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S/o. Xf- 
 
 -flC . <££-% Report No. 228 
 tOJ> . IM- 
 
 PARTIALLY SEPARABLE FUNCTIONS 
 
 by 
 
 THOMAS ANDREW SLIVINSKI 
 
 COO-IU69-OO61 
 
 
 y ' 
 
 -z 
 
 
 June 1 1967 
 
 
 ^ DEPARTMENT OF COMPUTER SCIENCE • UNIVERSITY OF ILLINOIS • URBANA, ILLINOIS 
 
Report No. 228 
 PARTIALLY SEPARABLE FUNCTIONS* 
 
 by 
 
 Thomas Andrew Slivinski 
 
 June 1, 1967 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 *This work was submitted in partial fulfillment of the Doctor of 
 Philosophy degree in Computer Science and supported in part by 
 Atomic Energy Commission Contract AT(ll-l) -IU69. 
 
■ID. S i 
 v 
 
 ACKNOWLEDGMENT 
 
 The author is very grateful to his advisor, Professor 
 S. Muroga, for his invaluable guidance, encouragement and constructive 
 suggestions in the preparation of this thesis. The author also wishes 
 to thank Professor D. E. Muller for his comments and suggestions. 
 
 The support of the Department of Computer Science, University 
 of Illinois is also gratefully acknowledged. 
 
 111 
 
TABLE OF CONTENTS 
 
 CHAPTER PAGE 
 
 1 INTRODUCTION 1 
 
 1.1 Introduction 1 
 
 1.2 Basic Definitions and Examples 2 
 
 1.3 Statement of the Problem 5 
 
 2 PROPERTIES OF PSEUDO -SEPARABLE FUNCTIONS 7 
 
 2.1 Introduction 7 
 
 2.2 Elementary Algebraic Properties of Pseudo-Separable 
 Functions 7 
 
 2.3 Bounds on the Degree of Hyperplanes and Conditions 
 
 for Separation 19 
 
 2.k Assummability as a Condition for Pseudo-Separability 26 
 
 2.5 Simultaneous Realizability and Pseudo-Separability 42 
 
 3 GENERAL PROPERTIES OF MARGINALLY SEPARABLE FUNCTIONS 58 
 
 3.1 Introduction 58 
 
 3.2 Basic Properties of Marginally Separated Functions 59 
 
 3.3 A Necessary and Sufficient Condition for 
 
 Quasi -Separability 68 
 
 3.4 Marginal Separation and the Degree of Supersummability 76 
 
 k SYNTHESIS OF PARTIALLY SEPARABLE FUNCTIONS 8$ 
 
 k.l Introduction 89 
 k.2 Integer Programming Algorithm for Pseudo-Separability 89 
 
 4.3 Threshold Synthesis of Pseudo-Separable Functions 96 
 k.h Integer Programming Algorithm for Marginal 
 
 Separability 99 
 
 4.5 Threshold Synthesis of Marginally Separable Functions 104 
 
 5 SUMMARY 108 
 
 LIST OF REFERENCES 110 
 
 VITA 111 
 
PARTIALLY SEPARABLE FUNCTIONS 
 
 Thomas Andrew Slivinski, Ph.D. 
 Department of Computer Science 
 University of Illinois, 1967 
 
 Threshold logic has been the center of much interest in recent 
 years. This attention has "been generated both by demands for logical 
 elements in computer circuits which can perform more complex Boolean 
 operations than "AND" and "OR" and by the desire to develop a single 
 unified theory in which the mathematical models for these various 
 logical elements appear as special cases. 
 
 This dissertation presents an extension of the concept of 
 linear separation to generate a wider class of functions. This 
 generalization is affected by modifying the requirements for separation 
 to allow both true and false assignments to reside on the same hyperplane 
 (pseudo-separation) or within the same neighborhood of a hyperplane 
 (marginal separation). Functions which can be separated in either of 
 these ways are called' partially separable functions. 
 
 Some of the theoretical algebraic properties of partially 
 separable functions are investigated and the relationship between 
 these and the corresponding properties for separable functions is 
 explored. Necessary and sufficient conditions for each type of 
 separation are also discussed. A parameter which shows whether a given 
 function is separable, pseudo-separable, or marginally separable is 
 introduced. 
 
Integer linear programming algorithms which are guaranteed 
 to minimize the number of true or false assignments on the same hyper- 
 plane for pseudo-separation or the number of assignments within the 
 same neighborhood of a hyperplane for marginal separation are presented. 
 A systematic procedure for synthesizing an arbitrary partially separable 
 Boolean function in a network of threshold elements based on these 
 algorithms is developed. 
 
CHAPTER 1 INTRODUCTION 
 
 1.1 Introduction 
 
 Linear separability of Boolean functions has been widely 
 studied in recent years. Techniques for deciding if a given function is 
 a threshold function have been developed and procedures for determining 
 threshold networks to realize an arbitrary function have been proposed. 
 This interest has been generated by two considerations. The first is 
 the demand for logical elements which express more complex Boolean 
 functions than the simple 'AND' and 'OR'. If a logical element can 
 perform more varied and complex logical operations, a smaller number of 
 such elements need be used in synthesizing an arbitrary Boolean function. 
 The second motivation has been the desire to incorporate the mathematical 
 models for various logical elements into a single unified theory. 
 Thus, much effort has been spent in the research of separable or 
 threshold functions [7] « 
 
 This paper represents an attempt to generalize the concepts 
 and properties of separable functions to include a wider class of 
 functions. As n, the number of variables, increases, the number of 
 threshold functions on n variables becomes very small when compared to 
 the total number of functions on n variables. Three percent of all 
 h variable functions are separable and only .002/o of all 5 variable 
 functions are separable [12] . Therefore, an extension of the theory 
 and ideas of separable functions to a more general class of functions 
 is needed. 
 
The generalization which this paper presents is a modification 
 of the actual requirements for separation. Instead of strict separation, 
 both true and false assignments are allowed to reside on the same 
 hyperplane or within the same neighborhood of the hyperplane. Functions 
 which can be non-trivially separated in either of these weaker senses 
 will be called "partially-separable" functions. 
 
 Very little appears in the literature about this type of 
 generalization. Robinson [10] discusses some theoretical properties 
 in devising a technique to determine if a function is separable using 
 determinants. Otherwise the present author is unaware of any other 
 work in this area. 
 
 1.2 Basic Definitions and Examples 
 
 Before any definitions are given, a few words about the 
 notation used in this paper are needed. Let f(x,, x p , . .., x ) be 
 an arbitrary Boolean function. Then f(x., x 2 , . .., x ) is a mapping 
 from (0, l) into {0,1} and will always be assumed to be dependent 
 on all n variables, x will be used to denote (x , x p , ..., x ) and 
 any n-tuple A = (a , a p , . .., a ) where each a. = 1 or 0, will be 
 termed an assignment to f (x) ■ If f (A) = 1, A is called a true 
 assignment and if f(A) = 0, a false assignment. When discussing 
 hyperplanes, [w; T] will represent the hyperplane W«x = T, where. 
 W* = (w.,, Wp, ..., w ). We will also refer to the hyperplane [W; T] 
 as a "structure" for a particular function f (x) . 
 
Definition 1.2.1; A Boolean function f(x) is said to be separable if 
 there exists some hyperplane [wj T] such that for any true assignment 
 A*, ^-A* > T and for any false assignment B*, W^B* < T. 
 
 This definition is equivalent to allowing either true or false 
 assignments to lie on the hyperplane [w; T], i.e., alternate requirements 
 tf't > T if t(t) = 1 and \t-t < T if f (a) = or ^-A* > T if f(t) = 1 and 
 vf-A* < T if f(3 =0 are equivalent. 
 
 Definition 1.2.2; A Boolean function f (x) is said to be pseudo- separable 
 if there exists some hyperplane [W; T] such that for any true assignment 
 A, W^'A > T and for any false assignment B, W*B < T. 
 
 Notice that both true and false assignments are allowed to lie 
 on the hyperplane [w; T]. 
 
 Example 1.2.1; Consider the function f(x) = x x p s/ x x, and the hyper- 
 plane [W; T] = [1,1,1,1; 2]. If A is any true assignment, then A must 
 either have more than 2 components a. with value 1 or else must be 
 (1,1,0, 0) or (0, 0,1,1). If A has more than 2 components with value 1, 
 then $.t> 3. If t is either (l, 1,0,0) or (0,0,1,1) then tf-£ = 2. In 
 either case, w*A > T. If Bis any false assignment, then F must either 
 have less than 2 components b. -with value 1 or else must be (1,0, 1,0), 
 (1,0,0,1), (0,1,1,0) or (0,1,0,1). In the first case W-B* < 1, while 
 in the second, w»B = 2. Therefore, for any false assignment w «B < T. 
 Hence x,x p ^ x x x k ^ s pseudo-separable . 
 
k 
 
 Definition 1.2.3: A Boolean function f (x) is said to be marginally 
 separable if there exists some hyperplane [w; T] and some non-negative 
 constant, A, called the margin , such that for any true assignment 
 A*, tf«A*> T - a/2 and for any false assignment % V?.I? < T + a/2. 
 
 Notice that we allow both true and false assignments to lie 
 in the region between the hyperplanes [% T - a/2] and [% T + a/2]. 
 
 Example 1.2.2: Consider the function f (x) = x-.Xp >/ x_x^. By Example 
 1.2.1 this function is pseudo- separable,, Therefore it must also be 
 marginally separable with margin a = 0. 
 
 Example 1.2.3° Consider the function f (x) = x-|Xp ^ x,x. x and the 
 
 structure [3,3,2,2,2; 6 l/2] with margin A = 1. If (3,3,2,2,2) -A; > 7, 
 
 — ^ 
 
 then A must either have a, = a. = 1 or a, = a^ = a- = 1. Therefore 
 
 A must be a true assignment. If (3,3,2,2,2) °B < 6, then either b = 
 
 — > 
 
 or b = 0, or one of b,, b. or b,_ is 0* In either case, B is a false 
 
 assignment. Therefore, for any true assignment (3,3,2,2,2) 'A > 6 and 
 for any false assignment (3,3,2,2,2) «B< 7» Hence x-^x^ v x x, x must 
 be marginally separable with margin A = 1° 
 
 Definition 1.2.4: A Boolean function f (x) is said to be a partially 
 separable function either if f(x) is pseudo-separable and W / (0,0,.. »,0) 
 or if f (x) is marginally separable and 
 
 n 
 
 a< 2livi. 
 
 i=l 
 
5 
 1.3 Statement of the Problem 
 
 Thus, by modifying the requirements for separation, a larger 
 
 class of functions is obtained. This can easily be seen by checking 
 
 the functions given in the examples in the previous section. These 
 
 functions are not separable, but are partially 'separable. The natural 
 
 question concerning the algebraic properties and necessary and sufficient 
 
 conditions for each type of separation arises. Along -with this comes 
 
 the desire to find an optimal structure, in the sense that a minimal 
 
 number of true and false assignments ]le on the same hyperplane (in the 
 
 case of pseudo-Separatior) or in the same neighborhood of the hyperplane 
 
 (in the case of marginally separable functions) . A function which is 
 
 not separable, but is partially separable -will have such assignments. 
 
 If we want to synthesize a partially separable function in a network of 
 
 threshold elements, then most of the assignments of f (x) can be realized 
 
 by a single threshold element having the structure [w; T] where f(x) is 
 
 partially separable, under [W; T], and if we use [w; T] as a threshold 
 
 element, then we must correct for false (or true) assignments lying on 
 
 the hyperplane by using additional threshold elements in the network. 
 
 Thus, if the number of such assignments is minimal, then the number of 
 
 these corrective elements in the network will also" be minimal. As a 
 
 special case, if all of these assignments are given as "don't care" 
 
 conditions in the actual specification of the network, then the function 
 
 can be realized by a single threshold gate. Therefore this paper will 
 
 be directed toward three goals: 
 
6 
 
 (1) A general description of the algebraic properties of 
 partially separable functions . 
 
 (2) Methods for reducing or minimizing the number of assign- 
 ments for which corrective elements are necessary in a 
 network. 
 
 (3) Techniques for the general synthesis of functions using 
 threshold elements. 
 
CHAPTER 2 PROPERTIES OF PSEUDO- SEPARABLE FUNCTIONS 
 
 2.1 Introduction 
 
 In this chapter some of the algebraic properties of pseudo- 
 separable functions are investigated. Some elementary properties and 
 ideas are presented in the first section. In the second part the con- 
 nection between pseudo-separable and separable functions is explored 
 with special emphasis on bounds for the number of points which lie on 
 the hyperplane. In the third section necessary and sufficient conditions 
 for a function to be pseudo-separable are developed based on the idea 
 of asummability. The relationship between simultaneously realizable 
 functions and pseudo-separable functions is discussed in the last 
 section, 
 
 2.2 Elementary Algebraic Properties of Pseudo-Separable Functions .. 
 
 In this section some elementary properties of pseudo-separable 
 functions are presented. Many of these properties are analogous to 
 properties of separable functions. However, because both true and false 
 assignments can lie on the same hyperplane, they are in general weaker, 
 In this section fix) is assumed dependent on exactly n-variables . 
 
 Definition 2.2.1 : The hyperplane LW; TJ is said to pseudo-separate the 
 
 function f (x) if for any true assignment A, W«A > T and for any false 
 
 -+-»—» — > -+ -> 
 
 assignment B, W«B < T. The number of assignments A. such that W°A. = T 
 — l l 
 
 is called the degree of [W; T] . 
 
8 
 Lemma 2,2.1: The structure [Oj 0] pseudo-separates every function. 
 
 n 
 Proof ; For any assignment A, 0»A = L o«a. = 0. Therefore every 
 
 i=l 1 
 assignment A must lie on the hyperplane LOj 0J. 
 
 Q.E.D. 
 
 Since [Oj 0] pseudo-separates every Boolean function, and 
 since this pseudo-separation is trivial in the sense that every assign- 
 ment must lie on [0$ 0], we would like to avoid this structure in our 
 future discussion,, 
 
 Definition 2.2.2 : A structure [Wj T] is called non-trivial if 
 W ^ (0,a,I'oi,0). A structure [Wj T] is called zero-free if all 
 w. / 0'U« 1,2, ...,n). 
 
 Lemma 2.2.2 : The degree of any non-trivial structure is strictly less 
 than 2 n , 
 
 Proof : If the degree 6f [W; T] is 2 , then every assignment A. must be 
 such that W°A. = T» Consider the assignments (0,0,.„.,0) and 
 (0,0, .„. ,0,1,0, . . o ,0) where this second assignment has 1 in position xj. 
 Because every assignment lies on the hyperplane [WJ T], 
 
 W°(0,0,...,0) = = W.(0,0,.. .,0,1,0, ...,0) = w.. 
 
 Hence all w. =0 and T = 0. Therefore the only structure of degree 
 
 2 n is [Oj 0). 
 
 Q.E.D. 
 
9 
 Definitipn 2.2. 3 l Let f (x) be an arbitrary Boolean function. Let 
 [W; T] be a non-trivial hyperplane which pseudo-separates f(x). An 
 assignment A is said to be segregated by LW; TJ if W-A f T. 
 
 Since [W; T] pseudo-separates f(x), this means that if A is 
 segregated by [¥; T] and true, then W-A > T and if A is false, W-A < T. 
 
 Using Definition 2.2. 3> Lemma 2.2.2 can be restated as follows: 
 
 Corollary 2.2.1 : Every non-trivial structure segregates at least one 
 assignment. 
 
 Notation 2.2.1 : If f (x) is any given Boolean function, the notation 
 
 "f(x; x. -* a,,...,x. -* a. )" will be used to represent the function 
 ' 1, 1 l k 
 
 which results when a (another variable or value) is substituted into 
 
 f (x) for x. , a^ substituted for x. and so forth. The notation 
 l 2 l 
 
 1 2 
 
 "f(x) > g(x)" means that for any assignment A, f(A) > g(A). If for some 
 assignment A, f(A) > g(A), and f(x) > jg'(x) we .Vill' wr'ite "f(x) > g(x)." 
 
 Definition 2.2.^ : f (x) is said to be positive in some variable x. if 
 f(x; x. -* 1) > f(xj x. ■* 0). f(x) is negative in x , if f(x 3 x. -> 0) > 
 f(xj x. -> l) . f(x) is called unatein x. if f(x) is either positive in 
 x. or negative in x.. f(x) is positive if it is positive in all 
 variables and negative if it is negative in all variables. f(x) is 
 called unate if it is unate in all variables. 
 
 Th 
 
 eorem 2.2.1 : Let f(x) be pseudo-separable under [Wj tj). Let x. be 
 
 any variable. Let A. be any assignment on the remaining n-1 variables. 
 
10 
 
 If f(A.: x. - l) = 1 and f(A. j x. -♦ 0) = 0, then w. > O If 
 i' i 11 i — 
 
 fit.; x. -* l) =0 and f(A. j x. - 0) = 1, then w. < 0. 
 i' i ii i — 
 
 Proof : For any variable x., it is well known that 
 
 f (x) = x.f (xj x. -*• l) v x.f(x.s x. -*• 0). 
 / i N > i i i' l 
 
 If f(A. | x. -»• 1) = 1 and f(A.$ x. -* 0) = 0, then since [W; T] pseudo- 
 separates f(x), 
 
 n n 
 
 Z w.a. < T and E w.a + w. > T. 
 j/i J J ~ tfi' J J X ~ 
 
 Combining these two expressions, w. > 0„ A similar.' proof is used to 
 
 show that if f(A.» x. -* l) = and f(A.$ x. -* 0) = 1 then w. < 0. 
 ii ii i — 
 
 QoE„D, 
 
 Corollary 2.2,2 : Let f (x) be a pseudo-separable function. If f (x) is 
 not unate in some variable x., then the weight corresponding to that 
 variable must be 0. 
 
 Proof ; If f(x) is not unate in x., then neither f(xj x. -*• l) > f(x; x„-» 0) 
 
 nor f(xj x„ -*• 0) > f(x; x. -* 1), Therefore there is a pair of assign- 
 or -* 
 ments A. and B. on the other n-1 variables such that 
 l l 
 
 (1) f(A. , x. «* l) = 1 and f(A. j x. -♦ 0) = 
 
 ii ii 
 
 (2) f(B. j x. -* 1) = and f($.', X. -♦ 0) = 1 
 
 i' l i- l 
 
 By Theorem 2.2.1, (l) implies w. > and (2) implies w. < 0. Therefore 
 
 v = 0. 
 
 Q.E.D. 
 
1] 
 
 Corollary 2.2.3 : If f(x) is pseudo-separable under a zero-free structure, 
 f(x) must be unate. 
 
 Corollary 2,2.^4-: If f(x) is pseudo-separable under [W; T] and positive 
 
 (or negative) in x., then w„ > (or w. < 0). . 
 
 i l — i — 
 
 Lemma 2.2.3 : Let f(x) be pseudo-separable under lW° T]. Then 
 f(x) x. -*x.) is pseudo-separable under 
 
 -*1 1 
 [w ± , w g , ...., * ±mml > ^ ± , \ +1 v v. M 'w n *,T - w 1 ] = [W ; T ]. 
 
 Further, the degree of [W ; T ] is the same as the degree of [Wj T]„ 
 
 Proof : Since [Wj T] pseudo-separates f(x), 
 
 n 
 
 (1) if f(x) = 1, then I w.x. > T 
 
 u n 
 
 (2) if f(x) = 0, then E w x. < T 
 
 x. can be expressed arithmetically as l-x. « 
 l i 
 
 n 
 
 Substituting x. for x. in f(x) and (l-x„) for x. in £ w.x. 
 
 n 
 (l)' if f(xs x. -* x.) = 1, then L w.x. +■ (l-x.) w. > T 
 i i j/i J J l' l - 
 
 i -* - n 
 
 (2) if f(x, x. ■+ x.) =0, then L w.x. + (l-x.) w. < T. 
 
 Let w. = -w, and subtract w. from both sides in (l)' and 
 li i 
 
 (2) V- then 
 
12 
 
 n 
 
 (l)" if f(x; x. ->x.) = 1, then E w x. + w. x.'>-T - w. 
 ' i i j^ J J i -i - ;. l 
 
 -♦ n 1 
 
 (2)" if f (xs x. -* x. ) =0, then E w.x. + w. x. < T - w. 
 i V tf ± 3 J i i " 
 
 Therefore, [wj T ] pseudo-separates f(x; x. =»"xT)„ 
 
 — * — »• — > 
 
 Let A he any assignment such that W'A = T„ Consider the 
 
 assignment A where a„ = a.(j 4 i) and a. = a . Then 
 
 -*1 -»1 n n 
 
 W"°A = E w,a. +- (l-a 4 ) (-w ) = E w,°a. + w„°a. - w. = T - w„ . 
 
 Therefore if A lies on hyperplane LW; TJ, A lies on hyperplane 
 
 [Wj T ]. Similarly if A is any assignment on [fj T ], then A will 
 
 lie on [W; T], Therefore there is a one to one correspondence between 
 
 assignments on these two hyperplanes so that they must have the same 
 
 degree. 
 
 Q.E.D. 
 
 The above lemma allows us to restrict our attention to positive 
 functions and to then generalize all results to unate functions. Hence- 
 forth, the term "positive function" will carry this extension implicity. 
 
 Lemma 2 2 U : If f (x) is pseudo-separated by [W$ T], then f (x) is pseudo- 
 separated by [-Wj -T] and the two hyperplanes have the same degree. 
 
 Proof : If f(x) = 1, then W-x > T or -(W-x) < -T and f-W)«x < -T. If 
 f(x) = 1, then f(x) = and j[-W)«x < -T. Further, if f(x) = 0, then 
 f(x) = 1 and W-3 < T or (-W) x > .?% Therefore [-Wj -T] pseudo-separates 
 f(x). 
 
13 
 
 To show that [W; T] and [-W; -T] have the same degree, let A 
 
 be any assignment such that W°A = T, Then (-W)°A = -T and A also lies 
 
 on [-W; -T]o Similarly if (-W)°A = -T, then Wo A = T. 
 
 Q.E.D. 
 
 Theorem 2.2,2 : If f (x) is pseudo-separated by [Wj T], then f (x) is 
 
 n 
 pseudo-separated by [W; Z w„ -T] and the degree of [Wj Tj is the same 
 
 i=l x 
 
 is the degree of [Wj Z w. -T] 
 
 i=l 
 
 n 
 
 i 
 
 Proof: Since f (x) = f(x) = f(xs x. -» x.. , x. -» x . . ... x -> x ) , the 
 ■ ' 1 12 2 n n ' 
 
 result follows from Lemmas 2»2.3 and 2„2 ^. 
 
 Q.E.D. 
 
 We would now like to investigate the relationship between 
 symmetric variables and their corresponding weights „ 
 
 Lemma 2-2.jp : Let f(x) be pseudo-separated by [W> T]. Let f(x) be 
 
 symmetric in x. and x.« Even if w. 4 w., f(x) is also pseudo-separated 
 i j i ' y 
 
 by [w^ w 2 , ..., w w w 1+1 , ..., w , w., w ..., w n ; T] = 
 
 y l+i- j-i- l- j- 
 
 [W ; T] and the degree of [w; T] is the same as the degree of [Wj Tj. 
 
 Proof : (l) Assume that [w ; T] does not pseudo- separate f(x). Then 
 there must exist some true assignment A such that W °A < T or some false 
 assignment B such that W -B > T. Assume that f (A) = 1 and VT-A < To 
 Since [W; T] pseudo-separates f(x), W°A > T. Let C be the assignment 
 which agrees with A in all variables other than the i and j 
 
lU 
 
 coordinates and c, = a., c, = a„. Then W-C = ¥ „A < T„ Therefore C 
 i 3 J i 
 
 must be a false assignment because [VT ; T] pseudo-separates f(x)„ 
 
 Since x. and x. are symmetric in f(x), A must also be a false assignment. 
 
 Therefore, we have a contradiction „ 
 
 (2) Let A be any assignment which lies on the hyperplane [Wj T]„ 
 
 Then the corresponding assignment C, which was defined in part (l) 
 
 will lie on hyperplane [W; T], Similarly if C is on [VT°, T], then A 
 
 must be on [w; T]„ Therefore, there is a one to one correspondence 
 
 between assignments on [Wj T] and [w $ T] and these two hyperplanes must 
 
 have the same degree „ 
 
 QoEoDo 
 
 Theorem 2 2,3' If f(x) is pseudo -separable under [W° T] and if x„ and 
 x. are symmetric in ftx), but w„ f- w., then there exists another 
 structure [w ; T ] with w. " tsw ; which, pseudo-separates f(x) ■■■'k^. 
 and whose degree is no greater than the degree of [Wj Tj„ 
 
 Proof ; Consider the hyperplane [VT ; T ] = [W +• w ; 2T] where [f; T] 
 was defined in Lemma 2„2<,5° We will first show that this hyperplane 
 pseddo-separates f(x)» Let x be any assignmento (W + w )«x = W°x +■ W °x, 
 If f (x) = 1, then W.x > T and W^x > T and (W 4- W 1 ) >x > 2T Similarly, 
 if f(x) = 0, then W*x < T and fa < T and (W 4- w~) -x < 2T„ Therefore 
 
 [W 4- w ; 2T] pseudo-separates f(x)„ Also, because w. = w., and 
 
 1 1 1 , _. - 11 11 
 
 w, = w., w. +■ w. = w„ 4- w. = w. 4- w.o Therefore w„ = w„ 
 iJ'iiiJOJ i z 
 
 Finally, we want to show that the degree of [W 4- w°, 2T], is less than 
 or equal to the degree of [Wj T]. Let x be segregated by [W$ T]„ If 
 
15 
 W-x > T, then W~-x > T and therefore, (W + W~) °x > 2T„ Similarly, if 
 W-x < T, then (W +■ F)»x < 2T» Thus x must also be segregated by 
 [W f W 1 ) 2T]. 
 
 We next define a partial ordering relation between different 
 variables of a function, The following theorem shows how this relation, 
 when it can be defined, relates the magnitudes of the corresponding 
 weights in any pseudo-separating hyperplane. 
 
 Definition 2.2,5 : Let f (x) be an arbitrary Boolean function. Let x, 
 
 X. * 
 
 and x, be any two variables, x„ > x. means that 
 J i - 3 
 
 f(x» x. -* 1, x. -* 0) > f(x: x„ -> 0, x. -*■ 1) 
 / l 3 ~ 1 
 
 and x > x. means that 
 i 3 
 
 f(x; x„ ■* 1, x. ■* 0) > f(xs x„ -* O.'x. -* 1) 
 ' i 3 i 
 
 Theorem 2.2. U : Let f(x) be pseudo-separated by [W; T]. If x. > x., then 
 
 -*- J 
 
 w, > w.. If w. = w.- then all assignments A such that 
 i - J i J 
 
 f(A; x. -» 1, x. -+ 0) = 1 and f(A; x. -» 0, x . -* l) - must lie on the 
 hyperplane [W| T], 
 
 Proof: Assume x. > x . . Then there must be some assignment B. . to the 
 i J i>0 
 
 other n-2 variables such that f(B. .; x. -* 1, x. -* 0) = 1 and 
 
 f(B. .) x. -* 0, x . -* l) = 0. Since [W; T] pseudo-separates f(x), 
 1 > 3 i J 
 
16 
 £ w °b f w. > T and Z w °b + w„ < T so that w. > w.o If 
 
 fa.i p p x _ p/i,j p p ° " x " ° 
 
 n n 
 
 w = w , then Z w b ■*- w. = Z w b f w, = T must hold and these 
 
 1 J p^,j pp x P^i.j pp a 
 
 true and false assignments must lie on the hyperplane [W| T]„ 
 
 Q.E.D. 
 
 The preceeding theorem is weaker than the corresponding result 
 for separable functions, where x„ > x„ implies w„ > w. in all structures 
 [7]. The next example shows that this weaker result is the strongest 
 property which we can guarantee „ 
 
 Example 2,2ol : Consider the function f (x) = x.x %s x x v x x, v x x „ 
 f (xj x =*• 1, x p -* 0) = x^ v x> while f (x$ x -» 0, x -> l) = x . There- 
 fore f (xj x.. -*• 1, x -* 0) > f (xj x„ -* 0, x -> l) and so x > x_„ 
 Consider the hyperplane [1,1,1,1$ 2] If A is any true assignment, then 
 A has more than 2 components of value 1 or else is one of (1,1,0,0), 
 (1,0,1,0), (1,0,0,1) or (0,l,l,0)o In either case (l,l,l,l) »A > 2„ 
 If B is any false assignment, then B has less than 2 components of 
 value 1 or else is (0, 0,1,1) or (0, 1,0,1) so that (l,l,l,l) °B < 2. 
 Therefore [1,1,1,1; 2] pseudo-separates f(x)„ Because w. = w = 1 
 and x > x , we may not strengthen Theorem 2 2,i+ to require that 
 
 x. > x. implies w„ > w.o 
 1 3 1 J 
 
 Another difference "between pseudo-separable and separable 
 functions is that pseudo-separable functions need not have all variables 
 related by this ordering, while for separable functions the relation 
 
IT 
 "<" simply orders the set of all variables [73. Consider the function 
 xx v xx,, which was shown in Example 1.2.1 to be pseudo-separable, 
 f (x; x -* 1, x -» 0) - x and f(xj x. -* 0, x -* l) = x, , so that neither 
 
 x l _ x -? nor x i - X V 
 
 Definition 2.2.6 : Let f (x) be an arbitrary Boolean function. Let 
 
 A = (a., a , ..., a ) and B = (b , b , ..., b ) be any pair of assign- 
 
 > -> 
 
 ments. Assume that A and B agree in position x. , x. , .... x. 
 
 12 p 
 
 (i.e., a. = b. , .... a. = b. ) and disagree in x . , x. , .... x. where 
 
 11 P P 1 2 d q 
 
 .i 
 
 -» -» .-*. 
 p + q = n. We will say that A and B are f(x; - comparable if either 
 
 f(xj x. -* a . , x . -+ a .,..., x . -*■ a ) > 
 
 J l J l J 2 J 2 J q J q 
 
 f(xj x. -* b . , x. -»• b . , . . . , x. -* b . ) 
 J l J l J 2 J 2 J q J q 
 
 or 
 
 f(xj x. •* b . , x. — b . , ..., x. -> b . ) > 
 J l J l J 2 ^2 J q J q 
 
 f(x; x. -* a. , x . -* a . , . . . , x . -+ a . ) . 
 
 J l J l J 2 J 2 ^q J q 
 
 The former case will be denoted by "A > B" and the latter by "A § B" . 
 
 If A > B and f(x; x. -* a . , x . -* a. , ..., x. -* a ) ^ 
 
 J l J l J 2 J 2 J q J q 
 
 f (x; x. ■* b . , x -♦ b .,..., x . -* b . ) , then we will write "A > B" . 
 J l J l J 2 J 2 J q J q 
 
 f (x) is called completely monotonic if every pair of assignments is 
 
 f(x) - comparable [8], [9], [11]. 
 
18 
 Theorem 2.2.5 ' Let f(x) "be pseudo-separable under [Wj T]. Assume that 
 f (x) is such that for two assignments A and B, A > B„ Then W°A > W°B„ 
 
 Proof: Assume that A and B agree on variables x_ , x^, .... x and dis= 
 1 2 q. 
 
 agree on variables x , n , x _,..., x where a , . = . . = a =1 and 
 & q+1' q+2' ' n q+1 p 
 
 a ,=.„.= a = 0(ri < p < n). Since A > B, there must exist some 
 p-t-1 n — — ' 
 
 -> -> 
 pair of assignments C and D which also differ on x , . , x r ,. „„., x 
 * q+1' q+2' ' n 
 
 where C agrees with A on these variables and f (C) = 1, while D agrees 
 
 with B and f (d) = 0, If W°A < W.B, then w ,_*■ w _-,f .. . + w < w , + ...+v 
 
 v ' ' q+1 q+2 p p+1 n 
 
 :\U~. 
 
 and because C and D differ on exactly this set of variables, W«D > W»C„ 
 
 This is impossible because f (S) = 1 and f (d) = 0. Therefore W°A > W.B. 
 
 Q.E.D. 
 
 Theorem 2.2.6 : Let f(x) be pseudo-separable under [W$ T]. Assume that 
 f (x) is completely monotonic and that for any pair of assignments A and 
 B, if A > B, then W°A > W°B„ Then f (x) is separable under [W$ T], 
 
 Proof ; Let A and B be any true and false assignments respectively such 
 that W«A = W°B = To Since f(x) is completely monotonic, A and B must 
 
 be f (x) - comparable. Because f (a) = 1 and f (b) = 0, A > B. Therefore 
 
 -+ -*■ -* -* 
 
 W»A > W°B, a contradiction. 
 
 Q.E.D. 
 
 The concept of complete monotonicity also leads to an important 
 difference between pseudo- separable and separable functions. Every , ; . 
 separable function mast be completely monotonic, but : an arbitrary pseudo- 
 separable function need not be so. To see this, again consider the 
 
19 
 pseudo-separable function x x 4s x^k" Let assignment A "be (l, 1,0,0) 
 and B be (1,0,1,0). These two assignments disagree on variables x and 
 x . f(xj x 2 -* 1, x -* 0) = x and f (x; x_ -> 0,x -> l) = x, , which says 
 that neither A > B nor B > A. 
 
 2.3 Bounds on the Degree of Hyperplanes and Conditions for Separation 
 
 In this section, the relationship between the number of assign- 
 ments on a pseudo-separating hyperplane and requirements for separability 
 of a given function are discussed. Bounds on the degree of pseudo- 
 separating hyperplanes are also investigated. 
 
 Theorem 2.3.1 : Assume that f(x) is pseudo-separated by both [W j T ] 
 
 and [W j T ]. Then f(x) is also pseudo-separated by [W + L T +■ T ] 
 
 and the degree of [W + W ; T + T ] is the number of assignments which 
 lie on both hyperplanes [W _j T J and [W ; ^ ]. 
 
 Proof : We will show two things. First, if [W ; T ] and [L T ] 
 pseudo-separate f(x), then [W. +■ W j T + IE ] also pseudo-separates f(x), 
 and second, if A is any assignment that is segregated by either 
 [W^ T ] or [W 2 j T ], then A is segregated by [W^ + ft ; T + T g J. 
 
 (l) Let A be any assignment. If f (A) = 1, then W-A > T^and 
 
 W-A > T and (% + W rt ) «A > T, + T„. If f (A) = 0, then W n -A < T n and 
 2-2 12' -12 ' 1-1 
 
 W 'A < T and (W + W ) 'A < T. + T . Therefore [W +■ W > T +■ T ] 
 pseudo-separates f(x). 
 
20 
 
 (2) Assume that A is segregated "by either [W , T ] or [W ; T ]. 
 
 Then either W «A / T or W* 2 °A / T^ Therefore (W^ f W 2 )°A / ^ + T 2 
 
 and A vill he segregated by [W + Wj T + T J. 
 
 1 * Q E„D« 
 
 When hyperplanes [W_ $ T_] and [W-j T ] have no assignments in 
 
 common or when all common assignments have the same value, then the 
 
 _— >• — ► = 
 hyperplane [W. + W j T f T J must separate the function. 
 1 2 1 ^ 
 
 Corollar y 2o3°l° Assume that f(x) is pseudo-separated by [W.^ °, T. ] and 
 [W_j T ]„ Let P he the set of all assignments which lie on hyperplane 
 [W. 5 T. ] and Q, he the set of all assignments which lie on [W p $ T ]„ If 
 PTIQ = 0, f(x) is separated by [W^ + W g j T ± + Tg], 
 
 Corollary 2o3°2: Assume that f(x) is pseudo-separated by [W j T. ] and 
 LW j T Jo If all assignments A„ which lie on both hyperplanes have the 
 same value, f(A„) = a, then f(x) is separable under [W- f W_^ T + T ]„ 
 
 Example 2o3°l ° We can illustrate this last result hy considering the 
 function which was defined in Example 2 2„1<, Let f (x) = 
 xx v x„x v x p x v x.x, „ We have already seen that it is pseudo- 
 separated hy [l,l,l,lj 2] The assignments which lie on this hyperplane 
 are (1,1,0,0), (1,0,1,0), (1,0,0,1), (0,1,1, 0), (0,1,0,1) and (0,0,1,1). 
 The first four are true and the last two are false „ If we can find a 
 hyperplane which pseudo-separates f(x) and segregates (0, 1,0,1) and 
 (.0,0,1,1), we will then he able to add the two structures to generate a 
 
21 
 
 new hyperplane which will separate f(x). Consider the hyperplane 
 [2,1,1,0; 2]. Let A he any assignment such that (2,1,1,0). A > 2, We 
 want to show that f (A) = 1. A must he such that a = 1 and either a = 1 
 or a - 1. In either case, f(A) = 1. If (2,1,1,0) »A < 2, then a = 
 and either a = or a = 0, therefore f(A) = 0. Therefore [2,1,1,0; 2] 
 pseudo-separates f(x). The assignments which lie on [2,1,1,0; 2] are 
 (1,0,0,0), (1,0,0,1), (0,1,1,0) and (0,1,1,1). The only assignments 
 common to both [1,1,1,1; 2] and [2,1,1,0; 2j are (l, 0,0,1) and (0,1,1,0) 
 which are both true. Therefore, [3,2,2,1; k"} must separate f(x) and 
 the only assignments which lie on this hyperplane are (1,0,0,1) and 
 (0,1,1,0). 
 
 Definition 2.3.1 : Let f(x) be a positive function. Let A = (a ,a p ,...,a ) 
 
 and B = (b ,b , . . . ,b ) be any pair of assignments, A and B are comparable 
 
 (as opposed to f(x) - comparable used in Definition 2.2.6) if for all 
 
 i = 1, 2, . . . , n, a, > b. or for all i, t>. > a.. We will denote the 
 
 i-i ' i — i 
 
 former case by A > B and the latter by A < B. If A > B and for some j, 
 a > b_, we will write A > B. A is an extremal true assignment if f(A) = 1 
 and for all B such that B < A, f (B) = 0„ A is an extremal false ass ignment 
 if f (A) = and for all B, B > A, f (b) =1. A is an extremal assignment 
 if A is either an extremal true or an extremal false assignment. If A 
 is not an extremal assignment, A is called non- extremal. 
 
 Theorem 2.3.2 : If f (x) is pseudo-separated by a zero-free structure 
 [W; T], then all non-extremal assignments must be segregated by [W; T], 
 
22 
 
 Proof ; If f (x) is pseudo-separable under a zero=free structure, then 
 
 f (x) must be unate "by Corollary 2<>2»3o We will assume that f (x) is 
 
 positive. Therefore, all w„ > by Corollary 2o2„U and because [Wj T] 
 
 is a zero-free structure, all w. > 0. 
 
 l 
 
 Let A be any non-extremal true assignment . (A similar proof 
 
 -*■ . 
 
 will hold if A is any non-extremal false assignment „ ) Then there exists 
 
 -> —»-»-* 
 another true assignment B such that B < A„ If A is not segregated by 
 
 — *• _ -> — > — > -» -*■ 
 
 [W$ TJ, then W-A = T„ Because B is also a true assignment, W°B > T 
 
 Therefore W° (A - B) < 0„ Since A > B. there must be at least one a. > b„< 
 
 — ' 11 
 
 Hence W» (A - B) > w„ „ Because all weights are positive, we have a 
 
 °"* r"* -\ 
 
 contradiction,, Therefore A must be segregated by LW$ TJ» 
 
 Q.E.D, 
 
 Theorem 2„3°3 ; let f (x) be a positive function., Assume that f (x) is 
 pseudo-separated by a zero-free structure [Wj T] Then the degree of 
 [W; T] is less than or equal to 2([pl) . 
 
 Proof : Since [Wj T] is zero-free, f(x) is unate and 2 ([~]) is the 
 
 maximum number of possible extremal assignments [7]° By the previous 
 
 theorem all non-extremal assignments must be segregated by [W; T]. 
 
 Therefore 2 ( [r] J is the maximum number of possible assignments which can 
 
 lie on [W> T}. 
 
 Q.E.D. 
 
 Theorem 2.3.^ : Assume :that f(x) is pseudo-separated by [Wj T], Let 
 
 -►■-+• -*•♦-+ -*■ -* _ 
 
 A , A , . „„, A, , B , B , ,.., B be all assignments which lie on LW3 TJ„ 
 
 J_ <— K i_ c. 6 
 
 Let [w j T ] be any hyperplane which contain^ A_ , A , „ 00 , A, and 
 
23 
 
 segregates B , B , . . „ , B 'i: ( [W ? T ] need not pseudo -separate f(x)i 
 
 ~* ~*1 -*• 1 
 
 There may exist true assignments C. such that W D C. < T and false assign- 
 ments D. such that W °D„ > T „ However such assignments may not lie on 
 
 [W> T]). Then there exists a hyperplane [W -, T ] which pseudo- 
 
 .-» -» -* =-* 
 
 separates f(x) and segregates B , B , „.., B together with all assign- 
 
 /J. c. e 
 
 ments which are segregated by [W> T]. 
 
 r~*l - l-i '~*\ 
 
 Proof : The problem nere is that I.W ; T J need not pseudo-separate f(x), 
 
 r ~* ~*1 1- 
 
 Consider the set of structures LmW + W f mT + T j where m is any positive 
 integer. We will show that for some m, the resulting hyperplane pseudo- 
 separates f(x), segregating B , B , „„,, B and all assignments segregated 
 by [W; T], 
 
 (1) Let D. , D , ooo. D be all false assignments such that 
 
 L d. S 
 
 — >~]_ "■* 1 -» -* -» 
 
 W. 'D. > T . Let C, , C_, . . „ . C^ be all true assignments such that 
 i 1' 2 t 
 
 -+1 -* 1 
 
 W °C. < T o Let M be the minimum positive integer such that for 
 
 all D. and C. , [^.C. - T 1 ] - M[T - W-C, ] > and [T 1 - W^D. j 
 
 - MLW.D. - TJ > 0„ Because all such D, and C, were segregated by 
 
 [W; T], W-C. > T and W°D> < T and so we have [T - (W»C. )] < and 
 
 i l l 
 
 [(W°D.) - T] < Oo Therefore, such a positive M can always be 
 found, 
 
 (2) We now show that [MW + W j MT + T j pseudo-separates f(x). 
 
 -» — » — * -*j_ — * 1 
 
 Let A be any assignment,, If W°A = T and w "A = T^ then 
 
 (MW + W )»A = MT + T , Therefore, if A is not segregated by either 
 
 [W; T] or [W 1 ? T 1 ], then A is not segregated by [MW + W^ MT + T 1 ], 
 
 "* -*• -* r^l 1- "? 
 
 If W°A = T and A is segregated by LW$ T j, then A is segregated by 
 
2k 
 [MW + VT f MT + T ] because if A is a true ^assignment, w °A > T 
 and (MW + W )*A > MT + T and if A is a false assignment ^T°A < T 
 and (MW + w )»A < MT + T . If A is a true assignment "which is 
 
 segregated by LWj TJ, then there are two different possibilities, 
 
 ~*1 ""*• ^ 1 ~tl °* 1 -»■ 
 
 either W^-A > T or W^A < T „ In the first case A must be segregated 
 
 by [MW + W 1 ; MT + T 1 ] because (MW + W 1 ) -A > MT + T . If W^A < T, 
 
 then A must be one of the C. m part (,!;„ 
 
 (MW + W 1 )-^ - (MT + T 1 ) = [(W^A) ? T 1 ] - MfT - (W° A) ] 
 
 and by the choice of M in part (l), this expression is always 
 positive. Therefore (MW + W;°A > MT + T and A is segregated 
 by this structure A similar proof holds if A is a false assign- 
 ment which has been segregated by [W$ T]. Hence [MW + W; MT + T ] 
 
 pseudo-separates f (x) and segregates all assignments segregated by 
 
 Q.E.D. 
 
 either [Wj T] or [W X j T 1 ] 
 
 Corollary 2. 3. 3 « Assume that f(x) is pseudo-separable under [W$ Tj and 
 that there exists another hyperplane [Wj T ] which segregates all 
 assignments which lie on [W; T], then f(x) is separable. 
 
 Proof : If [Wj T ] segregates all of the assignments which lie on 
 
 [Wj T], hhthefaii'! [MW f W; MT + T ] will separate f(x) since all assign- 
 
 ments lying on the hyperplane [Wj TJ will be segregated. 
 
 Q.E.D. 
 
25 
 
 Certain sets of assignments lying on a hyperplane are always 
 separable. One such set consists of only one true assignment and any 
 number of false assignments as the next theorem shows. 
 
 Theorem 2. 3. 5 - Assume that f(x) is pseudo-separable under [Wj T] and 
 that there exists only one true (false) assignment and any number of 
 false (true) assignments which lie on [Wj Tj. Then f(x) is separable. 
 
 Proof -; We need only show that there exists some hyperplane which 
 
 — > 
 segregates the one true assignment, A, and by applying Corollary 2. 3° 3, 
 
 the result follows. Assume that A = (a n , a OJ .... a ). Let 
 
 v 1' 2' n 
 
 a n = a_ = . . . = a =1 and a , = a n = ... = a =0. Let w. = 1 if 
 12 p p+1 p-t-2 n l 
 
 a. = 1 and w. = -1 if a. = 0. Let T. = p - 1. Clearly 
 
 i l l :. 
 
 -*1 -* . 1 -> r -»l i- 
 
 W «A = p > T = p - 1 so that A is segregated by IVT°, T J. For any 
 
 ■* -*1 -» 1 r -*l 1. 
 
 other assignment B, W -B < p - 1 = T , Therefore Lw \ T J segregates 
 
 the set of assignments which lie on [Wj T] and so f(x) is separable = 
 
 Q.E.D. 
 
 Definition 2.3.2 : Let {A 1 , A , . .., A J be a set of assignments. 
 
 [A , A , ..., A ] is called consistent if the rank of the matrix! 
 
 V - 
 
 u s) 
 
 i" . i •. ■; i , ' 
 
 K, 1, f(A ) 
 
 is the same as the rank of the augmented matrix! : : 
 
 \, 1, f(A k ) 
 
 Theorem 2.3.6 : Let (A^, A , . .., A k } be a consistent set of assignments, 
 
 -* . 
 Let A. = (a. , a. , ..., a. ). Then this set is separable if the rank 
 1 2 X n 
 
 of the augmented matrix in Definition 2.3.2 is less than n + 1. 
 
26 
 
 Proof : Consider a set of linear inequalities given "by W°A. - T = 
 
 for f(A. ) = 1 and W-A. - T = - 1 for f(A. ) = 0. If the rank of this 
 l i l 
 
 matrix is less than n + 1, and if the assignments are consistent, then 
 
 we can have a solution [W; TJ, This yields a segregating hyperplane for 
 
 I A, ) A , . . „ , A j „ 
 
 1 2 k Q.E.D. 
 
 Corollary 2o3.^ ' Assume that f(x) is pseudo-separable under [W; T]„ 
 
 -> -» -> . 
 
 Let [A , A , „ . „ , A } be the set of all assignments which lie on hyper- 
 
 1 (— K. 
 
 plane [W- T]. Then if {A.*, A , . .., A } is consistent and if the rank 
 
 I is less than n + 1, f (x) is separable. 
 
 a ll a i2 * ' ' a ln' 
 
 a kl a k2 * ' ' a kn 
 
 Proof : If this set is consistent and if the rank is less than n + 1, 
 then by the previous theorem, there exists some hyperplane [Wj T ] 
 which segregates this set. Since this set includes all assignments 
 which lie on [Wj T], f(x) must be separable by Corollary 2. 3° 3° Notice 
 that the above corollary also puts a lower bound on the number of con- 
 sistent assignments which can reside on the hyperplane „ 
 
 Q.E.D. 
 
 2.M Asummability .: as, ,'a Condition for Pseudo-Separabilit y • 
 
 In this section the idea of asummability is introduced and 
 from this property, a necessary and sufficient prerequisite for a function 
 to be pseudo-separable is developed. This condition is also used to 
 explore other algebraic properties of pseudo-separable functions „ 
 
27 
 
 Definition 2.4.1 : Let f (x) be an arbitrary function. f(x) is called 
 
 k-summable if there exist k true assignments A , A , . .., A and k 
 
 1 <— ri 
 
 false assignments B , B , ..., B , with possible repetitions, such that 
 
 k k 
 
 E A. = S B. . f(x) is called asummable if it is not k-summable for 
 
 . _ l . . l 
 
 i=l i=l 
 
 any k„ (See Example 2.4.1.) 
 
 The following theorem is due to Elgot [2] and Chow [l], and 
 
 the proof is omitted here. 
 
 Theorem 2.4.1: Let f (x) be an arbitrary Boolean function, f (x) is 
 separable if and only if it is asummable. 
 
 In developing a necessary and sufficient condition for pseudo- 
 separation, we would like to refer to the summability or asummability of 
 particular assignments. Therefore we introduce the next definition. 
 
 Definition 2.4.2 : Let A be any true (or false) assignment to f(x). 
 
 A is called summable if for some k, there exist k - 1 true (false) 
 
 assignments A , A , . .., A and k false (true) assignments 
 
 d 5 K k 
 
 B , B , ..., B , with possible repetitions, such that A + L A. = 
 
 1 Cm K 1 . ,-. 1 
 
 1=2 
 k 
 
 B. . Note that in such a case all of A n , A . .... A. and 
 i=l 
 
 —>—>—> — » 
 
 B , B , ..., B are summable. An assignment A is called asummable if it 
 
 1 £- K. 
 
 is not summable. 
 
28 
 
 Lemma 2.4.1 : Let f(x) be pseudo-separated by [Wj T]. If A is segregated 
 by [W; T], then A is asammable. Conversely, all summable assignments 
 must lie on the hyperplane [W$ T]. 
 
 -> , 
 
 Proof : Assume that A is a true assignment. (A similar proof can be 
 
 given if A is any false assignment) .Let W.A > T. If A is summable, 
 
 — > — * — * 
 then there exist true assignments A , A , . . . , K, and false assignments 
 
 b\ , B , . . . , B. such that A + E A. = Z B, . Therefore, 
 1 ' 2' k . i . , i 
 
 i=2 i=l 
 
 k k 
 
 W- (A + E A. ) = W° ( E B. ) . Because W-A > T and each W.A. > T, 
 . „ l . , l l — 
 
 i=2 1=1 
 
 k 
 
 W« (A + Zj A. ) > kT. Because each B:. is a false assignment, W-B. < T 
 l ■ • i i — 
 
 i=2 x 
 
 k 
 
 so that W- ( .E.. B.) < kT. Therefore, we have a contradiction. Hence, 
 i=l i — ' 
 
 -» ^-> _ 
 
 A can not be summable if it is segregated by LWj TJ. 
 
 Q.E.D. 
 
 Corollary 2.4.1 : Assume f(x) is pseudo-separable under a zero-free 
 structure [W; T]. Then all non-extremal assignments must be segregated 
 by [W; T] and hence must be asummable. 
 
 Proof : If f (x) is pseudo-separated by a zero-free structure, all non- 
 extremal assignments must be segregated (by Theorem 2.3.2) and hence 
 must be asummable (by Lemma 2.4.1). 
 
 Q.E.D. 
 
 Definition 2.4.3 ' f (x) is called super summable if, for some k, there 
 exist k true assignments A , A , ..., A and k false assignments 
 
 I c. K. 
 
 k k 
 
 B , B c , ..., B , with possible repetitions, such that E A. < E B„ . 
 
 1=1 1=1 
 
29 
 
 If we let each A. = (a. , a. , . .., a. ) and 
 
 1 i, i_ l 
 12 n 
 
 B. = (b. , "b. , .... b. ), then the above condition is fulfilled when 
 12 n 
 k k k k 
 
 Z a. < Z b. for j = l,2,...,n and for some j = p. Z a. < Z b. . 
 
 i=l j i=l j i=l p i=l p 
 
 Example 2.4.1: Consider the function f(x) = x n x_ v x x, x . (1,1,0,0,0) 
 12 j 4 ^ 
 
 and (0,0,1,1,1) are both true assignments while (1,0,0,1,0) and 
 (0,1,1,0,1) are both false. . Because (l, 1,0,0,0) + (0,0,1,1,1) = 
 (1,1,1,1,1) = (1,0,0,1,0) + (0,1,1,0,0), these assignments are summable. 
 Therefore, f(x) is not separable by Theorem 2.4.1. Now consider false 
 assignments (1,0,1,1,0) and (0,1,1,0,1). (1,0,1,1,0) + (0,1,1,0,1) = 
 (1,1,2,1,1) > (1,1,1,1,1). Therefore, f(x) is supersummable. 
 
 Definition 2.U.4 : Let f(x) be a supersummable function. f(x) is said 
 
 to be supersummable in variable x if for some k, there are k true 
 — c -P 
 
 assignments A , A , ..., A and k false assignments B , B , ..., B , with 
 
 _L c. K. _L c. K 
 
 k k k k 
 
 possible repetitions, such that Z A. < Z B. and Z a. < Z b. . 
 
 .,i.,i .,i.-.i 
 1=1 1=1 i=l p 1=1 p 
 
 In the previous example f (x) is supersummable in x and by 
 
 symmetry is also supersummable in x, and x . 
 
 Lemma 2.4.2 : If f(x) is a positive supersummable function, f(x) is 
 summable. 
 
30 
 
 Proof : Let true assignments A , A p , , . . , A and false assignments 
 
 k -> k 
 
 B, , B_, . ... B, be such that Z A. < Z B. . Let 
 
 12 k . , 1 . , i 
 
 1=1 1=1 
 
 A. = (a. , a. , .„., a, ) and B. = (b. , b. , .... b. ). Assume 
 i i, i ' i i i , i ' i 
 12 n 12 n 
 
 k k 
 
 Z a. = Z b„ for all j except j = p, , p., . .., p . Consider those 
 . , i . . , l . L d q 
 
 i=l j i=l j H 
 
 k k 
 assignments B for which b = 1. If Z b. - Z a. = s of these 
 
 . , l ■ i ■ l 
 
 i=l p x i=l p l 
 
 e 
 
 assignments are changed to the corresponding assignments with b = 0, 
 
 6p ] 
 
 -> k k_s s 
 and renamed B , then .Z a. = Z b. + Z b. Since f(x) is 
 e ' . - l . -, i . -. l 
 
 i=l p x i=l Pl i=l ?1 
 
 positive and B > B , f (B ) = implies f (B ) = 0. If this procedure 
 
 k 
 
 is repeated for each of p n , p^, .... p , the result will be Z a. 
 
 1 2 q . . l . 
 
 k k ^ k 
 
 Z b q for all j so that Z A. = Z B. q . 
 i=l X j i=l X i=l X Q.E.D. 
 
 Example 2.4.2 : Consider the function x x \y x x, . (l, 1,0,0) and (0,0, 1,1) 
 are true assignments and (0, 1,0,1) and (l, 0,1,0) are false assignments. 
 Because (l, 1,0,0) +■ (0,0, 1,1) = (l, 1,1,1) = (0,1, 1,0) + (l, 0,1,0), the 
 function is summable. Now assume that it is supersummable. Then there 
 are true assignments A , A , ..., A and false assignments B , B , ..., B 
 
 k -> k 
 
 such that Z A. < Z B. . Therefore, if we let 1 = (l,l,l, . . . ,l) , then 
 
 i=l x i=l x 
 
 k - - k 
 
 1- ( Z A.) < 1*( Z B.). Each true assignment A. has at least 
 
 . _ i . , i i — — ■ — — ■ 
 
 i=l i=l 
 
 k 
 
 2 components of value 1. Therefore* 1-A. > 2 and 1- ( Z A.) > 2k. 
 
 l — . , l — 
 
 i=l 
 
31 
 
 Since each B. is false, B. has at most 2 components of value 1. Hence, 
 
 k 
 1-B. < 2 and 1- ( E B.) < 2k. Therefore, it is impossible to have 
 i=l 
 
 k -* k 
 1- ( E A.) < 1- ( E B. ) and x.x. v x x, is not supersummable. 
 1-1 x 1=1 x 12 3 4 
 
 Lemma 2.4.3: If f(x) is positive and supersummable in x and f (x) is 
 pseudo-separated "by LW; TJ, then w = 0. 
 
 k k 
 Proof: If f(x) is supersummable in x , then for some k, E A. < E' B. 
 
 * i=l i=l 
 
 k k 
 
 and E a. < E b. , where A. = (a. , a. , .... a. ) are true assignments 
 . , i . , i 1 1-, i ! 
 i=l p i=l p 1 2 n 
 
 and B. = (b. , b. , .... b. ) are false assignments. If w > 0, then we 
 V X 2 P 
 
 k _> - k 
 
 have W- ( E A. ) < W. ( E B. ) because of w; > 0. Since each A. is true, 
 
 .,i . , l i- l 
 
 i=l i=l 
 
 k k 
 
 W-A. > T, so that W- ( E A. ) > kT. Similarly W- ( E B. ) < kT. This 
 l — . , l — .,i — 
 
 1=1 1=1 
 
 k k 
 
 contradicts W- ( E A. ) < W. ( E B. ) . Therefore, w =0. 
 
 i=l x i=l X P Q.E.D. 
 
 Theorem 2.4.2 : If f(x) is supersummable, f(x) can not be pseudo- 
 separated by a zero-free structure. 
 
 Proof : If f (x) is pseudo-separated by a zero-free structure, then it 
 
 must be unate by Corollary 2.2.2. Assume it is positive. Then because 
 
 f(x) is supersummable in some x , some w = by Lemma 2.4.3, a 
 
 p P 
 
 contradiction. 
 
 Q.E.D. 
 
32 
 
 Theorem 2.4.3 : Assume that f (x) is a positive Boolean function,, Then 
 f(x) is supersummable if and only if some non-extremal assignment is 
 summable. 
 
 Proof : (l) Let A be a summable non-extremal true assignment. Then 
 
 k - k 
 A + Z A = Z B.. Since A is not extremal, there is some true 
 
 1=2 i=l 
 assignment A such that A < A. Therefore A 4- Z A. < Z B. and f (x) 
 
 . i . , i 
 
 1=2 i=l 
 
 is supersummable, 
 
 (2) If f(x) is supersummable, f(x) is summable by Lemma 2.4.2. 
 
 Notice that the B used in the proof of Lemma 2.4.2 are non-extremal 
 
 assignments. Therefore a non-extremal assignment must be summable. 
 
 Q.E.D. 
 
 The following theorem presents the major result for pseudo- 
 separation by zero-free structures. 
 
 Theorem 2.4.4 : Let f(x) be a positive function. Then f(x) is pseudo- 
 separable under a zero-free structure if and only if f(x) is not 
 supersummable. 
 
 Proof : (l) By Theorem 2.4.2, the condition that f(x) not be super- 
 summable is necessary for f (x) to be pseudo-separable under a zero-free 
 structure. 
 
 (2) We now want to show that this condition is sufficient. 
 Because all segregated assignments must be asummable and all summable 
 
33 
 
 assignments must lie on the hyperplane by Lemma 2.4.1, we can consider 
 the following set of linear inequalities: 
 
 (a) W'A. > T + 1 if A. is asummable and true 
 
 i — l 
 
 (b) W-U. = T if U. is summable (either true or false) 
 
 J J 
 
 (c) W*B, < T - 1 if 1 is asummable and false 
 
 k — k 
 
 — > 
 
 Clearly, if we can find W and T which will satisfy the above inequalities, 
 
 Vre will then have a hyperplane which pseudo-separates f (x) . We can 
 combine inequalities of the form (a) and (b), (a) and (c), and (b) and (c) 
 to generate another set of linear inequalities. 
 
 (a) • W-(A. - U.) > 1 
 l j - 
 
 (b) ' W-(A. - B, ) > 2 
 
 i k — 
 
 (c)» W. (U. - B ) > 1 
 J k 
 
 A necessary and sufficient condition for this set of inequalities to be 
 
 inconsistent and therefore not to have any solution [Wj T] is that 
 
 there exists a. . > 0, b., > and c ., > such that .E. a. . + Z 2b.. 
 ij — lk — ik — . . ij . . lk 
 
 + E c., - > and 
 
 E a. . W- (A. - IJ.) + .E. b., W- (A. - B, ) •+ E c, W. (U. - % ) < 
 
 i,j 1J X J i,k lk X k j,k ^ J k " 
 
 (Fan, [2]). Notice that in order for the above system to be inconsistent 
 and all w. ^ 0, this last inequality must be either identically zero or 
 identically less than zero. Therefore, if a solution [W; T] does not 
 exist, 
 
3^ 
 
 .£. a.. (A. -U.)+ E b..(A. -B.) + E c to -\)<0 or 
 i,J i,k j,k 
 
 E a. . A. + E b., A. + E e U. < E a. . if. +■ E b.. S. + E c ., fi . 
 
 i,u 1J x i,k lk x j,k -> k = ~ i,j 1J J i,k lk k ; j, k J k k - 
 
 Therefore 
 
 E a. + E u < E u. + E b. (2.U.1) 
 i=i x j=i 3 j=i J i=i i 
 
 where 
 
 p = E a. .+ E b ,q= E c „ and r = E a . . . 
 i„j 1J i,k ik J,k o k i,d « 
 
 We now want to show that if condition (2.U.1) is fulfilled, 
 
 then fix) is supersummable. Let us divide U. into true assignments x. 
 
 i i 
 
 3, s <lz s _ 
 and false assignments Y.. Then we can write E U. = E x.+ E Y.. 
 
 J j=l J j=l J J=l J 
 
 r t , r-t . . 
 
 E-> y 1 ~*1 v ~*1 ~*1 
 
 U. = Z, x. f L Y. where x. are true and 
 
 3=1 J 3=1 J 3=1 J J 
 
 n s q.- s a 
 
 ~*1 V ""* V ~* V ~* 
 
 Y. are false* Substituting L x. + L Y. for L U. and 
 J 3=1 J 3=1 J 3=1 J 
 
 t -+1 r_t -tl r -» 
 
 E x. + E Y. for E U., we can rewrite (2.4.1) as: 
 
 3=1 J 3=1 J 3=1 J 
 
 E _ s _ q- s _ t _>i r -"t _*i p+q-r 
 
 E A. + E x. + E Y. < E x. + E YT + E B. (2.4.2) 
 
 i=l x 3=1 J 3=1 J 3=1 J j=l 3 i=l x 
 
 We want all true assignments on the left side of this expression and 
 
 — > 
 false assignments on the right. Each Y. is summable, therefore 
 
 g g 
 
 Y + L V = L Z where Z are true assignments and V are false, 
 jmm m m 
 
 m=2 m=l 
 
 g g . 
 
 Therefore, we can replace each Y. in (2.4.2) by E 7? - E V and add 
 
 <] , m m 
 
 u m=l m=2 
 
35 
 
 L V to both sides of this expression. Then we will have effectively 
 m=2 
 eliminated one false assignment from the left side of (2.1+.2). If we 
 
 repeat this process for each Y. and perform the corresponding substitution 
 
 J 
 -*1 
 and addition for each x., we will arrive at a summation of the following 
 
 J 
 
 form: 
 
 p q+d ^ P" 1 " 3, ■*■*■_> r4 "d 
 
 Z A, + Z T, < L B. + Z F. (2.U.3) 
 
 i=l x j=l J i=l x j=l J 
 
 -» -> -». -*• 
 
 where all A. and T. are true assignments and B. and F. are false assign- 
 1 J 1 j 
 
 ments. If (2.U„3) is an equality, we have a contradiction, since we 
 
 — »■ -> 
 have assumed that all of the A. and B. are asummable, Therefore, 
 
 1 1 ' 
 
 p q+d ^ p+q-r r+d 
 
 Z A, 4- Z T. < Z B. + Z F. and f(x) is supersummable,, 
 i=l X j=l J i=l X j=l J 
 
 Therefore, if f(x) is not supersummable, then f(x) is pseudo- 
 separable under a zero-free structure, 
 
 Q.E.D, 
 
 Corollary 2.^.2: If f (x) is positive and pseudo-separated by a zero- 
 free structure, then all asummable assignments can be segregated by some 
 hyperplane [W; T], 
 
 Proof : If f (x) is pseudo-separated by a zero-free structure, f (x) can 
 
 not be supersummable by Theorem 2,^.2. If f(x) is not supersummable, 
 
 the proof of Theorem 2„k„k guarantees the existence of a structure 
 
 which pseudo-separates f(x) and segregates all asummable assignments- 
 
 Q.E.D. 
 
36 
 
 If we combine the results of Theorems 2.^.3 and 2.k.k, we 
 have the following important corollary: 
 
 Corollary 2.U.3 : Let f (x) be a positive function, f (x) can be pseudo- 
 separated by a zero-free structure [Wj T] if and only if all of the 
 non-extremal assignments are asummable. 
 
 The previous results require zero-free structures. These are 
 more restrictive than non-trivial structures. The following example 
 exhibits a function which is supersummable, but can be pseudo-separated 
 by a non-trivial (but not zero-free) hyperplane. 
 
 Example 2.U.3: Consider the function f(x) = x n x^ v x x x v x_x x^. 
 — 12 135 2po 
 
 Since assignments (1,0,1,0,1,1) and (0,1,1,1,0,1) are false, and 
 (1,0,1,1,0,0) and (0,1,0,0,1,1) are true and (1,0,1,0,1,1) + (0, 1,1,1,0,1): 
 (1,1,2,1,1,2) > (1,1,1,1,1,1) = (1,0,1,1,0,0) 4- (0,1,0,0,1,1), this 
 
 function is supersummable. Consider the structure [1,1,0,0,0,0) l]„ 
 
 — > 
 The only assignments segregated by this structure have x = x . Let A 
 
 be any segregated assignment. If a = a - 1, then the assignment must 
 be true and (1,1,0,0,0,0) <A = 2 > 1. If a = a = 0, then A must be 
 false and (1,1,0,0,0,0) °A = < 1. Therefore, [1,1,0,0,0,0; l] pseudo- 
 separates f (x) . 
 
 Necessary and sufficient conditions for pseudo-separation of a 
 positive function by anon-trivial hyperplane appear to be much more 
 difficult than the zero-free case. However, much information can be 
 obtained by reducing this problem to the zero-free case by partitioning 
 
37 
 the set of variables x into two subsets, x , variables whose corresponding 
 weights are zero and x , variables whose weights are non-zero. 
 
 Notation 2.4.1 : Assume that a positive function f(x) is pseudo- 
 
 separated by [Wj T]. Assume that w = ... = w =0 and all w. > 
 
 for i > e + 1. Let x q - (x^ Xg, . .., x g ) and x^ = (* e+1 , x g+2 , . .., x^) 
 
 Then we have the following expansion because of the positiveness of the 
 
 function. 
 
 1) 
 
 
 ±u. 
 
 ) = 
 
 XL...X fi 
 
 1 e 
 
 ^', x x 
 
 — > 
 
 1, ... 
 
 
 
 V 
 
 x r ., 
 
 ,x 
 e 
 
 _ x f (x, x 1 - 
 
 ♦ 1, .. 
 
 , . , 
 
 Vl 
 
 - 1, 
 
 X 
 
 e 
 
 
 V . 
 
 '• • 
 
 v f(x"j X 1 
 
 - 0, , 
 
 
 , x^O). 
 
 
 Let 
 
 f 1 (x 1 ) = f(x, X;L - l,.. 4 ,x e - 1), 
 
 f 2 (x x ) = f (xj x x - 1, . . . , x e _ x -* 1, x g -» 0) , 
 
 , f, (x\ ) = f (x: x n -> 0, . . . , x -* 0) where k = 2 
 k 1 ' 1 e 
 
 e 
 
 Lemma 2.4.4 ; If [Wj T] pseudo-separates f(x) where w = ... = w = 
 
 — > ^ — > 
 
 and f (A ) / f (A ) for some s and t where A is an assignment of 
 
 (n - e) variables, instead of all variables, A., = (a ,_, . ... a ), 
 
 \ / > '1 e+1 n 
 
 then A must be extremal for every f.(x n ). 
 1 J x 1 
 
 Proof: Let A = (A , A n ) where A = (&., . . . , a ) is any assignment on 
 ■ — o 1 o 1 e 
 
 n 
 
 e variables. Then W-A = Z w.a„ is well defined. Since w. = . . . = w = 0, 
 
 .,11 1 e 
 
 i=l 
 
38 
 the first e coordinates do not matter. If f (A ) f f (A ) , then 
 
 S JL Xj 1 
 
 W*A = T because lWj TJ pseudo-separates f (x) . If A is not extremal 
 for f . (x ), then there exists B, such that f . (B ) = f . ( A ) and B > A 
 
 u J J ^ 1 ± 
 
 or B, < A n depending on whether A., is false or true for f ( x ) . If 
 11 1 J 1 
 
 f(A ) = 0, then f(B ) - and B > A so that W-B > W-A = T, a 
 
 contradiction. If f(A ) = 1, then f (B ) = 1 and A > B so that 
 
 W-B < W*A = T, a contradiction. 
 
 Q.E.D. 
 
 Many of the properties which were discussed earlier in this 
 section are applicable to the set of functions, f , f , . .., f „ 
 
 1 c- K 
 
 These will be merely stated without proof. After each the corresponding 
 theorem or lemma for single functions is given inside parenthesis. 
 
 Lemma 2.k.^ : If A is segregated by [Wj T] where w = . . . = w =0, then 
 A must be asummable in all f . (x ) and if A is summable in any f „(x ), 
 A must lie on the hyperplane [Wj T]„ (Lemma 2.4.1) 
 
 Th 
 
 eorem 2.k.5 - No f „ (x ) can be supersummable. (Theorem 2.4.2) 
 
 The orem 2.4.6 : Every non-extremal assignment of each , f . (x..) must, 
 be segregated by [Wj T] where w = ... = w =0. (Theorem 2„3.2) 
 
 Theorem 2.4.7: Let f(x) be a positive function„ Let (x. , x , .... x } 
 12 e 
 
 be the set of variables in which f (x) is supersummable. If 
 
 (— > . — > 
 
 A ; x , -*• 0, . . . , x -» 0) = 1 for some assignment A and if for each 
 o' e+1 ' n ' & o 
 
39 
 
 x.(e+l < i < n) there exists some assignment B . such that 
 J - - oj 
 
 f( V X e + 1^°^ ■•" Y 1 ^' T 1 ' X 3 + l- > °' ~" X n~* G) = °> 
 then f (x) is not pseudo-separable . 
 
 Proof : If [W; T] pseudo-separates f(x), w, = . .. . = w = by Lemma 2.4.3 
 
 .-->. -> 
 
 because f(x) is supersummable in all of these variables. Let A be the 
 
 -» 
 n-tuple whose first e components are A and whose last n-e are all 0. 
 
 Then f(A) = 1, but W-A = 0. For each x., (e +■ 1 < j < n), there exists 
 
 J 
 
 an assignment B . on fx_ , x^, .... x ] such that 
 & oj l 1' 2' ' e 
 
 f(B .; x , -> 0, .... x. , -* 0, x. -> 1, x. , -> 0, .... x -* 0) = 0. 
 v oj' e+1 ' ' jil ' d ' j+1 ' n 
 
 — » — » 
 
 Let B. be the n-tuple whose first e coordinates are B . and whose last 
 3 oj 
 
 th 
 n-e are 0, ..., 0, 1, 0, ..., with 1 in the j position. Then 
 
 W-A > W*B. "because f(A) = 1 and f(B.) = 0. Because W-A = and W-B. = w., 
 3 J 3 3 
 
 and since all w. > because f(x) is positive, w. = 0, Therefore all 
 J J 
 
 w. = and only the trivial hyperplane pseudo-separates f(x)„ 
 3 Q.E.D. 
 
 Since f (x) and f (x) are pseudo-separable together by Theorem 
 2.2.3, Theorem 2.4.7 can be extended to the next corollary. 
 
 Corollary 2.4.2 : Let f (x) be positive and supersummable in {x ,x , . . . ,x } 
 
 If f(A j x , -+ 1. .... x -> 1) = for some A and if for each 
 o e+1 n o 
 
 x.(e+l < j < n) there exists some assignment B . such that 
 J - - oj 
 
f( V Vi"* 1 ' •••/ Vi" 1 ' x j -* > Vi"* 1 ' ■■•' \^ z) =1 > 
 
 then f(x) is not pseurlo- separable o 
 
 Example 2.4.3 : Consider the function f (x) = x.x v x x, x . We showed 
 in Example 2. 4.1 that this function is super summable in x , x, and x . 
 Since (0,0,1,1,1) is a true assignment and (1,0,0,0,0) and (0,1,0,0,0) 
 are false assignments, f(x) is not pseudo-separable. 
 
 Henceforth, results under the condition that [Wj T] must be 
 zero-free will be presentedo 
 
 Theorem 2.4.8 : Let f(x) be an arbitrary positive function. Assume that 
 f (x) is completely monotonic and pseudo-separable under a zero-free 
 structure [W; T]. If there are only 2 true assignments A and B and two 
 false assignments C and D which lie on [Wj T], then f(x) is separable 
 
 Proof : If f(x) is asummable, then it is separable by Theorem 2„4„1. 
 Assume that f(x) is not separable. Since A and B are the only true 
 
 assignments which lie on [W; Tj, they are the only summable true assign- 
 
 . -> -» 
 
 ments by Lemma 2.4.1,, Similarly C and D are the only false summable 
 
 .-» ->-*->-» -> -» 
 
 assignments for f(x). Therefore, C + D = A + B. Assume that A and C 
 
 agree in positions x n , x n , . ..< x and disagree in x ., x , _.,..., x . 
 
 v 1' 2 q q+1 q+2' ' n 
 
 — » — » — > — > — > -> 
 If A t B = C + D, B and D must also disagree on x , , x _,„.., x and 
 ' q+1 q+2' ' n 
 
 agree on x., , x^, . ... x . Futher a. = d. for j = q+1, q+2, . ... n and 
 
 12 q 3 3 
 
 b. = c. on this same set of components because a. + b. = c. + d. and 
 3 3 3 3 3 3 
 
kl 
 
 Then we have that A = C and B = D . 
 
 q q q q 
 
 f(A : x , -» a ,, x _ -» a _,..., x -> a ) = 1 = f(A), 
 
 v q' q-t-1 q+1' q+2 q+2' ' n n J K ' ' 
 
 and f(A jx - -» c , , x ~ -» c _,..., x -> c ) = = 
 
 v q' q+1 q+1' q+2 q+2' ' n n y 
 
 f(C ; x , -* c _ , x -*c _,..., x -> c ) = f(C). 
 v q' q+1 q+1' q+2 q+2' ' n n y v ' 
 
 -» £ -» 
 This means that we can not have A < C. Also, 
 
 f (D S x l -* a , # x ~ ^ a ^, . „ . , x -» a ) = 
 
 v q' q+1 q+1' q+2 q+2' ' n n y 
 
 f(D -, x - -> d . , x -»d _,..., x -> d ) = f(D) = 0, 
 s q' q+1 q+1' qt-2 q+2' ' n n J v ' ' 
 
 f (D ix v^ c -, f x ^ "** c ^> • • ° > x -»■ c ) = 
 
 v q* q+1 q+1' q+2 q+2' ' n rr 
 
 and 
 
 f (B j x n ^ b ., x -» b _, . . . , x - b ) = f (B) = 1. 
 
 v q' q+1 q+1' q+2 q+2' ' n n y v ' 
 
 This means that we can not have A > C. Therefore A and C are not f (x) - 
 
 comparable and hence f(x) can not be completely monotonic. 
 
 Q.E.D. 
 
 Lemma 2.4.6 : If f (x) is self -dual and A is any true (or false) assign- 
 -* 
 
 ment, then A is a false (or true) assignment. 
 
 Proof : f d (x) = f(x). Therefore, if f(A) = 1, then f(A) = 1 or f(A) = 0. 
 
 Q.E.D. 
 
 Lemma 2.U.7 : If f(x) is self -dual, then any assignment A is summable if 
 
 — * 
 
 and only if A is summableo 
 
h2 
 
 — » 
 Proof : If A is true and summable, then there are true assignments 
 
 — > -> — * —►->-> 
 
 A , A , . .., A and false assignments B , B , . .., B such that 
 
 k _> k - 
 
 A + Z A. = Z B. . Because A = (l, 1, 1, . .., l) -A, 
 i=2 x i=l x 
 
 -»k^ k _, k -* k -* -* 
 
 (A + Z A ) = (k, k, . . . , k) - Zl. -X Therefore, Z B. = Z A. + A. 
 
 i=2 i=2 i i=l x i=2 x 
 
 Since f(x) is self -dual all A, A , A , . .., A are all false assignments, 
 ->-*-» -^ -* 
 
 while B. , B_, .... B are all true. Therefore, A is summable. 
 1 2 v 
 
 k Q.E.D. 
 
 Theorem 2.k.Q : If f (x) is self -dual and pseudo-separable under some 
 
 — > 
 zero-free structure, then any assignment A is segregatable if and only if 
 
 —* 
 
 A is segregatable. 
 
 Proof : We know that if f (x) is pseudo-separated by some zero-free 
 structure, then all of its asummable assignments can be segregated by 
 
 some hyperplane [W; T] by Theorem 2.^.4. A is summable if and only if 
 
 — — > — 
 
 A is summable by Lemma 2.4.7« Hence A and A must be segregated together 
 
 by this structure. 
 
 Q.E.D. 
 
 2.5 Simultaneous Realizability and Pseudo-Separability . 
 
 In this section the concept of simultaneous realizability is 
 
 examined in relationship to pseudo-separability. The major results of 
 
 this section are that the upper and lower bounds (see Definition 2.5*2) 
 
 for any refinement (see Definition 2.5.3) of a pseudo-separation are 
 
 simultaneously realizable. This idea is also applied to the question of 
 
 pseudo-separability for completely monotonic functions and to other 
 algebraic properties. 
 
U3 
 Definition 2.5.1 : Let g (x) and g p (x) be two separable Boolean functions. 
 
 We will say that g. (x) and g p (x) are simultaneously realizable , if there 
 
 exist some set of weights W and two thresholds T and T_, such that 
 
 g 1 (x) is separated by [W; T ± ] and g (x) by [W; T p ]. [7] 
 
 Definition 2.5.2 : Let f (x) be a pseudo-separable Boolean function on 
 exactly n variables. Let [W; T] be any structure which pseudo-separates 
 f(x). Let M be the minimum T = W°A. for all assignments that are 
 
 segregated by [W; T] (i.e. W.A. ^ T). Let < e < M. Let g(x") be the 
 
 — » — > 
 separable function on exactly n variables such that if W°A > T - e, then 
 
 g(A) = 1 and if W°A < T - e then g(A) = 0. Let h(x) be the separable 
 
 function on exactly m variables such that if W°A > T -f e, then h(A) =1 
 
 and if W-A < T + e, then h(A) = 0. g(x) is called an upper bound for 
 
 [¥j T] and h(x) is called a lower bound for [Wj T], If [Wj T] is the 
 
 structure which segregates all segregatable assignments of f(x), then 
 
 we will call g(x) the least upper bound and h(x), the greatest lower 
 
 bound for f (x) . 
 
 Notice that in the above definition, € is defined small enough 
 
 so that for any assignment A of f(x), h(A) = and g(A) = 1 if and only 
 
 — » — > 
 if W°A = T. To facilitate the definition of € for each distinct hyper- 
 
 plane [Wj T], we will refer to the function M(-WjT). This is meant to 
 
 represent the minimum T - W«A. for all A. segregated by LW$ TJ. 
 
 Example 2.5.1 : Consider the function x x s/ x o x ), ^ e saw ^" n ■$^ a . m .P.^. e . 
 1/2..1, that this function was pseudo-separated by [1,1,1,1$ 2]. The 
 upper bound for this structure is x x v x x v x x, v x x, , which has 
 
hk 
 
 structure [1,1,1,1; 1.5]. The lower "bound for this structure is separated 
 by the hyperplane [1,1,1,1; 2.5] and hence must be 
 
 xxx v xxx, sy xxx, v xxx,. The true assignments which lie on 
 [1,1,1,1; 2] are (l, 1,0,0) and (0,0, l,l). The false assignments which 
 lie on [1,1,1,1, 2] are (l, 0,1,0), (0,1, 0,1), (l,0,0,l) and (0,1, 1,0). 
 Since (1,1,0,0) + (0,0,1,1) = (l,l,l,l) = (1,0,1,0) + (0,1,0, l) = 
 (l, 0,0,1) + (0,1, 1,0). All six of these assignments are summable. 
 Therefore [l, 1,1,1; 2] segregates all asummable assignments (and in that 
 sense is the optimal pseudo-separating structure for x x ^ x x, ). 
 Hence x x v x x s/ x x, v x x \/ x x. v x x, is the least upper bound 
 
 1 X 2 X 3 V X 1 X 2 X 4 v X 1 X 3% " X 2 X 3^ 
 
 and x^x^x^ s/ x-.x^x, ^ x x„x, v x.x x. is the greatest lower bound 
 
 Definition 2.5-3 ' let g(x) and h(x) be two separable functions on exactly 
 n variables. Assume g(x) > h(x) . g(x) and h(x) are called linearly 
 distinguishable if there exists some hyperplane [W; TJ such that g(x) is 
 the upper bound of [W; T] and h(x) is its lower bound. In such a case 
 [W> T] is called the distinguishing hyperplane of g(x) and h(x) . 
 
 Theorem 2.5.1: If f(x) is pseudo-separated by [W; T], then f(x) can be 
 written in the form f (x) = h(x) v f (x)g(x) where h(x) is the lower bound 
 of [W; T] and g(x) is the upper bound. Conversely, if two separable 
 functions g(x) and'h(x) are linearly distinguishable, then for any 
 function f (x), h(x) v/ f (x)g(x) will be pseudo-separated by the 
 distinguishing hyperplane for g(x) and n (x), [W; T]. 
 
U5 
 Proof : Let f(x) be pseudo-separable under [Wj T]„ Let h(x) "be the lower 
 bound of [W; T~J and g(x) be the upper bound. Define f (x) as follows: 
 for any assignment A, if h(A) / g(A), then f (A) = f(A) and if 
 h(A) = g(A), then f (A) - 1. Consider the function h(x) v f (x)g(x). 
 For any assignment A, if f (A) = 1, then W°A > T. If W.A > T, then 
 W°A > T + € because < € < M(W, T) and therefore h(A) = 1. Hence 
 h'(A) s/ f 1 (A)g(A) - 1. Tf W°A = T, then because W«A > T - €, g(A) - 1. 
 Since W-A < T - e, h(A) = Therefore, h( A) f g(A) and f X (A) = 1 = f(A). 
 Hence h(A) v f 1 (A)g(A) = 1. Similarly, if f(A) = 0, then h(A) v/ f 1 (A)g(X)=0 
 Therefore f(x) = h(x) v f (x)g(x), 
 
 Now assume that h(x) and g(x) are linearly distinguishable 
 and let f (x) be any Boolean function. We Claim that h(x) v^ f (x)g(x) is 
 pseudo-separated by LW$ T], the distinguishing hyperplane for h(x) and 
 g(x). Let f(x) = h(x) v» ^(xJgCx). 
 
 If A is any true assignment to f(x), then either h(A) = 1 or 
 f 1 (A) = g(A) = 1. If h(A) = 1, because h(x) is the lower bound of 
 [W; T], W.A > T f€ and A will be segregated by [Wj T] ( If f 1 ® = 
 g(A) = 1, then W-A > T - € because g(x) is the upper bound of [Wj T°J and 
 W°A = T because < € < M(w, T) . In any case, W=A > T. A similar proof 
 is used to show that if f(B) = Q, then W-B'< T. Therefore, [W$ TJ pseudo- 
 separates h(x) v f (x)g(x) for any Boolean function f (x) if [W; T] is 
 
 the distinguishing hyperplane for h(x) and g(x)„ 
 
 Q.E.D. 
 
 Corollary 2, 5« 1 : If two separable functions, h^x) and g(x) are linearly 
 distinguishable, then for any f (x), h(x) v f (x)g(x) is pseudo-separated 
 
U6 
 by the distinguishing hyperplane [Wj T] of h(x) and g(x), and for any 
 assignment A, A is segregated "by LW) TJ if and only if h(,A) and g(A) 
 are equal. 
 
 Proof : If h(x) and g(x) are linearly distinguishable, then for any 
 
 f (x), [W; T] pseudo-separates f(x) = h(x) ^ f (x)g(x) by Theorem 2.5.1. 
 
 Assume that A is segregated by [W; T]. If f(A) = 1, then W-A > T and 
 
 because < e < M(W, I), W°A >T+e>T-e. Therefore, h(A) = g(A) = 1. 
 
 If f (1) = 0, then W«A < T and again because < e < M(W, T) 
 
 W-A < T - e < T + €; arid so g(A) = h(A) = 0. Hence, if A is segregated 
 
 by [W; T], then g(A) = h(A) . Now assume g(A) = h(A). If g(A) = h(A) = 0, 
 
 then W'A < T - € because g(A) = 0. Therefore, A is segregated by [Wj T]. 
 
 Similarly, if g(A) = h(A) = 1, then W-A > T + € and so A is again 
 
 segregated by [W; T]. 
 
 Q.E.D. 
 
 Definition 2^.k : Let f(x) be pseudo-separable under [W, j T ]. Assume 
 that h (x) and g (x) are the lower and upper bounds for this hyperplane. 
 If there exists another hyperplane [W ; T ] which pseudo-separates f(x) 
 with lower bound h p (x) and upper bound g p (x) such that h (x) > h (x) 
 and g p (x) < g,(x), [VI ) T p ] will be called a refinement of [W. \ T ]. 
 
 Example 2.5.2 : Consider the function x x v x x^ ^ x i x h v x o x v ^ e 
 saw in Example 2.2.1 that this function is pseudo-separated by 
 [1,1,1,1; 2]. In Example 2.3-1, we showed that [3,2,2,1; k] separates 
 this function, and therefore also pseudo-separates it. The upper bound 
 
hi 
 for [1,1.1,1; 2], we saw in Example 2.5.1, is x x v x x <• x x, v x x v 
 v xx, v xx, = g (x). The upper bound (and least," . upper bound for 
 f(x)) for [3,2,2,1; VJ is x x x 2 v x^ s/ x^ v x x = g^x) < g^x) . 
 Further, the lower bound for [1,1,1,1; 2] is X..X-X v x x x, -v x x x, 
 v xxx, = h (x) and the lower bound of [3,2,2,1$ k] is 
 h 2 (x) = x^ v x.x v x g x x^„ Because g^x) > gg(x) and h^x) < h^x) , 
 [3,2,2,1$ U] is a refinement of [1,1,1,1; 2] for x x v x x v x x, v x x 
 
 Lemma 2.5»1 « Let f (x) be an arbitrary Boolean function on exactly 
 
 n variables. Then f(x) is separable under [W; T] if and only if either 
 
 the lower bound h(x) = f(x) or the upper bound g(x) - f(x)„ 
 
 Proof : If f(x) is separable under [W$ T], then we have one of the 
 following 3 pairs of conditions: 
 
 (1) if f (A) = 1, then W.A > T 
 
 and if f (A) = 0, then W«A < T. 
 
 (2) if f (A) = 1, then W-A > T 
 
 and if f (A) = 0, then W-A < T. 
 
 (3) if f (A) = 1, then H > T 
 
 and if f (A) = 0, then W-A < T. 
 
 In case (l), there is no assignment which lies on [W; T], 
 Therefore h(x) = g(x), and [Wj Tj must also separate h(x) and g(x). 
 Since g(x), h(x) and f(x) are on exactly n-variables fi(x) = g(x) = f(x)„ 
 In case (2), we claim that f(x) = g(x) . Clearly if f(A) = 1, then 
 
U8 
 
 W-A > T and g(A) = 1 because g(x) is separated by [Wj T - e]. If 
 
 f (A) = 0, then W-A < T - e because < € < M(W, T) and therefore 
 
 g(A) = 0. Therefore f(x) = g(x) . In case (3), if :•.:" f(A) = 0, then 
 
 W-A < T and h(A) = 0, while if f(A) = 1, W-'A>T':+ e and h(A) = 1. Therefore 
 
 f(x) =h(x).;If f(x) = h(x) or g(x), then [Wj T + e] or [Wj T - e] must 
 
 separate f(x), and because < £ < M(W, T), [Wj T] also separates f(x). 
 
 Q.E.D. 
 
 In Example 2.3-1, we showed that x x v x x ^ x x v x x, was 
 separable under [3,2,2,1; k] . We also noted that true assignments 
 (l, 0,0,1) and (0, 1,1,0) resided on this hyperplane. Therefore this is a 
 separation of f (x) in the sense of case (2) and g(x) must be equal to 
 
 x l x 2 v X 1 X 3 v x l x i+ v X 2 X 3' 
 
 We would now like to show that the upper and lower bounds of 
 
 any refinement are both simultaneously realizable with the upper and 
 
 lower bounds of the original hyperplane. 
 
 Theorem 2.5.2 : If [W ; T ] is a refinement of [W j T ] for f(x), h (x) , 
 h Q (x), g. (x) and g p (x) are all simultaneously realizable. 
 
 Proof : We will demonstrate that we can always determine p, q, T , T , T 
 and T. such that [pW +■ qW ; T ] separates h (x) , [pW + qW„; T ] 
 separates h (x) , [pW + qW ; T_] separates g (x) and [pW + qW 2 ; T^] 
 separates g (x) . This means that we must find p, q, T , T , T , and 
 T. such that: 
 
U9 
 
 (1) if h (x) = 1, then (pW + qW 2 ) -x > T^ 
 if h (x) = 0, then (pW + qW )-x < T?" 
 
 (2) if h 2 (x) =1, then (p^ + qW 2 )-x > 1^ 
 if h 2 (x) = 0, then (p^ +qW 2 )-x < T^ 
 
 (3) if gg(x) - 1, then (p^ + qW 2 )-x > T^ 
 if g 2 (x) = 0, then (pW^ +■ qW 2 )»x < T^ 
 
 (h) if §1 (x) = 1, then (pW x + qW 2 )«x > tJ 
 
 if g x (x) - 0, then (pW x f qW 2 )-x < tJ 
 
 We will assume that h (x) < h (x) < g_(x) < g (x) . This 
 implies that T. < T < T < T and the above problem can be reduced to 
 finding p, q, T , T , T and T, such that: 
 
 (a) if h (A) = 1, then (pW + qW* 2 ) • A > T?" 
 
 (b) if h (A) = and h (A) = 1, then T^~ > (pW + qW ) "A > T 2 
 
 (c) if h (A) = and g g (2) = 1, then T 2 > (pW^ + qWg) ' A > T^ 
 
 (d) if g 2 (A) = and g;L (A) = 1, then T^ > (p^ f qW 2 ) -A > tJ 
 
 (e) if g x (A) = 0, then (pW^ f qW £ ) »A < tJ. 
 
 We will use the following notation. Let Q.(s,t,u,v) denote the 
 
 j 
 
 maximum of W.-A where h (A) = s, h (A) = t, g p (,A) = u g (A) = v and 
 r.(s,t,u,v) the minimum of W.-A where h (A) = s, h (A) = t, g p (A) = u, 
 g (A) = v. Using this notation, the five conditions above become: 
 
 (1) p0 1 (0,0,0,0) + q0 2 (0, 0,0,0) < tJ 
 
 (2) P 1 (0,0,0,l) + q0 2 (0,0,0,l) < T^ 
 
 (3) p0 1 (0,0,l,l)+ q0 2 (0,0,l,l) < T 2 
 
50 
 
 (k) p0 1 (0,l,l,l) + q©-g(0,l,l,l) < T^ 
 
 (5) pT^O, 0,0,1) + qp 2 (0,0,0,l) > tJ 
 
 (6) pI^O, 0,1,1) + q? 2 (0, 0,1,1) > T^ 
 
 (7) pr i (0,l,l,l) + qF 2 (0, 1,1,1) > T 2 
 
 (8) p^d,!,!,!) + 0F 2 (1,1,1,1) > T^ 
 
 1111 
 Now eliminate T , T , T and T . 
 
 (i)* P [r x (o, 0,0,1) - q 1 (o,o,o,o)] + q[r 2 (o,o,o,i) - o 2 (o,o,o,o)] >0 
 (2)' p[r l (o, 0,1,1) - o 1 (o, o,o,i)] + q [r 2 (o, 0,1,1) - o 2 (o, 0,0,1)] >0 
 (3)' P [r x (o, 1,1,1) - q i (o, 0,1,1)] + q[r 2 (o,i,i,i) - e 2 (o,o,i,i)] > o 
 
 (1+)' pC^d, 1,1,1) - Q 1 (0, 1,1,1)] + q[r 2 (l, 1,1,1) - Q 2 (0, 1,1,1)] > 0. 
 
 From the knowledge of h (x), h (x), g„(x) and g (x) we now 
 
 calculate the Q.'s and r.'s. If g. (A) =1 and h (a) = 0, then W -A = T. 
 J 3 1 I'll 
 
 because T - € < W -A < T + e , and < e < M(W, T'). Similarly, if 
 g (A) = 1 and h (A) = 0, then W 'A = T . Hence f 
 
 1^(0,0,0,1) = 0^0,0,0,1) = ^(0,0,1,1) = Q 1 (0,0,1,1) = 1^(0,1,1,1) = 
 1 (0, 1,1,1) = T x and 1^(0,0,1,1) = 9^0,0,1,1) = Tg- 
 
 Because g (x) is separated "by [W, j T - £ ] where 
 < e < M(W , T ) the maximum W -A such that g,{A) = is less than 
 T - €. . Similarly, the maximum W -A such that g p (Aj = is less than 
 T - € where < € < M(W , T ). Because h (x) is separated by 
 [W x ; T 1 + 6 1 ] and h 2 (x) by [W^ T g + 6 g ] if h^A) = 1, then 
 W -A > T + € and if h (A) = 1, W -A > T + £ . Therefore: 
 

 1 (0, 0/0,0) < T x - € 1 and 1^(1,1,1,1) > ^ +- g 
 
 9 2 (0, 0,0,1) < T 2 - e 2 and r g (0,l,l,l) > T g + e 2 
 
 9 2 (0, 0,0,0) < T 2 - e 2 and r 2 (l,l,l,l) > T g + e 2 
 
 Q (0,1,1,1) * T + e o sad rjD,.0,Q,l) < T - e o . 
 
 Then 
 
 (a) P [r 1 (o, 0,0,1) - 9^0,0,0,0)] > p^ -i't^ r i >},^ ;■ x 
 
 (b) P [r 1 (o, 0,1,1) - 9(0,0,0,1)] = p [t 1 - t 1 ] = 0. 
 
 (c) p[r i (0,l,l,l) ^ 0^0,0,1,1)3 = p[T 1 - T x ] = and 
 
 (d) P [r x (i, 1,1,1) - 9 L (o,],i,i)] > p [t x f e 1 - T x ] = pe r 
 
 Similarly, 
 
 (e) q[r 2 (0, 0,1,1) _ 9 2 (0, 0,0,1)] > $[T g - (T g - e g )!| = qc^ and 
 
 (f) q[l 2 (0, 1,1,1) - 9^0,0,1,1)] > q[T 2 + € ? - T g ] = qe g . 
 
 With this information, for any positive p and q conditions 
 (2)* and (3)' are automatically satisfied. Thus, only inequalities (l)' 
 and (h) ' need be considered. (l)' becomes 
 
 (i)" P € 1 + , q [r 2 (o, 0,0,1) - 9 2 (0, 0,0,0)] > 
 
 and ('4) ' becomes 
 
 (h) n p€ 1 + q[r 2 (l, 1,1,1) - 9 2 (0, 1,1,1)] > 0. 
 
 Since e > 0, both of these conditions can be satisfied by fixing q > 
 and by increasing p. Let q = 1. Then choose p large enough so that 
 
52 
 
 pe i > l^ 1 ' 1 ' 1 ' 1 ) - e 2 (o,i,i,i)l ^d pe 1 > |r 2 (o,o,o,i) - 9 2 (o,o,o,o)|. 
 
 Since such a positive integer can always be found, the weights pW + W 
 
 can he used to realize all four functions, h (x), h p (x), g p (x) and g (x) . 
 
 Q7E.D. 
 
 Example 2.5-3 : Consider the function given in Example 2.5.2, 
 
 f(x) = x..x_ ss x n x s/ x^,x„ v x.x, o To calculate a common set of weights, 
 '12132314 & ' 
 
 consider r (0,0,0, l), 9 (0, 0,0,0), r (l, 1,1,1) and 9 (0, 1,1,1). Since 
 [W.^ T 1 ] = [1,1,1,1; 2], g]L (x) = x x x 2 v x.^ v X] x^ n, x^ ^ y^a^x^ , 
 
 h 1 (x) = x^x v X 1 X 2 X 1 + v X 1 X 3 X 4 v X 2 X 3 X l4° Because [Wgj. T g ] = 
 
 [3,2,2,1; 4], g 2 (x) = x^ v x x x v x^ v x 2 x and h 2 (x) = 
 
 xx v xx \s x x x, o The only assignments A such that g (A) = 1 and 
 
 g (A) = are (0,1,0,1) and (0, 0,1,1). Therefore r (0,0,0,1) = 
 
 (3,2,2,1). (0,1,0,1) = (3,2,2,1)^(0,0,1,1) = 3. If g 1 (A) = 0, then A 
 
 has at most one component of value 1 and hence 9 (0,0,0,0) = 3 where 
 
 the maximum occurs for A = (l, 0,0,0). Similarly r (l, 1,1,1) = 5 
 
 = 9 (0,1,1,1) . Therefore, for any positive p, pW + W will yield a 
 
 common set of weights for these four functions. Let p = 1, then 
 
 W x +■ W 2 = (4,3,3,2) and T* = 8, T 2 = 7, T^ = 6 and tJ = 5. 
 
 Theorem 2. 5-3 - If two positive functions, h(x) and g(x) are linearly 
 distinguishable, all assignments A such that g(A) = 1, h(A) = must 
 be true extremal assignments for g(x) and false extremal assignments 
 for h(x) . 
 
53 
 Proof: If h(x) and g(x) are linearly distinguishable, for any k(x), 
 f (x) = h(x) s/ k(x)g(x) is pseudo-separated by the distinguishing hyper- 
 plane [W; T] by Theorem 2.5.1. Assume that g(A) = 1 and h(A) = 0, but 
 
 ~* f~*\ ~*1 
 
 A is not extremal true for g(x). Then there is another assignment A 
 
 .— ►» -»l -* /->. 
 
 such that g(A) = 1 and A < A. Since k(,x) can be any function in : .. 
 
 Theorem 2.5-1, let k(x) be such that k(A) = k(A ) = 1. Then A is not 
 
 extremal true for f(x) and hence must be segregated by [W; T]. Therefore 
 
 h(A) = g(A) = 1 by Corollary 2. 5.1> a contradiction. Therefore A must 
 
 be extremal true for g(x). A similar proof shows that A must be extremal 
 
 false for h(x) . 
 
 Q.E.D. 
 
 — > 
 
 Definition 2. 5.5 - An extremal true assignment A is called extrinsic if 
 
 -* -» — > 
 
 there exists some other extremal true assignment B such that A and B 
 
 are f(x) -comparable and A > B. An extremal false assignment C is called 
 
 — » 
 extrinsic if there exists another extremal false assignment D such that 
 
 C and D are f(x) -comparable and C < D. An extremal assignment which is 
 not an extrinsic assignment is called an intrinsic assignment. [7] 
 
 We now wish to extend Theorem 2.5-3 to show that all assign- 
 ments on which the values of g(x) and h(x) differ are intrinsic assign- 
 ments. 
 
 Theorem 2.5.k : If h(x) and g(x) are linearly distinguishable and g(A) = 1 
 while h(A) = 0, then A is intrinsic in both h(x) and g(x). 
 
5** 
 
 Proof : A must "be extremal in "both h(x) and g(x) by Theorem 2.5-3. 
 
 If A is not intrinsic in g\x), then there is another assignment B 
 
 g 
 such that g(B) = 1 and A > Band W- A > W-B by Theorem 2.2.5. Since 
 
 g(A) fi h(A), W-A = W-B by Corollary 2.5.1. Assume that A and B agree 
 
 in positions x n , x^., . . . , x and disagree in x . . , x _, . . . , x with 
 1 2' ' p pfl' p+2' n 
 
 a = a = ... = a - l and a = a o = . . . = a = 0(p < q < n). 
 p+1 p+2 q q+1 q+2 n K * — * - ' 
 
 Because W«A = W-B, w , + w _+...+■ w = w , + w _+...+ w . 
 
 ' p+1 p+2 q q+1 q+2 n 
 
 — » — » 
 Therefore for any pair of assignments C and D which agree in :: 
 
 — * -> 
 
 x. , x_, .... x and which are such that C agrees with A on 
 1 2 p 
 
 — * — >■ — * — > — > — > 
 
 x ,, x „,..., x and D agrees with B on these variables, W«C = W°D. 
 p+1' p+2' ' n e ' 
 
 Hence it is impossible to have g(x: x ..->.£. n , x , • -* a .... x -* a ) 
 ^ &v ' p+1 p+1' p+2 p+2' ' n n y 
 
 > g(xt x ,, -» b -,,-x rt -* b _, ...,x -> b ) . Therefore A K B, a 
 &v ' p+1 p+1' p+2 p+2' ' n n - ' 
 
 contradiction. Similarly if B > A, then the contradiction A > B would 
 
 arise. Therefore A must be intrinsic in both h(x) and g(x). 
 
 Q.E.D. 
 
 Theorem 2.5.5 - h(x) and g(x) are linearly distinguishable if and only 
 if h(x) v k(x)g(x) is pseudo-separable for any k(x) and for some k(x) 
 exactly those assignments A such that h(A) = g(A) are segregated by 
 the distinguishing hyperplane [W; T]. 
 
 Proof : If h(x) and g(x) are linearly distinguishable, the result 
 
 follows from Theorem 2.5-1 and Corollary 2.5.1. If f(x) = h(x) ^ k(x)g(x) 
 
 is pseudo-separable under [W; T] and if all assignments A such that 
 
 h(A) = g(A) are segregated, then h(x) must have structure [Wj T + e] 
 
 and g(x), [W; T - e] where < € < M(W, T) by Corollary 2.5-1 and 
 
 Theorem 2.5.1. Therefore h(x) and g(x) must be linearly distinguishable. 
 
 ' ."Q/E.D. 
 
55 
 
 Corollary 2.5.2 : If h(x) and g(x) are linearly distinguishable then 
 no assignment A. such that h(A. ) = 1 is summable in a sum including 
 assignments B. such that g(B.) = 0. 
 
 Proof : We know that g(x) > h(x) . Therefore if h(Aj_) = 1, then g(A. ) = 1 
 and if g(B. ) = 0, then h(B. ) = 0. Let k(x) he any Boolean function. If 
 g(x) and h(x) are linearly distinguishable then f (x) = h(x) v k(x)g(x) 
 is pseudo-separated by [Wj T], the distinguishing hyperplane for h(x) and 
 g(x) by Theorem 2.5.1- Also, all assignments C such that g(C) = h(C) 
 
 are segregated by [Wj T] (by Corollary 2.5-1) • Therefore all assign- 
 
 — > _— » — » — > — > 
 
 ments A. must be segregated by |_W; TJ and W-A. > T. Therefore each A. 
 
 must be asummable by Lemma 2.U.I. 
 
 Q.E.D. 
 
 Lemma 
 
 2.5-2 : Let f (x), f (x), ..., f v (x) be simultaneously realizable 
 
 functions. Then there exists a set of weights W and thresholds T. 
 such that each f.(x) has structure [Wj T.] (i = l,2,...,k) and W.A. is 
 distinct for each assignment A.. [2] 
 
 — > 
 Proof : Assume that we already have a set of hyperplanes of the same W 
 
 but with different thresholds, T., under which each f . (x) is separable, 
 
 but W-A = W-B for some A and B. Let A = (a , a., ..., a. ) and 
 
 1 2 n 
 
 B = (b., , b^, .... b ). Since A 4- B there is some coordinate x., where 
 * 1' 2 n ' j 
 
 a . = 1 and b = or b . = 1 and a . = 0. Let M be the minimum difference 
 3 3 3 3 
 
 -» -> 1 
 
 between the distinct W-A. 's and the T. 's. Replace w. by w . = w. 4- e 
 
 i i 3 3 3 
 
56 
 where < € < M. Call the new structures [w~; T.], where w = w (p ^ j), 
 
 w. = w. + e and T. = T . . Then W-A will be different from W-B. Since 
 
 — > — » 
 
 < € < M, any W°A„ ' s which were already distinct are still different 
 
 under [W> T.]. Similarly the W°A 's which were previously less than 
 
 T.'s will remain less than T.'s. Cbntinuing this replacement in this way 
 
 -*L1 
 
 a set of weights W will be eventually constructed so that each 
 
 W 1 " -A. is distinct and each f . (x) is separable under [w s T. ] 
 
 (i'= ; l,2,...,k). 
 
 Q.E.D. 
 
 Assume that f(x) is pseudo-separable under [W .°, T ] with 
 upper bound g, (x) and lower bound h (x) „ There exists some set of weights, 
 W such that WT°x is distinct for all x and such that [WTj T ] 
 
 separates h (x) and [W j T. ] separates g (x) by Lemma 2„5o2 If 
 1 l g 1 
 
 W i T p ] is a refinement of [VL $ T.], then its bounds g p (x) and h p (x) 
 
 are simultaneously realizable with h_ (x) and g (x) . Hence by applying 
 
 -*11 ->11 -* 
 
 the same lemma, a weight vector W can be found for which all w *x 
 
 are distinct and under which h (x), h (x^gJx), and g_ (x) can be 
 
 represented respectively as [WTj T ], [W_; T ], [W p $ T ] and 
 
 c - "-i *- o "P 
 
 [WTj T ] where T > T, > T > T Any assignment which is segregated 
 *- S-> "■•] ftp So ^>i 
 
 "by CW ) T p ], but not by [W > T.. ] will then be in the range 
 
 1 -tL -* 1 1 -tL -> 1 
 T < W p *x < T or T > W p -x > T . Therefore any assignment in the 
 
 g l 2 g 2 1 2 
 
 ""*1 ~* 1 /~*\ 
 
 range T < W°x < T must be false for f (x) and any assignment in the 
 
 g l d g 2 
 
 1 -+1 -* 1 ,-+\ 
 
 range T, > W *x > T, must be true for f (x) „ 
 
 h ! 2 h 2 
 
57 
 Theorem 2.5.6 : If f (x) is pseudo-separated "by [Wj T] with upper 
 
 bound g(x) and lower bound h(x) and if a refinement of [W; T] exists, 
 
 -»1 ,--*l n 
 
 then there must exist a set of weights W such that lW ; T J separates 
 
 /~*\ •- _> 1 -i /~*\ ~* ~~*1 ~* 
 
 h(x), LW °, T J separates g(x) and the assignment A such that W »A is 
 g 
 
 -* — » — * -*] -± ->2. ~* 
 
 the maximal W°x < T. and B such that w -B is the minimal W -x > T 
 
 h g 
 
 must be true and false assignments for f(x) respectively. 
 
 — > — > —> 
 Therefore, if some set of false assignments B, , B , . . . } B 
 
 — > — » -» 
 
 and true assignments A , k , . . . , A is segregatable, then there must 
 
 exist a set of weights such that W*A. > W«C. > W«B„ where the C„ are 
 
 111 1 
 
 all assignments which are summable and hence not segregatable. 
 
58 
 CHAPTER 3 GENERAL PROPERTIES OF MARGINALLY SEPARABLE FUNCTIONS 
 
 3*1 Introduction 
 
 In this chapter we will discuss a type of separation where 
 a mixture of true and false assignments are allowed to lie between two 
 parallel hyperplanes. This type of separation will be called marginal 
 separation. This is more general than pseudo-separation. 
 
 We saw in Corollary 2.4.1 that in order for a function 
 f(x) to be pseudo- separable under a zero free structure, all of its 
 non-extremal assignments must be asummable. An arbitrary positive 
 function need not have this property. We saw in Example 2.4,1 that 
 the function f (x) = x x sy x-XiX. has summable false assignments 
 (1,0,1,0,1) and (0,1,0,0,1) and because this last assignment was not 
 extremal, x,x_ v x x.x is not pseudo-separable, While all non-extremal 
 assignments are not asummable, this function does possess some asummable 
 assignments. We now would like to extend the theory that we have 
 developed to include any positive function which has asummable assign- 
 ments. This desire leads to the concept of marginal separation. The 
 class of marginally separable functions is also a generalization of 
 asummability as follows: If A. are true assignments for positive f(x) 
 and B. are false assignments, and if 
 
 k k k k 
 
 T, t. < E!., then W*' ( Z A*. ) < W • ( £ f . ) 
 
 i=l i=l i=l i=l 
 
 if W is to be zero free. Thus some vf-B. must be greater than some 
 
59 
 W'A. where B. is false and A. is true. The case is impossible if f(x) 
 
 is a pseudo- separable function. Therefore, if we wish to extend our 
 
 theory to all positive functions, we must allow w°A. < w«B. for some true 
 
 assignments A. and false assignments B. . In marginal separation, we 
 
 define some constant /\, called the margin, and require that all such 
 
 assignments be in the: 'range 
 
 T - a/2 < ^°~&- < ^'^-. < T + V 2 ° 
 In this way we can localize all such assignments between two parallel 
 hyperplanes, [w; T + a/2] and [W; T - a/2] and can segregate all other 
 assignments . 
 
 In the first section of this chapter we will put on a priori 
 bound on this margin and will examine the properties of functions 
 which can be realized under this restriction. In the second section 
 we will develop necessary and sufficient conditions for this type of 
 separation. In the last section we will not demand any prior bound on 
 A, but will develop bounds which will allow the segregation of all 
 asummable assignments. As before we will be dealing with Boolean 
 functions on exactly n-variables. We also will assume that all w. =}= 
 so as to eliminate the problems encountered when w. =0. 
 
 3»2 Basic Properties of Marginally Separated Functions 
 
 In this section the basic idea of marginal separation is 
 reintroduced. The concept of a quasi -separable Boolean function is 
 defined and some of the basic algebraic properties of qua si -separable 
 functions are discussed with particular emphasis on the relationship 
 between the function and the weights of the quasi -separating hyperplane. 
 
6o 
 
 Definition 5«2.1: Let f(x) be an arbitrary Boolean function. Let 
 [wj T] be an arbitrary hyperplane and A> a constant. We -will say 
 that [w; T,A] marginally separates f(x) if for any assignment A, 
 
 if t(f) = 1, then W*»A*> T - A/2 
 and 
 
 if t(t) = 0, then W*-A* < T +A/2. 
 
 The constant A will be called the margin of [w; T,a] and [W*; T,A] will 
 be, called a marginal structure. 
 
 Lemma 3»2.1: Let f(x) be an arbitrary Boolean function. Let [W*; T] 
 be an arbitrary hyperplane. Then there exists some A > such that 
 [w; T,a] marginally separates f (x) . 
 
 n 
 Proof: Let E |w. | = Z. 
 
 i=l 
 
 (1) If T > 0, let A = 2(T + Z) . Then T + a/2 = 2T + Z and 
 T - a/2 = Z. For any assignment k, - Z < W*'A < Z. Therefore for any 
 assignment A,T - A/2 < "•'&* < T + A/ 2 and [»j t >a] marginally separates 
 
 (2) If T < 0, let A = 2(Z - T) . Then by the above reasoning 
 we have that [w; T,A] marginally separates f (x) . 
 
6i 
 Lemma 3«2.2: Let A > be any constant. Then there exists some hyper- 
 plane [w; T] such that [w; T,a] marginally separates f(x) . 
 
 n 
 Proof; Let w be any weight vector such that E |w.|< a/2. Then, because 
 
 i=l X 
 
 each assignment A must lie in the range - a/2 < w A < a/2, if we let 
 
 T = 0, -we will have that [w; 0, a] marginally separates f (x) . 
 
 Q.E.D. 
 
 The two preceeding lemmas show that A and [w; T] must be related 
 in some way for the concept of marginal separation to be meaningful. In 
 
 this first section, we will use the relation a < I w . I where w . is 
 
 7 - ' mm ' mm 
 
 the minimum weight of w. This means that if A is any true assignment and 
 B any false assignment, then in order to have marginal separation of 
 
 this type tf.I? - \?.t < K» . I. 
 
 ° * ' min 1 
 
 Definition 3»2.2; A Boolean function f (x) is called qua si -separable 
 under [w; T,a] if [w; T,a] marginally separates f(x) and if A < | w . \ . 
 The pair of parallel hyperplanes [W*; T - a/2] and [% T + a/2] will 
 be called quasi -separating hyperplanes if A < j w . |. 
 
 As special cases, if a = 0, we have pseudo- separability and 
 if A < 0, we have separability. We will now examine some algebraic 
 properties of qua si -separable functions. 
 
 Lemma $.2.3> Let f(x) be qua si -separable under [w; T,a] • Let x. be 
 
 J 
 
 any variable and A. any partial assignment in the other n-1 variables. 
 
 J 
 
62 
 
 If f(A*.; x. -> l) =1 and f(A*.; x^ -> 0) =0, then v. > and if 
 •1 3 - J o, J 
 
 f (A*.; x -> 0) =1 and f(A.) xj * l) #. 0; (then w. < 0. 
 J 3 J J -, J 
 
 Proof: As in Theorem 2.2.1, we can write 
 
 f(x) = x.f(x; x. -> l) v 1 v/ x.f(x; x. -> 0) . 
 J J J J 
 
 n 
 
 (1) If f (A*.; x. -» 1) = 1, then Z v. a. + v > T - a/2. 
 
 3 3 ■*■ -*• J 
 
 n 
 
 (2) If f(A*.j x. -*0) = 0, then I v. a. < T + a/2. 
 
 J J l l - *-* 
 
 Combining (l) and (2), w . > - A. 
 
 J 
 
 Now assume w. < 0. Then we have© > w. > - A. Therefore, 
 
 3 ~ ~ 3 - 
 
 A > J w. | a contradiction because it was assumed that [w; T,A] quasi- 
 J 
 
 separates f (x) . A similar proof holds if f(A,; x. -* 0) =1 and 
 
 f(A*.; x. -> 1) = 0. 
 
 J J Q.E.D, 
 
 Theorem ^.2.1; If f(x) is quasi -separable, then f(x) must be unate. 
 
 Proof: If f(x) is not unate in x., then neither 
 
 - — : — J 
 
 f (x; x -> 1) > f (x; x -> 0) 
 
 J J 
 
 nor 
 
 f (x; x -> 0) > f (x; x -» l) 
 
 J J 
 
63 
 
 Thus there must exist a pair of assignments A. and B. on the other 
 
 J J 
 
 n-1 variables such that 
 
 (l) f(A.; x. -> 1) = 1 and f(A.; x. -» 0) = 
 
 J J J J 
 
 (2) f(H\; x. ->0) =1 and f(B > .; x. -> l) = 0* 
 
 (l) implies w. > by Lemma 3»2.3 and (2) implies v. < 0. 
 
 J J 
 
 Therefore we have a contradiction. Hence f(xj if it is not unate, 
 
 can not be quasi -separable. 
 
 Q.E.D, 
 
 Corollary 3«2.1: Let f(x) be positive (or negative) in some variable 
 
 x. and quasi -separable under [w; T,A]. Then the corresponding weight 
 J 
 
 w. must be positive (or negative). 
 
 J 
 
 Proof: If f(x) is positive in x. then f(x; x. -> l) > f (3c; x. -» 0) . 
 
 J J J 
 
 Therefore there must exist some assignment A. on the remaining n-1 
 
 j 
 
 variables such that f(A.; x. -» l) =1 and f(A.: x. -»0) =0. 
 
 J J JO 
 
 Therefore w . > by Lemma 3.2.3. 
 
 J Q.E.D. 
 
 Theorem 3*2.2: If f(xj is marginally separable under [w; T,a], then 
 f(x; x. ->x.) is marginally separable under 
 
 [w x , w , . .., ^ i _ 1 ^ -y ± , v ±+1 > •"> v n 3 T - v 1 ,a] = [^; T - v ± ,A] 
 
 Proof:' ' If . [W; I:/ A] marginally separates f'(x),' then- for any x 
 
6k 
 
 (1) If f(x) = 1, then tf-x > T - a/ 2 » 
 
 (2) If f (x) * 0, then ^«x < T + A/2.' 
 
 As in the proof of Theorem 2.2,3, substitute x. for x. in f(x) and 
 ( 1-x . ) for x . in vf °x. Then 
 
 (l) If f(x) x. ->x.) = 1, E w.x.-:+ w.(l - x.) > T - a/2< 
 
 . _ n 
 (2) If f (x; x. -» x.) = 0, Z w jXj + w. (l - x.) < T + a/2. 
 
 Subtracting v. from both sides in the linear inequalities: 
 
 (l) If f(x; x. -*x.) = 1, L w.x. - w.x. > T - a/2 - w, 
 v/ v ' 1 i' ' j j 11— ^ : 
 
 -» - n 
 
 (2) If f(x; x. ->x.) = 0, £ w.x. - w.x. <■ T + a/2 - w . , 
 
 v/ v ' 1 i' ' j j i i — ■ Hr 1 
 
 Therefore, [W'; T - w.,a] marginally separates f(x; x. -> x^) . 
 
 Q.E.D, 
 
 Lemma ^.2.k: If f(x) is marginally separable under [w; T,a], then 
 f(x) is marginally separable under [-W; -T,a] • 
 
65 
 
 Proof: If f(x) = 1, f(x) = and V?.x > T - ^2 or (-w) -x < - T + /±/2. 
 
 If f(x) = 0, f(x) = 1 and W*-x < T + a/ 2 or (-tf)«x > - T -a/2. 
 
 Hence [-w; - T,a] marginally separates f (x) . 
 
 Q.E.D. 
 
 Theorem 3>2»3: If f(x) is quasi-separable under [w; T,a], then f (x) 
 
 n 
 is quasi -separable under [W; Z w. - T,a] . 
 
 i=l 
 
 d - - 
 Proof: Since f (x) = f(x), "we can combine the results of the Theorem 
 
 3.2.2 and Lemma J>,2,h. 
 
 Q.E.D, 
 
 Definition 3*2. 3- Let f(x) be an arbitrary Boolean function. Let 
 [W; T,a] be any structure "which marginally separates f (x) . Let A 
 be any assignment to f (x) . We "will say that A is segregated by the 
 structure [% T,a] if f(A*) = 1 and W**A* >T + a/2 or f(t) = and 
 W C A < T - a/2. If there exists some structure which marginally 
 separates f (x) and segregates A, "we "will call A segregatable. 
 
 The concept of segregated assignment defined here, is very 
 similar to that used in pseudo-separability. In both instances if an 
 assignment A is segregated, we can determine f (a) by only examining 
 W*A and T. Notice that if an assignment is not segregated, then we 
 have either W*A = T in the case of pseudo-separation or 
 T - a/2 < W«A < T + a/2 for marginal separation and we need 
 additional information to determine f (a) . 
 
66 
 
 Lemma 3*2.5: If f(x) is quasi -separable under [w; T,a], then all non- 
 extremal assignments must be segregated by [w; T,a]« 
 
 Proof: f(x) is unate by Theorem 3»2ol. Therefore, we can assume all 
 
 w. > 0. Let A be any non-extremal true assignment. There must be some 
 
 other true assignment R such that A > Bo Since all w. > 0, W»A > W-B. 
 
 Since A* > B*, vf-A* - ^»B*> w . . If A* is not segregated by [% T,A], 
 
 T - A/2 < V?.A*< T + A/2. Also ^«B*> T - a/2 since if is true. Therefore, 
 
 A> w . , a contradiction. Similarly, every non-extremal false assignment 
 
 must be segregated by [Wj T,a]. 
 
 Q.E.D. 
 
 Theorem 3»2A: Let f(x) be a positive function. If f(x) is marginally 
 separated by [w; T,a] and if all non-extremal assignments are segregated 
 by this structure, then f(x) is quasi -separable. 
 
 Proof: Assume that A> v. for some i. There must exist a true assignment 
 
 A = (a., a p , ..., a ) and a false assignment, B = (b_, b ? , .«,., b ) such 
 
 that ^»A* = T - a/2 and tf-lf = T + a/2 or else A could be reduced. If 
 
 a. = or b. = 1, then the corresponding assignments A with a. = a. for 
 
 J4=l and a. ' = 1 or B ' with b. ' = b (j/i) and b. ' = will not be . . 
 
 segregated by [^ T,A] because A> w. and ^-A* = T - a/2. This would 
 
 contradict the original assumption that all non-extremal assignments are 
 
 segregated by fv?; T,A], Therefore a. = 1 and b„ =0. Consider the 
 
 transformation w ' = w (p=/i), w. ' = w. + €. and T' = T + e./ 2 where 
 p p v i 11 r 
 
 €. > 0. 
 
 1 
 
67 
 Because we want e. to be as large as possible and still main- 
 tain marginal separation, we can increase e. until we find either a true 
 assignment C such that "W* « A = ~W' if € or a false assignment D such that 
 W' *D = W' *B. An important point here is that c. = and d. = 1» If 
 
 c. = 1, then the transformation w.' = w. + £. has the same effect on 
 l ' ill 
 
 C as on A so that w*C = W°A and therefore 6. can be increased without 
 
 i 
 
 disturbing the marginal separation. Similarly if d. =0, then 
 
 W' .D = W'D = W'B = w ' °B and e. can be increased. Therefore, assume 
 
 i ' 
 
 that we have found a true assignment C with c. = such that 
 
 i 
 
 w' «A = w ' *C . Then 6. can not be increased further or else [w'; T',A] 
 
 will not marginally separate f (x) . Consider A and €. at the point that 
 
 V? ! •(? •= \t' 't and c . = 0. If A < e. ,'■ then because ^' «A* = T - a/2 + e. 
 i ■ — i' ^ l 
 
 and W f •]? = !?•!? = T + &J2, ^' -t > vf' «b\ Therefore, for all true 
 
 assignments A. and false assignments B . y W'-A. > w ' «B. . Hence 
 
 i ° li—i 
 
 [w'; T - a/2 + e. ] pseudo- separates (or separates) f (x) . So assume 
 
 that A > e. . Then tf» .]? > V?* -A*. Let A' = tf l ^ - tf 1 "A*. If A* > w. ' , 
 
 l — l 
 
 we employ the same argument used above to get a contradiction. Since 
 
 C has c. =0, the corresponding assignment C' where c : . ' =: c. for j/i, c. = 1, 
 
 must be true and non-extremal. Since W ' *(f = T + £./2 - A'/ 2 and 
 
 A' > w i ' , T + €./2 - a/2 < tf' "(ft < T + e . /2 + a/2 . Since 
 
 ^••(f 1 = tf«lf + €., T - S./2 - A'/ 2 < $'(?' < T - €./2 + A'/ 2 « Since 
 
 A' =A-£.,T-a/2< tf-cf' < T + a/2 - €. or T - a/2 < tf.(f* < T + a/2 . 
 
 Therefore C 1 is not segregated by [W; T,a], a contradiction since it is 
 
 non-extremal. Therefore A' < w. '. 
 
68 
 
 Consider f(x) under this structure. Clearly, [w'| T',A'] 
 
 marginally separates f(x) since e. was increased only until we were in 
 
 danger of destroying this property o Also A' < u : «. . If w. : is the only 
 
 weight such that A> w., we are done. If not, we repeat the procedure 
 
 for each such weight until a marginal structure [W M j T",A"] is found 
 
 which quasi-separates fix) . 
 
 Q.E.D. 
 
 Example 3»2.1; We would like to illustrate a very important fact about 
 marginal separation, namely that an assignment need not be asummable to 
 be segregated by even a quasi -separating structure. Consider 
 f(x) = x,Xp v x,x^x . In Example 1.2o2, we showed that this function 
 is marginally separable under [3,3>2 5 2, 2; 6 l/2,l] Since l/2 <w . =2, 
 this structure quasi-separates the function. Consider the false assign- 
 ment (l,0,0,0,l). This assignment is summable because (1,0,0,0, l) + 
 (0,1,1,1,0) = (1,1,1,1,1) = (1,1,0,0,0) + (0,0,1,1,1). But (3,3,2,2,2) ■> 
 (1,0,0,0,1) = 5 < 6 = T - A/2. Therefore (1,0,0,0, l) is segregated by 
 [3,3,2,2,2; 6 1/2,1]. 
 
 3«3 A Necessary and Sufficient Condition for Quasi-Separability 
 
 In this section we will develop a necessary and sufficient 
 criterion for quasi -separability. The condition which is presented 
 here is analogous to the supersummability conditions for pseudo-separable 
 functions which were introduced in Section 2.4-. 
 
69 
 Definition 3-5«l* Let f(x) be an arbitrary function, f(x) is called 
 
 extremely summable if there exists k true non-extremal assignments 
 
 —>—>■—> — > — > •=* 
 
 A,, A p , . .., A, and k false extremal assignments EL, B p , .«.., B, such 
 
 k k 
 
 that Z A*. = Z st.:. 
 
 1 1 
 
 i=l i=l 
 
 Lemma 3»3«1« If f (x) is a positive function, then f (x) is extremely 
 summable if and only if there exists k true extremal assignments 
 A-,, Ap, . .., A, and k false non-extremal assignments B, , Bp, ..., B, 
 
 k k 
 
 such that E A. = Z B. . 
 i i 
 
 i=l i=l 
 
 Proof: If k such true extremal assignments and k such false non-extremal 
 assignments exist and f (x) is positive, then we can, by changing some 
 components of the B. and A. from to 1, change each A. into A'.' and 
 
 B. "into BJ , where each B.' is extremal false and each A', is non- 
 
 i V i i 
 
 k k 
 extremal true. We can maintain Z A.' = Z B] if we "change the same 
 
 i=l i=l 
 
 component in one B. and one A. at a time. Therefore f(x) must be 
 extremely summable. 
 
 Conversely, if f(x) is extremely summable, then we can find 
 
 k true non-extremal assignments A,', AX, .... A,' and k false extremal 
 
 1 2' ' k 
 
70 
 k k 
 assignments B 1 , B' , ..., B' such that £ A' = Z B' . If we repeat 
 
 1 c. K 1 1 
 
 1=1 i=l 
 
 the process outlined in the first half of this proof, except changing 
 
 components from 1 to in both A* 1 and B*' at once, we will find k 
 
 — > — * — > 
 true extremal assignments A., A ? , ..., A, and k false non-extremal 
 
 k k 
 
 ;ignments B 1 ,, BJ , . .., B' such that A A„ = Z B ' . 
 1 ^ K 1=1 1 -i=l X 
 
 Example 3«3«1' Consider the function f (x) = x* x p vioc^x-x^ v XpX,_X/- 
 
 (1,0,1,1,0,0) and (0,1,0,0,1,1) are extremal true assignments. 
 (1,0,0,0,1,1) and (0,1,1,1,0,0) are false assignments. Since 
 (1,0,1,1,0,0) + (0,1,0,0,1,1) = (1,0,0,0,1,1) + (0,1,1,1,0,0) = 
 (1,1,1,1,1,1), these assignments are summahle. Since (1,0,1,1,0,0) < 
 (1,0,1,1,0,1) which is false and (0,1,0,0,1,1) < (0,1,1,0,1,1) which is 
 also false, both (1,0,1,1,0,0) and (0,1,0,0,1,1) are non-extremal false. 
 Therefore this function is extremely summable. 
 
 We can also relate the concept of extreme summability with the 
 idea of supersummability . Because we want to employ extreme summability 
 for quasi -separable functions in much the same way as we used super- 
 summability for pseudo- separable functions, and because we wish to 
 extend the range of functions realized, we expect extreme summability 
 to be a weaker property than supersummability. 
 
71 
 
 Theorem 3*3-1: If f(x) is positive and extremely summable, f (x) is 
 supersummable. 
 
 Proof: Let A... A., ....A, be non-extremal true assignments and 
 — — ■ 12k to 
 
 -» -* -» 
 
 B , B , . .., B be extremal false assignments such that 
 12 k 
 
 k k 
 
 A, = L B.„ Replace each A. with a corresponding extremal true 
 i=l i=l 
 
 ■• k k k 
 
 assignment U. « (A< > U 4 ) . Then Z U 4 < Z A. = Z B. and f(x) 
 is supersummable. 
 
 i=l x i=l X 1=1 
 
 Q.E.D. 
 
 Example 3° 3° 2: Consider the function X..X v x x, x . As was shown in 
 
 Example 2.1+.2, this function is supersummable with false assignments 
 
 (0,1,0,1,1) and (1,0,1,1,0) summing to (1,1,2,1,1) and true assignments 
 
 (1,1,0,0,0) and (0,0,1,1,1) summing to (1,1,1,1,1). The only extremal 
 
 true assignments for this function are: (1,1,0,0,0) and (0,0,1,1,1) „ 
 
 The only non-extremal false assignments are: (0,0,0,0,0), (0, 0,0,0,1), 
 
 (0,0,0,1,0), (0,0,1,0,0), (0,1,0,0,0), (1,0,0,0,0), (0,0,0,1,1), 
 
 (0,0,1,0,1), (0,0,0,1,1), All other assignments are either false 
 
 extremal assignments or true assignments „ Suppose there exists 
 
 a f p non-extremal false assignments B_ , B_, . ... B_ _, such that 
 
 12 Cft-p 
 
 a+p 
 Z B. = a(l,l,0,0,0) + p(0, 0,1,1,1). Let 1 = (1,1,. ..,1). Then 
 1=1 X 
 
72 
 
 OS-0 
 
 "?•( Z 1^) = ti[cc(l, 1,0,0,0) + p(0,0, 1,1,1,)] = 2a + 3p„ Since each 
 
 i=l 
 
 _V ->-> 
 
 B. has at most two coordinates of value 1, 1*B. < 2 
 
 l ' i — 
 
 ' . Of? 
 or l'( Z B.) < 2« + 20. Therefore 2a+ 20 > 2a + 30. This implies 
 i=l 1 
 
 a . 
 
 assignments such that Z B = Ol( 1,1, 0,0,0), which JLs impossible. . There- 
 
 , i=1 
 
 fore x.x v x^x.x^ is Supersummable hut not extremely summable. 
 
 Theorem 3«3»2s A necessary and sufficient condition for a positive 
 function f (x) to be quasi -separable is that f (x) not be extremely 
 summable. 
 
 Proof: (l) If f(x) is positive : and quasi -separable, then there is 
 
 some marginal structure [w: T,4i with A< w „ which marginally 
 
 mm 
 
 separates f (x) . If f (3c) is extremely summable, there exists k 
 
 — > — > — * 
 extremal true assignments A , A ? , <.<,<>, A. and k non-extremal false 
 
 k -> k 
 assignments B,, B c , . «.,, B, such that Z A, = Z B. . 
 
 12 k 1=1 X 1=1 x 
 
 Let U. be an extremal false assignment which corresponds to B. . 
 
 k k k 
 
 Then \? > t . The tf° ( Z A*. ) = tf ° ( Z ? . ) < tf« ' ( Z % ) • Since 
 
 1=1 2 1=1 1=1 
 
73 
 
 if. > t . , tf-T?. > W*.B* + w . . 
 i 1' i — min 
 
 k k k 
 
 Therefore \f' ( Z if.' > W*« ( E B*. ) + k v . 4- (E t) +kw . . Because 
 j , X — . , i, min . _ i mm 
 i=l i=l ' i=l 
 
 each A. is a true assignment, w«A, > T -A/2. Since f(x) is positive, 
 
 k -* 
 all v. > 0- Therefore W*' ( £ A.) > kT - kA/2. Similarly, 
 
 issl x 
 
 k 
 
 W*«( Z if. ) < kT + kA/2. Therefore kw . < kA and f(x) is not quasi- 
 ._ n i — ' min — 
 
 separable . 
 
 (2) We will now show that if f (x) is not extremely summable, 
 then f(x) is quasi -separable. If f(x) is quasi-separable, "we know that 
 all non-extremal assignments must be segregated by the marginal structure 
 by Lemma 3«2.5« Therefore only extremal assignments can reside in the 
 range T - a/2 < W*«A*. < T + A/2. Also, if f(x) is positive and quasi- 
 separable, then all w. > by Corollary 3.2.1. Therefore if 
 A > B, W«A > w.B*. Conversely if all extremal assignments satisfy these 
 conditions, then the function is quasi -separable by Theorem 3«2.4. There- 
 fore, the problem of determining sufficient conditions for a positive 
 function to be qua si -separable is the same as finding a marginal structure 
 [W*; T,a1 such that: 
 
 vf'tf. > T -A/2 if \f. is extremal and true. (3.3^l) 
 
7^ 
 V?°V^ < T + A/2 if V* k is extremal and false and (3..3-1) 
 
 A < w . . 
 
 man 
 
 Combining these inequalities w o (u. - v. ) > - w . or 
 
 ^•(lf. - ^. ) + w . > 0. 
 x j k' mm 
 
 A necessary and sufficient condition that such a set of 
 
 inequalities not "be consistent and therefore not have a solution, is 
 
 the existence of a., > such that Z a., > and 
 
 Ea., $'(\f. - f. ) + w . ] < a [3] 
 .. ,Jk J k min - 
 
 J,k 
 
 Let 2 be the assignment with zeros in all positions except the coordinate 
 
 which corresponds to w . <> If more than one such position exists (i.e., 
 
 min 
 
 ■y . v . . . v all correspond to v . ) write 2 = — (fT.n + a © +...+ Z_ ) 
 i ■ • ' ■? > x i mm p x X d p ' 
 
 X l X 2 p 
 
 Then we can rewrite the above expression as 
 
 Yj a,, V?°(tf. - V*,) + . X a.,w»S<0. Since we have assumed all 
 . , j K. j K J /K. J K 
 
 w.> 0, this expression must be identically zero or identically less 
 
 than 0. Therefore 
 
75 
 
 Za (I? - f) + Eal<0 (3. 3.2) 
 
 jk Jk J k j,k jk 
 
 or 
 
 ^k<V*^*°' 
 
 Then E pa., (ft: + t) < Z pa ,t. 
 
 j,k Jk J " J,k jk k 
 
 pt?. + pZ* = p!?. +2., + Z*„ + . , . + Z*. . Let I?. + zf. = A . . Since 
 
 f(x) is positive and if. is extremal true, each A. must be non-extremal 
 
 J jq 
 
 P _> 
 
 and true. Therefore pU. + pZ = E A. and (3.3.2) becomes 
 
 J e=l je 
 
 V a ( I A J < v ^ # 
 J,k Jk e=l J j,k Jkk OOO; 
 
76 
 
 We can always make the relation in (3-3»3) an equality by 
 
 — -. 
 changing components in A. from to 1 and still he guaranteed that the 
 
 resulting assignment is non-extremal true. This means that 
 
 L A. =■-■ T, v. for some q and f(x) is extremely summable. Therefore, 
 i=l x i*l. x 
 
 if f (x) is not quasi -separable, then (3«3»l) are inconsistent and hence 
 
 (3«3«3) which implies f(x) is extremely summable. 
 
 Q ;E .D . 
 
 Example 3»3«3 : In Example 3«3»1> f(x) = x^-v x.. x x. w x ? x x •- was shown 
 
 to be extremely summable. Therefore this function is not quasi -separable. 
 In Example 3.3.2.x n x^v/ x X|,x c was shown to be quasi-separable, but; not 
 
 i. 2 5 *r J V. .?. T..1 T ..'. ■. >G .: i q '.;.'..-,' j_~ .' ' ..3Xi } . i> :.\C ,, 
 
 pseudo- separable. Therefore there exists positive functions which are 
 quasi-separable, but not pseudo -separable. 
 
 J.h Marginal Separation and the Degree of Super summability 
 
 This section is devoted to an analysis of marginal separation 
 without any a priori bound on the margin \ However, later we will 
 obtain lower bounds on A in terms of the weight vector W for specific 
 functions, as we did in the previous sections. The degree of super- 
 summability tells us how -^ and W are related. In particular it gives 
 a lower bound on A as a multiple of the minimum weight (i.e., 
 
77 
 
 A >T w . where F is the degree of supersummability) . Before we 
 — mm 
 
 examine this property, we prove some general results about marginal 
 separation. 
 
 Theorem 3»^»1' Let f(x) be a positive function. Then there exists a 
 marginal structure :[w; T, A] which marginally separates f (x) and which is 
 such that for any assignment A, A is segregated by [W; T, A] . if and cl 
 only if A is asummable in f (x) . 
 
 Proof: Because we require that all assignments segregated by [W; T, a] 
 must be asummable and all asummable assignments must be segregated by 
 [w; T,a], the problem of finding such a structure can be reduced to 
 determining [w; T,a] such that 
 
 (1) tf-A* t > T + A/2 if A*, is asummable and f(A*. ) = 1. 
 
 (2) \t-t. < T - A/2 if If. is asummable and f(lf.) = 
 
 (3) T - A/2 < tf'C*. < T + A/2 if (f is summable. 
 Combining these inequalities, 
 
 (1)' If- (2^ - 1) > A 
 (2)' tf.(t ± - C^) >0 
 
 (3)' ^.(c^ -?,) >o 
 
78 
 This system is inconsistent if and only if there exists non-negative 
 numbers a.,, h , and c. such that Z a A «* and 
 
 ij IK jK » i» lj 
 
 Z • w-C^ - S ) - Z b lk ».ff 1 -? k ) ♦ Z c jk (c k - B k ) <o [33.OAD 
 
 i/j i/k 3,k 
 
 (3«^.l) must he identically or identically less than 0. 
 Hence, it can he rewritten as. 
 
 Z . ff -i ) ♦ Z b (X ~c .) ♦ Z c (8 k -B ) <0 
 
 i,j lj x 3 i,k lk * k J,k 3k k d 
 
 or 
 
 Z a X. ♦ Z b X «- Z c C < Za .5. + Z b C + Z c B 
 i,j iJ i i,k ik x d,k 5k k i,J id J i,k lk k j,k ^ J 
 
 We can rewrite this last expression as: 
 
 I X. 4- 2 C < ^"f. + Z C. ,':,•) (3.^.2) 
 
 i=l 1 i=l X i=l x i=l 1 
 
 where 
 
 p = Z a . + Z b , q «= Z c and r « Z b . 
 i,j 10 i,k 1K J,k Jk i,k llc 
 
 Because Z a. .A > 0, some a. . must be positive. 
 
 As in the proof of Theorem 2»k k, we again denote the C„ which 
 
 -* •>♦ -♦ 
 
 are true as D. and the C. which are false as E. and write 
 11 1 
 
 Zc. = Z D. + L E J . Similarly, we can write Z C. = Z D ! + Z E. 
 . . i . - i . . i "' . . i . . i . , i 
 
 i=l i=l i=l i=l i=l . i=l 
 
 where D. are the true C. and E. are the false. Substituting these 
 
 equalities into (3A2) 
 
79 
 
 L A. + Z D. + L E. < 2 B. + Z D. + Z I 13 4.3) 
 . , i . , l . , i - . , i . - i . -, i 
 
 i=l i=l i=l i=l i=l i=l 
 
 To arrive at an expression with only true assignments on the left side 
 and false assignments on the right, we use the fact that each E, and 
 
 D. is summable. Thus E. = Z V 1 - L Z 1 and D. = Z Z 1 - Z V 1 , 
 i l . , m m l ., m _ m ' 
 
 m=l m=2 m=l m=2 
 
 -*d ~*i ' -h -*i ' 
 
 where the Z and Z are false summable assignments and the V and V 
 mm mm 
 
 are true summable assignments. Substituting the corresponding equality 
 
 into (3-^-3) for each E. and D. we get 
 
 p q+d p+q-r r-t-d 
 
 Z A. 1- Z T. < Z B. + Z F. 
 . - 1 . , l — . , l . , i 
 1=1 1=1 1=1 1=1 
 
 -> -> -* — * 
 
 where A. and T. are true assignments and B. and F. are false. Since 
 11 11 
 
 p f- and p + q - r ^ (because some a. . > 0),this implies that some 
 
 -*- J 
 
 of the A. and B. must be summable or supersummable. If it is super- 
 11 
 
 summable, it is accordingly summable by Lemma 2.4.2. This, however, 
 
 -* 
 contradicts the original assumption that the C. were the only summable 
 
 assignments. Therefore, the above system of inequalities always has 
 
 a solution. 
 
 Q.E.D. 
 
 The marginal structure which is guaranteed to exist by this 
 theorem will not in general be optimal in the sense that a maximal 
 number of assignments need not be segregated. We can in general find 
 another structure which not only segregates all asummable assignments, 
 but some of the summable assignments as Example 3-2,1 shows. 
 
80 
 Definition 3.^-1 : Let f (x) be an arbitrary Boolean function on exactly 
 n variables. Assume that [Wj T, A] marginally separates f(x). Let 
 M(Wj T, A) be the minimum of (T - A/2) - W»A. for segregated false 
 assignments and W'A - (T ¥ A/2) for segregated true assignments. Let 
 g(x) be the separable function on n variables such that if ' ^ 
 W'A > T - A/2 - e, then g(A) = 1 and if W'B < T - A/2 - e, g(B) = 
 where < € < M(W, T, A) , Let h(x) be the separable function on n 
 variables such that if W«A > T f A/2 f c, then h(A) = 1 and if 
 W-B > T + A/2 + e, then h(B) = where < € < M(W, T, A). g(x) is 
 then called the marginal upper bound for [W; T, A] and h(x) is called 
 the marginal lower bound . 
 
 Notice that because < € < M(W, T, A), if g(A) = 1 and 
 h(A) = 0, then T - A/2 < W*A < T f A/2, 
 
 Lemma 3°*+«l: Let h(x) and g(x) be any pair of simultaneously realizable 
 functions. Assume h(x) < g(x) . Then there exists some [Wj T, A] such 
 that h(x) is the marginal lower bound of [W; T, A] and g(x) is its 
 marginal upper bound. 
 
 Proof : Let h(x) have structure [W; T ] and g(x) [W; T ]. Since 
 
 h(x) < g(x), T. > T . Let A = T,_ - T - 2c, where < e < minimum 
 " h g h g 
 
 (¥-A. - T u for h(A.) = 1 and T - W.A, for g(A.) = 0) . Let 
 i h l h i i 
 
 T = T t l/2 A + e. Then h(x) is the marginal lower bound for 
 g 
 
 [W; T, A] and g(x) is the marginal upper bound, 
 
 Q.E.D. 
 
$1 
 
 Theorem 3.4.2 : If g(x) and h(x) are the marginal upper bound and lower 
 "bound for [Wj T,A ], then for any function k(x), h(x) v g(x)k(x) is 
 
 > — > 
 
 marginally separated by [Wj T,A ]. Conversely, if f(x) is marginally 
 separable under [Wj T,A ], then there exists some function k(x) such 
 that f (x) = h(x) v g(x)k(x) where g(x) and h>(x) are the marginal bounds 
 for [W; T, A] and any assignment A is segregated "by £W; T, a] if and only 
 if g(A) and h(A) are equal. 
 
 Proof : If f(x) = h(x) v k(x)g(x) and h(x) has structure [W; T + a/2 + e] 
 while g(x) has structure [W; T - a/2 - e] where < e < M(W, T, a), then 
 we claim [Wj T, A] marginally separates f (x) and segregates all assign- 
 ments such that g(x) = h(x). Let A be any true assignment. Then either 
 h(A) - 1 or g(A)k(A) = 1. If h(A) - 1, W-A > T + A/2 f e and A is 
 segregated by [Wj T, A]. Also, because h(A) = 1 and g(x) > h(x), g(A) = 1. 
 If g(A) = 1 (and h(A) - 0), then W-A > T - a/2 - e and since < e<M(W,T,A), 
 W-A > T - A/2. Since h(A) = 0, W-A < T + a/2. Therefore .. 
 T -A/2 < W-A < T + A/2 and A is not segregated by [W; T, A]. A 
 similar arguement shows that if f (B) = 0, then h(B) = must hold. If 
 g(B) = 0, W-B < T -A/2. If h(B) = and g(S) = 1, then T - A/2 < W-B 
 ^ T + A/2 and B is not segregated. Therefore f (x) is marginally separated 
 by [Wj T, A] and all segregated assignments are such that g(A) = h(A). 
 
 If f (x) is marginally separated by [WJ T, A], pick the marginal 
 lower and upper bounds to be h(x) and g(x) respectively. Then f(x) must 
 be of the form h(x) v k(x)g(x) where k(x) is such that f(A) = 1 and h(A) =0, 
 k(A) = 1 and if f(S) = and g(B) = 1, k(A) = 0. Since O <e < M(W, T,A ) 
 
82 
 
 and h(x) and g(x) have structures [W; T + A/2 f e] and [Wj T„.- £/2 - e] 
 
 respectively, all segregated assignments will be such that h(A) = g(A). 
 
 Q.E.D. 
 
 Corollary 3.4.1 : If f(x) is any positive function, then f (x) can he 
 written as h(x) v k(x)g(x) where h(x) and g(x) are simultaneously 
 realizable and for any assignment A, if h(A) = 1, A is asummable and 
 true in f (x) and if g(A) = 0, A is asummable and false 
 
 Proof : If f (x) is positive, there exists some marginal structure 
 [W; T, A] which marginally separates f (x) and segregates only asummable 
 assignments by theorem 3°^°1° Then f(x) can be written as h£x) v k(x)g(x} 
 where h(x) is the marginal lower bound of [W; T, A] and g(x) is its 
 marginal upper bound by Theorem 3*^.2. If h(A) = 1, then g(A) = h(A) =1 
 because g(x) > h(x)« Similarly if g(A) = 0, h(A) = 0. Because all 
 assignments A such that g(A) = h(A) are segregated by [W) T, A] ("by- 
 Theorem 3° 4. 2) and because all segregated assignments are asummable 
 
 (by Theorem 3. 4,1), the result follows 
 
 Q.E.D. 
 
 Another important result about marginal separation is that 
 there exists a structure [w; T,a] which segregates only one assignment, 
 and for each distinct assignment a structure exists which segregates only 
 that assignment. 
 
 Theorem 3«^Q: Let f (x) be an arbitrary Boolean function and let A be 
 any assignment. Then there exists some marginal structure [W; T,a] which 
 segregates only A. 
 
83 
 
 Proof: Assume that f(Aj = 1. Let A = (a,, a~, .... a ). Let 
 
 n / ^ -. \ .. -| 
 
 h(x) = IT x. i where x. = x. and x. = x. . Let g(x) be identically 
 . -, i ii ii 
 
 i=l 
 
 equal to 1. Then f(x) = h(x) v f(x)g(x). Clearly h(x) and g(x) are 
 
 simultaneously realizable, Therefore* by Theorem J>,k.2, we can find 
 
 [W; T, A] which marginally separates f (x) and also segregates only 
 
 assignments such that g(A) = h(A). Because g(x) = 1, and A is the only 
 
 assignment such that h(A) = 1, it is the only assignment segregated by 
 
 [Wj T, A]. 
 
 Q.E.D. 
 
 .-*. -> -> -> 
 
 Let f(x) be a given function. Let A , A , . , . , A be any 
 
 1 £- K. 
 
 -> -> -» 
 
 set of k true assignments and B , B , . .., B any set of k false assign- 
 
 k k 
 
 ments. If these two sets are summable, i.e. if Z A. = E B., the 
 
 • -i i • n i 
 1=1 i=l 
 
 i:. :•.;.-.. ■:; i ■: W. . '.': k k 
 
 function can not be separable. If Z A. = Z B. and some of the 
 
 . , l . ., l 
 i=l i=l 
 
 -* -> 
 
 A. or B, are non-extremal, then the function is supersummable and hence 
 
 -> -> 
 not pseudo-separable. If all of the A. or B. are non-extremal, then the 
 
 function is extremely summable and therefore not quasi-separable. We 
 
 would now like to develop a lower bound on A, the margin, in terms of 
 
 W for specific functions. Thus, we introduce the concept of the degree 
 
 of supersummability. 
 
 Definition 3.^.3 ' Let f(x) be an arbitrary positive function. Let 
 {A , A ..., A J be a set of k true assignments and {B , B , ..,, B ) 
 k false assignments. These sets are called k-comparable if 
 
81+ 
 
 k k k k 
 
 Z A. > Z B. or Z B. > Z A. . 
 
 i=l 1_ i=l X i=l x ~i=l X 
 
 Notice that if A. = (a. , a. ..... a. ) and 
 
 1 i_f i ' ' i 
 
 12 n 
 
 B. = (b . , b. , . .., b, ) then two sets are k-comparable if 
 
 J D l J 2 J n 
 
 k k k k 
 
 Z a. > Z b. for all j or Z a. < Z b. for all j. 
 , i . - . , l . , , i . — . i i . ° 
 
 i=l j i=l j i=l j i=l j 
 
 Definition 3.^.^ s Let f (x) be an arbitrary positive function . Let 
 
 k k 
 8, be the maximum possible It ( Z B. - Z A.) for all k-comparable 
 
 _^ i=l i=l 
 
 sets of assignments, where 1 = (l,l, ...,l). Let r be the maximum 8,/k for 
 
 K. 
 
 all k. F will be called the degree of supersummability for f(x). 
 
 Lemma 3° h.2 : The degree of supersummability is attained by summing 
 only extremal assignments. 
 
 Proof : Assume F is the degree and that r is attained as ! =~ . • ' 
 
 k k 
 
 k r = 1: ( L B. - L A.). If some B. or A. is non-extremal, then the 
 . , l . , l 11 
 
 i=l i=l 
 
 -+ -> 
 
 degree can be increased by substituting for A. or B. the corresponding 
 
 extremal assignment. In such a case, T is not maximal, a contradiction. 
 
 Q.E.D. 
 
 Lemma 3.^.3 - Assume T > 0. Then the assignments which are summed to 
 
 obtain this degree are summable. 
 
 k k 
 
 Proof: Let kT = Is ( Z B. - Z A.). Then since T > and 
 
 . , i . , i 
 
 i=l i=l 
 
 k -+ k -* 
 {a\ , a\, ..., A, } and {B n , B„, ... B. } are k-comparable, Z B. > Z A., 
 1' 2' k 1' 2' k . , i . , l 
 
 i=l 1=1 
 
85 
 
 k k 
 Assume that Z bi ^ Z a. . If we change x components in the B.'s 
 
 — > 
 
 from 1 to and call the resulting assignments B'. , then 
 
 J 
 
 k _> k _ k k 
 
 Z B'. rl Z A. and Z b ! = £ a . . Since f(x) is positive 
 
 i=l 3 " i=l J i=l X j i=l ^ 
 
 and B. > B., B. is a false assignment. We repeat this procedure for 
 J J J k k k k ^ 
 
 each component such that £ b! > £ a. . Then £ "BY = Z A, 
 
 .,i.,i . , l . , l 
 i=l p i=l p i=l i=l 
 
 — > 
 where BV are the resulting false assignments. Therefore, all true 
 
 assignments are summable. Similarly all of the false assignments 
 
 — »• 
 
 B. must be summable. 
 
 3 Q.E.D. 
 
 Example 3.^.1: Consider the function x n x_ s, x x,x_. We have already 
 1 c. 3^5 
 
 shown that this function is supersummable since (l,l,0,O,0)+(O,0,O,l,l) 
 = (1,1,1,1,1) < (1,1,2,1,1) = (1,0,1,1,0) + (0,1,1,0,1). Therefore 
 the degree is at least 1/2. We have also shown in Example 3.3-2, 
 that this function is not extremely summable. Since all assignments 
 used in achieving the degree must be extremal and since the only 
 true extremal assignments are (l, 1,0,0,0) and (0,0,1,1,1), the degree 
 must be achieved as 
 
 p+q _ 
 
 (p-: + q)r = l-[ Z B - p(l, 1,0,0,0) -q(0, 0,1,1, l) ] . 
 i=l 
 
 Again we note that each extremal false assignment has at most 3 ones 
 
 p+q _ 
 
 as components. Therefore, 1- ( Z B.) < 3p + 3q while 
 
 i=l X 
 
 1- [p(l, 1,0,0,0) 4- q(0,0,l,l,l)] - 2p + 3q. Therefore (p + q)r : p or 
 P. 
 
 p+q* 
 
 r <-— -. Therefore the degree of this function is 1/2 < r < 1. since q / 
 
86 
 
 We will later show that this bound on V is sufficient to determine 
 what type of separation is possible* 
 
 Theorem 3-4.4 : If T < 0, then a positive function, f (x) is separable. 
 
 Proof : If T < 0, then whenever {£. , A , . .., X) and (B , B , . .., B. ) 
 
 12 It k ^12 k ^k 
 
 are comparable sets of true and false assignments, Z A. > E B. . 
 
 i=l 1 i=l x 
 
 Therefore, f (x) can not be summable for any k and this implies that 
 
 f(x) is separable by Theorem 2.4.1. For incomparable sets of true 
 
 and false assignments which are not considered in computing r, of 
 
 course, equality can not hold. 
 
 Q.E.D. 
 
 Theorem 3«4.5 : If r =0, then a positive function f (x) is pseudo- 
 separable. 
 
 Proof : If r = 0, then whenever {A A , . . . , A.) and {B , ]L, . . . , B } 
 
 k k 
 are comparable sets of true and false assignments, Z A. > Z B. . 
 
 i=l x ' i=l 1 
 Therefore, f (x) can not be supersummable and this implies that f (x) 
 
 is pseudo-separable by Theorem 2.4.3. For incomparable sets of true 
 
 k k 
 and false assignments, the inequality Z A. < Z B. can not hold. 
 
 1=1 " i=1 ' Q.E.D. 
 
 Theorem 3-4.5 : If r > and [W$ T, A] marginally separates a positive 
 
 function f(x), A > Tw . . 
 ' — mm 
 
87 
 
 k k k k 
 
 Proof: Let T = 1- ( E B. - £ A.). Then E B. > E A.. If 
 
 i=l i=l 1=1 i=l 
 
 k 
 [W; T, A] marginally separates f(x), $• ( E B. ) < kT + kA/2 and 
 
 i=l X 
 
 k k h. k -> 
 
 W- ( E A.) > kT - kA/2. Let E B. - Z A. = (c , c , . .., c ) 
 
 i=l i=l i=l 
 
 n n 
 
 where all c. > and some c. > 0. Then clearly, E c.w. > E c.w 
 
 J . =1 i i - i=1 i min 
 
 k k 
 
 Since W- ( E B. ) < kT + kA/2 and W- ( E A. ) > kT - kA/2, .. 
 
 i=l x i=l x 
 
 k k n n n 
 
 W-( E B - E A. ) = Ewc <kA. Therefore, kA>.Ew.c >Ecw. 
 . . i . t i . , l i - -*•=, i l -, n i mm 
 
 i=l i=l i=l i=l i=l 
 
 n c. n c .. 
 
 E — rv . . Since T = E — -. 
 i=l- kmn i=l k ' Q.E.D. 
 
 or a > E -rrv . . Since T = E —r-, then A > T w . 
 k min . , k — mm 
 
 Notice that if some w. 4 w.. then E c.w. > E c.w . and 
 
 l ' ,v . , i I . , i mm 
 u i=l "■ i=l 
 
 A > r w . , 
 " min 
 
 Theorem 3* ^-.6 : If T < 1, then a positive function, f (x) is quasi- 
 separable. 
 
 Proof : If f(x) is extremely summable, then there exists k true 
 
 extremal assignments A , A , . . . , A and k false non-extremal assign- 
 
 12 k k k 
 
 ments B , B , ..., B such that L A. = L B. . Since each B. is 
 -Lc. K. _ i . _ i i 
 
 1=1 1=1 
 
 non-extremal and false, there is another false assignment U. > B. . 
 
 ^k^k 1 ^ 1 
 
 Since U, > B. , 1- (U, - B. ) > 1. Therefore, 1- ( E U. - E B. ) 
 l i' l l — ' . , i . , i 
 
 i=l i=l 
 
 k -> k 
 
 = l-( E U. - E A.) >k. Hence r > 1. Thus if V < 1, f(x) is not 
 
 i=l x i=l X 
 
 extremely summable and by Theorem 3.3-2, f(x) is quasi-separable. 
 
 Q.E.D. 
 
83 
 
 The above theorem gives the reason that the bounds on I? 
 which were found in Example 3«^.1> are sufficient to determine the 
 type of separation for this function. 
 
89 
 CHAPTER 4 SYNTHESIS OF PARTIALLY SEPARABLE FUNCTIONS 
 
 4.1 Introduction 
 
 In this chapter methods for realizing arbitrary partially 
 separable functions in networks of threshold elements are presented. 
 In the first section an algorithm to minimize the number of true 
 assignments (or false assignments) lying on a pseudo-separating 
 hyperplane is discussed. In the second section a procedure for 
 network synthesis for pseudo- separable functions is given. Algorithms 
 to minimize the number of assignments not segregated by a marginally 
 separating structure are developed in the third section. In the last 
 part a general synthesis procedure for marginally separable functions 
 is outlined. 
 
 4.2 Integer Programming Algorithm for Pseudo-Separability 
 
 Before any discussion about an algorithm which minimizes 
 the number of true assignments (or false assignments) lying on the 
 pseudo-separating hyperplane is presented, we should explain why -a 
 hyperplane which has the minimum number of true (or false) assignments 
 will be optimal for our network for a given function. The complete 
 procedure for network synthesis will be given in Section 4.3. (This 
 does not mean that the network discussed in Section 4.3. has the 
 minimum number of threshold elements. For a synthesis which guarantees 
 the optimum number of elements see Muroga's synthesis by integer 
 linear programming. [7] It requires in general many more inequalities 
 
90 
 
 than the synthesis of the present paper)* • Our primary concern is to 
 find a network of threshold elements to realize a pseudo- separable 
 function f(x), using the pseudo-separating hyperplane as the threshold 
 
 structure for one of the threshold, element s. jj 
 
 The network, which we propose to use, employs the pseudo- 
 separating hyperplane as a threshold element and then uses other 
 threshold elements to correct for either false (or true) assignments 
 which lie on this hyperplane. An important point is that we must 
 correct for all such true assignments or all such false assignments, 
 but not both. To see this, assume that f(x) is pseudo-separated by 
 [W; T]. Let the upper bound of [w; T] be g(x) and the lower bound 
 be h(x) . Assume that true assignments A,, A„, ..., A, and false 
 assignments B., IL, *.., B lie on [W*; T]. In using [w; T] as a 
 threshold structure, we have two alternatives. If k > e we can 
 assume that all assignments which lie on [W; T] are true for f(x) 
 and then add extra threshold elements to correct for those which are 
 actually false. This corresponds to representing [w; T] by its upper 
 bound g(xj and then correcting for B.. , B_, . .., B . If e > k we can 
 assume that all assignments on [w; T] are false and then correct the 
 network for those which are actually true. This corresponds to repre- 
 senting [w; T] by h(x) and adding extra threshold elements for 
 
 — * — i — i 
 
 A,, Ap, ..., A, . Either alternative is permissable since both of the 
 resulting networks will realize f (x) . When h(x) or g(x) is realized 
 as a threshold element, we then need extra elements to correct for 
 only the true or false assignments on [w; T]. Therefore, we want a 
 pseudo-separating hyperplane where the number of such extra elements 
 is minimal. 
 
91 
 Because we have the choice of correcting for either true or 
 
 false assignments, it would seem that we must devise two algorithms, 
 one to find [w ; T ] where the number of true assignments on this hyper- 
 plane is minimal and another to find [w ; T ] where the number of 
 false assignments is minimal. Then we would choose the hyperplane 
 which requires the least number of extra elements in our network. 
 Actually, we can combine both of these algorithms into a single 
 algorithm to determine [W; T] for which the total number of assignments 
 on this hyperplane is minimal, as follows: 
 
 Lemma 4.2.1: Let f(x) be pseudo -separated by [w ; T ] . Let 
 A, , Ap, ..., A, be all true (or false) assignments lying on [w ; T ]. 
 If [w ; T ] minimizes the number of true (or false) assignments lying 
 on it, then every hyperplane which pseudo-separates f(x) must contain 
 
 A* A* A* 
 
 1' 2' ' ' "' k* 
 
 Proof: Assume [w ; T ] minimizes the number of true assignments lying 
 on it. If some hyperplane [VL ', T ] segregates at least one of these A. 
 ( [w. ; T, ] may contain true assignments not on [W ; T ]), then 
 [W + W ; T + T ] will segregate all true assignments segregated by 
 either [vf 1 ; T 1 ] or [#; T^] (by Theorem 2.3.1). Hence, [#" + #j T 1 + T^] 
 will contain less true assignments than [W , T ] } a contradiction. 
 Similarly, if false assignments A,, A~, ..., A, lie on [w ; T ], then 
 
 no pseudo-separating hyperplane can segregate any A. . 
 
 Q.E.D. 
 
92 
 
 Theorem U.2.1: Let f(x) be pseudo- separated by [W% T ]•, [V?" 1 ; T 11 ] 
 
 and 
 
 1 m l- 
 
 [Tf;' T]. Assume that the number of true assignments on [w ; T ] is 
 
 minimal for all pseudo-separating hyperplanes. Assume that the number 
 of false assignments on [W j T ] is minimal for all pseudo-separating 
 hyperplanes and that the total number of assignments on [w; T] is 
 minimal. Then every assignment that is segregated by either [W ; T ] or 
 [W ; T ] must be segregated by [Wj T], i.e.. if true assignments 
 A, , A , ..., A lie on [W j T J and false assignments B , B c , . „ . , B 
 
 r~€.l 11-, -* ~* -»-»-» -> 
 
 lie on [W j T ], then only A , A , ..., A , B , B , ..., B will lie 
 on [w$ T]. (Note the following: this theorem does not assert that 
 the number of false assignments lying on [W ; T ] or the true assign- 
 ments lying on [w -, T ] is minimal. ) 
 
 Proof : Xf [W ; T ] minimizes the number of true assignments lying on 
 
 - * ~* — * — 1 1 — *L1 11 
 
 it and if A , A , . .., A lie on [W -, T ], then neither [W~ j T ] nor 
 
 [w; T] can segregate any of these A. (by Lemma U.2.1). Similarly, 
 
 neither [wj T ] nor [W; T] can segregate any B. . Therefore [W; T] 
 
 -> -» 
 can not segregate any A. or B. . Consider the hyperplane 
 
 [w + w ) T +■ T ]. This hyperplane pseudo-separates f(x) and 
 
 - * ~* i 
 
 contains all A. and B. by Lemma 4.2.1. Also it segregates all assign- 
 
 "~ 1 1 " - ' , *11 11 
 ments that are segregated by either [W ; T ] or [W j T ] by 
 
 Theorem 2.3.1. Therefore, it must segregate all assignments except 
 
 A , A , ..., A , B , B , ..., B . Because these assignments must also 
 
 lie on Iwj T], [w j T] must also segregate all other assignments. If 
 
93 
 
 — > — »]_ — >n i 11 
 
 [W; T] did not segregate all other assignments, then [W +■ W ; T + T ] 
 
 — > ; 
 
 would contain less assignments than [Wj T] contradicting the assumption 
 
 that the number of assignments on [W> T] is minimal. 
 
 Q.E.D. 
 
 r~tl It 
 
 As stated in the theorem, [W > T J, while segregating the 
 largest number of true assignments, need not segregate the most false 
 
 assignments. In addition to B , B , . .., B , there may exist other 
 
 — »}_ ]_ ~~ *11 11 
 
 false assignments on [W j T ]. Similarly, [ w \ T ] may contain more 
 
 true assignments than A , A , ..., A . However, LW; TJ is guaranteed 
 
 to contain both the least number of true assignments and the least 
 
 number of false assignments. 
 
 Therefore, when we need to solve two problems, one to minimize 
 the number of true assignments on the pseudo-separating hyperplane and 
 the other to minimize the number of false assignments, it is sufficient 
 to solve one problem to minimize the total number of assignments there. 
 
 We now introduce a definition to facilitate our discussion 
 of the threshold elements used in our synthesis procedure. 
 
 Definition U.2.1 : Let f(x) be a separable function. Assume that 
 [W$ T] separates f(x). [Wj T] is called a threshold structure for 
 f(x) if for any assignment A, f (A) = 1 implies W-A > T and f (a) = 
 implies W«A < T - u where u, a positive constant called the gap of 
 f(x), has been predetermined. 
 
 The gap u is necessary for the reliable operation of the 
 
 — > 
 
 threshold element with structure [Wj T] and is predetermined. [7] 
 
9 k 
 By regarding w./li and T/U as a new structure, u may always be normalized 
 to 1. In the rest of this section we will assume that the gap is 1. 
 
 let us now discuss the actual algorithm to minimize the total 
 number of assignments on a pseudo-separating hyperplane for a given 
 function. 
 
 Let f (x) be an arbitrary Boolean function. We want to develop 
 and algorithm based on integer linear programming to find the optimal 
 pseudo-separating structure for f(x). In other words, we want to find 
 [W; T] such that 
 
 -» — > — » 
 
 (1) if f(A.) = 1, W.A. > T 
 
 l l — 
 
 -> -> -> 
 
 (2) if f(B.) = 0, W-B. < T 
 
 (3) the number of assignments which lie on [Wj T] is 
 minimal. 
 
 Consider conditions (l) and (2). These can be rewritten as constraints 
 using slack variables a. >~ and a . > as follows: 
 
 (1)' if f(A\) = 1, W«A. - a. = T 
 
 l ' i i 
 
 (2)' if f(B*.) = 0, WoB. +a. = T 
 
 Now let us turn to the problem of choosing an objective 
 function to minimize the number of assignments which lie on the hyper- 
 plane [wV T]. 'To minimize the number of such assignments, -consider ,tha.t 
 if A., or B resides on [w; T], then the corresponding a. (or a.) must be 
 
 J -^ J 
 
95 
 
 0. Thus, we want to introduce into the objective function a penalty for 
 
 each assignment of CC. = or a. =0. Let J. be a variable with the 
 
 l j l 
 
 following properties: a. +• J. ^ .1 for all i = 1, 2, . .., 2 and 
 
 J. = or 1, tvhere lisMihe) 'normalized gap). We then add a penalty 
 
 to the object function if J. 4 (or a. = 0). 
 
 ° i ' l n 
 
 n 2 
 
 Define the objective function as S |w. |+ F( E J.) where 
 
 i=l 1 i=l X 
 
 n 2 n n 
 
 F » E lw. J Then our procedure will be to minimize F(Z -J. )+ E |w. I 
 
 i=l 1 i=l 1; i=l x 
 
 under the constraints 
 
 (1) if f(£.) = 1, f-1. - a. = T 
 
 l ' i i 
 
 (2) if f (B ) = 0, W-B* + a, = T 
 
 J' 9 J 
 
 (3) a. $i 0, a. > formal!, i and j 
 
 (U) a. + J. > 1 for all i and j 
 11 — 
 
 (5) j. = v or 1 for all i and j. 
 
 2 n 
 If E J. is minimized, then the number of assignments on 
 i=l X 
 
 the hyperplane is minimized because if J. = 0, OL. must be positive 
 
 — » 
 and the corresponding assignment A. is not on the hyperplane. Therefore 
 
 the resulting hyperplane [Wj'.T] must segregate the most assignments and 
 
 hence must be optimal. 
 
 Because we know that [0; 0] pseudo-separates every Boolean 
 
 function (by Lemma 2.2.1), we can use [Oj 0] as an initial feasible 
 
96 
 solution to our problem and hence we do not need artificial variables. [7]. 
 
 k.2) Threshold Synthesis of Pseudo-Separable Functions 
 
 In this section we will use the algorithm which was presented 
 in the previous section to develop a synthesis procedure for pseudo- 
 separable functions . 
 
 The basic threshold configuration which we will use in this 
 synthesis is shown in Figure 4.3.1* when the threshold element JI repre--:- 
 sents the pseudo-separating structure, and also has additional inputs 
 from the elements Q , fi , ..., Q , which represent corrections:' for 
 either the true assignments on this hyperplane or the false assignments. 
 
 Figure U.3.1; Configuration for Synthesis of Pseudo-Separable Function 
 
97 
 Let f(x) be an arbitrary pseudo- separable function. Let [w; T] 
 
 be the pseudo -separating hyperplane which segregates the greatest number 
 
 of assignments. (This hyperplane can be found by using the algorithm 
 
 outlined in the last section) . Let h(x) be the lover bound of [W; T] 
 
 and let g(x) be its upper bound. Assume that true assignments 
 
 A*, , A* 2 , . .., At and false assignments K, Bp, . .., B lie on [w; T]. 
 
 Let e > k. Then in order to minimize the number of extra elements 
 
 ft., ft p , . .., ft , we will represent [w; T] by its lower bound h(x) and 
 
 will correct for all true assignments A.., A p , . .., A, . (Actually, we 
 
 have the alternative of deleting all true assignments on [w; T] in 
 
 h(x) , and using the resulting partially specified function to represent 
 
 II . For notational ease, however, we will simply use h(x) .) 
 
 The problem of determining an optimal threshold structure 
 , , n 
 
 [w ; T ] for h(x) (optimal, in the sense that £ | w. | is minimal for 
 
 i=l 1 
 the gap u or by normalization for the unit gap) has been investigated 
 
 previously. Muroga, Toda, and Takasu present a clear discussion of 
 
 this problem. [8] Therefore, the procedure will only be outlined here. 
 
 (1) Convert h(x) into a positive function p(x) 
 
 (2) Solve the following linear programming problem for [ton; T ] 
 
 n 
 Minimize £ w. under the constraints 
 i=l X 
 
 (a) all w p > (^.5.1) 
 
 (b) if v (t) = 1, \P-t> T p 
 
 (c) if p(ft = 0, fc*.B*<T P - u. 
 
 (3) Form [w; T ] by negating all weights corresponding to 
 variables which were changed in converting h(x) into a positive function 
 and subtracting them from T P . 
 
98 
 Having found [w ; T ], threshold element IT in our network is 
 now determined as having structure [w , a , Op, . .., ou; T ] where a. 's 
 correspond to inputs from ft. 's. 
 
 We now show how to construct each extra element £2. . Let A. 
 
 i l 
 
 be any true assignment lying on [w; T]. Let A. = (a.,, a. p , . .., a. )« 
 
 t 4. /"N (2a. ,-1) (2a. -l) . -1 - , 1 
 Let m. (x) = x_ ll ... x in where x. = x. and x. = x.. 
 i 1 n J J J J 
 
 Assume that g. of the a., coordinates are 1. Define [w. : T.] for ft 
 
 & i ij i' i J i 
 
 for gap u as follows: 
 
 (1) if a. . = 1, let w.' . = 1 
 
 (2) if a. . = 0, let w.' . = -1 
 
 ij ij 
 
 (3) let T.' = q. . 
 ' i l 
 
 then [WJ; T.'j separates the function m. (x) with unit gap. If B is any 
 other assignment , then W*.' •]? < T .' - 1. Therefore Jj/tl; TJ ] has the 
 normalized gap, 1. To assure the gap to be u, we multiply w.' and T.' by u. 
 Let W. = uw.' and T. = uT.' . Then [W. ; T. ] is a threshold structure and 
 
 i i 
 ft. will have gap u. 
 
 The only other thing which must be discussed before the network 
 in Figure ^.3.1 is completely specified, is the weights which the outputs 
 of ft , ftp, ..., ft will have when input into n . These weights must be 
 large enough so that if the output of some ft. is 1, then n will also out- 
 put 1. In other words, if we let the input weight corresponding to output 
 of ft. be a., then we must have: 
 
 [a. - (Z |w h |)] >T h or a. > Z |w h | + T h . 
 
 i=l i=l 
 
 (Actually, we need only sum w. which are negative, but the Labove lower 
 bound on a. is sufficient to guarantee the operation of the circuit) . 
 
99 
 
 We now will show that the network we have designed realizes the 
 function f (x) . Let B be any false assignment to f (x) » Since [W; T] 
 pseudo- separates f (x) and h(x) is- the upper bound, h(B) =0. Since each 
 element SI. corresponds to one true assignment on [w; T], the output from 
 each of these elements will be (T.< Therefore, the element II will output 0. 
 If A is any true assignment, there are two possibilities, either h(Aj = 1, 
 or h(A) =0. If h(Aj = 1, then II must output 1 because it realizes h(x) . 
 If h(A) = and A is a true assignment, then g(A) = 1 and A must reside 
 in the hyperplane [W*; T] (by Corollary 2.5.1) and hence one of the ft. 
 must have output 1. Because 
 
 h n h 
 a. > T + £ | w 4 j t then n must also have output 1. 
 
 i=l 
 
 Therefore, this network realizes f (x) . 
 
 In most of this work we have assumed that there were more false 
 assignments on [w; T] than true assignments. However if there are more 
 true assignments than false, we can repeat the same procedure, with slight 
 modification, using the upper bound instead of the lower bound and then 
 correcting for false assignments instead of true. 
 
 h-.k Integer Programming Algorithm for Marginal Separability 
 
 The procedure for synthesizing a marginally separable function 
 in a network of threshold elements which we will use in very similar to 
 the technique discussed earlier in this chapter for pseudo -separable 
 functions. This approach entails representing the marginal structure 
 which segregates the largest number of true (or false) assignments by 
 its marginal lower (or upper) bound and then adding extra elements to 
 
100 
 
 correct for the true (or false) assignments which were not segregated 
 by the original structure . 
 
 Marginal separation differs from pseudo- separation in that it 
 does not possess the property of Theorem ^.2.1, i.e., we have no guarantee 
 that by finding a marginal structure which segregates the largest total 
 number of assignments we automatically can find no other structure which 
 segregates more true assignments but less false ones (or more false assign- 
 ments and less true ones) . Therefore, if we want to minimize the number 
 of extra elements we must add to the network, we must solve two problems. 
 The first problem will be to m i ni m i ze the number of true assignments not 
 segregated by the marginal structure and the second will be to minimize 
 the number of false assignments not segregated. 
 
 We now discuss an algorithm which will find [wl ; T-.,A,] which 
 maximizes the number of segregated true assignments. Later we will show 
 how this algorithm can be modified to .maximize the number of segregated 
 false assignments. 
 
 Let f (x) be an arbitrary Boolean function. We want to devise 
 an algorithm to find [W.; T,,^] which segregates the largest number of 
 true assignments for f (x) . If [W.; T-.,^] marginally separates f(x), 
 then we must have: 
 
 ■ 
 
 (1) if t(t ± ) = 1, then \'t ± £ T - Aj/2 (i = 1, 2, ..., k) 
 
 (2) if f (I? ) = 0, then W^-lt < 0^ + Aj/2 (i = k + 1, ..., 2 n ) 
 
 Let W- = (w..-., ..., w.. ) and let (? = (c n , ..., c ) be any 
 
 1 11' ' lir 1' ' n 
 
 assignment. Let w' = (v,,, ..., w ,&-/2) and C' = (c.., ..., c , 5) 
 
101 
 
 where 5 = 1 if f(C^) = 1 and 5 =-1 if f(C?) = 0. Using this notation, the 
 above requirements become: 
 
 (1) if f(t.) = 1, then ft 1 'tl > T n -and 
 
 l 1 i — 1 
 
 (2) if f(lf. ) = 0, then fl •#! < T n 
 x ' i' ' 1 l — 1 
 
 As we did in the previous algorithm in Section ^.3> we now add 
 
 slack variables a. > for all i. Our constraints then become: 
 i — 
 
 (1) if f(t.) = 1, tf'-A*. - a. = T for 1, 2, ..., k 
 
 i ' 1 i i l ' ' 
 
 (2) if f(]?.) = 0, tf'.it + a. = T n for i = k + 1, ..., 2 n 
 
 (3) a. > for i = 1, 2, ..., 2 n 
 
 To minimize the number of true assignments not segregated 
 by the marginal structure, we must penalize the objective function for 
 any true assignments such that ¥ »A. < T + A,/2. ; Such assignments must 
 have jc a. < A* • As we did for pseudo-separable functions, we introduce 
 a variable J. for each a.(i = 1, 2, . .., k) with the following properties: 
 
 a i + ('^1 + 1 ^ J - - \ + 1 and J - = ° or 1 « 
 
 An important fact about such a procedure as this is that 
 
 we can predefine A, at the outset and still be guaranteed that we will 
 
 find a marginal structure which segregates the most true assignments, 
 
 because if [w..; T ,A, ] segregates the most true assignments and A'. r 
 
is any other number, then 
 
 IF 1' A' 1' 
 
 A' : 
 
 102 
 
 ■will then segregate exactly 
 
 those assignments segregated by [w, ; T ,A, ] • Therefore, -we can let 
 
 A, = 1. Then J. is defined so that a. + 2J. > 2 and J. = or 1. 
 T. i 11-1 
 
 k n 
 
 We then add F( E J.) to our objective function where F » E |w . | 
 i=l x i=l - LL 
 
 Because [0; 0] pseudo-separates every function (by Lemma 
 2.2.1), the marginal structure [0; 0,0] marginally separates every 
 function and hence can be used as an initial feasible solution. 
 Therefore, there is no need for artificial variables. [7] (Kenneth 
 Breeding has formulated a similar procedure for a similar problem) . 
 
 Our mixed integer programming algorithm which will determine 
 [w j T ,l] segregating the greatest number of true assignments is 
 
 K n 
 
 to minimize F( E J.) + E |w n . I under the constraints 
 . , l , ' li ' 
 
 i=l n=l 
 
 (1) tfl-tl -a. = T if f(t.) = 1 (i :=.1,.2, , k) 
 
 (2) tfl'tl + a. = T n if f(lf.) = (i = k+1, ..., 2 n ) 
 
 1 i i 1 i 
 
 (3) a. > for i = 1, 2, ..., 2 n (^.l) 
 (k) a. + 2J. > 2 for i = 1, 2, ..., k. 
 
 l — 
 
 (5) J. = or 1 for i = 1, 2, ..., k. 
 
103 
 
 k 
 Again note that "by adding F( E J.) as a penalty to our 
 
 i=l x 
 
 objective function we can minimize the number of true assignments 
 
 n 
 not segregated by the .marginal structure by minimi zing E J. because 
 
 i=l 1 
 J. = 1 if and only if A. is not segregated. 
 
 As we have stressed, the algorithm outlined in (^A.l) 
 ■will determine the marginal structure which segregates the largest 
 number of true assignments. No claim of any sort is made for the 
 number of false assignments segregated. Therefore, in order to find 
 the juarginal structure [W_, T n ,A_] which segregates the most false 
 assignments, we must solve another mixed integer programming problem 
 which is obtained by modifying this algorithm. 
 
 If we want to minim ize the number of -false assignments not 
 segregated by the marginal structure, we must penalize the objective 
 function for any false assignments such that « 'if. > T fi - A-,/2. -A 
 false 'assignment B. will not be segregated if < a. < A.. Therefore, 
 
 the variable J. must be introduced so that a. + (<v + l)«T. > A. + 1 and 
 
 n 
 J. = or 1 for i = k + 1, ..., 2. Hence the algorithm to minimize 
 
 the number of non-segregated false assignments is to: 
 
 c n 
 
 d n 
 
 minimize F( Z J.) + Z w . under the constraints 
 
 i=k+l i+I 
 
 (1) $^l - a..="T ,if i\t ± ) = 1 (i = 1, 2, _., k) 
 
 (2) ^.g: + a. = T Q , if f(B\) = (i = k + 1, ..., 2 n ) 
 
104 
 
 (3) a. > for all i = 1, 2, ■'-'..;, 2 n (UA.2) 
 
 (4) a. + 2J >>2 for i * k+1, . .., 2 n 
 
 (5) J. = or 1 f or i = k+1, ..., 2 n 
 
 4.5 Threshold Synthesis of Marginally Separable Functions 
 
 The synthesis procedure in this section is very similar to 
 the procedure for threshold synthesis of pseudo-separable functions 
 outlined in Section 4.3. The basic configuration of the circuit -which 
 we will use in our threshold network will be the same as before . The 
 reader is directed to Figure 4.3.1 for this configuration. Once again, 
 II represents the marginal separating structure as a threshold element 
 and the elements Q, , Q p , . .., 0, represent corrections for either the 
 true or false assignments not segregated by this marginal structure, 
 depending upon which assignments not segregated have a fewer number. 
 
 Let f (x) be an arbitrary marginally-separable function. 
 Let [w.,; T, ,A, ] be the marginal structure which segregates the most 
 true assignments and [w n ; T ,A~.] the structure which segregates the 
 largest number of false assignments. (These structures can be found 
 by using the algorithms (4.4.1) and (4.4.2).) Assume that true assign- 
 ments A. ,. A~, ..., A are not segregated by [w, ', T .,&. ] and false 
 assignments E_ , B_, ..., B are not segregated by [W ; T q ,aJ. 
 
105 
 
 Assume that k> e. Then in order to minimize the number of extra 
 corrective elements ft . , we use [W-j ^-i'^i) an ^ we will represent 
 
 this marginal structure "by its marginal lower bound h (x) and then 
 
 — > — > — > 
 correct for A, • A„, .... A in the network, 
 1 d e 
 
 Having determined that -we will represent [w ; T-,A,] by 
 
 its marginal lower bound h (x) and add extra elements for 
 
 — > — > — > 
 
 A,, A p , . .., A , the rest of the procedure is exactly the same as we 
 
 outlined in Section h .3 and will be omitted here. 
 
 Notice that if e >> kj then we must use [^L; T n ,/V] and 
 we must represent this marginal structure by its marginal upper 
 bound . g n (xj and then correct for false assignments BL , B p , . .., B, 
 by adding extra elements, Q. . 
 
 There is another approach to synthesis of marginally separable 
 functions. Assume that some procedure persists which yields the ; '■ 
 structure [W, ; T,,,^] for which the total number of assignments of 
 f(x), not segregated, is minimal. (Such an algorithm can be obtained 
 by combining the mixed integer linear programming algorithms (U.^.l) 
 and (h.k.2).) lUse two threshold elements, one corresponding to the 
 marginal upper bound of [w ; T , A,], g, (x) , and the other corres- 
 ponding to its marginal lower bound nix) to separate all of the 
 assignments segregated by [\f ' ; T,,Al. Let f , (x) be the partially 
 specified function which results from f (x) when all assignments 
 segregated by [w ; T,,A,] are deleted, (all assignments A where 
 g, (A) = h (a) ) . Repeat the procedure to find an optimal marginal 
 
io6 
 
 structure for f , (x) , [w p ; T , A~J by solving the mixed integer linear 
 programming. Again construct two threshold elements corresponding to 
 the marginal upper and lower bounds of [w p ; T p , A-,]. 
 
 If all assignments for f(x) are segregated by either 
 [W, ; T ,A.]or fw p j T p ,Ap] "we will terminate the procedure. If not, 
 find f p (x) by deleting all assignments segregated by [w p ; T p ,/V,] from 
 f , (x) and repeat the procedure. In this r, way a finite chain of marginal 
 structure can be found -with the property every assignment of f (x) is 
 segregated by at least one structure in the chain (by Theorem 3-^«3)« 
 
 Figure 4.5.1; shows how the pair of threshold elements g. and 
 
 h. corresponding to [ w . °, T.,A. ] will be connected to the pair of -J. 
 
 elements g.,, and h. n corresponding to [w. ., ; T. ...A. -. ] . We want 
 D i+1 l+l * ° l+l l+l' i+I 
 
 to weigh the inputs of g. and h in such a way sq that if the 
 
 output of g. is the same as the output of h., then both g. and h. 
 
 will have the same output. Let g. have structure fw ;T. , ] and 
 
 h. n have structure [w : T. n ]. Let the input weight a. corres- 
 l+l i+I l 
 
 n . 
 ponding to g. be such that a. ' <■- £. |w. j and let the input weight 
 1 x 3=1 J 
 
 corresponding to h. be B. = | a. |. Then if g. = h. = 1, because of the 
 
 negation (expressed with a short bar in Figure 4.5.1) of output of 
 
 g- }&• = and both g. , and h. n will output 1. Similarly if g. = h. = 0, 
 •^i'^i & i+l i+I * J °i l 
 
 then both g. n and h. , will output 0. 
 i+I i+I 
 
107 
 
 Figure U.|?.l : Configuration for Synthesis of Marginally Separable 
 Function. 
 
i'oa 
 
 CHAPTER 5 SUMMARY 
 
 The major effort in this paper has been to explore the 
 theoretical properties of pseudo and marginal separation. Most of 
 this investigation has been directed toward the algebraic properties 
 which a Boolean function must have if it is to be pseudo or marginally 
 separated. 
 
 The basic concepts of pseudo-separation and marginal separation 
 were introduced in Chapter 1. 
 
 In Chapter 2 some general properties of pseudo-separable 
 functions were presented. The concept of a supersummable function was 
 introduced and used to determine necessary and sufficient conditions 
 for pseudo-separability. The relationship between simultaneously 
 realizable functions and the upper and lower bounds for a hyperplane 
 were also discussed as well as a method for determining better bounds 
 if they exist. The correspondence between separable and pseudo- 
 separable functions was also discussed. 
 
 Chapter 3 was devoted to the algebraic properties of margin- 
 ally separable functions. The idea of quasi-separation was introduced 
 and a necessary and sufficient condition for this type of separation 
 was presented in terms of extreme summability. The concept of the 
 degree of supersummability was also discussed and the relationship 
 between this parameter and the various types of separation was explored. 
 
 Integer linear programming formulations to find optimal 
 segregating structures for an arbitrary partially separable function 
 
109 
 
 were discussed in Chapter k. Synthesis procedures based on these 
 algorithms for realizing a function in a network of threshold elements 
 were also introduced. 
 
 This work represents an attempt to generalize the concept of 
 a threshold element and the idea of total separation. It serves two 
 purposes. First, it presents a new approach to the problem of 
 threshold synthesis in terms of a more general type of separation. 
 Second, it provides better insight into the properties of separable 
 functions by allowing them to be examined as special cases of more 
 general types of separated function. 
 
110 
 LIST OF REFERENCES 
 
 [I] Chow, C. K. , "On the Characterization of Threshold Functions", 
 Switching Circuit Theory and Logical Design , pp. 31-38* Sept. 1961. 
 
 [2] Elgot, C. C, "Truth Functions Realizable by Single Threshold 
 Organs," Switching Circuit Theory and Logical Design , pp. 225-21+5, 
 Sept. 196l. 
 
 [3] Fan, K., "On Systems of Linear Inequalities," Linear. Inequalities 
 and Related Systems, Annals of Mathematical Studies #38, Princeton 
 University Press, 195^ 
 
 [U] Hadley, G., Linear Programming , Addison-Wesley Publishing Company, 
 1962. 
 
 [5] Hadley, G. , Non-linear Programming , Addison-Wesley Publishing 
 Company, I96U. 
 
 [6] McNaughton, R., "Unate Truth Functions", IRE Transactions on 
 Electronic Computers , Volume EC -10, pp. 1-6, March I96I. 
 
 [7] Muroga, S., "Threshold Logic", Lecture Notes, Department of 
 Computer Science, University of Illinois, 1966. 
 
 [8] Muroga, S., Toda, I., and Takasu, S., "Theory of Majority 
 Decision Elements", Journal of the Franklin Institute , Volume 271* 
 pp. 376-418, 196l ' 
 
 [9] Paull, M. C, McCluskey, E. J., "Boolean Functions Realizable with 
 Single Threshold Devices", Proceedings of IRE , pp. 1335-1337, July i960. 
 
 [10] Robinson, A., "Some Remarks on Threshold Functions", IBM Report, 
 1963. 
 
 [II] Winder, R. 0., "Threshold Logic", Ph„D. Dissertation, Princeton 
 University, 1962. 
 
 [12] Yen, Y. , "Multi-threshold Logic", Ph.D. Dissertation, University 
 of Illinois, 1966. 
 
Ill 
 
 VITA 
 
 Thomas Andrew Slivinski was born in Evergreen Park, Illinois 
 in 19^+2. He received a B. A. degree in Mathematics from Lehigh 
 University Bethlehem, Pennsylvania, in June 19^3 • ^ n September 1963, 
 he began his graduate study at the University of Illinois, Urbana, 
 where he received a M. S. degree in Mathematics in February 1965. 
 
 Since September 19^3^ he has been working as a research 
 assistant in the Digital Computer Laboratory, the University of 
 Illinois, Urbana. 
 
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