LIBRARY OF THE 
 
 UNIVERSITY OF ILLINOIS 
 
 AT URBANA-CHAMPAIGN 
 
 r\o.2&T-292 
 
 cop. 2 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/designprocedures288whit 
 

 /rt 
 
 tzZ^X- 
 
 Report No. 288 
 
 DESIGN PROCEDURES FOR 
 CELLULAR LOGIC ARRAYS 
 
 by 
 Willard Francis Whitaker III 
 
 September 20, 1968 
 
 JHE LIBRARY. OF. i 
 
 NOV 9 1972 
 
 UNIVERSITY OF ILLINOIS 
 AT URBANA-CHAMPAiGfcJ 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 
 I \\ Mq:lHiiW*]aiMl?L«lKS WMMH 
 
Report No. 2^8 
 
 DESIGN PROCEDURES FOR 
 CELLULAR LOGIC ARRAYS* 
 
 by 
 
 Willard Francis Whitaker III 
 
 September 20, 1968 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 Submitted in partial fulfillment of the requirements for the degree 
 of Master of Science in Electrical Engineering, September 1968. 
 
Ill 
 
 ACKNOWLEDGEMENT 
 
 The author wishes to thank his advisor, Professor 
 Sylvian Ray, Mrs. Donna Stutz for typing the thesis, and John 
 Otten for preparing the drawings . 
 
IV 
 
 TABLE OF CONTENTS 
 
 Page 
 
 1. INTRODUCTION 1 
 
 2. DESCRIPTION OF THE ARRAY AND NOTATION 3 
 
 3. COMBINATIONAL CIRCUITS 7 
 
 Decomposition of Functions 7 
 
 Algorithm for Single-Output Combinational Circuits 10 
 
 Proof That Algorithm Terminates 12 
 
 Upper Bounds on Size of Array lU 
 
 Example l6 
 
 h. SEQUENTIAL CIRCUITS 20 
 
 Form and Decomposition of Functions 20 
 
 Algorithm for Sequential and Multiple-Output Circuits 22 
 
 Examples 23 
 
 5. CONCLUSION 31 
 
 BIBLIOGRAPHY 33 
 
LIST OF FIGURES 
 
 Figure Page 
 
 1 Basic Cell Inputs and Outputs 1+ 
 
 2 Numbering of Cells k 
 
 3 Generation of f from u, v, and w 8 
 
 k Example l._ Combinational Circuit 19 
 
 Example 2. -Sequential Circuit 26 
 
 6 Example 3-Feedback Shift Register 30 
 
1 . INTRODUCTION 
 
 Several types of cellular logic arrays have "been developed. 
 These arrays consist of a number of identical cells which are con- 
 nected to perform logical functions. The cells usually have parameters 
 which determine the function to be produced in each cell and the inter- 
 connections among the cells. 
 
 One of the more significant of the basic cells which have 
 been designed is the cutpoint cell developed by R. C. Minnick. This 
 cell produces combinational and sequential functions of two variables. 
 In his article, Minnick describes the cutpoint cells and presents an 
 algorithm for constructing an array of these cells to produce a given 
 function. 
 
 H. A. Finkelstein, in seeking an improvement over the cutpoint 
 
 cell, has designed cells which can implement functions of three variables 
 
 2 
 in one cell. However, he did not develop an algorithm for programming 
 
 an array of these cells to produce more complicated functions. 
 
 Minnick, R.C., "Cutpoint Cellular Logic", IEEE Transactions on 
 Electronic Computers, December I96U - 
 
 2 
 Finkelstein, H. A. , "Design of Digital Conrputer Circuits Using a Basic 
 
 Logic Cell" S M.S. Thesis, University of Illinois, Department of Computer 
 
 Science Report No. 287 May 2k , 1968. 
 
The purpose of this thesis will be to develop procedures 
 for programming an array of cells similar to Finkelstein' s hexagonal 
 cells. It would be desirable to have the capability of forming an 
 array to produce any combinational or sequential logic function. 
 The procedures should specify an array which makes reasonably 
 efficient, but not necessarily optimum, use of the cells. The pro- 
 cedures should completely specify the array, so that a computer can 
 be used to carry out the procedures and control the array without 
 further intervention. 
 
2. DESCRIPTION OF THE ARRAY AND NOTATION 
 
 The cell used is a hexagonal cell similar to those designed 
 by Finkelstein. This cell has three inputs and three outputs, as 
 shown in Figure 1. It has the advantage over a four-terminal square 
 cell that every cell in an array can have the same input-output con- 
 figuration while allowing for three inputs to each cell. Also, in 
 the examples given by Finkelstein, usually fewer hexagonal cells 
 than square cells are required to realize a given function. 
 
 The cells are connected into an array as shown in Figure 2. 
 
 The cells are numbered according to coordinates i,j. The coordinate 
 
 system begins at the lower right-hand corner of the array and uses i 
 
 as the vertical coordinate and j as the horizontal coordinate. The 
 
 output will be produced in cell 1,1 at the lower right-hand corner of 
 
 the array. The inputs to cell i.j are A.,, B.., and C.., and the 
 
 ij ij ij 
 
 outputs are D. . , E. . . and F. . . 
 ij ij ij 
 
 The set of functions realizable in each cell has been modified 
 
 to facilitate use of the algorithm which will be presented. These 
 
 functions are listed in Table 1. Each cell is specified by a five-element 
 
 vector Z. . = (Z..., Z.._, Z.._ s Z.._, Z.._,). The first three components 
 ij iJA' ijB' ijC' ijD* ijE' * 
 
 of Z. . specify whether the input or its complement is to be used, or 
 the input is to be inhibited. 
 
B 
 
 A * 
 
 ■►D 
 
 FIGURE I. BASIC CELL INPUTS AND OUTPUTS 
 
 r v^( 1 r >f< 1 r V< 1 r >r^ 
 
 \ 
 
 \ 
 
 \ 
 \ 
 
 \ 
 
 1 — v 
 
 i =4 
 
 i = 3 
 
 \ 
 
 \ 
 
 \ 
 
 V 
 
 • -\- 
 
 i = 2 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 i = I 
 
 J + 
 
 \» \* \<° \" 
 
 FIGURE 2. NUMBERING OF CELLS 
 
If Z... = 0, the A input is to be inhibited. If Z = 1, 
 
 1 J A X J A 
 
 the A input is to be used as is. If Z = 2, The A input is to be 
 
 1 J A 
 
 complemented. The B and C inputs are specified similarly. In Table 1 
 
 expressions such as (A, A) are used. This expression means that A 
 
 to be used if Z. ,. = 1, and A is to be used if Z. JA = 2. If Z. . = 0, 
 ljA ljA ijA 
 
 the expression is to be deleted. The F output is C if Z. .„ is or 1, 
 
 ijC 
 
 and is c" if Z. ,_ = 2. 
 
TABLE 1: FUNCTIONS OF CELL 
 
 Z. JT ^ or Z.,„ Function generated at D or E 
 
 ljD iJE 
 
 1 
 
 2 1 
 
 3 (A,A)(B,B)(C,C) 
 
 k (A,A) + (B,B) + (C,C) 
 
 5 (A, A) © (B,B) © (C,C) 
 
 6 (A,A)(B,B)(C,C) + (A,A)(B,B)(C,C) 
 
 7 (C,C)(A,A) + (C,C)(B,B) 
 
 8 (A,A)(B © C) + (A,A)(B,B)(C,C) 
 
 9 (A,A)(B © C) + (A,A)(B,B)(C,C) 
 Flip-flop 
 
 inputs next output 
 
 (A, A) (B,B) (C,C) D = Y E = Y 
 
 1 
 
 
 
 1 
 
 y 
 
 y 
 
 1 
 
 1 
 
 1 
 
 
 
 l 
 
 
 
 
 
 1 
 
 i 
 
 
 
 
 
 1 
 
 1 
 
 y 
 
 y 
 
 d 
 
 d 
 
 
 
 y 
 
 y 
 
3. COMBINATIONAL CIRCUITS 
 
 Decomposition of Functions 
 
 Since three-input cells are being used, it is desirable 
 that each function be decomposed into three functions. One of 
 these three must be an input variable or its complement, since 
 only these are available at the C input. 
 
 Any function of n variables can be expressed in terms of 
 one input variable and two functions of n-1 variables. 
 
 f(x l5 ...,x n ) =x n g(x 1 ,...,x n _ 1 ) + xh(x 1 ,...,x n _ 1 ) 
 
 Proof: Express f in sum-of -products form and factor out all 
 
 occurences of x and x . 
 n n 
 
 f(x l5 ...,x n ) ^^(x^...^) + x n v(x 1 ,...,x n _ 1 ) + w(x 1 ,...,x n _ 1 ) 
 
 = xu + xv+(x + x )w 
 n n n n 
 
 = x (u + w) + x (v + w) 
 
 n n 
 
 = x g(x , . . . ,x . ) + x h(x_, ,. . . ,x ) 
 n A* n-1 n 1 n-1 
 
 for some functions of n-1 variables u, v, w, g, and h such that 
 
 g = u + w and h = v + w. u, v, and w have no common terms. 
 
 Figure 3 illustrates how f can be generated in this manner. 
 
♦ f 
 
 FIGURE 3. GENERATION OF f FROM u, v, AND w. 
 
Basically, the algorithm implements single-output combina- 
 tional functions in a tree structure with cell assignments to fit 
 the tree to the restrictions of the array. The functions are decomposed 
 in the manner shown above. However, if u, v, or w is zero, a simpler 
 arrangement is used than that shown in Figure 3- The only case where 
 both outputs of a cell are used is illustrated in Figure 3, and both 
 outputs are w. Therefore, it is possible to let both outputs of each 
 cell be the same. The output function will be f. .. 
 
 A push-down stack, L, will be used to store the locations of 
 branch points to be returned to. The top location of L is L . The 
 location stored in L will be the last one stored there which has not 
 yet been read out . 
 
 Each point where an input variable is found at the A or B 
 input to a cell corresponds to a branch of the array. The branch 
 consists of all cells connected in a path from this point to the 
 output, plus any cells that drive this input. The branches are numbered 
 in the order of implementation by the algorithm. This numbering corre- 
 sponds to the location of the branch in the array, numbered from upper 
 right to lower left. 
 
10 
 
 Algorithm for Single-Output Combinational Circuits 
 
 1. Initially, i = 1, J = 1, f = f (x , . . . ,x ) . 
 
 2. If C. is specified: 
 
 a. If Z... . . J 2, C. - C... , . 
 
 i+l,j-l,C ij i+I, j-1 
 
 b. If Z.^ n . _ = 2, C. . = C. , , _ 
 
 i+I, j-1, C ij i+I, j-1 
 
 3. If C, n , _ is not specified: 
 
 i+l, J-1 
 
 a. If some x^ appears in every term of the minimal S of P 
 
 expansion of f . , let C. . = x. (if more than one such x_ , use 
 
 the largest k). 
 
 t>. Otherwise, choose from the input variables on which f . . 
 
 ij 
 
 depends the one with the largest k, and let C. . = tl. 
 
 ij k 
 
 k. If cell i+1, j is unused, go to 9. 
 
 5. Iff., does not depend on C. j5 let f. .., - f. ,; 
 
 ij ij i,J+l iJ 
 
 Z. . = (1,0,0,U,U); set j = j+1; and go to 2. 
 
 6. If g. . j and h., ± 0, let f. ,._ = f. ,; Z. . = (l,0,0,U,M; 
 
 ij ij i.J+1 ij ij 
 
 set j = j+1; and go to 2. 
 
 7. If f . . = C or f . . = c": 
 
 ij iJ 
 
 a. If f . . = C, let Z.. = (0,0,1,U,U). 
 
 ij ij 
 
 b. Iff.. = C, let Z. . = (0,0, 2,1+, U). 
 
 ij ij 
 
 c. If L is empty, stop. 
 
 d. Set i,j = L and go to 2. 
 
 8. (Either g or h is zero.) 
 
 a. If g. i 0, let Z = (1,0,1,3,3), and f ± J+1 = & ± y 
 
 b. If h. . t 0, let Z. . = (1,0,2,3,3), and f. _ , = h. .. 
 
 ij iJ i.j+l ij 
 
 c. Set j = j+1, and go to 2. 
 
11 
 
 9. (Cell i+1, j is unused.) 
 
 If f,, does not depend on C. j5 
 
 let f = f ; Z = (0,1,0,U,U); set i = i+1; and go to 2. 
 
 1+-L j J 1 J 1 J 
 
 10. If neither g. . = nor h = 0, go to 13. 
 
 J- J J- J 
 
 11. If f . . ± C. , and f . . j C. . , go to 12. 
 
 a. If z is unspecified and cell i+1, j-1 is unused, 
 
 1 , J — -L 
 
 go to 11. d. (C.., or C. . is input to B. n . and C. is 
 unspecified. ) 
 
 b. If f = C , let Z = (0,0.,1,U,U). 
 
 -L J J- J J- J 
 
 c. If f . . = C. ,, let Z. . = (0,0,2,U,U) 
 
 d. If L is empty, stop. 
 
 e. Set i,j = L and go to 2. 
 
 12. (Either g. . or h.. is zero.) 
 
 a. If g j 0, let Z = (0,1,1,3,3) and f = g . 
 
 b. If h. . i 0, let Z. . = (0,1,2,3,3) and f. , . = h. .. 
 
 c. Set i = i+1 and go to 2. 
 
 13. If w. . = 0, let Z. . = (1,1,1,7,7); f. .... = g. .; 
 
 ij ij i,J+l ij 
 
 f . = h ; set L = i,j+l; set i = i+1; and go to 2. 
 
 1 + -L»J IJ 1 
 
 Ik. Let z.j = (1.1.1.7.7). andf 1+1) J+1 =v... 
 
 a. If u.j f 0, let f. >J+2 = u.j; Z. jJ+1 = (1.1.0.4.1.); 
 and set L = i,j+2 
 
 b. If u. . = 0, let Z. . , = (0,1,0,U,U). 
 
 ij i»j + l 
 
 o. If v. . 4 0, let f. +2jJ = v. j; Z. +lj . = (l.l.O.U.U); 
 set L = i+1, j+1; set i = i+2; and go to 2. 
 
 d. Let Z . = (l,0,0,U,U); set i = i+1, j = J+l; and go to 2, 
 
 1+-L , J 
 
12 
 
 Proof That Algorithm Terminates 
 
 I. Any function of n variables can be produced if any function of 
 
 n-1 variables can be produced and if all previous branches of the 
 array terminate. 
 
 1. 
 
 '. (x , . . . ,x ) is to be produced at the output of cell i,j. 
 1 , j x n 
 
 2. If C is not specified, it will be specified so that f. . 
 
 i.J i » J 
 
 depends on C . 
 
 3. If cell i+1, j is not previously used (this will be the case 
 until the first branch terminates, since i does not decrease 
 except when a branch terminates) 
 
 a. If f. . does not depend on C. . , i is increased by 1. 
 
 i.J i » J 
 
 Either a cell will be reached where f. . depends on C. , . 
 
 or a cell will be reached where i is greater than for any 
 
 previously used cell (since all previous branches terminate), 
 
 and C. . will be specified so that f. . depends on it. 
 i.J i.J 
 
 b. If f. . depends on C . ., the A and B inputs to cell i.j are 
 
 i.J i.j 
 
 functions of n-1 or fewer variables. 
 h. If cell i+1, j has been previously used 
 
 a. If f. . does not depend on C . , j is increased by 1. 
 
 i.J i » J 
 
 Eventually some cell will be reached such that f . . depends 
 
 i >J 
 
 on C. ., since all previous branches terminate. 
 i.j 
 
 b. If g. . or h. .is zero when f. . depends on C. ., the 
 
 i.J i.J i,J i.J 
 
 inputs to cell i,j will be functions of n-1 or fewer variables. 
 
 c. Otherwise j is increased by 1. f. = f. .. Eventually a 
 
 1 . <] + -L i.J 
 
 cell will be reached either satisfying the conditions of part b 
 
13 
 
 or such that cell i+l,j is unused, assuming that all 
 previous branches terminate. Cell i, j+1 is previously- 
 unused, since i does not decrease in any branch. 
 5. In 3 and k t eventually a cell will be reached whose output is 
 
 f. . and whose inputs are functions of n-1 or fewer variables. 
 i.J 
 
 II. Every Branch of the Array Terminates 
 
 1. For any cell i,j in the first branch, cell i+1 , j is not 
 previously used, and C. .is not previously specified, since 
 
 i does not decrease until the first branch terminates. There- 
 fore, the first branch terminates. 
 
 2. Any function of one variable is produced in at most one cell. 
 
 3. If all previous branches terminate, any function of n variables 
 can be produced in a finite number of cells from functions of 
 n-1 variables. Since any function of one variable can be 
 produced in one cell, all branches used in producing a function 
 of n variables terminate. (Each terminates if all previous 
 branches terminate.) 
 
 k. Since the first branch terminates, and every branch terminates 
 if all previous branches terminate, all branches terminate. 
 
lit 
 
 Upper Bounds on Size of Array 
 
 Any function of n variables can "be produced from three 
 functions and an input variable in at most four cells , as in 
 Figure 3. Therefore, if any function of n-1 variables can be 
 produced in q cells (other than cells which only provide connec- 
 tions), any function of n variables can be produced in at most 
 3q + h cells. Let q be an upper bound on the number of cells 
 
 EL, 
 
 (other than connecting cells) required to produce a function of k 
 variables. Then q = 1, and q = 3q + h. If q =3 - 2, 
 then a = 3q + h = 3(3 n ~ -2)+U=3 n -2, and any function 
 of n variables can be produced with 3-2 cells other than 
 connecting cells. 
 
 From Figure 3 it is seen that no more than three cells 
 (other than connecting cells) in each branch are required in 
 reducing a function of n variables to functions of n-1 variables. 
 Therefore the number of cells other than connecting cells in each 
 branch is no greater than 3n. There are no connecting cells in 
 the first branch, since no cells or C inputs have been previously 
 specified. 
 
 The number of connecting cells required in each branch is 
 limited by the number of cells in the previous branches . The number 
 of connecting cells required will be no greater than the total number 
 of cells in the longest previous branch. (This number of cells will 
 extend the branch to a point where the previous branches do not inter- 
 fere.) If the upper bound, p , on the number of cells in branch k is a 
 
15 
 
 nondecreasing function of k, the number of connecting cells will 
 be no greater than p , and p can be p + 3n. p = 3n. If 
 p = 3n(k-l), then p = 3nk. By induction, p = 3nk for all k. 
 
 The number of branches in the array can be no greater than 
 3 , since this is the greatest number of functions of one variable 
 used in implementing a function of n variables. Therefore, the 
 total number of cells in an array which implements a function of n 
 variables is no greater than 
 
 _n-l n-1 _ 
 
 k Z x 3nk = 3n ^ k = 3n ^- (l + 3 n_1 ) = § 3 n (l + 3 n ) 
 
 The number of cells in a line in either the horizontal or vertical 
 direction can be no greater than the maximum number of cells in a branch 
 (n3 ), for no branch can extend farther than this from the output cell. 
 
 These bounds can be used as a preliminary check to assure 
 that an actual array is large enough to implement a given function, or 
 they can be used in partitioning an array when more than one function is 
 to be generated. The actual number of cells required for implementa- 
 tion, however, may be much smaller than these bounds indicate. 
 
l6 
 
 Example 
 
 1. Combinational Circuit 
 
 f = X 2 \ X 5 + X 2 X 3 X 5 + X l X 2 X 3 *k + X l X 2 X l+ X 5 + X 2 X 3 X 5 
 step 
 
 1. i = 1, j = 1, f xl = f(x lS x 2 , x 3 , x^, x ) 
 
 2. C. , . , is not specified. 
 
 1+1, J-l 
 
 3. x appears in every term of f , so C = x . 
 
 h. Cell i+lj j is unused. 
 
 9. f depends on C . 
 
 10 - g ll = X 3 V h H = X U x 5 + X l X 3 X h + x l X U x 5 + X 3 X 5 
 13. w n = 0; u xl = g 1]L ; v^ = h^; Z^ = (1,1,1,7,7); 
 
 f 12 == S ll' f 21 =: h ll' L T = 1,2; and i = 2. 
 
 2. C. in . n is not specified. 
 
 i+l, J-l 
 
 3. No )c appears in every term of f ; let C = x . 
 k. Cell 2, 2 is unused. 
 
 9. f 21 depends on C^ , 
 
 10 - §21 = X U + X l X 3 V h 21 = X l X 3 X U + X l X I+ + X 3 
 
 13 - w 21 s v 3 V° 
 
 lU. Z 21 = (1,1,1,7,7); Z 22 = (1,1,0,4,1+); Z^ = (1,1,0, M); 
 
 f 23 = U 21 = V f Ul = v 21 = X l X h + X 3 ; 
 f 32 = W 21 = X l X 3 V L T = 2 >3; L T = 3,2; i = k. 
 2. ( i = U , j = l) C . is not specified. 
 
 1+-L , J — -L 
 
 No 
 
 x^ appears in every term of f. ; let C. = x. 
 
IT 
 
 h. Cell 5,1 is unused. 
 9« fi depends on C, . 
 
 10. g Ul = x 3 ; h Ul = x ± + x 3 . 
 
 13 ' w Ul = x 3 
 
 Ik. z kl = (1,1,1,7,7); \ 2 = (o,i,o,U,U); z 51 = (1,1,0,U,U) 
 
 u Ul = 0; V U1 = f 6l = *1 5 f 52 = W U1 = x 3 ; L T = 5 ' 2; i = 6 ' 
 
 2. (i = 6,j = l) C. in . _ is not specified. 
 
 l+l, j-1 
 
 3 * f 6l = V C 6l = x r 
 
 k. Cell 7,1 is unused. 
 
 10 ' g 6l = °' h 6l = X - 
 
 11. f^. = C^ n ; Z. . _ is unspecified, and cell i+1, j-1 is unused. 
 
 61 61 i,j-l 
 
 x is input to B . i,j = L T = 5,2. 
 
 2. C^- is not specified. 
 
 3. f 52 =x 3 =C 52 
 
 U. Cell 6,2 is unused. 
 
 9. fj-p depends on C 
 
 10. g 52 = l; h 52 = 
 
 11. f = C ; Z 51 is specified. Z = (0,0,1,U,U); i,j = L T = 3,2, 
 2. C Ul = x^; Z Ulc = 1; C^ = C^ = x^ . 
 
 k. Cell U,2 has been used. 
 
 5. f 32 = x 2 x 3 x h 
 
 6. g 32 = 0; h 32 = x ± x 3 
 
 3, 
 
 Z 32 = (1,0,2,3,3); f 33 = h 32 = x ± x^ J = 3. 
 
 2. (i = 3, j = 3); C, = C is unspecified. 
 
 3. C 33 =x 3 
 
 h. Cell i+,3 is unused. 
 
18 
 
 9- f-,-, depends on C . 
 
 10. h 33 = x^; g 33 = 
 
 12. Z 33 = (0,1,2,3,3); f U3 = X 1 ; i = k. 
 
 2. (i = k, j = 3) C 52 = x 3 ; Z 52C = 1; C^ = x^ 
 
 k. Cell 5,3 is unused. 
 
 9. f^ does not depend on C^. f = f^ = x^ Z^ = (0 ,1 ,0 ,U ,U) ; i = 5 
 
 2. (i = 5, j = 3) C>- is not specified. 
 
 3 - C 53 =x l 
 
 k. Cell 6,3 is unused. 
 
 9- f^- depends on C . 
 
 10. g = 0; h = 1. 
 
 11. ff = C ; Z is specified. Z = (0 ,0,2 ,U ,1+) ; i,J = L T = 2,3. 
 
 2 * C 3 2 = V Z 3 2C = 2; C 23 = °32 = V 
 
 U. Cell 3,3 has been used. 
 
 5 ' f 23 =X U=^23 
 
 7. Z 23 = (0,0,2,U,1*); i,j = L T = 1,2. 
 
 2. C 21 = x 5 ; Z 21C = 1; C^ = x y 
 
 h. Cell 2,2 has been used. 
 
 5 - f 12 = X 3 X 5 
 
 6- S 12 = X 3' h 12 = °* 
 
 8. Z 12 = (1,0,1,3,3); f 13 = g 12 = x 3 ; j = 3. 
 
 2. (i = 1, j = 3) C is not specified. 
 
 3. C 13 =x 3 
 
 k. Cell 2,3 has been used. 
 
 5. f _ = x depends on C . 
 
 T. f 13 = C 13 ; Z = (0,0,1,U,U). L is empty, stop. 
 
19 
 
 z> 
 o 
 <z 
 
 o 
 
 _l 
 < 
 
 h- 
 
 UJ < 
 
 <£ ? 
 
 Ll O 
 O 
 
 UJ 
 
 _l 
 
 CL 
 
 < 
 X 
 
 UJ 
 
20 
 
 U. SEQUENTIAL CIRCUITS 
 
 Form and Decomposition of Functions 
 
 For sequential circuits it will "be assumed that the inputs 
 
 are x_,,...,x , the internal variables are y_ , . . . ,y , and the outputs 
 In 1 m 
 
 are f_,...,f . The next state of y, is Y. . (y. (t) is replaced by 
 1 q k k k 
 
 Y (t+At).) The following functional dependence will be assumed: 
 
 K. 
 
 Y k = Y k (x lS ...,x n , yi ,... 9 y m ). 
 
 The function is decomposed in a manner similar to that used 
 for combinational circuits, except that when the function to be generated 
 is a next-state variable, the internal variable is used rather than an 
 input variable. 
 
 \ = \ ^ h - 
 
 where g and h are independent of y . This function can be implemented 
 in a flip-flop having the application table shown: 
 
 y Y g h 
 
 d 
 
 1 d 1 
 
 1 1 1 d 
 1 d 
 
 It is seen that g corresponds to a reset input, and h to a set input. 
 A g = 0, h = 1 input causes the flip-flop to change state. There is 
 also a clock input, which must be 1 for the flip-flop to change state. 
 
21 
 
 Each internal variable is considered as a separate output 
 and input. Each desired output function is produced in an array 
 which has the x ' s and y 's as inputs. Several output functions 
 can be produced using internal variables in common. Each y is 
 
 K. 
 
 produced in an array having x , . . . , x and y y , y , . . . ,y 
 
 1 n 1 K-l a+1 m 
 
 as inputs. These arrays may actually be different sections of one 
 array. Provision must be made for connection between input and 
 output y ' s . It is expected that in an actual implementation of this 
 algorithm there would be a program which would allocate space in the 
 array to the various functions and connect them in such a way as to 
 make efficient use of space and reduce the number of external connections 
 required. The manner in which these connections are made will not be 
 considered here, however. 
 
 For the y outputs, Y = y g + y h. The cell 1, 1 is 
 specified as a flip-flop with f = g and f = h. C is the clock 
 input for clocked circuits, or 1 for unclocked circuits. The g and h 
 functions are generated as combinational functions of x , . . . ,x , 
 
 y i'""' y k-l' y k+l'"*' y 
 
 m 
 
22 
 
 Algorithm for Sequential and Multiple-Output Circuits 
 
 1. Call outputs f_....,f ; inputs x, ,...,x ; and internal 
 
 c 1 q 1 n 
 
 variables y„ «....y • Express f, ' s and Y ' s in terms of 
 1 m k k 
 
 X ' s and y ' s . 
 
 2. Apply the combinational algorithm to each f , 1 <_ k <_ q, 
 
 if f, ^ x or y for any p. Let f, , = f , and use x, , . . . ,x and 
 
 kpp 11 k 1 n 
 
 y, ,. . . ,y as inputs. 
 J- m 
 
 3. For each y fc , let Y k = y k g + y fc h = y k u + y fc v + w. 
 Let Z = (1,1,1,0,0). 
 
 a. If w = 0, u 4 0, v 4- 0, let f ±2 = g, f gl = h. 
 Set L = 1,2 and i = 2. 
 
 b. If g = 0, set f = h, and i = 2. 
 
 c. If h = 0, set f = g and j = 2 
 
 d. If w 4 0, set f = w. 
 
 If u 4 0, set Z 12 = (l,l,0,i+,l+); ^ = u; and -L T = 1,3. 
 
 If u = 0, set Z 12 = (0,1,0,U,1+). 
 
 If v 4 0, set Z 21 = (l,l,0,i+,U); f = v; L T = 2,2; and 1 = 3 
 
 If v = 0, set Z = (l,0,0,U,U); i = 2;~ and j = 2. 
 
 e. Go to step 2 of the combinational algorithm and execute it 
 
 until it stops, then return to this step (3), and apply it 
 
 to each y, . 
 k 
 
 h. The output where each y is generated is to be connected to all 
 
 K. 
 
 inputs where it is required. 
 
23 
 
 Examples 
 
 2. Sequential Circuit 
 
 step 
 
 1. I 1 = x x J x + x 2 Y 2 = ^ x 2 y 2 + x 2 y x * ^ ^ 
 
 f l = y l + y 2 
 
 2. (i) f n = y 1 + y 2 
 
 (3) C u = y 2 
 
 (9) f,, depends on C . 
 
 (10) 
 
 s ll = 2 h ll = y l 
 
 (13) 
 
 v ll = y l u ll = y l v ll = ° 
 
 (1U) 
 
 Z 11 = (1,1,1,7,7) Z lg = (1,1,0,1+,!+) 
 
 
 f 22 = y i f 13 = U ll = y i L T = 1 ' 3 
 
 Z 21 = (1,0,0,1+,!+) i = 2, j = 2 
 
 (2) C is not specified. 
 
 (3) C 22 = y ± 
 
 do) g 22 = i, h 22 = 
 
 (ii) z 22 = (0,0,1,1+,!+) 
 (2) (i = 1, J = 3) f 13 = y x 
 
 C 13 = °22 = y i 
 
 do) g 13 = h 13 = 1 
 
 (11) Z = (0,0,2,1+,M l is empty, return 
 
2k 
 
 3. Y 1 =x 1 y 1+ x 2 
 
 z n = (1,1,1,0,0) 
 
 f 22 = w = x 2 
 
 Z 12 = (0,1,0, U,U) 
 
 f 31 = v = x x Z 21 = (1,1,0, U,U) L T = 2,2; i = 3 
 
 (3) C 31 =x 1 
 
 (ll) f .. = C_, Z. , _ is unspecified and cell i+1, j-1 is 
 31 31 i,J-l 
 
 unused. Input x to B . i = 2, j = 2 
 
 (2) C is unspecified. 
 
 (3) C nn = x n 
 
 22 2 
 
 (ii) f 22 = c 22 z 22 = (0,0,1, M) 
 L is empty, return. 
 
 3. Y 2 = x i x 2 y 2 + X 2 y i + X l y i 
 
 U = X l X 2 v = W = x 2 y i + X l Y l 
 Z u = (1,1,1,0,0) f 22 = v 
 
 Z 12 = (l,l,0,li,U) f l3 = u L T = 1,3 
 
 Z 21 = (1,0,0,1+,U) i = 2, j = 2 
 
 (2) C is not specified. 
 
 (3) C 22 = y ± 
 
 . (12) g 22 =0 h 22 = x i + x 2 
 
 Z 22 = (0,1,2,3,3) f 32 = x x + x 2 
 
 (2) C, is not specified. 
 
 (3) C 32 = x 2 
 
25 
 
 (10 
 
 (iu 
 
 (2 
 (3 
 
 (11 
 (2 
 
 (11 
 
 (2 
 (9 
 
 (2 
 
 (U 
 (6 
 (8 
 (3 
 (10 
 (11 
 
 '32 
 
 = 1 
 
 32 1 
 
 z 32 = (1,1,1,7,7) 
 
 f U3 = v Ul = x l 
 
 U 32 = X l f 3U = X l Z 33 = (1 ' 1 '°' U ' U 
 
 i - U, J = 3 
 
 L T = 3 ' U Z i+2 = t 1 ' ' ' 1 ^) 
 C is unspecified. 
 
 C k3 = X l 
 
 f k3 = C k3 Z k3 = (O.O.l.M) i.J = L T = 3, 1 * 
 
 c 3 4 = C U 3 = x i 
 
 f 3 U " Su Z 3 L = < » - 2 "^> 
 
 i,J = L T = 1 >3 
 
 C 13 = °22 " 3Tl 
 
 f does not depend on C . 
 
 f 23 = f 13 Z 13 = (O' 1 ' ^^) i = 2 
 
 C 23 = °32 = X 2 
 
 Cell 3,3 has been used. 
 
 £ = 
 fe 23 
 
 "23 := X l 
 
 Z 23 = (1,0,2,3,3) f 2k = x 1 
 
 C 2U " X l 
 
 3 2U 
 
 = 
 
 h^i = 1 
 2U 
 
 f 2l| = °2U Z 2U = t°.°- 2 - U ^) 
 L is empty, return 
 
 k. Make the necessary connections 
 
26 
 
 FIGURE 5. 
 EXAMPLE 2: SEQUENTIAL CIRCUIT 
 
27 
 
 3. Feedback Shift Register 
 
 step 
 
 
 i. fl = 
 
 •/ *1 *^Q *3 li 17*11/ 
 
 Y i = 
 
 y 5 
 
 \- 
 
 </ -| ^ J [T " ~ J -| r/-- J-l J"- 
 
 V 
 
 y 2 
 
 h = 
 
 y 3 
 
 Y 5 = 
 
 Vk 9 y 5 = y U y 5 + y li y 5 
 
 2 - f ll 
 
 " f l 
 
 (3) 
 
 C ll = y l. 
 
 (10) 
 
 g ll = ° h ll = y i y 2 y 3 
 
 (12) Z ±l = (0,1,2,3,3) f 21 = h 
 
 (2) (i = 2, j = l) C is not specified. 
 
 (3) C 21 = y 3 
 
 (10) s =0 h = y y 
 v i & 21 21 ^ jr 2 
 
 (12) Z 21 = (0,1,2,3,3) f 31 = h 21 i = 3 
 
 (2) C. , . _ is not specified. 
 
 i+l, J -1 
 
 (3) C 31 = y 2 
 
 ( 10) g 3i =0 h 31 =y 1 
 
 (12) Z 31 = (0,1,2,3,3) f Ul = y 1 i = k 
 
 (3) C Ul =y l 
 
 (11) f 4l = Sl 
 
 Z. is unspecified and cell i+l,J-l is unused, 
 
 1 , J ~ -L 
 
 y is input to B . L is empty, return. 
 
28 
 
 3. Y a = J± y 5 + y x y,. 
 
 Z = (1,1,1,0,0) w^O, u=0,v=0 
 z i2 = (o,i,o,U,U) z 21 = (1,0,0, U,U) 
 f 22 = w = y 5 i = 2, j = 2 
 (3) C 22 = y 5 
 (9) f 22 depends on C^. 
 (11) f 22 = C 22 Z 21 is specified 
 z 22 = (0,0,1,U,U) 
 L is empty, return. 
 
 3- Y 2 = y 1 y 5 + F x y 5 
 
 Z u = (1,1,1,0,0) w^O, u = 0,v = 
 
 z i2 = (o,i,o,U,U) z 21 = (1,0,0, U,U) 
 
 f 22 = w = y x y 5 + F 2 y 5 
 
 (3) C 22 = y 5 
 
 do) g 22 = y ± h 22 = yi 
 (13) w 22 =0 Z 22 = (1,1,1,7,7) 
 
 f = cr =y f = h =V 
 
 23 6 22 ^1 32 22 *1 
 L T = 2,3 i = 3 
 
 (3) C 32 = y ± 
 
 (11 ) Z is unspecified and cell k t l is unused. 
 
 y 1 is input to B^ i = 2, j = 3 
 
 (2) (f = y ) C is unspecified. 
 
 (3) C 23 = y x 
 
29 
 
 (11) f 23 = C 23 Z 23 = (0,0,2, U,U) 
 
 L is empty, return. 
 3. y_ and y, are implemented similarly to y . 
 3. Y 5 = y 5 y u + y" 5 y^ 
 
 Z = (1,1,1,0,0) w = 0, u ^ 0, v^O 
 
 f 12 = g = y k f 21 = h = y u 
 
 L T = 1,2 i = 2 
 (3) C 21 =y u 
 
 do) g 21 = 1 h 21 = 
 
 (ll) Z. is unspecified and cell i+1, j-1 is unused, 
 
 1 5 J — -L 
 
 Input y^ to B xl . i,j = L T = 1,2 
 (3) C 12 =y u 
 
 do) g 12 = h 12 = 1 
 
 (ii) f 12 = c 12 = y^ z 12 = (0,0,2, k 9 k) 
 
 L is empty, return. 
 U. Make the necessary connections. 
 
50 
 
 FIGURE 6. EXAMPLE 3: 
 FEED-BACK SHIFT . REGISTER 
 
31 
 
 5. CONCLUSION 
 
 In this thesis procedures have been demonstrated which 
 can be used for programming an array of hexagonal cells to produce 
 combinational and sequential logical functions. It has been shown 
 that these procedures are capable of specifying finite arrays for 
 any such functions. However, these procedures do not generally provide 
 an optimum result. It appears that further investigation could produce 
 improvements both in procedures and in the design of the array. 
 
 The algorithms developed here do not use the full capabilities 
 of the cells. It appears that more efficient use of the cells usually 
 requires a more complicated algorithm. The cells used here can be 
 specified in ways not used by the algorithms, such as selecting from 
 the full set of functions of the cell and generating different 
 functions at the two functional outputs of the cell. An algorithm 
 might be devised which would make a choice from several possible 
 decompositions of a function, depending on the nature of the function. 
 It would also be desirable to consider what would be the best order in 
 which to factor out input variables from the function. For multiple- 
 output arrays, an implementation which shared circuitry among the 
 outputs, wherever possible, would be more efficient. 
 
32 
 
 Other types of arrays could be more efficient. The input 
 and output configurations could be different. For example, a four- 
 input cell might use a function decomposition like that used here, 
 with u, v, w, and x functions as inputs. A second layer of cells, 
 consisting of only connecting circuitry, could provide connections 
 between any two cells in the array, and eliminate the need for some 
 external connections, while relaxing the geometrical restrictions on 
 implementation. Equivalently , a two-layer interconnection structure 
 could be provided in a single layer of cells by using more connections 
 on each cell. This arrangement would be particularly useful for 
 multiple- out put circuits and for sequential circuits. 
 
 There appears to be considerable promise in the use of 
 cellular arrays for implementation of logical functions. It is 
 hoped that the results of this thesis will be useful in the develop- 
 ment of more efficient arrays and procedures for their use. 
 
33 
 
 BIBLIOGRAPHY 
 
 Finkelstein, H.A. /'Design of Digital Computer Circuits Using 
 a Basic Logic Cell", M.S. Thesis, University of Illinois, 
 Department of Computer Science Report No. 287 May 2U, 
 1968. 
 
 Maitra, K.K. , "Cascaded Switching Networks of Two-Input Flexible 
 Cells", IRE Transactions on Electronic Computers, Vol. 11, 
 p. 136, April 1962. 
 
 Minnick, R.C., "A Survey of Microcellular Research", Journal of 
 the Association for Computing Machinery, Vol. lU, No. 2, 
 p. 203, April 1967. 
 
 Minnick, R.C., "Cobweb Cellular Arrays", Proc . AFIPS Fall Joint 
 Computer Conference, Vol. 27, Pt . 1, pp. 327-3^1. 
 
 Minnick, R.C., "Cutpoint Cellular Logic", IEEE Transactions on 
 Electronic Computers, Vol. 13, pp. 685-698, December 196U. 
 
 Schneider, P.R. and D.L. Dietmeyer, "An Algorithm for Synthesis 
 of Multiple-Output Combinational Logic", IEEE Transactions 
 on Computers, Vol. C-17, No. 2, p. 117, February 1968. 
 
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