = 
 
 
 
 
 X. = 
 
 In correlating these intervals with the mode functions defined in 
 Chapter 4, it is convenient to partition the intervals over which X. =0, 1 
 into a set of disjoint subintervals . The intervals -3/2 < y T < - V3 an( 3- 
 
 5/3 < y T < " 3/2 are partitioned into a series of subintervals having end 
 
-124- 
 
 points ax . and bx . . These end points are defined by Eqs . 5:2 and 5:3 
 
 J J 
 
 which are valid for < j < n . 
 
 10 
 ax . 
 
 J 
 
 J 
 
 - 3/2 ♦ £ 2 ' 2k ' 1 
 k=l 
 
 x. 10 
 
 bx . 
 J 
 
 - 3/2 - ^ s"^- 1 
 
 (5:2) 
 
 (5:3) 
 k=l 
 
 The intervals 3/2 < y* < 5/3 and 4/3 < y < 3/2 are partitioned into a series 
 
 of subintervals having end points ax. and bx . . These end points are defined 
 
 J J 
 
 AL 
 
 by Eqs . 5:4 and 5 '• 5 and are valid for < j < n . 
 
 01 , 10 fez \\ 
 
 ax . = 3 + ax . (5:4} 
 
 J u 
 
 , 01 10 /_ _» 
 
 bx . = 3 + ^x . (5:5) 
 
 J J 
 
 We can now make the following observations. NL.(l, 0, YT J =1 
 
 over the interval - 4/3 < Vt < - 1. Assuming that a. . = b. . = 1 in Eq. 4:l8 
 
 - L & ij ij 
 
 and 4:19, AL. .(l, 0. Y^ . ) = 1 over ax 10 n < y* < ax 10 and EL. .(l, 0, YT , ) = 1 
 
 ' ij ■ ' l+l' j-1 — J L j ij ' l+l 
 
 10 * 10 / i \ / * 
 
 over bx . < y T < bx . n . NL.(0, 1, YT , ) = 1 over 5/3 < y T < 2. Assuming 
 j — L j-1 i v ' i+l' ' — "'L 
 
 01 ,01 , n n _Jj x 01 ^ * ^ 01 
 
 that a . . = d . . = 1 . AL..(0, 1 , YT , ) =1 over ax . . < y T < ax . and 
 
 ij ij ' ij v ' » i+l' j-1 - J L j 
 
 BL. .(0, 1, Y L J = 1 over bx 01 < yf < bx 01 n . As defined by Eqs. 4:31 and 
 ij v l+l' j - J L j-1 
 
 4:32 it is also clear that AL. = 1 over - 3/2 < y T < - 4/3 and 3/2 < y T < 5/3; 
 and BL. = 1 over - 5/3 < y < - 3/2 and 4/3 < Y T < 3/ 2 - 
 
 J. Ij Li 
 
 It follows that every restricted, minimal L.(\. ,, y. ., YT , 
 
 J ' i x l+l' ^i+l l+l 
 
 assumes a value of 1 over a unique set of subintervals in the range - 2 < 
 
 * / o T j.i n / • 10 ,10 01 , -01 , .. 
 
 y T < 2. In the most general case (i.e. when a. ., b. ., a. . and b. . are arbitrary 
 J L & ij' ij' ij ij 
 
 Boolean functions) a subinterval in the ranges - 5/3 < Y T < ~ V3 and V3 < Y L 
 
 < 5/3 ma Y contain only one value 
 
•125- 
 
 The set of symmetric , minimal L. defined in Chapter k can be easily- 
 visualized in terms of the intervals over which X. = 1. Every symmetric, 
 minimal L. has a value of 1 over - k/j> < y < and 5/3 < y T < 2; and may have 
 
 i L L 
 
 a value of 1 over any combination of the interval pairs 
 
 i 10 ^ # . 10 01 - * . 01 x 
 
 ( ax . ., < y7 < ax . : ax . n < y T < ax . ) and 
 j-l-^L j j-l-^L j y 
 
 , 10 . * ^ , 10 . 01 ^ * ^ , 01 n 
 
 ( ^ <y L <^.r bx j <y L <bx j-i } - 
 
 The three symmetric, minimal L. defined as NL., SL. and EL. have a 
 
 ' i i l i 
 
 value of 1 over the intervals given below . 
 
 NL. = 1 over - k/j < y T < and 5/3 < y* < 2. 
 i L L 
 
 SL. = 1 over - 3/2 < y T < and 3/2 < y T < 2. 
 
 1 — L ■— " Li 
 
 EL. = 1 over - 5/3 < y* < and k/j < y* < 2. 
 
 1 L L 
 
 As applied to multiplier recoding, the set of recoding functions 
 which exhibit a property called arithmetic symmetry is of interest. This 
 property is defined below and is not limited to minimal L.. 
 
 Definition 5-2 : The Set of Arithmetically Symmetric Left Directed Mode Functions 
 
 Let rtC denote the set of L.(X. , , y . . , Y^ . ) in which Y\ . may 
 u a i v i+l' J i+1' i+l y l+l J 
 
 include y. to the left and right of y. and whose corresponding recoding 
 functions, F T (y), have the property of arithmetic symmetry. F T (y) = y' 
 has the property of arithmetic symmetry if and only if F (-y) = -y' . 
 
 Li 
 
 Theorem 5-1: The necessary and sufficient condition that L. e oL is for 
 — J 1 a 
 
 L. ^ L D . 
 
 1 1 
 
-126- 
 
 Proof : Part I 
 
 •f D 
 
 We assume first that L. e^L and show that L-j = L.. Since 
 
 1 a L 1 
 
 L- e c£ > F (-y) = -y' . This implies that if +1 appeared in y' ; then a -1 
 
 appears in -y' and conversely. Note that -y can be represented "by the 
 
 diminished radix complement of y with X = 1. 
 
 To interpret this in terms of a restriction on L . we examine the 
 
 "basic recoding equation y! , = y. _ + X. , - 2\.„ Of the six allowed combina- 
 & ^ J l+l J l+l l+l l 
 
 tions of (y. . , \. . , A..), y! _ = for (0, 0, 0) and (l, 1, l): y! _ = 1 for 
 w i+l' i+l' i" J l+l \ t > / \ } ) t> j 1+1 
 
 (010) and (100); and y! = -1 for (Oil) and (lOl). When y is replaced by -y, 
 the first element in each combination is complemented. If yl , =0 for y, 
 then y! =0 for -y. To insure this, the second and third elements of the 
 combination must also be complemented. This also insures that +1 is replaced 
 by -1 and vice versa. It follows that 
 
 \. = L.(\. ., y. ., Y^ _) = (L.(\. ., y. ,, Y L .)) 
 
 i i v i+l' J i+1' i+l 7 v i v i+l' "'i+l' l+l" 
 
 where Y. implies that every element of the set is complemented. Since 
 
 L.(\. ., y. .,■£.)= [L.(\. ., y. ., Y 1 . ) D ] , it is clear that X. = 
 
 l i+l' ''i+l' i+l L i v i+l' J i+1' i+l' J ' l 
 
 L.(\. . , y. . , YT n .) = L. (\. n , y. n , YT . ) . Note that X = 1 as required. 
 
 l i+l' ''i+l' i+l' i v i+l' ''i+l' i+l' n 
 
 Part II 
 
 If L. = L., then L. = L. . This implies that if y is replaced by 
 
 ii'ii 
 
 its diminished radix complement, then X must also be replaced by its dual. In 
 doing this X = 1 so, in effect, y has been replaced by -y. As shown above, 
 the basic recoding equation replaces y'. . by -y'. ,. Therefore, if F T (y) = y', 
 
 1+1 1+1 1j 
 
 then F (-y) = -y ! . It follows that any L. which satisfies the condition 
 
 Ju 1 
 
 L. - L. must be an element of c< . 
 li a 
 
-127- 
 
 It has already "been shown that SL. = SL. . The canonical, minimal 
 recoding which SL. produces is easily shown to have the property of arithmetic 
 
 X 
 
 symmetry. NL. and EL. are clearly not elements of 
 
 The minimal L. e cZ in which "T7 . includes only those y. to the left 
 1 a l+l J ° j 
 
 of y . , can be readily interpreted in terms of y' . The interval map of every 
 i+1 L 
 
 restricted, minimal L. that is self -dual can he constructed by the following 
 
 procedure. For every y (i+l) in the interval h/j> < y < 5/3 assign X. a value 
 
 of or 1 as desired. Over the intervals < y < k/ 5 and 5/3 < y T < 2, X. 
 
 has the required values of and 1 respectively, The value of X. that is 
 
 assigned to every y'(i+lj in the interval - 2 < y < is now determined. 
 
 For every y in the interval < y < 2 there is y^ in the interval -2 < 
 Li — L L "- 
 
 *D < . We require that X. be associated with y T ^ if A., is associated with 
 J L * i °L 1 
 
 y . This requirement is automatically satisfied for those intervals over which 
 
 the value of X. is fixed for a restricted, minimal L.. Therefore, to insure 
 
 1 ' 1 ' 
 
 that L. = L. it is only necessary to guarantee that the requirement is 
 satisfied for the intervals in which the value of X. may be arbitrarily chosen. 
 It is clear from this interpretation that SL. is the only symmetric, 
 
 minimal L., as defined in Chapter k, that is also an element of _ 
 
 5.2.2 The Right Directed Case 
 
 Definition 5~3 ° The Arithmetic Value of a Partial Right Directed Sequence. 
 
 Let every partial sequence, p. , , X (n+A), for n finite or infinite, 
 
 1--L K 
 
 -* 
 be assigned a unique value, y_., as defined below. 
 
 n 
 
 n+A 
 
 **M ' y i " 2p i-i + I y i +j 2 " J (5:6) 
 
■128- 
 
 *, 
 
 This definition of yd) is used to obtain the following interpre- 
 tation of Table 4-4 as extended in section 5»1« 
 
 Case I: 
 
 -2/3 
 
 < 
 
 y R d, 
 
 > < 
 
 
 
 p i 
 
 = 1 
 
 
 
 - 3/4 
 
 < 
 
 y R d 
 
 ) < 
 
 - 2/3 
 
 p i 
 
 = o, 
 
 1 
 
 
 - 5/6 
 
 < 
 
 y R d 
 
 ) < 
 
 - 3/4 
 
 p i 
 
 = o, 
 
 1 
 
 
 - 1 
 
 < 
 
 y R d 
 
 ) < 
 
 - 5/6 
 
 p i 
 
 = 
 
 
 Case II: 
 
 - 2 
 
 < 
 
 y R d 
 
 > < 
 
 - 1 
 
 p i 
 
 = 1 
 
 
 Case III: 
 
 1 
 
 < 
 
 y R d 
 
 ) < 
 
 2 
 
 p i 
 
 = 
 
 
 Case IV: 
 
 5/6 
 
 < 
 
 y R d. 
 
 i < 
 
 1 
 
 p i 
 
 = 1 
 
 
 
 3/4 
 
 < 
 
 y R d. 
 
 > < 
 
 5/6 
 
 p i 
 
 = o, 
 
 1 
 
 
 2/3 
 
 < 
 
 y R d, 
 
 i < 
 
 3/4 
 
 p i 
 
 = o, 
 
 1 
 
 
 
 
 < 
 
 y R d, 
 
 i < 
 
 2/3 
 
 p i 
 
 = 
 
 
 As in the left directed case, it is useful to partition the intervals 
 over which p. =0, 1 into a set of disjoint subintervals . The intervals - 3/4 
 < y R < - 2/3 and - 5/6 < y R < - 3/4 can be partitioned into a series of sub- 
 
 R 
 
 intervals having end points ex. and dx . . Equations 5*7 an d 5°8 define ex. 
 
 J J J 
 
 and dx 11 for < j < n*. 
 
 ex 
 
 11 
 
 
 k=l 
 
 J 
 
 dx 11 = - 3/4 
 J 
 
 -2k-2 
 
 -2k-2 
 
 (5^7) 
 
 (5:8) 
 
 k=l 
 
-129- 
 
 The intervals 3A < y D < 5/6 and 2/3 < y^ < 3A can be partitioned into a 
 — n — r\ 
 
 series of sub intervals having end points ex. and dx . . Equations 5^9 an( i 
 
 J J 
 
 5:10 define ex. and dx . for < j < n . 
 
 J J 
 
 ex°° = 3/2 + exf ( 5 :9) 
 
 J J 
 
 dx°° = 3/2 + dx 11 (5:10) 
 
 J J 
 
 Note that NR.(l, 1, Y?) = 1 over the interval - 2/3 < y * < . By 
 11 K 
 
 setting c 11 = d 11 = 1 in Eqs. i+:52 and k:5k, CR. .(l, 1, Y R ) = 1 over ex 11 , < 
 
 y* < exf and DR (l, 1, Y?) = 1 over dx^ 1 < y* < dx^ 1 . NR (0, 0, Y*) = 1 
 
 K J IJ 1 J — K J-l 1 1 
 
 over 5/6 < yj < 1. If we let c°° = &°.°. = 1 in Eqs. l+:58 and k:60, CR. .(0, 0, Y?) 
 
 K ij ij ij 1 
 
 = 1 over ex . < y* < ex . and DR. . (0, 0, Y. ) =1 over dx . < y^ < dx . . . By 
 J - - 1 - — R J ij 1 J — R J-l 
 
 Eqs. 1^:73 and ^:7^, it is apparent that CRi = 1 over - j/k < y* < - 2/3 and 
 3A 1 Yr 5 5/6; and DR. = 1 over - 5/6 < y* < - 3A and 2/3 < y* < 3A- 
 
 Every restricted, minimal R. has a value of 1 over a unique set of 
 
 subintervals in the range - 2 < y < 2. The set of symmetric, minimal R. 
 
 — K 1 
 
 includes those R. which have a value of 1 over - 2 < y^ < -1. - 2/3 < y„ < 
 
 1 — J R ' ' ^R 
 
 and ^6< y* < 1; and may have a value of 1 over any combination of the interval 
 
 t 11 , * . 11 00 , * . 00x , , , 11 . * , 11 ,00 
 pairs (cx.^ < y R < cx^ ; cx.^ < y R < ex. ) and (dx.^ < y R < dx . ; dx._ ± 
 
 * 00 \ 
 
 < y_. < dx . ). The symmetric, minimal R. defined as NR., SR. and ER. have 
 — "'R j ' 1 11 1 
 
 the following interval maps . 
 
 NR. = 1 over - 2 < y* < - 1, - 2/3 < y* < and 5/6 < y* < 1. 
 
 SR. = 1 over - 2 < y* < - 1, - 3/^ < y* < and 3A 1 Y^ < 1- 
 
 ER. = 1 over - 2 < y*.< - 1, - 5/6 < y* < and 2/3 < y* < 1. 
 
 Right directed recoding functions which possess the property of 
 arithmetic symmetry are of interest in the study of division processes. This 
 
■130- 
 
 property is not limited to minimal R., so the following definition and theorem 
 are stated in rather general terms . 
 
 Definition 5-k : The Set of Arithmetically Symmetric Right Directed Mode 
 
 Functions 
 
 Let U\ denote the set of R.(p. . , y., YV) in which Y. may include 
 a i^i-l' 1 l l 
 
 y. to the left and right of jr., and whose corresponding recoding functions 
 J -*- 
 
 F T3 (y) have the property of arithmetic symmetry. F 1D (y) = y' has the property 
 of arithmetic symmetry if and only if F_(-y) = -y 1 . 
 
 (R. 
 
 X 
 
 D 
 
 Theorem 5-2: The necessary and sufficient condition that R. e \J\ is for 
 ■ — — - — ■ ° l a 
 
 R. = R 
 i l 
 
 Proof: The proof of this theorem follows the same pattern as the one given 
 for Theorem 5-1° 
 
 In Chapter k it was shown that SR. s SR. , so SR. e Q\ . Since 
 
 i i' i a 
 
 NR. = ER. neither NR. nor ER. are elements of 0\ . 
 11 11 a 
 
 The restricted, minimal R. which are self -dual and therefore elements 
 
 ' l 
 
 
 of \J\ are represented by a unique set of interval maps. The procedure used 
 
 3. 
 
 to construct these maps is identical to one used for the minimal L. ecx . It 
 
 ^ i a 
 
 is clear from this set of maps that SR. is the only symmetric, minimal R., as 
 
 defined in Chapter k, that is also an element of _ 
 
 <R. 
 
 5° 3 Minimal Multiplication Algorithms 
 
 In this section we consider two distinct classes of binary, minimal 
 multiplication algorithms. The multiplier, yeY, is assumed to have a finite 
 
-131- 
 
 binary representation in the range - 1 < y < 1. We let - 1 < M < 1 and 
 
 - 1 < P . < 1 denote the multiplicand and product. 
 n+1 — 
 
 We assume a binary computer that is capable of forming P with the 
 following recursion relationship. Let P . represent the partial product at 
 the (n-i) step, then 
 
 P . = 2" 1 P . _ + y! _ M 
 n-i n-i-1 J l+l 
 
 (5:11) 
 
 In this equation y! . =y. , + X. . - 2X. where X. =L.(\. , , y. , , Y^ , ) and 
 
 ^ J i+1 J i+1 l+l l 11 l+l ^i+l' l+l 
 
 Y. , may include y. on either side of y. 
 l+l J i+l 
 
 Theorem 5 -3 : If P =0, then Eq. 5:11 yields P = y M in n+1 steps. 
 
 Proof : Initially, i = n-1 so that y' is the first multiplier digit recoded 
 
 with X =0. 
 n 
 
 P-, = y* M 
 
 1 "'n 
 
 P~ = 
 
 2 y" M + y* n M 
 J n J n-1 
 
 Assume that 
 
 P n-i-l 
 
 n-2 
 
 2 J- n+2 y' 
 
 n+i-j 
 
 J = i 
 
 M 
 
 Then 
 
 n-i 
 
 n-2 
 
 V 2J" n+1 y' 
 L^ J n+i- 
 
 J=i 
 
 M + 
 
 y n+(i-l) - (n-2) 
 
 M 
 
 Ey replacing j by j + 1 in the first term and then absorbing the second we 
 obtain 
 
 n-i 
 
 n-2 
 
 j-n+2 , 
 
 y n+(i-l)-j 
 
 J = i-1 
 
 M 
 
 It follows that for i = -1 
 
n+1 
 
 n-2 
 
 ■132. 
 
 2 " n+2 rU-i 
 
 In this expression let j = n - k - 2 so that 
 
 n+1 
 
 2_k y; 
 
 k=0 
 
 M = y' M , 
 
 Since y' is a recoding of y, y = y' and P = y M. 
 
 Definition 5-5 ; Minimal Multiplication Algorithm 
 
 A minimal multiplication algorithm is defined by Eq. 5:11 with 
 
 y! = y. + k. -2k. . where X... = L. , L.el and L. (k. . . y . _ , Y. n ) = 
 J i 1 1 l-l 1 1' 1 ^ i v 1+1' ''l+l' 1+1' 
 
 L. , (X. . , y. _, Y L J for < i < n - 1. 
 
 l+l N i+l' 'l+l' 1+1 y — — 
 
 It follows that every L.eoL defines a minimal multiplication 
 
 algorithm. Since a general L. is a function of k. . and all of the digits 
 
 to 1 l+l 
 
 of y, most of the algorithms so defined are not practical. 
 
 The simplest L., SL . , is most attractive and yields the canonical, 
 
 A 
 
 minimal recoding . An interesting "but impractical algorithm is defined by L. 
 
 which denotes the left directed mode function which is equivalent to the 
 
 A 
 
 simplest right directed mode function, SR. . L. is defined by 
 
 A A 
 
 l. = \. _ y . . y (*•■ n V y. n ) k. 
 1 l+l* 7 1+1 v x 1+1 v •'l+l' 1 
 
 (5:12) 
 
 where K. = ST. and is defined by Eq. + .'100,, 4:101 and 1+:102. Two examples 
 are now given to demonstrate these algorithms. Let y = 1.101001 = - 23'/ 6k and 
 let M = 0.101101 = +5/ 61+. 
 
 We consider SL. first. The arithmetic interpretation of SL. as 
 
 given in section 5-2.1 is used to recode y starting at the right. Since 
 
-133- 
 
 X 6 = °' y L^ = 1 + 0.001011. It follows that \^ = since < V-r(6) < 3/2. 
 
 Therefore y> = y^ = 1. Proceeding in this manner we obtain X, = X = and 
 
 consequently y = y and y, = y, . At y we note that y (3) = 1 + 0.011111 
 
 where three units have "been added to account for negative sign propagation as 
 
 y is shifted right. Since < y.(3) < 3/2, ^« = and y_, = y_. On the next 
 
 - 1 2 3 3 
 
 ^. 
 
 step y*(2) = 0.111111. Since 5 y T (2) < 3/2, \ = and y' = y . Continuing 
 Li J.J 1 2 2 
 
 we obtain 3/2 < y L (l) = 1 + 0.111111 < 2, so X = 1 with y' = y - 2 = 1. At 
 y , - 3/2 < y*(0) = -1 + 0.111111 < 0, so \ = 1 with y' = y +1-2=0 
 which satisfies the boundary condition X = y . It follows that y' = 
 0.101001 = - 23/61+. By using -M = 1.010011, the multiplication, P = y'M, is 
 accomplished with three additions. 
 
 0.000000101101 
 0.000101101 
 
 1.1010011 10J5 t \f k5 
 
 p 7 . 1.101111110101 
 
 In working with L. it is convenient to extend the definition of y . 
 i L> 
 
 This is accomplished by replacing the term (y. - 2X. ) in Eq. 5:1 with 
 the term (y • -, + 2y. - k-X . -, ) • Note that every extended partial sequence 
 still corresponds to a unique y . Substituting K. in Eq. 5:12 yields 
 
 -Li I 
 
 A 
 
 J i "i+l'i+l v "i+r 7 i+l- 7 i+2 "i+r 7 i+l J i+2 ~i v ~> ^ *i+l 
 
 " — - — L 
 
 L . =K^y<^ V \.^y, ,-,y, l0 V \, , n y, , n y, i0 l, (o, i, Y^,J y 
 
 \+l y i+l y i+2 L i (l > °' 4l } 
 
 where L.(0, 1, Y 1 ) and L.(l, , ^ ) are defined by Eqs . 4:100 and 4:101. 
 
 In terms of the extended definition of y T ; X. n y. j. _ L.(0, 1. YT , ) =1 
 
 J 1/ x+r i+l* 7 1+2 i N ' ' i+l y 
 
 over 4/3 < y* < 2 . an * X i+ iyi + iy i+ 2 L i (l ^ °' ^+1^ = 1 over " k ^ - y L < " lm 
 
 Likewise x ±+1 y ±+1 (y ±+2 V y i+2 ) = 1 over - 3 1 y L < - 2 and - 1 < y * < 
 
 while ~X-,-,y - +1 y • , p = 1 over 3 < y < 4. Thus in the interval - 4 < y < 4, 
 
 L. = 1 over - 3 < y* < - 2, - 4/3 < y* < 0, 4/3 < y£ < 2 and 3 < y£ < 4. 
 
A 
 
 Note that L. = over exactly those intervals obtained by taking the dual of 
 
 a A A D 
 
 each of the L. =1 intervals. It follows that L. = L. which also can "be verified 
 i 11 
 
 by Boolean operations. 
 
 A 
 
 We now recode y by L. using the arithmetic interpretation given above. 
 
 Since X^ = and y = 0, y*(6) = 1 + 0.001011. It follows that X = and yl 
 
 = yg since < yA6) < k/j. At the next step, 2 < y (5) < 3 so \. = and 
 
 y R = y . Proceeding in this manner we obtain y' = 0.011001 = - 23/6*4-. The 
 
 multiplication , P = y' M, is completed with one addition and two subtractions 
 
 of M. 
 
 0.000000101101 
 1.111010011 
 1 . 11010011 
 
 -1035 
 
 P ? = 1.101111110101 = -^ 
 If -y were used as the multiplier, SL. would yield -y' = 0.101001 
 
 A _ A 
 
 and L. would yield -y ! = 0.011001 because SL. and L. are both self -dual. 
 1 J J 11 
 
 Definition 5-6 ; Universal, Minimal Multiplication Algorithm (u.m.m.a.) 
 
 A u.m.m.a. is based in Eq. 5°H with X. = L. where L. is any element 
 
 ^ ' 11 1 
 
 of cC > a subset ofc-C. 
 
 Since a minimal L. is, in general, a function of y.to the left and 
 1 J 
 
 right of y. , the corresponding multiplication algorithms are not always 
 practical. To improve on this, we consider the class of semi-universal, 
 minimal multiplication algorithms as defined below. 
 
 Definition 5~7 ° Semi -Universal, Minimal Multiplication Algorithm (s-u.m.m.a.) 
 A s-u.m.m.a. is any u.m.m.a in which oC includes only restricted, 
 
 minimal L. . 
 
 1 
 
 I 
 
-155- 
 
 An easy way to visualize the class of s-u.m.m.a. is in terms of the 
 
 values of y T (i+l) for which \. = 1. Since YT . includes only those y. to the 
 J L l l+l j 
 
 left of y. , , the interval - 2 < y < 2 suffices for this purpose. We know 
 i+1 J-i 
 
 from section 5-2.1 that \. must equal 1 over - h/~$ < y T ( i+ l) < ° and 5/3 < 
 
 1 — Li — 
 
 y (i+l) < 2 and \. may equal 1 over - 5/3 < y (i+l) < - k/j and k/j < y (i+l) 
 
 L/ 1 Li L 
 
 < 5/3- For any s-u.m.m.a., X.( 0< i < n-l) must he 1 over the first pair of 
 intervals. For a particular s-u.m.m.a., \. = 1 over some set of values in the 
 second pair of intervals, hut the values included in that set may change with i. 
 
 The simplest s-u.m.m.a. is the minimal multiplication algorithm 
 based on SL.. In specifying the simplest s-u.m.m.a. we choose the set <?£. such 
 
 that the join of all L.e c£L involves the fewest digits of the multiplier. 
 
 « 
 It is apparent from results of section 4.2.2 that cAL should include SL. and 
 
 any L. formed by joining any combination of AL. . to NL. for 1 < i < n . Any 
 L. formed by joining BL. . to NL. must be excluded from cL. . Under these 
 conditions the join of all L. in ^X, is SL . . 
 
 In practice the minimal multiplication algorithm defined by SL. is 
 the easiest to implement because the multiplier digits are always precisely 
 known. This is not true of quotient digits so we can expect one or more semi- 
 universal, minimal division algorithms to have some practical value. 
 
 5-4 Minimal Division Algorithms 
 
 Two basic classes of binary, minimal division algorithms are considered 
 in this section. These algorithms produce a minimally recoded approximation 
 of the exact quotient. 
 
 Let the divisor, D, and dividend, 2P , satisfy l/2 < |d| < 1 and 
 < |2P I < |d). Let y = 2P /D denote the exact quotient which is guaranteed 
 to satisfy - 1 < y < 1. In this analysis yeY may have an infinite binary- 
 representation . 
 
-156- 
 
 We postulate a "binary computer that is capable of generating a 
 recoded quotient defined as 
 
 m 
 
 y : (m) = Y y± 2_1 (5:13) 
 
 i=0 
 
 where y. =y. + p. -2p. . and p. = R.(p. , , y., Y.). As used here, R. is 
 1 l l l-l M i l *i-l ''i' l' ' l 
 
 ■p 
 any right directed mode function in which Y. may include y. on either side 
 
 - 1 - J 
 
 of y. - During the i step of the division process, y! is determined and used 
 to form the next partial remainder, P. , as defined by 
 
 P. . = 2P. - y! D (5:1*0 
 
 i +1 1 J 1 \s i 
 
 ! 
 
 Theorem 5 -5 provides the rule for determining y. in general. 
 
 Theorem ^-H- i Let P denote the initial partial remainder, then 
 
 o 
 
 2P. 
 
 -^ = y*(D (5=15) 
 
 where y.(i) is defined by Eq. 5°&< 
 
 2P. P. 
 
 Proof ; We write Eq. 3 ilk as -~ = y. + p. - 2p . . + -~— . 
 1 D " l ^l l-l D 
 
 For convenience let 
 
 *7u - L y i +J 2 " J (5:16) 
 
 j=l 
 
 2P 
 
 Note that -=r- = -2p . + y + y_ (O) - y„(0) where p _ = 1 if y < and if 
 
 D -1 O K K -1 
 
 y > 0. Substituting this in the modified form of Eq. 5:1^ for i = yields 
 
 p 
 
 " 2d -1 + ^0 + y R (0) = y o + p o " 2p -l + - 
 
 Simplifying, rearranging, multiplying by 2 and substituting j + 1 for j in 
 3 
 
 y p (0) yields 
 
-137- 
 
 2P 
 
 — = y i " 2p o + y R (1) - 
 
 2P 
 It follows that -7— = y*(l). 
 
 JJ K 
 
 2P 
 
 * "X* .y.y. 
 
 We now assume -=— = y_,(i) = y. - 2p. + y_ (i) . By substituting 
 JJ Ja 1 1-1 n 
 
 this equation in the modified form of Eq. k-:lk and simplifying as before we 
 
 2P 
 
 i+1 "*/ \ 
 
 obtain — - — = y i+1 -2p j[ + y R (i+1). 
 
 2P. 
 Corollary : - 2 < -rp < 2 . 
 
 Proof: We know that < p. . < 1 since p. , = R. _(p. _. y. _, Y. .). The 
 
 — l-l — "i-l l-l i-2 l-l l-l 
 
 uniqueness of the representation of y was established in the infinite case 
 
 by ruling out all representations which terminate in an infinite series of 
 
 units. Therefore, < y_ (i) < 1. 
 
 — n 
 
 X3 
 
 Theorem 5-5 : If p . = R.(p. , , y . , Y.)% and R. is arbitrarily chosen from the 
 — - 1 11-I 1 1 ' 1 
 
 total class of right directed mode functions, then 
 
 y- = - 1 if - 2 <y*(i) < - 1; 
 
 y [ = - 1 or if - 1 <y*(i) < 0; 
 
 y[ = or 1 if <y*(i) < i; 
 
 y! = 1 if 1 <y£(i) < 2. 
 
 Proof : Any R. in the total class of right directed mode functions is defined 
 by Eq. J>:h and its characteristic function, S.(y., Y.). If R. is arbitrarily 
 chosen, the value of p. is not always uniquely defined. In essence, p. = 1 
 
-158- 
 
 if p. =1 and y. = 0; p. = if p. = and y. = 1: and p. =0, 1 if 
 l-l i "i "i-l J i ' K i ; 
 
 Pi-1 = y i" 
 
 If - 2 < y (i) < - 1, then examination of Eq. 5:6 shows that p. . 
 — R 1-1 
 
 must he 1. By adding 2p . , to this inequality we ohtain _< y. + y (i) < 1 
 
 l-l i R 
 
 which shows that y. must be 0. When (p. , y.) = (l, 0), p. = 1 by Eq. 3:^; 
 
 = -L 
 
 and y ± = y i + p ± - 2f> 1 _ 1 
 
 If -1 < y R (i) < 0, then Eq» 5:6 again shows that p. must be 1. 
 
 ,y y y y 
 
 Adding 2p . to this inequality yields 1 < y. - y_ (i) < 2 where y (i) is 
 i-1 — i R R 
 
 defined by Eq. 5°l6° Since < y D (i) < 1, y. must be 1. When (p. , y.) 
 
 — K 1 l-l i 
 
 = (1, l), p. = 0, 1 as defined by Eq„ ^:k with S. = 0, 1. It follows that 
 7-1= -1, 0. 
 
 If < y„(i) < 1, then Eq, 5:6 shows that p . . = since < 
 
 — R 1-1 — 
 
 y-D (.ij < 1. Therefore 0<y. + y (i) < 1, and y. must be 0. When (p. , y.) 
 R — i R i i-1 i 
 
 = (0, 0), p. = 0, 1 and y!_ = 0, 1. 
 
 If 1 < y*(i) < 2, then D . -, = and 1 < y. + y**(i) < 2. This 
 
 — R 1-1 — 1 R 
 
 implies that y. = 1. When (p. , y.) = (0, l), p. = by Eq. Jih and 
 
 If p . is defined by any right directed mode function, 
 
 R.(p. , , y., Y.), in which Y. includes only those y. to the right of y., then 
 1*1-1' i' i /; i J i 
 
 y. is uniquely specified over the interval -l<y(i)<l. 
 l — R 
 
 Proof : Since R. is a function of p . n , y. and y. to the right of y.. a unique 
 — l K i-1' J i "'.i °^ l' 
 
 * 
 
 value of p . is associated with every y_(i) in the interval - 1 < y_(i) < 1. 
 i R — R 
 
 */ \ 
 If - 1 < y^li) < 0, we know that p. , - 1 and y. = 1. Given the means to 
 — ^R ' 'l-l l 
 
 determine the set of y R (i) i- n this interval for which p. = 1, the value of 
 
 y! = y. + p.- - 2p. _ is uniquely determined for all y*(i) in the interval . A 
 J l l l l-l ^ J R 
 
 similar argument applies for the interval < y*(i) < 1. 
 
 ~ R 
 
-139- 
 
 It is Important to note that if Y. includes digits to the left of 
 
 i 
 
 y., then the quotient digits generated prior to y. must be taken into account 
 
 in the determination of y.. No attempt is made to develop a procedure for 
 handling R. of this type. 
 
 Theorem 3-6 : Let any right directed mode function, R., "be defined such that 
 
 •p 
 Y. includes only those y. to the right of y. . Equation 5:1*+ then yields 
 
 Y'(m) =y - 2- (m+l) y*(m + l) ( 5 :1 7 ) 
 
 in m + 1 steps where y. is determined by knowledge of the p. = R.(p. ,y.,Y.) 
 
 2P 
 associated with the value or approximate value of — — = y (i). 
 
 D K 
 
 Proof : By using Theorem ^-h, Eq. 5:li+ may be written as 
 
 y*(i + l) = 2y*(i) - 2y: ( 5 :l8) 
 
 If i = 0; y*(l) = 2y*(o) - 2y^ . 
 
 If i = 1; y*(2) = 2 2 y*(0) - 2 2 y^ - 2y^ . 
 
 Noting that y~(0) = y we assume 
 
 J+l 
 
 y*(i) = 2 1 y - ) y. 2 
 3=0 
 
 Then 
 
 y*(i + D =2 i+1 y- I y. 2^ -2y[. 
 
 By replacing j by j-1 and absorbing the final term we obtain 
 
 i+1 
 $1*1) - 2 1+1 y - £ y 2 J+1 . 
 
 o=o 
 
-140- 
 
 By setting i = m, multiplying by 2 ^ ' and substituting y'(m) as defined 
 by Eq. 5-:13, we obtain Eq. 5:1?. 
 
 The preceding theorems deal with R. as any element of the total 
 
 class of right directed mode functions. As such, R. may or may not be a 
 
 minimal mode function. We now limit the discussion to R.(p. , , y„, Y.) e (K 
 
 l -i-l' i/ i WK - 
 
 R 
 in which Y. is restricted to those y. to the right of v.. 
 
 l J J i 
 
 Theorem 5-7° If p . = R.(p. , , y., Y.) and R. is an arbitrary element of C/C 
 l li-l l 11 
 
 X) 
 
 for which Y. is restricted, then 
 l ' 
 
 y[ = - 1 if -2 < y*(i) < - 5/6 ; 
 
 Y- = - 1, if -5/6 < y*(i) < - 2/3 ; 
 y[ = if -2/3 < y*(i) < 2/3 ; 
 y\ = 0, 1 if 2/3 < y*(i) 1 5/6 ■ 
 
 i ' ' ' "R 
 
 y. 
 
 1 if 5/6 < y R (i) < 2 . 
 
 Proof: Since R. is an arbitrary element of the restricted subclass of 
 — l 
 
 (R, 
 
 p. 1, if - 2 < y R (i) < - 1, - 2/3 < y R (i) < and 5/6 < y*(i) < 1; and 
 
 p. = 0, 1 if - 5/6 < y R (i) < - 2/3 and 2/3 < y R (i) < 5/6. This is taken 
 directly from the arithmetic interpretation of Table k-k as given in sec- 
 tion 5„2.2o The proof that the rules for choosing y. are correct follows 
 the same pattern as the proof of Theorem 5 - 5- 
 
 2P. 
 
 The Corollary to Theorem 5-4 states that - 2 < — =■ < 2. These 
 
 limits apply to a general right directed recoding. For the restricted olass 
 
-11(1- 
 
 of minimal R., these limits are narrowed as indicated by Theorem 5-8. 
 
 Theorem 5-8 : If p. is defined "by a particular element, R., of the restricted 
 
 (R 
 
 subclass of u\- such that 
 
 y! = -1 if -2 < y*(i) < - v ; 
 y[ = if - v < y*(i) < z ; 
 
 y[ = i if z < y R U) < 2 , 
 
 where 2/3 < w, z < 5/6, then the true limits' en y R (i) are 
 
 -2w < y*(i) < 2z . 
 
 Proof ; Because of the restriction on 2P ; - 1 < y o (0) < 1. Starting with 
 — ~ ~~~ O R 
 
 this interval we generate a series of new intervals using Eq. 5°l8 an l "the 
 rules for choosing y.. We stop when the new interval has the same outer 
 limits as its predecessor. 
 
 The intervals - 1 < yt!(0) < - v, - w < y*(o) < z and z < y*(o) < 1 
 
 r R (0) < - w, - w < y R (~, . - JR , 
 
 map into < y*(l) < 2 - 2w, - 2w < y*(l) < 2z and 2z - 2 < y*(l) < 
 
 respectively. Only the new intervals created during this step need to be 
 
 analyzed in the next. These new intervals are - 2w < y_(l) < - 1 and 1 < 
 
 n — — 
 
 y RVl -W _ "•• *■*"- -—--"« ~*- "'^"^- i"*-fw j-*.*«v ■_ -r.. - j,_ 
 
 (l) < 2z . The first of these maps into 2 - Uw < y*(2) < and the second 
 — R — 
 
 maps into < y?t(2) < kz - 2. Since - 2w < 2 - kw and Kz - 2 < 2z for 2/3 
 
 •#. 
 
 < w, z < 5/6, no new intervals were created. It follows that -2w < y (i) 
 — — R 
 
 < 2z. 
 
 Definition 5-8 : Minimal Division Algorithm 
 
 A minimal division algorithm is defined by Eq. 5:1^ with y. = 
 
 y. + p. - 2p . .. where p. = R . , R.evfand 
 
 J 1 "i "i-I K i 1' 1 ' 
 
-142- 
 
 R. _(p. ., y., Y R ) = R.(p. _, y,, Y R ) for 1 < i < 
 
 i-I^i-I' J 1' i' i Xh i-l' ; r 1 — — 
 
 (R 
 
 Every R.e(A defines a minimal division algorithm; however, the 
 
 i 
 method of determining y. as outlined "by Theorem 5-5 an d its Corollary applies 
 
 6? 
 
 only to those R.e(7C for which Y. is restricted. The functions NR., SR. and 
 J l l i' l 
 
 (7? 
 
 ER. are included in this restricted subclass of LA . The interval maps 
 
 */ \ 
 relating p. to y (i) for these functions are given in section 5-2.2. 
 
 Two examples are now given to illustrate the minimal division 
 
 A . , A 
 
 algorithms defined by SR. and R. • Under the assumptions of section 4.4 R. is 
 
 equivalent to SR. • 
 
 V p i-i< r±> Y i> ■ Pi.i^i v (pi-i v V *± (5:19) 
 
 A 
 
 In this equation S. = SJ. is defined by Eqs . 4:95, 4:9b and 4:97 of 
 
 section 4.4. 
 
 Let 2P = 1.101111110101 = - i~H and D = 0.101101 = g| . 
 Since the division is exact, y' (6) must be a minimal recoding of 
 y = 1.101001 = - 23/64. Note that this division is the inverse of the multi- 
 plication example given in section 5- 3° 
 
 We consider SR. first. The interval map for SR. given in section 
 
 5,2„2 shows that p = 1 for - 2 < y R (i) < - 1, - 3/4 < y R U) < and 3/4 < 
 
 y R (i) < 1. By Theorem 5-5 and its Corollary y. must be uniquely specified 
 
 over the entire range of - 2 < y^Ai) < 2. The rules for choosing y. when 
 
 — K l 
 
 
 p. = SR. are given below, 
 H i 1 & 
 
 7[ = -1 if - 2 < y*(i) < - 3/4. 
 y- = if - 3/4 < y*(i) < 3/4. 
 y!_ = 1 if 3/4 < y*(i) < 2. 
 
-143- 
 
 Note that in the class of minimal division algorithms the "boundaries 
 
 on y (i) are sharp. In this case y. must be for y*(i) = ~5>/h - e for e 
 K i rv 
 
 arbitrarily small but not 0. If e = then y". must equal 1. If this is not 
 
 true, then one or more minimal R. v other than SR., govern the choice of y' near 
 
 the boundaries. Division algorithms, which permit this, belong to the class of 
 
 universal, minimal division algorithms which is considered later. To insure 
 
 that the boundaries are sharp all of the bits of D and 2P. must be used in 
 
 determining y (i). That is, precise knowledge of y*(i) must be available at 
 n K 
 
 every' step. For this reason the class of minimal division algorithms is 
 impractical. 
 
 In the present example, y!t(°) = y = ~qI • Therefore, y' = 
 yielding P ± = 2P - 0. It follows that y£(l) = - || so y^ = 0. 
 
 P 2 = 2P 1 - = kP Q and y*(2) = - f| so y^ = - 1 . 
 
 P^ = 2P 2 + D = (y*(2) + 1) D = - ^H and y*0) = - J so y^ = - 1 . 
 
 p^= (y*(3) +dd-5^ «* *>> =1 bo y ; = o. 
 
 P 5 = 2P^ - and y*(5) = | so y^ = 0. 
 P/- = 2P„ 
 
 Therefore , 
 
 > 6 = 2P 5 - and y*(6) = 1 so y g = 1 
 
 P ? = (y*(6) - 1) D = 
 
 — — 2^ 
 The recoded quotient y' (6) = 0.011001 = - zr which is identical to 
 
 the recoded multiplier that was obtained in the preceding section by using a 
 
 minimal multiplication algorithm based on L.. This was expected since L. is 
 
 equivalent to SR. - 
 
■Ikk- 
 
 We now develop the rules for choosing y. in the minimal division 
 
 A 
 
 algorithm based on R.. Equation 5*19 ma y be written as 
 
 P. = P- -, y. V P- i y. NR.(1, 1, Y R ) V p. - y. NR.(0, 0, Y R ) y 
 l l-l l l-l l l l T 'l-l i i ' ' i 
 
 pi-i ? i C V ' °> Y i> v p±.i 'i DR i(°' °- Y i ) • 
 
 From the results of section 5.2.2 p, =1 for - 2 < y„(i) < - 1, - 2/3 < 
 
 i — R — 
 
 y R (i) < and 2/3 < y R (i) < !• The corresponding rules for choosing yl are 
 
 given below o 
 
 y{ = ■ 1 if - 2 < y*(i) < - 2/5. 
 y!_ = if - 2/3 < y*(i) < 2/3. 
 
 y!_ = 1 if 2/3 < y*(i) < 2. 
 
 As before, y R (0) = y = - 23/6^ so y' = 0. P = 2P - and y R (l) = 
 - 23/32 so y[ = - 1. P 2 = (y*(l) + 1) D = (9/32) (>^M) and y*(2) = 9/l6 so 
 yg = 0. P^ = 2P 2 - and y*(3) = 9/8 so y^ = 1. P^ = (y*(3) - l) D and 
 y*(k) = 1/k so y' k = 0. P^ = 2P^ - and y*(5) = l/2 so y^ =0. P 6 = 2P 5 - 
 and y R (6) = 1 so y, = 1. Therefore P = (y R (6) - l) D = 0. 
 
 The recoded quotient y 1 (6) = 0.101001 = - 23y / 6^- is identical to the 
 recoded multiplier obtained in section 5 "by using the canonical, minimal 
 multiplication algorithm. As in the previous example, this was expected 
 
 A 
 
 because SL. is equivalent to R. . 
 
 If 2P were replaced by - 2P , the algorithm based on SR. would 
 yield - y'(6) = 0.011001 and the algorithm based on R. would yield -y' (6) = 
 
 — _ A 
 
 0.101001, This is a consequence of the fact that both SR. and R. are elements 
 
 ) 
 
■H5- 
 
 Definition 5~9 ° Universal, Minimal Division Algorithm (u.m.d.a.) 
 
 A u.m.d.a. is 
 of (A , a subset of (jt 
 
 A u.m.d.a. is based on Eq. 5°!^ with p. = R. where R. is any element 
 
 To limit the discussion, we consider a subclass of u.m.d.a. desig- 
 nated as the semi-universal, minimal division algorithms, s-u.m.d.a. A 
 typical element of this subclass is defined as follows. 
 
 Definition 5-10 ° Semi -Universal, Minimal Division Algorithm (s-u.m.d.a.) 
 
 A s-u.m.d.a. is any u.m.d.a. in which (j\ includes only restricted, 
 
 minimal R. . 
 
 1 
 
 A typical s-u.m.d.a. is defined by (j\ = (SR., R.). The corresponding 
 rules for choosing y. are given below. 
 
 y[ = - 1 if - 2 < y*(i) < -3/k. 
 
 y! = -1,0 if - 3/k < y*(i) < - 2/3. 
 
 Y ± = if - 2/3 < y*(i) < 2/3» 
 
 y[ = 0, 1 if 2/3 < y*(i) < 5A- 
 
 y[ = i if 3/^- < y*U) < 2, 
 
 This s-u.m.d.a. is equivalent to the Wilson-Ledley algorithm [15]- 
 In that algorithm the partial remainder is always normalized and then (depending 
 upon the range of the approximate partial quotient) multiplied by 2, 1 or 2 
 before adding or subtracting the divisor. If Q(j) represents the absolute value 
 of the partial quotient based on a normalized partial remainder, the shift rules 
 and the effective quotient digit before shifting are as follows. 
 
1 
 
 
 
 
 
 
 
 
 
 
 or 
 
 + 
 
 1 
 
 
 
 + 
 
 1 
 
 
 
 
 
 o, 
 
 + : 
 
 L ( 
 
 Dr 
 
 + 
 
 1 
 
 
 
 or 
 
 + 
 
 1 
 
 
 
 -146- 
 
 RANGE SHIFT EFFECTIVE y 
 
 1/2 < Q(j) < 2/3 2 
 
 2/3 < Q(j) < 3/4 2 or 1 
 
 3A < Q(J) < V3 1 
 
 4/3 < Q(j) < 3/2 2" 1 or 1 
 
 3/2 < Q(j) < 2 2" 1 
 
 Let [ y ( i ) | represent the absolute value of i partial quotient in 
 K 
 
 the s-u,m.d.a. defined above . If < |y (i) | < l/2 ; the partial remainder is 
 not normalized and y. = 0. This agrees with the Wilson-Ledley algorithm since 
 a quotient digit is inserted during each shift to normalize the partial 
 remainder. If 2/3 < |y*(i) | < 3/4, y[ = 0, + 1. If 3/4 < |y*(i)| < 4/3, 
 y. = + 1. Both of these cases agree directly with the Wilson-Ledley algorithm. 
 
 If 4/3 < |y (i)j < 3/2, then y. = + 1. In the Wilson-Ledley algorithm 
 we may shift the normalized partial remainder right one bit position at this 
 point* The corresponding range for |y R (i-l) J is derived from the above range 
 by assuming y. = 0. Thus 2/3 < |y*(i-l)J < 3/4 so that y! = or + 1. 
 If yl , =0, we have already observed that y. = + 1. If y. = + 1, then the 
 new range for y£(i) is - 2/3 < y R (i) < - l/2 so y. =0. Likewise, if y. = -1, 
 then the new range is l/2 < y R (i) < 2/3 so y! =0. 
 
 If 3./2 < |y R (i)| < 2, then y. = + 1. In the Wilson-Ledley algorithm 
 we always shift right in this case. The corresponding range for y (i-l) j is 
 again obtained by assuming y = 0. Thus 3.A < |y (i-l)| < 1 so y! = +_ 1. 
 This shows that a right shift of one bit position is necessary. In the given 
 s-u.m.d.a. ; we obtain 3/2 < |y*(i) | < 2 only if y!^ = + 1. If y\ ^ = + 1, 
 then the new range for y R U) hased on 3/4 < y*(i-l) < 1 is - l/2 < y*(i) < 
 so y! = 0c If yl = -1, the new range is < y R (i) < l/2 so y! =0. 
 
 
-147- 
 
 In the given s-u.m.d.a. , if y! n = + 1 and y! £ then yl must 
 equal + 1, and if y' = - 1 and y! £ then y! must equal - 1. Let j/k < 
 
 y*(i-l) < 2, then y! _ = + 1. It follows that + l/2 < y*(i) < 2 so y! = 0, + 1. 
 
 * R l-l — K l 
 
 Likewise, if - 2 < y*(i-l) < - 3A, then y! = - L Therefore, -2 < y*(i) < 
 — n 1-1 — n 
 
 l/2 so y' =0, - 1„ This is a characteristic property of the Wilson-Ledley 
 algorithm. 
 
 In the physical realization of any s-u.m.d.a. the amount of hardware 
 
 1 
 that is necessary to determine y. is a function of the width of the ■boundary- 
 intervals. The precision to which y^i) must "be known is clearly less for 
 larger widths. In the preceding example the width is l/l2. Theorem 5-7 shows 
 that a maximum width of l/6 is possible. To achieve this maximum width we con- 
 sider the s-u.m.d.a. defined by CM = (ER., NR.)„ The corresponding rules for 
 choosing y. are those given in the hypothesis of Theorem 5-7* 
 
 The S-R-T algorithm is equivalent to this s-u.m.d.a. for 3/5 < |D| 
 - 3/4 as verified by Freiman's statistical analysis [k]. In the S-R-T algorithm, 
 the shifted partial remainders, 2P., are always compared with the constant l/2. 
 The rules for choosing y! may be stated as follows. 
 
 •- 1 lf w - y R (1) < m 
 
 - ° if m- y R (1) < m 
 
 -x ■ x lf m- y R (1) < w 
 
 In order for this algorithm to yield minimally recoded quotients, 
 
 the conditions of Theorem 5-8 must be satisfied where w = z and 2/3 < z < 5/6. 
 
 Therefore 1 -. 1 > z__. and -r^T < 2z„ . It follows that a minimal quotient 
 2 1 D I — Mm I D I — Max 
 
 will be generated for 3/5 < |d| < 3/4„ 
 
-148- 
 
 The Metze modification of this algorithm [7] permits generation of 
 minimally recoded quotients over the entire normalized range of D. The rules 
 
 for choosing y! are similar to those for the S-R-T algorithm except that 
 
 I */ \ I K K 
 
 y„(i) is compared with -r=rr where K is chosen such that z r „. < —=— < z., 
 |J R V /( * \D\ Mm - D - Max 
 
 is always satisfiedo The range of |d| for a given K is therefore 6/5 K < 
 
 |d| < 3/2 K. Metze has shown that three constants will suffice to cover the 
 
 entire range of |D| but that four or more require less precision. 
 
 Another prediction technique that takes advantage of the s-u.m.d.a. 
 
 defined by (/[_. = (ER., NR„) is now considered. Note that y^Ci) may be compared 
 
 1 i K 
 
 with 3/4 with a precision of + 1/12. Let y R (i) represent the approximate value 
 
 */ \ 
 of y_(ij. The rules for choosing y! are given below. 
 a 1 
 
 y. = - 1 if - 5/3 < y R < - 3A • 
 y[ = if - 3/4 < y* < 3/4 . 
 y ± = 1 ^ 3A < y R < 5/3 • 
 
 A-& . . / 
 
 Provided the error in y (i) is never greater than 1/12 in absolute 
 
 R 
 
 value, the outer limits of + 5/3 are guaranteed. Let 
 
 A A 
 
 |2P. I = |2P. I + a and |d| = |d| + where |a| M&x = |p 
 
 Max 
 
 A A A A 
 
 We form 3/4 |d| and compare with |2P. | at each step. If |2P. | < 3/4 |d|, 
 
 y. = 0. It is necessary to insure that in the worst case this implies that 
 |2P. I < 5/6 |D|. Therefore, we let |2P.| - |a ^ < 3/4 |d| + 3A |p| M&x and 
 require that 3/4 |d| + 7/4 |a| < 5/6 |d|. This implies that |dL. < -W- • 
 In the worst case Id| = l/2, so Ice I., = IpL < l/42. The maximum precision 
 for both D and 2P. is therefore 2 „ A similar argument for the lower bound 
 of 2/3 |d| yields the same precision requirements. 
 
-149- 
 
 5-5 Interrelation of Minimal Multiplication and Division Algorithms 
 
 Minimal multiplication and division algorithms are defined by a 
 
 particular minimal mode functions, L. and R., where L. . = L. and R. = R. _ . 
 * '11 l-l i l i+l 
 
 for < i < n - 1. Because of the identity requirement, the corresponding sets 
 of minimal mode functions, L(y) and R(y), are completely defined when L. and 
 R. are specified. Under these conditions there exists a unique minimal R. 
 which is equivalent to a given minimal L. and conversely. This equivalence 
 was established by Theorems 3-5 an( i 3-6 and modified slightly in section k.k. 
 
 Definition 5-11 : Inverse Multiplication and Division Algorithms 
 
 If the product xy is formed by a multiplication algorithm in which 
 the multiplier, y, is recoded as y', then the inverse division algorithm 
 generates a recoded quotient, y' ' = xy/x, which is identical to y ! for all 
 yeY. 
 
 Theorem 5-9 • For every minimal multiplication algorithm there exists an 
 inverse minimal division algorithm and conversely. 
 
 Proof : Every minimal multiplication algorithm is uniquely defined by some 
 
 L.ejl where L. . = L. for < i < n - 1. If we extend this definition such 
 l l-l i — — 
 
 that L , = L. for n + 1 < i < 2n, then Eq. Ik 91 defines a J. which, when 
 
 n-1 j — — ' i 
 
 substituted in Eq. 3: 9. yields the equivalent minimal R. for < i < n - 1. 
 This R. uniquely defines a minimal division algorithm. By Definition 5-H 
 this division algorithm must be inverse to the given multiplication algorithm 
 since R. is equivalent to L.. By a similar argument the converse is also true, 
 
 Two examples of inverse minimal multiplication and division 
 algorithms were given in sections 5«3 an( i 5-^« In the first example, the 
 
-150- 
 
 canonical, minimal multiplication algorithm which is defined by SL. was shown 
 to be inverse to the minimal division algorithm defined by R. . In the second 
 example, the minimal division algorithm defined by SR. was shown to be inverse 
 
 A 
 
 to the minimal multiplication algorithm defined by L.. The equivalence of SL. 
 
 A A 
 
 to R. and SR. to L. was established in section k.h. 
 1 11 
 
 The practical consequences of inverse minimal multiplication and 
 division algorithms are nil since the minimal division algorithms require a 
 full length comparison of divisor and shifted partial remainder. If the minimal 
 division algorithm is not based on a restricted, minimal R., some or all of 
 the quotient digits that have been generated must also be examined . Likewise, 
 if a minimal multiplication algorithm is based on an unrestricted, minimal L., 
 some or all of the previously used multiplier digits must be examined at each 
 
 A 
 
 step. The minimal multiplication algorithm defined by L. provides an example 
 
 of this. 
 
 Under the condition that L. _ = L. and R. = R. _ for < i < n - 1, 
 
 l-l l l l+l - - 
 
 A 
 
 it is conjectured that SL. and its equivalent R. are the only simple examples 
 
 of equivalent minimal mode functions which are also restricted. If this 
 
 conjecture is true, then the only simple minimal multiplication and division 
 
 A 
 
 algorithms that are inverse and restricted are defined by SL. and R. 
 & ' i l 
 
 respectively. 
 
 It is theoretically possible to define universal, minimal multiplica- 
 tion algorithms which are inverse to universal, minimal division algorithms 
 
 L 
 
 process. Once this is known, Eqs. ki^l and 4:92 provide the link between the 
 
 and conversely. In order to do this it is necessary to know the minimal L 
 
 or minimal R.e CX. that is used at each step of the multiplication or division 
 
 
 equivalent L. and R. . In practice this information is not available. The 
 practical semi -universal, minimal division algorithms allow R. to range over 
 
-151- 
 
 the set (A. and do not require that it "be precisely defined at each step. 
 For this reason, any practical s-u.m.d.a., such as the Wilson-Ledley algorithm 
 or the Metze modification of the S-R-T algorithm, can not be used to define 
 an inverse universal, minimal multiplication algorithm in the sense of 
 Definition 5-11. 
 
6. CONCLUSIONS 
 
 6.1 Summary 
 
 The purpose of this investigation was to analyze arithmetic recodings 
 in general, minimal "binary recodings in particular and to show how the latter 
 are used to define fast multiplication and division algorithms. A brief 
 summary of the investigation is given below. 
 
 The analysis of general arithmetic recodings was conducted in 
 Chapter 2. A set of recoding functions was developed which transforms a radix 
 representation having digits, y„, in the range < y. < r - 1 into an alge- 
 braically equivalent radix representation having digits, y!, in the range 
 
 ! 
 
 1 - r < y. < r - 1, It was shown that this transformation can be defined at 
 — J l — 
 
 each digital position by y! = y. + m. - rrn. where the mode digits, m., have 
 
 a value of or 1 and are chosen such that the range restriction, 1 - r < 
 
 y\ < r - 1, is always satisfied. A set of mode functions was introduced which 
 
 defines each m. in terms of the y. digits and controls the characteristics of 
 i l D 
 
 the recoded representation. A one-to-one correspondence was shown to exist 
 between recoding functions and mode functions „ Left and right directed 
 recoding functions and mode functions were recognized and defined in terms of 
 the order in which the recoded digits are generated. The existence of 
 equivalent left and right directed mode functions was established. 
 
 The results of Chapter 2 were specialized to include only the binary 
 case in Chapter 3 an d all following chapters. As a consequence of this, the 
 left and right directed mode functions assumed a Boolean form. It was also 
 demonstrated that every set of left directed mode functions has an equivalent 
 set of right directed mode functions and conversely. Sets of left and right 
 directed mode functions were defined as being equivalent if and only if their 
 
 -152- 
 
-153- 
 
 corresponding sets of recoding functions produce identical recodings of the 
 original representation. 
 
 In Chapter k, the added constraint that the recoded representation 
 must contain a minimum of 1 and 1 digits was applied. The introduction of a 
 suitable set of theorems concerning the optimum choice of the next mode digit 
 ultimately led to the explicit definition of restricted, minimal left and 
 right directed mode functions. In this context a left directed mode function 
 was defined to "be restricted if the set of independent variables was void of 
 any y. to the right of the digit "being recoded. A restricted right directed 
 mode function was similarly defined. Three principal mode functions of this 
 type were defined and investigated for both the left and right directed cases. 
 The simplest, minimal left directed mode function was shown to yield the 
 canonical, minimal representation. A method of obtaining all minimal, left 
 and right directed mode functions was suggested. 
 
 In Chapter 5, minimal left and right directed mode functions were 
 used to define multiplication and division algorithms which generate minimally 
 recoded multipliers and quotients. Algorithms defined by a single minimal 
 mode function were classed as minimal algorithms. Those defined by two or 
 more minimal mode functions were classed as universal or semi-universal, 
 minimal algorithms depending on whether or not the mode functions are un- 
 restricted or restricted. Two examples of minimal multiplication and division 
 algorithms were given. Multiplication and division algorithms were defined to 
 have an inverse relationship if they produce identical recodings of all 
 multipliers and quotients which are algebraically equal. Inverse minimal 
 multiplication and division algorithms were studied in terms of the minimal 
 left and right directed mode functions which define them. 
 
-15k- 
 
 6.2 Results and Conclusions 
 
 An arithmetic recoding which converts a unique integral radix repre- 
 sentation into a redundant integral radix representation was effectively 
 analyzed as a one-to-many mapping with algebraic value preserved. In the 
 "binary case it was shown that the characteristics of a minimal representa- 
 tion are governed Toy a set of Boolean mode functions. We may conclude that 
 an appropriate choice of these mode functions can be used to control other 
 properties of the recoded representation in addition to, or in place of, a 
 minimum of nonzero digits „ It was also demonstrated that for every set of 
 left directed Boolean mode functions an equivalent set of right directed 
 Boolean mode functions can be created and conversely. 
 
 A deterministic approach was successfully used to derive the Boolean 
 form of restricted, minimal left and right directed mode functions. A similar 
 approach might prove useful in the derivation of mode functions which guarantee 
 other characteristics of the recoded representation. 
 
 It was shown that multiplication algorithms which use minimally 
 recoded multipliers can be uniquely defined in terms of minimal left directed 
 mode functions. In an analogous manner, minimal right directed mode functions 
 were shown to uniquely define division algorithms which generate minimally 
 recoded quotients. The minimal multiplication algorithm defined by the 
 canonical, minimal left directed mode function was shown to be the simplest to 
 implement since the digits of the multiplier are precisely known. The semi- 
 universal, minimal division algorithms which are defined by a set of restricted, 
 minimal right directed mode functions were found to be most practical. This 
 is due to the fact that the choice of the next quotient digit as + 1 or can 
 
 be based on whether or not the predicted partial quotient, Q. , satisfies y < |Q.| 
 
 J J 
 
 where 2-/3 5 f — 5'/ 6. Regardless of where y falls in this range, the recoded 
 
-155- 
 
 quotient is guaranteed, to have a minimum of nonzero digits. The Wilson-Ledley 
 algorithm and the Metze modification of the S-R-T algorithm were shown to "be 
 semi -universal, minimal division algorithms. 
 
 It was previously known that multiplication algorithms which recode 
 multipliers in a nonredundant manner have corresponding inverse division 
 algorithms which produce an identical nonredundant recoding of the quotient . 
 On the "basis of this investigation, we may conclude that this inverse relation- 
 ship still exists when the multiplication and division algorithms involve 
 redundant recodings. In the case of minimal recodings, the inverse relation- 
 ship was shown to depend on the equivalence of the minimal left and right 
 directed mode functions which define the multiplication and division algorithms 
 It is conjectured that there are no simple minimal multiplication algorithms 
 which have simple inverse minimal division algorithms and conversely. 
 
 6.3 Areas for Further Study 
 
 Many extensions of this investigation are possible. An analysis of 
 the other properties which binary recoding functions possess would be of 
 interest from a practical standpoint. For example, if the time required to 
 change the multiple of the multiplicand or divisor that is fed to the adder 
 is large and the difference in the time to add a fixed multiple or zero is 
 small, a minimal derivative recoding may offer more advantages than a strictly 
 minimal recoding. An approach similar to the one employed in Chapter k might 
 be used to obtain mode functions which guarantee a recoding having this and/or 
 other properties. A greater knowledge of the recoding properties associated 
 with a given class of mode functions would permit selection of the most 
 efficient multiplication and division algorithms for a given computer topology. 
 

 -156- 
 
 BIBLIOGRAPHY 
 
 1. Avizienis, A., "Signed-Digit Number Representations for Fast Parallel 
 Arithmetic/' IRE Transactions on Electronic Computers , Vol. EC-10, No. 3 
 (September 196l), pp. 389-^00. " 
 
 2. Birkhoff, G. and Mac Lane, S., A Survey of Modern Algebra . (Macmillan 
 Company, New York, N. Y., 1953), PP° 356-368. 
 
 3o Burks, A, W., Goldstine, H. H. and von Neumann, John, Preliminary 
 
 Discussion of the Logical Design of an Electronic Computing Instrument . 
 Part 1, Volo I, Second edition revisedo A Report of the Institute for 
 Advanced Study, Princeton, New Jersey, (September 19^+7) • 
 
 k. Freiman, C. V., "Statistical Analysis of Certain Binary Division Algorithms," 
 Proceedings of the IRE. , Volo k9 (January 1961), pp. 91-103. 
 
 5. Lehman, M., "High Speed Multiplication," IRE Transactions on Electronic 
 Computers , Vol. EC-6 (September 1957), pp. 204-205 . 
 
 60 MacSorley, 0. L., "High-Speed Arithmetic in Binary Computers," 
 Proceedings of the IRE , Volo k9 (January 196l), pp. 67-91. 
 
 7. Metze, Go, A Class of Binary Divisions Yielding Minimally Represented 
 Quotients , File No. 466 of the Digital Computer Laboratory, University 
 of Illinois, Urbana, Illinois, (July 1962)0 
 
 8. Metze, G. and Robertson, J. E., "Elimination of Carry Propagation in 
 Digital Computers," Proceedings International Conference on Information 
 Processing, Paris, France; June 13-23, 1959 - 
 
 9. Reitwiesner, G. ¥. , "Binary Arithmetic," Advances in Computers , F. L. 
 Alt, Editor, Academic Press, Inc., New York, N. Y., i960. 
 
 10. Robertson, J. E., "A New Class of Digital Division Methods," IRE 
 Transactions on Electronic Computers , Vol. EC-7 (September 1958), 
 pp. 218-222. 
 
 11. Robertson, J. E., Theory of Computer Arithmetic Employed in the Design 
 of the New Computer at the University of Illinois . Notes for the 
 University of Michigan Engineering Summer Conference "Theory of Computing 
 Machine Design," Ann Arbor, Michigan; June 13-17> I960. 
 
 12 o Staff of the Digital Computer Laboratory, On the Design of a Very High - 
 Speed Computer . Report No. 80 of the Digital Computer Laboratory, 
 University of Illinois, Urbana, Illinois, (October 1957), pp. 17^-220. 
 
 13. Tocher, K. D. , "Techniques of Multiplication and Division for Automatic 
 Binary Computers," The Quarterly Journal of Mechanics and Applied 
 Mathematics, Vol. 11 (August 1958), pp. 364-384. 
 
-157- 
 
 1k. Weinberger, A. and Smith, J. L., "Shortcut Multiplication for Binary 
 
 Digital Computers/' National Bureau of Standards Circular 59^ > Sec. 1, 
 pp. 13-22. 
 
 15. Wilson, J. B. and Ledley, R. S., "An Algorithm for Rapid Binary- 
 Division." IRE Transactions on Electronic Computers , Vol. EC-10, 
 (December, 1961), pp. 662-67O. 
 
APPENDIX 
 
 The set of lemmas contained in this appendix are used to simplify 
 many of the Boolean functions derived in Chapter k and also appear in the 
 proof of several theorems. 
 
 Lemma A-l: 
 
 Vq y P±( 2 J +1 ) y P+ 2 J J[ y P±(2k+l) y p±2k 
 
 n* j-1 
 
 "Vq y p±(2j+l) Jl _ ± y p±(2k+2) 
 
 Proof : For convenience only the relative portion of the indices is retained 
 in the inductive argument which follows. 
 
 i i-i 
 
 V y 2j+l y 2j i' y 2k+i y 2k = y y i y y 2 y 3 
 j=0 k=0 
 
 Let H( q ) . "\3r y % y sk+2 . 
 
 j=0 ° k = -1 
 
 q-1 j-1 
 
 Assume that y Y^^ J J^ +1 7 2k = H(q-l) 
 
 J -U K-U 
 
 Then, V , r % y 2k+i y 2k . H(a-l) V 
 
 j=0 " " k=0 
 
 ^ 
 
 Note that 
 
 y 2q+i y 2q !_} y 2k+i y 2k 
 
 q-1 
 
 h(o) V y 2a+1 y 2a _TT y Pk+1 y ?k = h(o) v y y -XT y 2k+1 y 
 
 q-1 
 
 2q+r2q jM 2k+r 2k * ' J 2q+r 2q 
 
 -158- 
 
 2k+r 2k 
 
 y o 
 
■159- 
 
 q-1 
 Assume that H(r) V y 2a+1 y 2a % 7^7^ 
 
 2q+r 2q 
 
 k=0 
 
 H(r) V V V 
 
 K ' ,y 2q+l' y 2q 
 
 q-1 
 
 li yoiri.n« 
 
 -k=r+l 
 
 2k+r 2k 
 
 r r-l 
 
 TT 7 
 
 J '-k = -1 
 
 2k+2 
 
 q-1 
 
 Then, H(r + 1) V y y TT y 2k+1 y 2k 
 
 k=0 
 
 = H(r) V y 2q+1 y 2q 
 
 _ y 2k+i y 2k 
 
 Ha = r+1 
 
 r-l 
 
 .TT y 2k+2 
 u k = -1 
 
 y 2r+3 ,11 . y 2k+2 
 k = -1 
 
 = H(r)V 
 
 7r,„,^y, 
 
 / q-i 
 
 ( TT 
 
 y mi- 1 1 « ov / ^oyL? » y^ 
 
 7T y ; 
 
 " 2q+r 2q V . 1 1 ^ 2k+r 2k / ^ 2r+3 ' ^ 2r+3 ' • 2k+2 
 
 \k= r+2 ' J k = -1 
 
 q-1 
 
 = H(r+l) V v V 
 V ; ,y 2q+l' y 2q 
 
 
 k = r+2 
 
 y 2k+i y 2k 
 
 r 
 
 TT 
 
 J Ha = -1 
 
 2k+2 
 
 If q - 1 = r - i, then H (»-l) V y y Jf y 2k+1 y 2k 
 
 k=0 
 
 = H(q-l) V y 
 
 q-1 
 
 2q+l 
 
 k = -1 
 
 y 2k + 2 = H((l) 
 
 The lemma follows 
 
 Lemma A-2: 
 
 V^ y p±2j TT y p ± (2k+l) y p ± 2k 
 
 n* 
 
 = V w 
 
 TT 
 
 P"P+2J k <J Q J P±(2k+l) 
 
-i6o- 
 
 Proof: An inductive argument follows 
 
 2 j-1 
 
 V y 2j N y 2k+i y 2k y o y i y 2 ^ y y i y 3 y + * 
 
 j=l k=0 
 
 Let h( 4 ) - V Vaj TT y 2k+ i 
 
 j=l ° k=0 
 
 q J-1 
 
 Assume that y y 2 j TT y 2k+ i y 2k = H ^)' 
 
 q+1 j-1 
 
 Then, -y y ff y 2k+1 y 2k = H(q) V y 
 
 j=l ^ J k=0 ^ +1 ^ 
 
 2j I' • 7 2k+l"2k '2q+2 JL y 2k+i y 2k 
 
 Note that 
 
 q 
 
 TT 
 
 k=0 
 
 H(1 > v y 2a+ i TT y 2k+1 y 2 k ■ *M v V 24+2 [ft y 2 * + i y 2 kj 
 
 q 
 
 rr 
 
 ■k=2 
 
 irl 
 
 TT: 
 
 •k=0 
 
 2k+l 
 
 Assume that H(r) V y JT y 2k+1 y 2k 
 
 ^ k=0 
 
 = H(r) V y y 
 
 TT . y Pk+ iyp k it y 
 
 o J 2q+2 . ' I _ * 2k+r 2k ' I ^ 2k+l 
 »-k = r+1 J u k=0 
 
 Then, H(r + 1) V y JT y 2k+1 y 2k 
 
 k=0 
 
 H(r) V v y 
 K J v ,y o ,y 2q+2 
 
 q ~i r r 
 
 I' . y 2k+l y 2k .11 y 2k+l 
 L k = r+1 J ^k=0 
 
 V 
 
 y o y 2r+2 
 
 r 
 
 S) y2k+i 
 
 = H(r) V y 
 
 L y 2k/ 
 
 y 2q+2 IjL y 2k+i y 2k/ y 2r+3 y 2r+2 ^ y 2r+2 .1 1_ * 2k+l 
 ^ v k=r-2 - 1 k=^> 
 
 TT y ; 
 
 k=0 
 
 = H(r+1) V y y 
 
 TT y 2k+i y 2k /' y 2k+l 
 ^ L k = r-2 J ^k=0 
 
 rr-l 
 
 If a - r + 1, then H(a) V y 2q+£ -Jf y 2k+1 y 2k 
 
 q 
 
 k =o 
 
 ■ H(4) V y y 24+2 TT ^ ■ H(q + 1) . 
 
 k=0 
 
-161- 
 
 The lemma follows . 
 
 Lemma A-3: 
 
 . n * j- 
 
 ^ y p±2j T]q y p±(2k+l) y p±2k 
 
 n* _ j-1 
 
 V Vp±2j Jo y P±( 2k+1 ) 
 
 Proof : The proof is accomplished by taking the dual of the complement of 
 the equation given in the hypothesis of Lemma A-2. 
 
 Lemma A-k-: 
 
 V y p±(2j+l) k U_ 1 y p+(2k+2)^ 
 
 n* j-1 n* 
 
 = ~Y y P±2j ][ y P±(2k + D v TT y p± ( 2 j + i) 
 
 Proof: An inductive argument follows 
 
 V y 2J+1 k T_ x y 2k+2 y = y G ^ y i y 2 ^ y i y 3 y i. ^ y i y 3 y 5 
 
 X ^ S F ~ v i 
 
 Let H (q) = Y y 2 . Jf y, k+1 V JT ^ +1 
 
 Assume that ^7^^ /TT\ y 2k+ 2 j = H <*> ' 
 
 Jrl 
 
 k = -1 
 
 tten > (^ ^ 2j+ l jjf^ w) ■ l^ 2 a + , V .T 
 
 L "2q+3 ' V y 2k+2 
 
 k = -1 
 
 H(q) 
 
■162- 
 
 Note that 
 
 k = -1 J ,i=0 ° k=0 l ^ > ^=2 J 
 
 y 2q+3 . Y 
 
 k = -1 
 
 Assume that 
 
 y 2q+3 . V . y 2k+2 
 ^ ^ k = -1 
 
 H(r) 
 
 = V y 2J TT y Pk+1 ^ 
 
 j=0 d k=0 
 
 2k+l L 2q.+3 
 
 ,-v" 
 
 k=r 
 
 2k+2 
 
 H(r) . 
 
 Then, 
 
 Y 2q+3 * k V _ y 2k+2_ 
 
 H(r+l) 
 
 = V y 2 j TT y 2 k + i v 
 
 j=0 J k=0 
 
 _ ^ _ " 
 
 y 2q+3 ^ . V y 2k+2 
 k=r 
 
 H(r) V 
 
 -\V i r- 
 
 /2q+3 V y 2k+2_ [ y 2r+2 V y ^ 
 
 ^2r+3_ 
 
 q. 
 
 r 
 
 Tl y 2k+l 
 k=0 
 
 ■ "VT y 2J ft y 2k+1 V [y 2q+3 V V ^kj H ( r+1 > V 
 
 j=0 k=0 -"=•-' k = r _i j 
 
 y 2r+2 H(r) V 
 
 r 
 
 . V . y 2k+2 
 
 -k = -1 
 
 _ y 2r+2 V y 2r+3 
 
 r 
 
 k 7T y 2k + i 
 
 y 2j TT 
 
 j=0 J k=0 
 
 ffi ^ v 
 
 r+1 
 
 r+1 r+1 
 
 r+1 j-1 
 
 / 2q+3 v k y * 
 
 1-3 V , y 2k+2 
 
 : = r+1 
 
 H(r+l) V 
 
 ^ ^ k = r+1 J 
 
 If q. = r + 1, then 
 
 y 2q+3 V . V _ y 2k+2 
 k = -1 
 
 H(q) 
 
•163- 
 
 _q r _ J- 1 _ 
 
 j=0 ^ J k=0 
 
 y 2q+3 * y 2q+2 
 
 H(q) 
 
 q+1 _ j-l q+1 
 
 = X " 2j K " 2k+iV S " 2j+i = H(4+1) 
 
 The lemma follows 
 
 Lemma A- 5: 
 
 n* 
 
 j-l 
 
 ^ Vp±2j TT y p± 
 
 n* 
 
 j-l 
 
 y 
 
 v Y V(2.i-D TT y p+2 k v IT 
 
 y. 
 
 ^ "p+(2j-l) ^ ■ 7 p±2k ' 1^ 
 
 Proof: An inductive argument follows 
 
 V v 2J g , 2k+1 
 
 = y o V y x V y 2 y 5 V y^ 
 
 Let H(4) - y Q ^ "$" y 2j -i it y 2k v TT y 2j 
 
 J =1 k=l j -1 
 
 Assume that (^T yj^ Jj y^j . H (q) 
 
 Then, / -y y y TT y 2k+1 
 
 \ j=l ° k=0 
 
 y o y y 2q+2 ^ VI y 2k+l 
 k=0 
 
 H(q) 
 
 Note that 
 
 y o V y 2q+2 V ^ y 2k+1 
 
 H(2) 
 
 y Q vV y 2 j-i JT y 2k v 
 
 J =1 k=l 
 
 y 2q+2 ^ Vp y 2q+l_ 
 
 H(2) 
 
-I6i4- 
 
 Assume that Yq V y V y y 
 
 '- k=0 
 
 H(r) 
 
 = y o ^ V y 2 ^ f y 2k V 
 
 j=l u k=l 
 
 y 2 4+ 2 V "^ y 2 k + lj « r > 
 
 Then, 
 
 Y o * y 2q+2 * Vq y 2k+l 
 
 H(r+l) 
 
 y 
 
 o V V y 2 j_! TT y 2k v 
 
 j=l ^ J k=l 
 
 _ ^ _ 
 
 y 2q+2 * n V y 2k+l 
 . k=r 
 
 H(r) V 
 
 y 2q+2 y .V y 2k+l 
 k=0 
 
 y 2r+l V y 2r+2 
 
 TT 
 
 J k=l 
 
 y 
 
 2k 
 
 j =1 d k=l 
 
 y 2q+2 . V y 2k+l 
 ^ k = r+1 
 
 H(r+l) V 
 
 y . H(r) V 
 
 J 2r+l v ' 
 
 ~ r 
 
 I ' y 2k+l 
 -k=0 
 
 y V y 
 
 " y 2r+l ,y 2r+2 
 
 TT 
 
 J k=l 
 
 y 
 
 2k 
 
 J + ± - J" 1 - 
 
 = y v V y 2 j-i TT y 2k \/ 
 
 j=l u k=l 
 
 y 2q+2 V ,V 1 y 2k+l 
 
 ■ * k=r+l 
 
 H(r+l) ^ 
 
 r+1 
 
 r+1 
 
 V y 2i-l l y 2k 
 j=l d k=l 
 
 r+1 _ J-l _ 
 
 = y o v V y 2 j-i TT y 2k V 
 
 j=l J k=l 
 
 y 2q+2 " . V y 2k+l 
 - k = r+1 
 
 H(r+1) 
 
 If q = r + 1, then 
 
 y o V y 2g + 2 V V ^^1 ' 
 '' k=0 
 
 = y V "^ yjy.! jjj y 2k v (/ 2q+2 V y 2q+1 ) h( 4 ) 
 
 (; 
 
 q+1 _ jj^ q+1 
 
 TT y?*v TT y 2J - h(i + d 
 
 The lemma follows 
 
-165- 
 
 Lemma A -6: 
 
 V Vp±2J J[ y p+(2k+l) 
 
 * 
 
 , v v 
 
 -p+(2j-l) J^ ^p+2k 
 
 T 
 
 k=L 
 
 n* 
 
 VTT 
 
 y 
 
 0=1 
 
 P+2j 
 
 Proof : The proof is accomplished "by taking the dual of the complement of 
 the equation given in the hypothesis of Lemma A-5 
 
 Lemma A-7 : 
 
 n* 
 r=0 V(2j + 1) k = _ ± 
 
 n* 
 
 v„ V(2j + D jn _ n v( 2 k +2 ) v y v P ±2j J y P± (2k + D 
 
 = y. 
 
 j=o x 
 
 y 
 
 V y. 
 
 p+2j * ,y p+(2j 
 
 -J 
 
 Proof : Let H(n*) represent the left side of the above equation and consider 
 the case n* = 2. 
 
 H(2) = y o ( yi V y 2 V Y^V 7 k V y 5 ) 
 
 Let I(q) = y, 
 
 S£ (y 2j ^ y 2j + l). 
 
 Assume that H(q) = l(q)« 
 
 It follows that Hfo+1) = 1(0 V y o y 2q+5 fl- y a+? V y^ Tt y 2k+1 
 
 q 
 Ho 
 
 q 
 
 TT 
 
 k=0 
 
 Note that l(2) V y y fT V \/ v V 
 
 J o- y 2q+3 .IL y 2k+2 V ' y o' y 2q+2 .|L 
 k=0 k=0 
 
 q 
 
 q q. _ 
 
 = 1(2) v y o y jr y 2k+2 v y y 2q+2 IT y 2k+i 
 
 k=2 k=3 
 
 
•166- 
 
 Assume that l(r) V yj JT y 2k+2 V VJ^ JT y 2k+1 
 
 k=0 k=0 
 
 : I ^ r ) V ^0^2^+3 7' y 2k+2 y y o y 2q+2 . TT y 2k+l 
 
 k=r k = r+1 
 
 q. 
 
 TT 
 
 k=0 
 
 Then I(r + 1) V jj^ TT y 2k+2 V YJT Jf 
 
 q. 
 
 TT 
 
 k=0 
 
 iW V y y 2r+2 V y y 2r+J V 
 
 y o y 2q+3 y 2r+2 . . y 2k+2 
 
 u k = r-1 
 
 V yy „ l0 y, 
 
 o'"2q+2 J 2r+3 [_ M p J 2k+1_ 
 
 TT „ y , 
 
 Kr+l)V y Q y 7f y 2k+2 V y Q y Tf y 2k+1 
 
 v k = r+1 k = r+2 
 
 Let r+1 = q 
 
 H(q+D = i(q) v y y 2q+3 y 2q+2 V y Q y 2q+2 
 
 H(q+1) = I(q+1) 
 
 The lemma follows .