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VM 
 
 -UIUCDCS-R-75-728 
 
 NOR NETWORK TRANSDUCTION BY GENERALIZED GATE 
 MERGING AND SUBSTITUTION PROCEDURES 
 
 (Principles of NOR Network Transduction 
 Programs NETTRA-G3 and NETTRA-G^) 
 
 by 
 
 H. C. Lai 
 Y. Kambayashi 
 
 June 1975 
 
UIUCDCS-R-75-728 
 
 NOR NETWORK TRANSDUCTION BY GENERALIZED GATE 
 MERGING AND SUBSTITUTION PROCEDURES 
 
 (Principles of NOR Network Transduction 
 Programs NETTRA-G3 and NETTRA-G^) 
 
 fey 
 
 H. C. Lai 
 Y. Kambayashi 
 
 June 1975 
 
 Department of Computer Science 
 
 University of Illinois at Urbana-Champaign 
 
 Urbana, Illinois 
 
 This work was supported in part by the National Science Foundation under 
 Grant No. NSF GJ-^0221. 
 
10, w 
 
 fi ZS^33 
 
 
 ACKNOWLEDGMENT 
 
 The authors are grateful to Professor S. Muroga for his guidance 
 and careful reading of this report. The authors would like to thank Mr. 
 J. N. Culliney for his valuable discussions throughout the period of this 
 research. 
 
IV 
 
 ABSTRACT 
 
 Based on the concept of Compatible Sets of Permissible Functions 
 (CSPF) discussed in the previous paper [l], several NOR network transduction 
 procedures aiming at reducing the number of gates in a given network are 
 introduced. These procedures can be classified into two types: gate 
 merging procedures and gate substitution procedures. 
 
 In a given NOR network, two gates can be merged to one if the 
 intersection of their CSPF's is not empty and a function contained in that 
 intersection can be realized by connecting the outputs of some gates already 
 existing in the network to a new NOR gate as its inputs. This process is 
 called gate merging. Several procedures of gate merging are discussed. 
 On the other hand, a gate substitution procedure replaces each 
 fan-out connection of a certain gate with a set of connect able functions 
 
 based on CSPF's (possibly each fan-out connection with a different set of 
 
 connectable functions). A generalization based on possibly-connectable 
 
 conditions is also discussed. 
 
 Two procedures, one for each of gate merging and gate substitution 
 
 were implemented as computer programs NETTRA-G3 and NETTRA-Qlf respectively. 
 
 Computational results showed that these procedures are efficient and 
 
 effective in reducing the number of gates in a NOR network which has 
 
 relatively many redundant gates. 
 
TABLE OF CONTENTS 
 
 1. INTRODUCTION ± 
 
 2. GATE MERGING AND ITS GENERALIZATIONS 1+ 
 
 3. GATE SUBSTITUTION AND ITS GENERALIZATIONS 30 
 
 k. COMPUTER PROGRAMS AND EXPERIMENTS kf 
 
 U.l Program for Gate Merging Uy 
 
 k.2 Program for Gate Substitution Uo, 
 
 ^4.3 Experiments by Computer Programs 55 
 
 5. CONCLUSIONS ^ 
 
 REFERENCES 55 
 
1. INTRODUCTION 
 
 The design of NOR (or NAND) networks with many gates is one of 
 important problems in logic design. The network transduction methods [1] 
 appear promising to solve this problem ("transduction" means trans formation 
 and re duction ). 
 
 The concept of CSPF's (Compatible Sets of Permissible Functions) 
 was introduced in [1]. Several network transduction methods based on 
 CSPF's were discussed and applied to the simplification of NOR networks [2-5]. 
 In this paper, using CSPF's, we will generalize two known network transduction 
 methods (gate merging and gate substitution) discussed in [7] and [8]. The 
 possibly-connectable condition will be introduced for further generalizations. 
 
 The concepts and symbols defined in [1] will be used throughout 
 this paper. The following definitions and theorems introduced in [k] will 
 also be used. 
 
 Definition 1.1 A gate or an input terminal, v., is said to be 
 
 connect able to a gate v. with respect to a set of permissible functions 
 
 of v., G(v ), if (1) the output function of v. remains in G(v.) after adding 
 3 3 3 3 
 
 connection C , and (2) the network obtained after this input addition is 
 loop- free. 
 
 Definition 1.2 Connection C. . is said to be disconnectable from a 
 
 gate v with respect to a set of permissible functions of v., G(v ) if the 
 J 3 3 
 
 output function of v. remains in G(v.) after removing C. . from the network. 
 
 3 3 i,J 
 
Definition 1.3 The set of connectable functions to gate v with 
 
 - j 
 
 respect to G(v.), K(v.), is the set of all functions such that the addition 
 
 J J 
 
 of any subset of these functions to v. as its inputs keeps the function 
 
 realized at v. remain in G(v. ). 
 -] 3 
 
 Theorem 1.1 A gate or an input terminal, v., is connectable to v. 
 with respect to set G(v.) if and only if the following conditions are 
 
 J 
 
 satisfied: 
 
 (1) For all d such that f^(v. ) = 1, 
 
 G<; d) (v.) = or *. 
 c 3 
 
 where f (v. ) is the d-th component of the output vector 
 of gate v. . 
 
 (2) v. is not contained in S(v ) where S(v ) denotes 
 
 1 3 3 
 
 the set of successor gates of gate v..- 
 
 Theorem 1.2 Connection C. . is disconnectable from gate v with 
 
 13 fa .3 
 
 respect to set G(v.) if and only if the following condition is satisfied: 
 
 J 
 
 For all d such that f^ d '(v. ) = 1, either 
 
 G (d) (v.) = * or 
 
 J 
 
 V f (d) (v) = 1 
 veIP(v.) 
 
 v/v. 3 
 
 ' 1 
 
 where ^ denotes logical OR operation and IF(v.) denotes the set of all 
 
 3 
 
 immediate predecessor gates of gate v.. 
 
 J 
 
 Theorem 1.3 The set K(v.) of all connectable functions to a gate 
 v. with respect to G(v.) is given by 
 
 'J J 
 
 IC d ^(v.) == for all d such that G^(v.) = 1. and 
 3 3 
 
 K^ d '(v.) := * for all other d's. 
 J 
 
NOR network transduction procedures based on the above connectable 
 and disconnectable conditions are discussed in [k]. These conditions are 
 also essential for the gate merging and gate substitution procedures to 
 be discussed in Sections 2 and 3, respectively. The "possibly-connectable" 
 condition will be introduced in Section 2 in order to generalize these 
 procedures. 
 
 Two computer programs designated NETTRA-G3 and NETTRA-QJ+ have been 
 prepared and tested. Program NETTRA-G3 realizes a gate merging procedure 
 whereas program NETTRA-Cx4 realizes a gate substitution procedure. Section k 
 will be devoted to the discussion of these programs along with computation 
 results. 
 
2. GATE MERGING AND ITS GENERALIZATIONS 
 
 In this section generalizations of a transduction method known as 
 gate merging will be discussed. The "possibly-connectable" condition 
 will be introduced in order to generalize these procedures. 
 
 Figure 2.1 shows a simple example of gate merging. The networks 
 in Figure 2.1(a) and (b) realize function f, 
 
 I = X, Xp ^y X_ X~ . 
 
 In the network in Figure 2.1(a) we can add redundant inputs x p and x to 
 gates Vr and v , respectively. By adding these redundant inputs we can 
 merge gates V/- and v , deriving the network in Figure 2.1(b). This kind 
 of transformations was discussed in [6] and [7]. 
 
 Definition 2.1 Gates v. and v. are said to be merge able if the 
 
 l j a 
 
 following conditions are satisfied: 
 
 (1) v. ^ S(v. ) and v. £ S(v.), where S(v. ) denotes 
 
 J -L -L J -L 
 
 the set of all successor gates of v. and S(v.) 
 
 i ^ 
 
 is similarly defined. 
 
 (2 ) Each output terminal v _, . realizes a function in set 
 ' * n+R+i 
 
 z. f for i = 1, 2 f ..., m, after the following network 
 modifications : 
 (2-1) Add connections from all gates and input 
 
 terminals in IP(v. )-IP(v. ) to gate v., where 
 
X. 
 
 Vs >o- 
 
 V 
 
 x^ 
 
 ^i>^r 
 
 5 J - 
 
 (a) A given network 
 
 v. 
 
 Vr 
 
 x P -y 
 
 v... 
 
 X, 
 
 2 ~\ v, 
 
 l . 
 
 *i- 5 
 
 V 
 
 V, 
 
 8 
 
 O f 
 
 (b) Simplified network 
 
 Figure 2.1 Example of Simple Gate Merging 
 
IP(v.) denotes the set of all immediate 
 
 predecessor gates of v. and IP(v. ) is 
 
 3 1 
 
 similarly defined. 
 (2-2) Add connections from all gates and input 
 terminals in IP(v. ) - IP(v.) to gate v.. 
 If two gates v. and v. are mergeable, then we can replace these 
 two gates by gate v. . satisfying 
 
 IP(v. .) = IP(v. ) U IP(v.), 
 ij i 3 
 
 IS(v. .) = IS(v. ) U IS(v.). 
 " ij i J 
 
 This gate, v. . is called the merged gate. Condition (l) of Definition 2.1 
 is a loop-free condition for the network after merging two gates v. and v . 
 
 Conditions whether we can add an input to a gate without changing 
 output functions of the network were discussed in [6], [7] and [8]. The 
 condition given by Lai, Nakagawa and Muroga [8] is the best among these 
 conditions. The condition is as follows. "When on each path from gate v. 
 to an output terminal there exists a gate which has an input connection 
 from v. (gate located outside all these paths or input terminal), we can 
 
 J 
 
 add a connection from v. to v. without changing that output of the 
 network. If this property holds with all outputs of the network then 
 we can add a connection from v. to v. without changing any output of the 
 network. " However, in order to use this condition we have to check all 
 possible paths from gate v. to all output terminals. We can generalize 
 the procedure using CSPF's. 
 
 Theorem 2.1 Gates v. and v. are mergeable if the following 
 conditions are satisfied: 
 
(1) v. £ S(v. ) and v. £ S(v. ), 
 
 (2) g c (v.) na c (y.) /0, 
 
 (3) f^)-^) e G c (v.) nG c (v.), 
 
 where symbol • denotes the logical AND operation for each component of 
 
 f(v. ) and f(v.). 
 i 3 
 
 Proof If the function realized by the merged gate v. . is contained 
 in G (v ) n G (v ) then each output terminal v _, . realizes a function in 
 set z i (i=l, .. .,m). We only need to prove that f(v. )«f(v.) is the output 
 
 J 
 
 function of the merged gate v. . . As gates v. and v. realize f(v ) and 
 
 f(v ), respectively, the following relations are satisfied because v and v 
 3 i 3 
 
 are NOR gates : 
 
 V f(v) = f (v ,), 
 
 v£lP(v. ) x 
 l 
 
 V f(v) = f(v.). 
 
 veIP(v.) J 
 3 
 
 The merged gate realizes function 
 
 ( V t{v)K V f(v)J 
 veIP(v. ) veIP(v.) 
 
 J 
 
 = f(v ) V f(v ) = f(v. )-f (v.). Q.E.D. 
 
 If we want to simplify a network by gate merging, we have to 
 examine for each pair of gates whether the conditions of Theorem 2.1 are 
 satisfied. The following lemma gives a sufficient condition for 
 condition (2) of Tneorem 2.1. 
 
 Lemma 2.1 If there exist two gates v. and v. satisfying 
 
 IS(v. ) = IS(v.) 
 i 3 
 
8 
 then 
 
 G (v. ) DG (v.) / 0. 
 c 1 c .i ' ' r 
 
 Proof : For simplicity we consider the case where v. and v. are connected 
 to one gate v . There are three cases as follows: 
 
 (1) G^ d) (v ) = *, then G (d) (v. ) = * and G (d) (v.) = ♦. 
 
 C JS C 1 C J 
 
 (2) G {d \v ) = 1, then G^ d) (v. ) = and G (d) (v.) = 0. 
 
 C K CI C J 
 
 (3) G^ d) (v k ) = 0, then G^ d) (v.) = 1 and G^ d) (v.) = *; 
 
 G (d) (v. ) = * and G (d) (v.) = 1; 
 C x c v J J ' 
 
 or G (d) (v. ) = * and G (d) (v.) - * 
 
 c i c v i \ J 
 
 depending on ordering r and other inputs 
 connecting to v (see [1] for ordering). 
 Since there exists no d such that 
 
 G (d) (v. ) = 1 and G (d) (v.) = 0; 
 
 c X C x J J ' 
 
 or G (d) (v. ) - and G (d) (v.) = 1, 
 CI c J 
 
 the theorem statement holds. The above argument can be easily generalized 
 to the case when IS (v. ) consists of more than one gate. 
 
 Q.E.D. 
 Procedure MGC : Merging of gates using CSPF's. 
 
 Step 1 Select two gates v. and v. such that 
 — i J 
 
 v.. i S(v.), 
 
 v. I S(v. ), and 
 
 G (v. ) DG (v.) / <L. 
 
 CI c .1 ' ~ 
 
Step 2 If f(v. )-f(v.) e G (v. ) HG (v.), then go to Step 3; 
 
 3 
 otherwise terminate the procedure. 
 
 Step 3 Replace gates v. and v. by a new gate v. . satisfying 
 
 IS(v. .) = IS(v. ) U IS(v.), 
 10 l 3 
 
 IP(v. .) = IP(v. ) U IP(v.). 
 13 i v 3 
 
 CSPF's for all gates should be calculated before applying the 
 
 procedure. However, if we calculate CSPF's for v. and v. after we select 
 
 v. and v we can get larger G (v. ) and G (v.) using any ordering r 
 ■L 3 c i c J 
 
 satisfying 
 
 r ( v a ) > r( \) 
 for every v i S(v ) U S(v ) U (v., v.) and every v e S(v. ) U S(v.) U {v., v.} 
 As a result we have a greater possibility of merging of these two gates. 
 We can easily prove that this procedure gives no worse result than the 
 procedure using the condition given by [8]. 
 
 The following example shows how Procedure MGC works. 
 
 Example 2.1 : Figure 2.2(a) shows a network which realizes a 
 four-variable function f. 
 
 f = (l 0000 0011 110 1) 
 
 Functions realized at input terminals and gates are as follows : 
 
 f (v x ) =(0 000 0000 1111 1111) 
 
 f(v 2 ) =(0 000 1111 0000 1111) 
 
 f(v 3 ) =(0 011 0011 0011 0011) 
 
 f(v^) =(0 101 0101 0101 010 1) 
 
 f (v^) =(1111 1111 0000 000 0) 
 
 f(v u ) =(0 000 0000 1100 110 0) 
 
LO 
 
 x„ 
 
 V 
 
 x,.— v 9 
 
 v, 
 
 x. 
 
 X 2--| y 10> '** \ v 7 >c- 
 
 -g> x ^=5^r>- 13^> — 
 
 5 >° — o v. 
 
 13 
 
 (a) A given network 
 
 1^02 
 
 6 V 
 
 x 2 _Jv 10 V- 
 
 X, 
 
 Xp r~ —\, 
 
 ^o_Jx 3 q V8 v 
 
 ^y° — z^_ \>-° 
 
 V 
 
 13 
 
 (b) Network obtained after applying 
 procedure GMC 
 
 Figure 2.2 Example 2.1 
 
11 
 
 f(v 1Q ) =(1111 0000 1111 000 0) 
 
 f (v ) =(1000 1000 1000 100 0) 
 
 f(vg) =(0 000 0000 1100 000 0) 
 
 f(v ) =(0 000 1010 0000 0010) 
 
 f(v 6 ) =(0 111 0111 0000 000 0) 
 
 f(v ) =(1000 0000 0011 110 1) 
 
 Let us calculate CSPF's. Ordering r is as follows: 
 
 r(v ± ) = i-if for i = 5, 6, . .., 12, 
 
 r(v i ) = i+8 for i = 1, 2, . .., h. 
 
 Since v is an output terminal, 
 
 g c (c 5 ) = (1 0000 0011 110 1) = f. 
 
 As v is a single output gate, 
 
 G (v_) =(1000 0000 0011 110 1). 
 c p ' 
 
 CSPF's for inputs of v_ are calculated as follows: 
 
 G (v r ) =(1000 0000 0011 110 1) 
 c 3 
 
 f(v 6 ) = (0Q©@ 0®1Q 0000 000 0) 
 
 f(v ) = (0 @0®0 0000 ooQo) 
 
 f (vq) =(0 000 0000 QQ 0) 
 
 ( d ) 
 For any d such that G^ J (v ) = at least one input should be 1. These 
 
 selected inputs are encircled. For any d such that G^ d '(v ) = 1, the d-th 
 
 c p 
 
 component of every input must be 0, so CSPF's for inputs of v are: 
 
 5 
 
 G c (C 6 ) = (0 111 * 1 * 1 "00 00*0), 
 G c (C ) = (0*** 1*1* **00 0010), and 
 G c (Cg ) =(0*** **** 1100 00*0). 
 
12 
 
 As y<, v and Vq are single output gates, 
 
 °c ( V " G c<V' G c ( V ■ G c (v 75>' c<V = °e ( V' 
 
 CSPF's for inputs of v- are as f ollows : 
 
 G (vg) = (0 111 * 1 * 1 **00 00*0), 
 f ( Vl ) =(0 000 0000 1 lQ@ CD© 1Q), 
 f(v ) =(©000 1000 1000 100 0). 
 Using ordering r we select "1" inputs shown by circles. 
 
 G (c l6 ) = (* * * **11 11*1), 
 G (c ,) =(1000 * * * * * * ****) = G (v ) 
 In a similar manner, we can calculate CSPF's as follows: 
 
 G (v-0 =(1*** 0*0* * * l * **0*), 
 
 c 10 ' ' 
 
 G (v ) = (*** * 0*0* * * * * 1*0*), 
 c 11 " 
 
 G (c „) = (1 * * * * * * * 00** * * * *), 
 G (c )=(**** 1 -x- * * *-■*** 0***), 
 
 = (l * '* * ]_ * * * 00** 0** *). 
 
 Now let us check if we can merge gates v and v, p . Based on the 
 
 results just obtained, we have 
 
 G c (v ) G c (v 12 ) =(1000 10*0 00** 0***), and 
 
 f(v ).f( V]2 ) =(1000 1000 0000 000 0). 
 
 Therefore v and v^ are mergeable. By merging two gates v and v , the 
 
 network in Figure 2.2(b) is obtained. 
 
 If we use the condition given by [8], then we cannot merge these 
 
 two gates. The reason is as follows: 
 
 (l) We can add x to v as its input because v^ is the 
 
 only one immediate successor of v^ and v r has x, 
 
 9 o 1 
 
 as an input. 
 
13 
 
 (2) We can add x to v^ as its input because all 
 
 immediate successors of v n (which are v Q and v ) 
 
 y o 11 / 
 
 have x as inputs. However, we cannot conclude 
 whether or not we can add x, to v because there 
 exists a path v^ -» Vg -* v on which there is no 
 gate which has an input connection from x» . 
 Even if two gates are not mergeable we can sometimes make them 
 mergeable by adding redundant connections. 
 
 Example 2.2 The network shown in Figure 2.3(a) realizes function f 
 f = (0 1 1111 1111 000 1). 
 • For gates v and v , we have 
 
 fO n ) =(1010 1010 0000 000 0), 
 
 v( Vj2 ) -(1010 0000 1010 000 0), 
 
 G c ( v n) = (1*1* 1*** 000* * * *), 
 
 G c (v 12 ) = (1010 000* **** * * * *) # 
 
 As G c^ v ll^ flG c ( ' V 12' ) = ^ holds, we cannot merge these two gates. But if 
 we can change the 5-th component of G c (v n ) from 1 to *, then we can merge 
 these gates. 
 
 The 5-th components of the inputs of v^ are 
 
 f(5) (V = 0, f (5) (v 9 ) = 0, and f (5) ( Vll ) = 1. 
 
 As G c ^ = lf G c ^11^ is re q uired "to be 1. If we can find a gate v, 
 satisfying the following conditions, then by adding a connection from 
 
 v to v. we can change G^^v ) from 1 to *. 
 1 o c 11 
 
 (1) v. is connectable to v.. 
 
 1 o 
 
 (2) f (5) (v.) = 1. 
 
 l 
 
 (3) v\ i s ( v £) (loop-free condition). 
 
 i 
 
Ik 
 
 As 
 
 G (v,) = (* * * 0000 0000 **10) 
 holds, the set of connectable functions for Vs is given by 
 
 K(vs) = (* * * * *-*-*■* **-**■ * * *). 
 v satisfies the above conditions 
 
 f(v ) =(0 101 1111 0000 000 0). 
 The network in Figure 2.3(b) is obtained by adding a connection from v 
 
 10 
 
 to V/-. 
 
 In this network 
 
 G (v__ ) = (1 * 1 * * * * * 000* * * *). 
 c 11 ' 
 
 o the condition 
 
 ((tuWtjj) * s c (t u ) ns^) 
 
 is satisfied. By merging v and v „ the network in Figure 2.3(c) is 
 obtained. 
 
 In Procedure MGC the input set for the new gate v. . is the union 
 of input sets for v. and v.. However, we can permit a different input set 
 for the new gate. Such a generalization is given in the following 
 procedure. 
 
 Procedure GMGC . A generalized merging of gates using CSPF's. 
 
 Step 1 Find two gates v. and v. such that 
 £ — i J 
 
 G (v. ) n 3 (v.) = G (v. .) / 6. 
 
 Step c Consider an imaginary gate v. . whose CSPF is G (v. .). 
 c — ij c 13 
 
 Calculate the set K(v. .) of all connectable functions to v. .. 
 Step 3 Select the set U of gates and input terminals, y, 
 satisfying the following conditions: 
 
15 
 
 (a) A given network 
 
 (b) Intermediate result 
 
 (c) Network obtained by gate merging 
 
 Figure 2.3 EKample 2.2 
 
16 
 
 (a) For all v e u, f(v) e K(v. .). 
 
 (b) For all v e U, v ft S(v. ) U s(v.) 
 (loop- free condition). 
 
 ste P ^ Connect all gates and input terminals in U to gate 
 v. .. If the output function is contained in G (v. ), then 
 
 -LJ C 1J " 
 
 go to Step 5; otherwise v. and v. cannot be replaced by- 
 
 gate v. . . 
 10 
 
 Step 5 As 
 
 V f(v) e G (v. .) 
 veU c ^ 
 
 holds, using the disconnectable condition select a new input 
 set U' (which is a subset of U) such that 
 
 V f(v) € G (v ). 
 ve U ' J 
 
 ste P 6 Replace v. and v. by gate v which has input connections 
 
 from all the gates and input terminals in U*. 
 
 In this procedure after the selection of v. and v. we can disconnect 
 
 all inputs of v and v . If we calculate CSPF's after the selection of v 
 x J i 
 
 and v CSPF's will become larger and it gives a greater possibility of 
 merging. In such a case Step 1 of Procedure GMGC is replaced by the 
 following step: 
 
 Step 1 (1-1) Select two gates v. and v.. 
 
 i J 
 
 (1-2) Disconnect all inputs from v. and v.. 
 
 i d 
 
 (1-3) Calculate CSPF's using ordering r 
 
 satisfying the following condition: 
 
 r(v ) - 1, r(v.) = 2. 
 
 (1-U) If G (v . ) ns(v) = fi, then select 
 (-1 c j 
 
 another pair of gates; otherwise go 
 to Step 2. 
 
17 
 
 Procedure GMGC has the following differences from Procedure GMC : 
 
 (1) Sometimes we can replace v. and v by v even if 
 
 1 3 ±3 
 
 f(v )-f(v ) e G (v ) na (v.) is not satisfied. 
 
 J <- X C J 
 
 (2) The conditions, 
 
 v. i S(v ) and v. ^ S(v.), 
 j - 1 - i j 
 
 are not necessary. 
 
 (3) If v e S(v ) (or v € S(v.)), the set of immediate 
 
 <J ± i 3 
 
 successors of v. sometimes becomes smaller than the 
 
 -*-u 
 
 set IS(v. ) U IS(v.). 
 i v 3 
 
 (k) Sometimes there are isolated subnetworks after removing 
 
 all inputs of v. and v . 
 1 3 
 
 The following two examples show the cases (3) and (k), respectively. 
 Example 2.3 . Figure 2.k shows networks realizing function f 
 f = (1 0000 0010 110 1). 
 As in the network in Figure 2.U(a) we have 
 
 f (vg) - f(v lx ) =(0 111 1111 0000 000 0), 
 we can merge v g and v^ The result is shown in Figure 2.U(b). In this 
 network the connections from V]J to v 1Q and from x^ to v are redundant 
 so these connections can be removed (these connections are shown by dotted 
 lines in Figure 2.k{\>)). In this example, 
 
 IS(v xl ) Uis(v 8 ) = {v 10 ,v 5 }, but 
 IS(v 8 ^) = {v 5 }. 
 Thus IS(v i;L ) U IS(vq) = IS(v Q ) is not satisfied. The connection from 
 v ll t0 v 10 becomes redundant due to the removal of the connection from v 
 to v Q (this removal changes CSPF for v, to a larger set). 
 
18 
 
 
 x? 
 
 V. 
 
 X 
 
 *i-l3/ ^ 
 
 V 10 
 
 =&■ 
 
 \ 
 
 V, 
 
 X-, 
 
 v, 
 
 V 7 ^— -) v 5 
 
 (a) A given network 
 
 x 
 
 1 
 
 *a v i2 
 
 J 
 
 Xi 
 
 X 
 
 M^°-% 
 
 V 
 
 10 
 
 % 
 
 Xi 
 
 V 
 
 ;■:,- 
 
 v r 
 
 r 
 
 ! v 5 
 
 (b) After applying Procedure GMG-C (dotted lines 
 show redundant connections ) 
 
 Figure 2.k Example 2.3 
 
19 
 
 Example 2.k. The network shown in Figure 2.5(a) realizes function f 
 f = (1 1 1 0111 0001 000 1). 
 Functions realized at all input terminals and gates are as follows: 
 f (Vj) = (0 000 0000 1111 1111) = x 
 f(v 2 ) =(0 000 1111 0000 1111)=, 
 f(v 3 ) =(0 011 0011 0011 11) = x 
 f (v. ) = (0 101 0101 0101 10 1) = Xl 
 f(v 13 ) = (1111 0000 1111 000 0) 
 f ( v !2) =(0 000 1100 0000 110 0) 
 f ( v u ) =(1010 0010 0000 000 0) 
 f ( v 10 ) = (0 000 1000 1010 1010) 
 f(v 9 ) = (1100 0100 0000 000 0) 
 f(vg) =(0 001 0000 0000 000 0) 
 f(v ? ) =(0 000 1000 1100 110 0) 
 f(v 6 ) = (0 000 1000 1010 1010) 
 f(v 5 ) = (1110 0111 0001 000 1) 
 Here we want to replace ^ and v^ by a new gate, y^^. Let us choose 
 ordering r as follows: 
 
 r(v 1Q ) = 1, r (v xl ) = 2, r^) = 3j r (v 6 ) = k, 
 r(v ? ) = 5, r(v 8 ) = 6, r(v 9 ) = 1, r^) = 8, 
 r(v x ) = 9, r(v 2 ) = 10, r(v 3 ) = 11, r (^) = 22. 
 Disconnect all inputs of v 10 and v^ and calculate CSPF's as follows: 
 o c (v 5 ) = (1110 0111 0001 000 1) 
 G c ( v £)=(0 00* *000 * * 1 Q * * 1 0) 
 G c (v ? ) =(000* 1000 11*0 11*0) 
 G c (vg)=(0 001 *000 * * * ***o) 
 
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21 
 
 G (c /-) = (1 * * * * * * * * * o * * * o *) 
 
 G (c g) = (* * 1 * * * i * * * o * * * o *) 
 
 G (c Q J = (1 1 * * 1** 00** 00**) 
 c ?> { 
 
 G ( c ^ q)=(11*0 * * * * * * * * * * * *\ 
 c 9> o ' 
 
 G (c nn Q ) = (* * 1 * * -x- * * * -x- •* •*•*-* *-) 
 c 11, o y 
 
 G (v ) =(11*0 01** 000* 000*) 
 
 C y 
 
 G (v ) =(0 0** 10** * * * * * * * *) 
 G (v,, ) =(**10 * * 1 * **o* * * o *) 
 
 = (0 010 101* **o* * * *), 
 K(v ) = (* * * 0*0* * * * * ****). 
 After we removed all inputs of v 1Q and v we have obtained the network 
 shown in Figure 2.5(b). There is an isolated subnetwork consisting of two 
 gates (v^ and v ). We can use any function realized by a new network 
 using at most two NOR gates as an input of v (i.e., the isolated 
 subnetwork is ignored and we try to realize a new network of at most two 
 gates which realizes functions useful for us). If we use the following 
 functions as the inputs of v 1Q 1± , then v realizes a function in 
 
 G c (v i0,ll> : 
 
 Thus, the network in Figure 2.5(c) is obtained. 
 
 If the isolated networks resulting from the disconnection of the 
 inputs of v ± and v consist of t gates, we discard these isolated subnetworks 
 and construct a new network of at most t gates such that the output functions 
 at the gates in this network can be used as the input set to the new gate 
 
22 
 
 under consideration. However, such a procedure is very difficult to 
 formulate so we restrict the set of available functions as inputs of v. . 
 as follows : 
 
 (1) external variables, 
 
 (2 ) functions realized by gates not contained in 
 
 S(v. ) U s(v.) U [v., v.), 
 l j i j 
 
 (3) t functions which are realized by t separate 
 NOR gates (with input from input terminals). 
 
 For a further generalization of gate merging we define the 
 possibly-connectable condition as follows. 
 
 Definition 2.2. Gate v. is said to be possibly-connectable to v. 
 if the following conditions are satisfied: 
 
 (1) f (v. ) fi K(v.), where K(v.) is the set of 
 
 -k d J 
 
 connectable functions to gate v., 
 
 (2) K(v ) OG (v.) f p, 
 
 (3) v. i S(v ). 
 
 Condition (l) shows that v. is not connectable to v.. Condition (2 
 
 1 
 
 shows that v. can be connectable to v. if v. realizes a proper function 
 in G (v. ). Condition (3) is a loop-free condition. 
 
 Table 2.1 shows possible combinations of K (v.), G (v.) and 
 
 J L» _L 
 
 f (v.) when condition (2) is satisfied. 
 
 If for every d the combination of K (v.), G^ '(v. ) and f^ (v.) 
 
 j c x 1 
 
 is shown by some row except the third row of Table 2.1, then v. is 
 
 connectable to v.. In order to satisfy condition (l), there must exist 
 
 at least one d such that the combination of K (v.), G (v.) and f (v.) 
 
 j ' C 2. i 
 
Table 2.1 
 
 23 
 
 K (d) (v.) 
 3 
 
 G (d) (v.) 
 c i' 
 
 f (d) (v.) 
 1 
 
 
 
 
 
 
 
 
 
 * 
 
 
 
 
 1 
 
 
 
 
 
 
 
 ■x- 
 
 1 
 
 1 
 
 
 * 
 
 
 
 
 
 1 
 
 1 
 
 2 
 
 3 
 
 k 
 
 5 
 6 
 
 7 
 
2k 
 
 is represented by the third row of Table 2.1. For such a d if we can find 
 
 v. satisfying the following conditions, then by connecting y to v. , f Vv ) 
 & k 1 v i / 
 
 changes to (corresponding to the second row of Table 2.1). 
 
 (1) f(v k ) e K(v.), 
 
 (2) f (d) (v k ) = 1, 
 
 (3) v i S(v.) US(v ) U (v ,v ) 
 (loop- free condition). 
 
 If we can find such a v for every d satisfying the combination shown in the 
 
 third row of Table 2.1, then v. becomes connectable to v. by adding input 
 
 connections from these v 's to v.. 
 
 k i 
 
 The possibly- connect able condition can be used in gate merging. 
 Suppose gates v. and v. are to be merged to gate v. . . In order to realize 
 a function contained in G (v. . ) by gate v. ., functions which are modified 
 based on the possibly- connectable condition may be used as inputs to gate 
 v. .. The following procedure is a generalized procedure of gate merging 
 using the possibly-connectable condition. 
 
 Procedure GMGPC . A generalized merging of gates using the 
 psosibly-connectable condition. 
 
 Step 1 - Step 3 The same as Procedure GMGC. 
 
 Step k Connect all gates and input terminals in U to 
 
 gate v. . . If the output function is contained in 
 
 G (v. . ), then go to Step 5; otherwise go to Step 7. 
 c ij 
 
 Step 3 - Step 6 The same as Procedure GMGC. 
 
 Step 7 Calculate an additional requirement vector H 
 
 for v. . . 
 
25 
 
 C H (d ^ = for all d such that G^(v. ) = 1 
 IT ' = * for all d such that G^(v ) = * 
 
 c ±y 
 
 or v t^ d \v) = 1. 
 veu 
 
 JP d ^ = 1 for all d such that G^(v ) = 
 
 c ±y 
 
 and v f^(v) = 0. 
 veU 
 
 Step 8 Find a gate v satisfying 
 
 (a) G c (v h ) n K (v..) /-0, 
 
 (b) There exists at least one d such that 
 H (d) = 1 and f (d) (v h ) = 1. 
 
 (c) v h 4 S(v.) U S(v.) 
 
 (a) and (c) show that v is possibly- connectable to v 
 
 n ±y 
 
 ste P 9 By adding inputs which are connectable to v , 
 
 tr 
 
 try to make v h connectable to v... if ^ becoraes con nectable 
 
 to v h , then U - U U {v h }. Update CSPF's and functions 
 
 realized at all the gates. 
 
 Step_10 Repeat Step 8 and Step 9 for all gates satisfying 
 
 conditions in Step 8. 
 
 Step_ll Connect all gates and input terminals contained in U to 
 
 gate v if the output function is contained in G (v ), then 
 
 c i j 
 
 go to Step 5; otherwise v and v cannot be replaced by gate v 
 
 d id" 
 
 Note that by each application of Step 9 the network is modified 
 
 (new connections are added). The following example shows how Procedure 
 GMGPC works. 
 
K 
 
 Example 2.5 . The network in Figure 2.6(a) realizes function f 
 f = (1 0000 0010 110 1). 
 Funtions realized at input terminals and gates are as follows : 
 
 f( V] 
 f(v g 
 f(v. 
 
 f(v, 
 
 f(v 
 
 f(v 
 f(v 
 
 12 
 
 11 
 
 10 
 
 f(v 9 
 f(v 8 
 
 f(v„ 
 
 f v, 
 
 f v. 
 
 0000 0000 1111 1111 
 
 0000 1111 0000 1111 
 
 0011 0011 0011 0011 
 
 0101 0101 0101 0101 
 
 1100 1100 0000 0000 
 
 0000 0000 1100 1100 
 
 1010 0000 0010 0000 
 
 0011 0011 0000 cooo 
 
 0101 1111 0000 0000 
 
 0101 0000 1101 0000 
 
 0000 1010 0000 0010 
 
 1000 0000 0010 1101 
 
 We will try to merge gates Vn and v . For the calculation of 
 
 CSPF's, we can remove inputs of Vo and v Q (see Figure 2.6(b)). CSPF's 
 for gates are calculated according to an ordering r given as follows : 
 
 r(vg) = 1, r(v 9 ) = 2, r(v^ = 3, 
 r(v 6 ) = h, r(v 7 ) = 5, r ( v 1Q ) = 6, 
 r(v 11 ) = 7, r( V]L2 ) = 8, r( Vl ) = 9, 
 r(v 2 ) - 10, r(v 3 ) = 11, r(v h ) = 12. 
 The CSPF's for all the gates in the network shown in Figure 2.6(b) 
 are as follows : 
 
X 
 
 1 — \ V. 
 
 x. 
 
 13 
 
 x, 
 
 X 
 
 3~W 
 
 11 
 
 x. 
 
 Xi 
 
 V. 
 
 10 
 
 ■^=£>1 
 
 x— 1 V 8 
 
 '1 ~1 v 
 
 V 
 
 27 
 
 of 
 V 13 
 
 (a) A given network 
 
 x. 
 
 x. I 12 
 
 v 
 
 V 
 
 11 
 
 X 
 
 k 
 
 V 
 
 10 
 
 V 
 
 \ — -\ 
 
 K>' 
 
 v 9 
 
 r^ 
 
 13 
 
 (b) Network after removing the inputs of v and v 
 
 o c 
 
 X 
 
 k- 
 
 X- 
 
 x^ V 
 
 Xi 
 
 X, 
 
 3 H 12 
 
 11 
 
 X. 
 
 D~l V 
 
 x, 
 
 X 
 
 i v 7 > 
 
 10 
 
 L 
 
 V- 
 
 y v 
 
 x. 
 
 8,9 
 
 13 
 
 (c) Network obtained by gate merg: 
 
 mg 
 
 Figure 2.6 Example 2.5 
 
28 
 
 G (v c ) -(1000 0000 0010 110 1) 
 c v 5 
 
 (vs) = (0 * * * 1*1* * * * 0010) 
 
 c 
 
 '6 
 
 G (v) =(0 1*1 **** 1101 00*0) 
 G (v Q ) = (0*** * 1 * * **o* 00*0) 
 
 CO 
 
 G (v ) =(0*1* ***1 * * o * 00*0) 
 
 C y 
 
 G (v nrt ) = (10*0 0*0* 0010 * * *) 
 c 10 ' 
 
 G (v ) =(0*** 0*0* 1*0* 1*0*) 
 
 G(v ) = (l * * ■* i*** o*** 0***) 
 
 For the new gate Vn Q , 
 
 G (vo n ) = G k) OG (v_) 
 c o, 9 co c 9 
 
 =(0*1* *1*1 * * * 00*0) 
 
 K(v ^)=(**0* *0*0 * * * * * * * *) 
 Of 9 
 
 So v , V/-, v and v are connectable to Vn Q . According to 
 Step 1+ of Procedure GMGPC, 
 
 u = ( v^, v 6 , v _, v^} , and 
 
 V f(v) =(1010 0101 0000 000 0). 
 
 veU 
 
 This function is not contained in G (v Q „). Therefore, we have to use 
 
 c o, 9 
 
 possibly- connectable connections. 
 
 In Step 7> H is calculated as follows: 
 
 H=(l*00 *0*0 * * -x- -x- -x -x- -x- *) # 
 
 Then in Step 8, we find that v and v^ satisfy the conditions 
 required. 
 
 In order to make v no connectable to v Q „, we first calculate the 
 
 set of connectable functions to v, p as follows: 
 
 Since G (v no ) = (l * * * i * * * o*** 0***), 
 c ±d 
 
 K(v )=(o*** o * * * * * * * -x- -x- -x -x), 
 
29 
 
 v^ is connectable to v^. If we connect v. to v^, the output of v 
 becomes 
 
 f(v J2 ) =(1000 1000 0000 000 0). 
 This function is connectable to Vn q and its first component is 1 which is 
 required by H. 
 
 In Step 11, we recalculate the function corresponding to the new 
 set U of connectable functions. 
 
 u = < v i> v 6> V v n' v i2^ 
 
 V f(v) =(0 011 0101 0000 000 0) 
 veU 
 
 It is contained in G (v Q „). 
 
 c o, 9 
 
 We then try to obtain a reduced set U' of U as follows: 
 
 G c (vq g ) = (0 * 1 * * 1 * 1 * * * 0*0), 
 
 f(v ) =(0 000 0000 1111 1111), 
 
 f(v 6 ) = (0 000 1010 0000 0010), 
 
 f(v ) = (0 101 0000 1101 000 0), 
 
 f(v 11 ) = (0 000 0000 1100 1100), 
 
 f(v 12 ) = (1000 1000 0000 000 0). 
 
 As V 6' V 7 and v ll are disconne ctable, the input set U r for v ft Q is 
 
 {v-j^, v^). The network obtained is shown in Figure 2.6(c). 
 
30 
 
 3. GATE SUBSTITUTION AND ITS GENERALIZATIONS 
 
 In this section, a netowrk transformation procedure by gate 
 substitution which was discussed in [8] will be generalized using CSPF's. 
 A further generalization of this transduction method by gate substitution 
 is also discussed using the possibly-connectable condition. 
 
 If the disjunction of outputs of some gates and external variables 
 is found to be identical to the output of a gate v., then we can 
 substitute the sum of these outputs of gates and/or external variables 
 for the output connections of gate v. [8] . 
 
 A simple example is shown in Figure 3.1. Here, we assume that 
 the disjunction of outputs of v.,, v. ? and external variable x is 
 identical to the output of v., that is, 
 
 f(v dl ) V f ( v a - 2 ) V x^ = f(y ± ). 
 
 Therefore, the connections from v. to y _ and y _ can be replaced by the 
 ' i kl k2 
 
 connections from v.-., v. p and x„ to v ,, and v ? as shown in Figure 3.1(b). 
 If we use CSPF's this procedure can be generalized as follows. 
 Procedure SGC . Substitution of a gate using CSPF's. 
 Step 1 Calculate CSPF's for all gates in a given network. 
 Step 2 Select one gate v. . If we can find a set U of 
 gates and input terminals satisfying the following conditions: 
 
 (a) v ^ S(v. ) for every v € U, 
 
 (b) V f(v) e G (v i ), 
 
 V€U 
 
31 
 
 (a) Original network 
 
 (b) After gate substituti 
 
 on 
 
 Figure 3.1 A Simple Example of 
 Gate Substitution 
 
32 
 
 then go to Step 3; otherwise repeat Step 2 until all 
 gates are considered. 
 
 Step 3 Substitute connections from gates and input 
 terminals in u for the output connections of gate v. . 
 
 l 
 
 To calculate a set U in Step 2, we can use the following method: 
 
 (2-1) Calculate a set H(v. ) which is defined by 
 
 H(v ) = fl K (v), 
 veIS(v. ) 
 
 where K(v) is the set of connectable functions to gate v 
 
 with respect to the CSPF for v (i.e., G (v)). 
 
 (2-2) Let Ube a set of all gates and input terminals 
 
 satisfying 
 
 v <ji S(v. ), and 
 
 f (v) e H(v. ) for every v e U. 
 
 (2-3) If v f(v) € G (v.), then go to Step 3; 
 
 veU c 1 
 otherwise repeat Step 2 until all gates are considered. 
 
 It should be noted that a set Umay contain redundant elements, so 
 we can eliminate some of them using disconnectable conditions (Theorem 1.2) 
 after the substitution. 
 
 Instead of substituting outputs of gates and input terminals for 
 gate v , we can generalize the procedure by substituting outputs of gates 
 and input terminals for each output of gate v. (possibly a different set 
 of outputs of gates and input terminals for each output of v. ). A simple 
 example is shown in Figure 3.2. Gate v of the network shown in Figure 
 3.2(a) has three outputs connecting to v,-, v and Vn. These connections, 
 however, can be replaced by connections from v , v and v , 
 
33 
 
 x. 
 
 X,. 
 
 X 
 
 1 
 X 3 
 
 (a) A given network 
 
 "1-1 S 
 
 (b) After the substitution for each of the output 
 
 connections of v. 
 
 12 
 
 Figure 3.2 A Simple ©cample of Procedure SOGC 
 
3h 
 
 respectively. So we can remove gate v^ (see Figure 3.2(b)). The following 
 
 procedure is a generalization of Procedure SGC. 
 
 Procedure SOGC . Substitution of outputs of a gate using CSPF's. 
 
 Step 1 Select one gate v. . 
 
 Step 2 Calculate CSPF's for all gates, input terminals 
 
 and output connections of v. . An ordering r satisfying 
 
 the following conditions may give a good result: 
 
 r(v. ) = 1, and r(v.) < r(v ) for every v. e S(v. ) and 
 
 every v ^ S(v. ). 
 
 Step 3 For each output connection c. . of v., calculate 
 £ — ij l 
 
 a set U of gates and input terminals. 
 
 U - {vlv e V n U V T , v 4 S(v. ), 
 f(v) € K(v )}. 
 
 J 
 
 Step k If the following condition is satisfied then 
 
 substitute connections from all elements in U to v 
 
 3 
 
 for c. . 
 
 V f(v) e G (c ). 
 ve U c 1J 
 
 Disconnect some of these new connections by applying 
 
 disconnect able conditions to gate v.. 
 
 Step 5 Repeat Step 3 and Step k for all output 
 
 connections of gate v. . If there exists an output 
 
 connection which cannot be substituted, then v. cannot 
 
 i 
 
 be removed. If so, repeat the procedure on the original 
 network by selecting another gate. 
 
35 
 
 If we want to reduce computation time, we can calculate CSPF's for 
 gates, input terminals and connections before applying the procedure. In 
 this case, Step 2 is unnecessary, but the result may not be as good as 
 the result using Step 2. 
 
 The possibly-connectable condition (Definition 2.2) can be used in 
 
 Procedure SOGC. An example is shown in Figure 3.3. If we apply Procedure 
 
 SOGC to gate v 10 of the network in Figure 3.3(a) we cannot remove this 
 
 gate. However, if a redundant connection from v to v is added (see 
 
 Figure 3.3(b)), then we can substitute the output of v for c , and 
 
 11 10, 6 
 
 the output of Vj2 for c 10 ^, A procedure using the possibly-connectable 
 conditions is as follows. 
 
 Procedure SOGPC . Substitution of outputs of a gate using the 
 
 possibly-connectable condition. 
 
 Step 1 - Step 3 The same as Procedure SOGC. 
 
 ste P ^ If "the following condition is satisfied, then 
 
 substitute connections from all the gates and/or 
 
 input terminals in U to v. for c. . and go to Step 10: 
 
 J j- j 
 
 V f(v) € G (c ). 
 ve U c X J 
 
 If this condition is not satisfied, go to Step 5. 
 ste P 5 Calculate an additional requirement vector H 
 for c. . such that 
 
 -'-J 
 
 IT d ^ = 0, for all d such that G^^c ) = 
 
 E { = * for all d such that G^(c ) = * 
 
 ( ° J 
 
 or y f (dj (v) = 1. 
 
>2_j 
 
 1 — l v 
 
 12 
 
 x 
 1 
 
 X 3 
 
 j 
 
 10 
 
 a> 
 
 i 
 
 U^V^>- 
 
 ■o v 
 
 13 
 
 x,, _fll 
 
 X, 
 
 -) v 8 
 
 'a) A given network 
 
 X, 
 
 1 lv 
 
 JV 
 
 4'9 
 
 X-, 
 
 ^ 
 
 x l -.- 
 
 x i 
 
 4 = 
 
 X 
 
 10 
 
 V/ 
 
 X, 
 
 x? \>- 
 
 X 
 
 =)v 
 
 2 — I v c 
 
 4, A X. 
 
 1 V 5> 
 
 V 
 
 X, 
 
 V) — I 
 
 y 
 
 13 
 
 (b) A redundant connection from x is added to v 
 
 2 \ v 
 
 ^4v> 
 
 >: 
 
 1 " i v n ' 
 X U J ^ 
 
 x 
 
 x 3 H v ic 
 \ -L — 
 
 y 
 
 XT 
 
 / 
 
 Vi ) 
 
 4^ 
 
 -O V 
 
 13 
 
 ^3 - 
 
 X, —j "11 
 
 :-:.- 
 
 H 
 
 / 
 
 x. 
 
 -3 
 
 \ V r 
 
 (c) Network obtained by gate substitution 
 
 Figure 3.3 A Simple Example for Procedure SOGPC 
 
37 
 
 IT ■ = 1 for all d such that G^(c ) - 1 
 
 c ±y 
 
 or v f^ d \v) = 0. 
 veu 
 
 ste P 6 Find a gate v h satisfying 
 
 (a) G c (v h ) HK(v ) / 0, 
 
 (b) there exists at least one d such that IT d ^ = 1 
 and f (d) (v h ) = 1. 
 
 (c) v h i S(y.). 
 
 (a) and (c) show that v is possibly-connectable to v.. 
 
 J 
 
 Step_7 By adding inputs to v h , try to make v connectable 
 
 to v . If v h becomes connectable to v., then U = U U {v } . 
 
 J h 
 
 Update CSPF's and functions realized at all the gates in 
 
 the network. 
 
 ste P 8 Repeat Steps 6 and 7 for all gates. 
 
 Step_9 If v f(v) e G (c ), then go to Step 10: 
 
 veu J 
 
 otherwise c. . can not be replaced. 
 
 ste P 10 Substitute connections to v. from gates and/or 
 input terminals in U for c. .. Disconnect some of the 
 new connections by applying the disconnectable conditions 
 to gate v.. 
 
 d 
 
 Ste P ^ Repeat Step 3 - Step 10 for all output connections 
 of gate v i# If there exists an output connection of v. 
 which cannot be substituted for, then v. cannot be 
 removed. If so, select another gate and apply Steps 3-10 
 to the original network again. 
 
V: 
 
 An application of this procedure is shown in the following 
 example . 
 
 Example 3.1 . Let us consider the network shown in Figure 3. Ma) 
 which is the first solution by Program ILLOD(NOR-B) based on the 
 branch-and-bound method [9*10] for the four-variable function f. 
 
 f = (1 1 1000 1011 0111) 
 Functions realized at input terminals and gates are as follows : 
 
 (0 000 0000 1111 1111] 
 (0 000 1111 0000 1111] 
 (0 011 0011 0011 11] 
 (0 101 0101 0101 10 1] 
 
 (ioio ioio ioio ioi o; 
 
 (1000 1000 0000 0] 
 
 (oioo oioo oioo oio o; 
 
 (0 010 0010 0000 ooo o; 
 (1001 0000 1011 0] 
 (0 110 0000 0100 0] 
 (0 000 0000 0000 10 0] 
 (0 110 0111 0000 0] 
 (1001 1000 1011 111] 
 First, we want to remove Vq so the following ordering r is used 
 in the calculation of CSPF's: 
 
 r(vg) = 1, r(v 5 ) = 2, r(v 6 ) = 3, r(v ? ) = k, 
 
 r(v ± ) = 1-k for i = 9, 10, . .., 13, 
 
 r(v ) = j+9 for j = 1, 2, ..., k. 
 
 J 
 
 f(v x ) - ( 
 
 f(v 2 ) = ( 
 
 f(v 3 ) = ( 
 
 f(v 4 ) - ( 
 
 f(v 13 ) - ( 
 
 f( vl2 ) - ( 
 
 f(v n ) = ( 
 
 f(v 1Q ) - ( 
 
 f(v 9 ) - ( 
 
 f(v 8 ) - ( 
 
 f(v 7 ) = ( 
 
 f(v g ) - ( 
 
 f(v 5 ) = ( 
 
39 
 
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 fe 
 
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^3 
 
 CSPF's are as follows: 
 
 G (vj = 
 c 5 
 
 As 
 
 G c (v 6 ) = 
 G c (v ? ) = 
 G c (v 8 ) = 
 
 W = 
 
 G c< v 10> = 
 G c (v lx ) = 
 
 G c ( Vl2 ) = 
 
 G c (v 13 ) = 
 
 1001 1000 1011 0111 
 
 0110 0111 0*00 *000 
 
 0**0 0*** 0*00 1000 
 
 0**0 0*** 0100 * 
 
 1001 *000 1011 0*** 
 
 0*10 **** 0*00 •**** 
 
 01*0 **** 0100 **** 
 
 100* 1000 **** o*** 
 
 10** **** 101* **** 
 
 K(v ) =(0**0 0*** 0*00 * 0), 
 
 only v 1Q is connectable to v , but as f (vg) > f (v ) the connection from 
 
 V 10 to v 5 is redundant « Only v lx satisfies the possibly-connectable 
 condition 
 
 G c (v i:l ) HK(v 5 ) 
 = (0 1*0 0*** 0100 * 0) / 
 Since f (v^) =(0100 0100 0100 1 0), if we can find a 
 function which is connectable to v and whose llj-th component is 1, 
 f ( v i:l ) wil1 become connectable to v after connecting this function to 
 v ll* The set of connectable functions for v is given by 
 
 K(v 1 ) = (* * * **** *o** ****) 
 
 so v 2 , v^ and v are connectable to v . As f 
 
 (1*0 
 
 (Vp) = 1, we can use the 
 
 connection from v . The new function realized at v becomes 
 
 f ( v n) = (° 100 oooo oioo oooo), 
 
and f(v,.,) e G (vj- So the connection from v to v can replace v p . The 
 
 result is shown in Figure 3.Mb). Next we check whether or not v, p can be 
 
 removed by substitution. During the calculation of CSPF's the connection 
 
 from v , to v is removed, and the network shown in Figure 3«M C ) is 
 
 obtained. The connection from v, p to v can be replaced by a connection 
 
 from v to v . The connection from v, p to v^ cannot be replaced by the 
 
 outputs of other gates and/ or input terminals. However, by adding a 
 
 redundant input from v~ to v,~, the output of v,_ can substitute for the 
 
 connection from v, p to v^. The result is shown in Figure 3«Md). The 
 
 network is proved to be a minimum network for f by Program ICCOD(NOR-B) 
 
 [10,11], Since Vp is a single-output gate during the first step, 
 
 Procedure GMGPC discussed in Section 2 would have given the same result. 
 
 In Procedure SOGPC, we try to substitute for all the output 
 
 connections of a selected gate v. , but sometimes we can substitute only 
 
 for part of these connections. If so, the CSPF for v. often becomes 
 
 larger than that in the original network. This may help to remove some 
 
 redundant inputs of v. . Especially when a single-output gate is connected 
 
 to v. and we can remove such an input from v. , the cost of the network 
 i i 
 
 will reduce. One example is shown in Example 3.2. 
 
 Example 3.2 . The network shown in Figure 3.5(a) realizes function f 
 f = (1 1 1000 1011 0111). 
 During the application of Procedure SOGPC to gate v , the connection from 
 v to Vn is replaced by the connection from v~ to Vp (see Figure 3.5 (b)), 
 
 because 
 
 G (vp) = (0 1*0 0*** 0100 * 0), 
 
J+5 
 
 o f 
 
 X H V 13 
 
 (a) A given network. 
 
 v 
 
 11 
 
 x^=) V 12 
 
 v 
 
 V 
 
 X3-I V 1C 
 
 X 
 
 8 
 
 V 
 
 (b) Connection C has been replaced by C 
 
 9,8 * J 3j 8' 
 
 -o f 
 
 V. 
 
 Ill 
 
 O f 
 
 (c) Redundant connection C has been removed. 
 
 Figure 3-5 Example 3.2, 
 
i*6 
 
 and 
 
 f(v ) v x V x , = (0 1 10 0). 
 9 <- 3 
 
 On the other hand, we cannot replace all output connections of v , so v 
 cannot be removed. However, the new CSPF for v is larger than before, and 
 as a result the connection from v to v satisfies the disconnect able 
 condition. The result is shown in Figure 3.5(c) which can be further reduced 
 (see Figure 3.3). 
 
hi 
 
 k. COMPUTER PROGRAMS AND EXPERIMENTS 
 
 Two computer programs NETTRA-G3 and NETTRA-G1+ using the generalized 
 procedures for gate merging and gate substitution, discussed in Sections 2 
 and 3, respectively, have been prepared and tested. The actual procedures 
 used in NETTRA-G3 and NETTRA-Gl* are described in Sections k.l and k.2, 
 respectively. Several examples are shown in Section U.3 to illustrate the 
 computer procedures. 
 
 k.l Program for Gate Merging 
 
 Three procedures for gate merging, i.e., MGC, GMGC and GMGPC have 
 been described in Section 2. As we discussed, procedure GMGC is more 
 powerful than MGC, since a pair of gates which can be merged by MGC can 
 always be merged by procedure GMGC. Similarly, procedure GMGPC is more 
 powerful than GMGC since it uses the possibly-connectable condition which 
 is a looser condition than the connectable condition used by procedure 
 GMGC. Procedure GMGPC, however, requires more computation time than 
 procedure GMGC in order to modify a possibly-connectable function to a 
 connectable connection (steps 8-11 in procedure GMGPC). 
 
 A FORTRAN program NETTRA-G3 based on procedure GMGC has been 
 developed. In NETTRA-G3, the gate merging is realized by subroutine 
 GTMERG which applies the following procedure to a given NOR network when 
 it is called by the subroutine MAIN in NETTRA-G3. 
 
V; 
 
 Procedure used in subroutine GTMERG 
 
 Step 1 Calculation of CSPF's 
 
 Call subroutine MINI2 to calculate a compatible set of 
 
 permissible functions for each v. e V_. The detailed 
 
 1 G 
 
 procedure of MINI2 is discussed in [2] and [3]. 
 
 Step 2 Selection of a pair of gates 
 
 Select two gates v. and v. such that j > i both according 
 
 to the ascending order of i and j . If all possible gate pairs 
 
 have been considered, return to the calling subroutine. 
 
 Step 3 Calculation of CSPF for the merged gate 
 
 Calculate G. (v. . ) = G (v. ) D G (v . ) and K( v. . ) . If G (v. . ) = 
 
 which means that v. and v. are not mergeable, go to Step 2. 
 
 Step k Check substitutability 
 
 If f (v. ) e G (v.) and v. t S(v.), go to Step 8. If 
 l cj l r j 
 
 f(v.) e G (v.) and v. k S(v. ), interchange the values of i 
 
 and j and then go to Step 8. 
 
 Step 5 Select connectable functions for v. . 
 * — ij 
 
 Calculate a set 
 
 U == {v k |v k e V, v k e K(v ±J ), v fe ^ S(v.) U S( Vj ) U {v.} U {v.}} 
 
 Step 6 Check realizability of the merged gate 
 
 If U f (v ) k G (v. . ), then v. . is not realizable with 
 
 TT k r c ij ij 
 v eU d d 
 
 k 
 
 functions in U only and go to Step 2. 
 
 Step 7 Making the merged gate 
 
 Disconnect all input connections of v. , and connect all v e U 
 
 to v.. (The function f(v. .) e G (v. .) is realized at v..) 
 
 1 1J c 1J X 
 
h9 
 
 Step 8 Connect v. to immediate successors of v. 
 
 ■3 
 
 Disconnect all output connections of v. and connect v to 
 
 i 
 
 all IS(v ). The resulting network still realizes the 
 
 J 
 
 specified functions. 
 
 Step 9 Update information on the new network 
 
 Call subroutine SUBNET to update the information on IS(v ) 
 
 k 
 
 IP(\), s (\)> f(\) for all v k e V. Go to Step 1. 
 
 This procedure is similar to procedure GMGC except Step k which is 
 
 not included in procedure GMGC. Since whether or not f(v ) e G (v ) or 
 
 1 c J 
 
 f( V £ VV is sa tisfied can be easily checked during the calculation of 
 
 G c (v..), adding Step k will skip the process of finding connectable functions 
 
 when the condition for substitution is satisfied. 
 
 As discussed in Section 2, if all input connections for v and v 
 
 1 J 
 
 are disconnected before the calculation of CSPF's and if a proper ordering 
 
 suitable for a specific pair of gates is used in the calculation of CSPF's, 
 
 then the CSPF of the merged gafp i p> r ( -,r 1 i * -u -, 
 
 c merged, gate, i.e., G^v ), could be larger and thus 
 
 the procedure would be more powerful. Howeyer, it is time consuming because 
 recalculation of CSPF's is required each time when a pair of gates is 
 selected. In Step 1 of subroutine GTMERG, a fixed ordering is used in the 
 calculation of CSPF's and the recalculation is required only after a pair 
 of gates has been merged to one. 
 
 Reference [ll] discusses the actual program in more detail and 
 proyides the necessary information on how to use the program. A program 
 listing of NETTRA-G3 is also attached in [ll] as an appendix. 
 
 k.2 Prog ram for Gate Substitution 
 
 A computer program NETTRA-Gl* based on procedure SOGPC is written 
 
50 
 
 in FORTRAN IV since SOGPC is the most powerful procedure among the three 
 procedures for gate substitution mentioned in Section 3. 
 
 Two orderings are used in this procedure. Ordering r n which is for 
 the calculation of CSPF's is based on the levels of gates and, when there 
 is a tie, the labels of gates. The gate v. which is under consideration to 
 be substituted is not included in this ordering because the calculation of 
 G (v. ) is not necessary (however, the CSPF of each of its output connection 
 must be calculated). 
 Ordering r n for v € V^, - {v_ . } : 
 
 -L Li X 
 
 first. 
 
 (1) r 1 (v,)<r(v.) if L(v.)<L(vJ, 
 
 (2) r (v. ) > r(v ) if j > k and L(v.) = L(v ) where L(v ) 
 
 is the level of gate v . 
 
 k 
 
 According to r , a gate v. with the smallest r (v. ) is selected 
 
 Ordering r , which is defined for input connections of a gate 
 
 v. € V - {v.} is based on the numeric order of the gate labels, treating 
 
 J Lt i 
 
 the output connections of v. differently as follows. This ordering is used 
 
 when an input of gate v. to be assigned 1 in order to satisfy G (v.) = 
 
 J c j 
 
 is chosen during the calculation of G (C n .) for c, . € C where C is the set 
 
 c hj hj 
 
 of all connections . 
 Ordering r : 
 
 (1) If v. e IS(v, ), r (c, ,) < r Q (c.,) for any c, . € C - {c..}. 
 
 J i 2 hj 2 ij hj ij 
 
 This makes the CSPF for c. . larger, and therefore the 
 
 ij 
 
 substitution for c. easier. 
 
 (2) r 2 (c hJ ) > r 2 (c kj ) for c kJ , c hJ e C - {c.j} -h>». 
 
 According to r_, a connection c. . with the smallest r (c, .) is 
 & 2' hj 2 hj 
 
 selected first. 
 
51 
 
 In order to help the understanding of the computer procedure for 
 calculation of CSPF's, the concept of intermediate CSPF's for a gate is 
 introduced. Since the CSPF's for connections (except the output connections 
 of v. , the gate to be substituted) are important only during the calculation 
 process of CSPF's for gates, they are not stored in the program to save 
 memory space. As the CSPF for a gate v. is calculated by 
 
 J 
 
 G c ( V = c., e C G c (c jk } vhere the operation of intersection is commutative, 
 jk 
 
 the intermediate CSPF for a gate v up through a certain connection c 
 
 J jk 
 
 based on an ordering r (ICSPF denoted by I^Cc )") is defined recursively 
 
 as follows : 
 
 I cr (c jk ) = ( *> *' ' *) if c - k € C and 
 
 r(v k ) = min Mv £ )|v £ 6 IS(v.)} 
 
 I ( c .. ) = I ( c , ) Hg ( c lifo e r 
 cr jk y cr v jh/ V jh/ 12 c jk L > 
 
 r(v k ) / min (r(vjv £ e IS(v )}, and 
 r(v h ) = max {r(v £ )|v £ e IS(v ) s r(v £ ) < r(v k )} 
 It is easy to see that I cr (c Jk ) 2 Gq (v.). I cr (c Jk ) is the 
 
 intermediate result during the calculation of G (v.) based on ordering r. 
 
 In the actual program, only one 1^ is stored for each gate. When all 
 
 output connections of a gate v have been taken into account, G (v ) is 
 
 J c j 
 
 obtained: 
 
 °c ( V = I cr (c jk ) nG c (c jk ) " here C j k £ C a*d 
 
 r(v k ) = max {r(v £ )|v 6 IS(v.)}. 
 
 According to the way each gate is selected, the ordering r in the 
 
 calculation of y^l'e is the same as ordering r ± described above. 
 
 Program NETTM-G>* contains a major subroutine PROCV which applies 
 
52 
 
 the following procedure to a given network when it is called. Step 3 of 
 this procedure is realized by another subroutine RQRNW which calculates 
 CSPF's for gates and output connections of a selected gate. 
 
 Procedure used in subroutine PROCV : 
 
 Step 1 According to the ascending order of i, select one gate 
 
 v. € V satisfying the following conditions: 
 
 (a) i > n + m. This condition means that v. is not connected to 
 
 l 
 
 any output terminal. 
 
 (b) |lS(v k )| | > 1 for every v k £ IP(v. ). This condition eliminates 
 the cases where the substitution for v. is unlikely. 
 
 If all gates have been considered, the procedure terminates . 
 
 Step 2 If there exists a v. € IS(v. ) and j > n + m such that 
 
 |lP(v )| = 1, then v. is redundant. Remove v. from the network 
 J J J 
 
 and connect y to v. for all v, e IP (v. ) and v, € IS (v.). Go to 
 h k hi k j 
 
 Step 1. 
 
 Step 3 Calculate CSPF' s for all gates and for all output connections 
 
 of v according to the following substeps. (This step is programmed 
 
 J 
 
 as a subroutine RQRNW. ) 
 
 3-1 G (v ) = z, for 1 < h < m. Initialize I 's. 
 c n+R+h h cr 
 
 3-2 Select a gate v. 4 v. and v. € v_ u V n according to ordering r n . 
 
 J 1 J (i U 1 
 
 If all gates have been considered go to Step k. 
 
 3-3 For all d such that G (d) (v.) =1, set G ^(c, .) = for every 
 
 c j c kj 
 
 c. , e C. For all d such that G (v.) = 0, set 
 
 k J c j 
 
 a. G c (d) (e kj ) = * ^r e^ £ C and f U) (v k ) = 0; 
 
 b. G (c ) = * for c. . e C and there exists a c, . e C such 
 
 c kj kj hj 
 
53 
 
 C# °c (c kj ) = 1 for c kj € C such that f (d) (v k ) = 1, and 
 r 2 (c k( .) = Mn ^ c £J ) | c £j e C, ^^(c^) = 1} (if there 
 
 exists no ChJ e C such that I (d) ( c ) = l) ; and 
 
 (d) 1 
 
 G c (c jl^ = * for all c„. e C and£/k. 
 
 IV £J , XUX c^X C £ . 
 
 (Vj) = *, set G c ^ CL; (c k ) :: * for every 
 
 For all d such that G ^ d ''" ^ •■ * --■ «■ " ^ d ^ 
 
 i 
 
 c. . e c. 
 
 3± Calculate 1^, (^J n G c (c kJ ) for c kJ € C - { c }. (The 
 
 ^r^j* is then ^Placed by 1^ (c^) fl G^.)). Go to Step 
 3-2. 
 
 Ste P h Se lect one connection c € C according to ordering r 
 
 J 2 
 
 If all output connections of v ± have been considered, go to Step 8. 
 
 Step__5 Calculate K(v ) from G (v.). Calculate a set 
 
 J c j 
 
 U= {v|v € V G U V T , v^ S(v. ), f(v) e K(vJ} 
 Calculate a set 
 
 W= {v|ve V Q U V I§ v^ S(v.), f(v) i K(v ), K(v.) fl G (v) / 0} 
 
 J <J c 
 
 §tep_6 If v V t f(v) e G c (c..), set c = c + {c £ .|v £ e U), update 
 G c (v £ ) for v £ € P(v.) by calling RQRGT, which is an entry point in 
 subroutine RQRNW, and then go to Step k. 
 
 Ste P 7 . Fi nd additional inputs to v. in order to replace c 
 according to the following substeps. 
 
 7-1 Calculate an additional vector H for c. . as follows: 
 
 ( d) 
 H =0 for all d such that G ^( c ) = 
 
 (d) ° 1J 
 
 H = * for all d such that G (d) (c ) = * or V f (d) f^ .. n 
 
 H =1 for all d such that G c (d) (c..) = 1 and V f (d) ( v ) = 
 
 If there is no d such that H (d) = 1, go to Step h. 
 
5U 
 
 7-2 Select a gate v € W such that H^ d) = 1 and f (d) (v ) = 1 for 
 at least one d. If all v € W have been considered, go to Step 9- 
 7-3 Find v e V € V satisfying the following conditions: 
 
 a. v £ i S( Vi ), 
 
 b. f J (v ) = for the d defined in Step 7-2, 
 
 c. r d) (vj = 1 for every d such that IT d ' = 0, r d ^( v ) = 1. 
 
 Jc n 
 
 If there is such a v , set C = C + {c., , c , }, update G (v) for 
 
 v £ p ( v h ) b y calling RQRGT, and update f(v) for v € S(v ) - S(v. ) 
 
 by calling UPDATE which is an entry point in subroutine EQENW. Go 
 
 to Step k. 
 
 If there is no such v., go to Step 7-2. 
 
 Step 8 Remove v. from the network by setting C = C - {c., |c, £ C}. 
 c — i ik' lk 
 
 Update the information on f(v ■ ) , IP(v ) , IS(v ) , P(v ) , S(v ) for 
 
 K. K. iC &. K. 
 
 v. e V T U -v by calling subroutine SUBMIT. Go to Step 1. 
 is. J. \i 
 
 Step 9 v. cannot be eliminated by this procedure. Restore the 
 
 original network and go to Step 1. 
 
 The above program is a rather simple version of the substitution 
 
 procedure. To increase the power of the program, more complicated orderings 
 
 may be used. For example, using the following ordering r instead of r 
 
 would likely make the substitution of v. € V„ easier. 
 
 l G 
 
 Ordering r : 
 
 r_(c. . ) > r (c, . ) if c. . , c n . £ c 
 
 3 ij 3 kj ij' kj 
 
 W >r 3 (c kj } if c hj' C kj £ c " {c ij } ' v h £ S(v i } and 
 
 \ * S(v i } 
 
 r 3 ( °hJ ) >r 3 (c kj } if C hj' c kj £ c " {c ij } ' v h' v k e S(v i } 
 
 or v, , v 4. S(v. ) , h > k. 
 h k l 
 
55 
 
 The elaborated procedure for the calculation of CSPF' s discussed in [k] 
 could also be used in this program. Another possible way to make this 
 program more powerful is to change Step 7-3. Instead of searching one 
 function, searching a set of functions whose disjunction satisfies the 
 required conditions will definitely increase the power of the substitution 
 procedure. These modifications, however, require additional computation 
 time, and therefore the implementation depends on the tradeoff between 
 efficiency and effectiveness. 
 
 Reference [ll] discusses the actual program in more detail and 
 provides the necessary information on how to use the program. A program 
 listing is also attached in [ll] as an appendix. 
 
 k.3 Experiments by Computer Programs 
 
 In order to compare the speed and effectiveness of the two procedures 
 implemented as subroutines GTMERG and PROCV, 1+0 single-output test networks 
 realizing 20 different h- and 5-variable functions were chosen as initial 
 networks for the procedures. In this case, two initial networks realize 
 each of the above 10 different U-variable functions and 10 different 5- 
 variable functions. For each function, one network was obtained by a 
 simple procedure for generating a 3-level network to realize a given 
 function. The other network was obtained by a "universal network" generating 
 program (for either k- or 5-variable functions) which can realize any given 
 (k- or 5-variable) function simply by adding a few connections. 
 
 Table U.3.1 and U.3.2 compare the results showing the size of the 
 transformed network (i.e., the number of gates and number of connections) 
 and the corresponding computation time in seconds on an IBM 360/T5J computer 
 
56 
 
 for each initial network and each transduction method. Table ^.3.1 is based 
 on the initial networks derived by the "universal network" generating 
 program, while Table H.3.2 is based on initial networks derived by the "3- 
 level" network generating program. Each function is represented by its 
 output function vector in a hexadecimal form (i.e., the output column in 
 the truth table is expressed in a hexadecimal expression). In Table U.3.1> 
 the column for the networks derived by the program NETTRA-G^ is blank. This 
 does not mean that NETTRA-GU is ineffective, but, rather, that NETTRA-GU 
 cannot transform a universal network. The reason is that in an n-variable 
 universal NOR network each gate except the output gates realizes only one 
 minterm of n-variable and therefore no gate can substitute for any other 
 gates. Since program NETTRA-G3 is combined with procedure MDTI2 which 
 removes some redundant connections while calculating CSPF's, the situation 
 is much different. However, this does not mean that NETTRA-G3 is always 
 more powerful than NETTRA-GU. This can be seen from the last column, which 
 shows the results obtained by applying NETTRA-GU on the results derived by 
 NETTRA-G3 alone. Among ten networks of 5-variable functions, NETTRA-G3 
 could improve six of them by one gate. Similar observations can be made 
 from Table U.3.2, which shows the results using initial networks derived 
 by the 3-level network generating program. For 5-variable function 
 z = (U9F363CD) £, NETTRA-G3 gives a solution with one less gate than the 
 solution given by NETTRA-GU, while for functions z = (BFD6c6dA) g and 
 z = (C6ET103E) £ NETTRA-GU gives better solutions than NETTRA-G3 does. 
 As discussed in Section 3, the procedure based on GMGC, i.e., 
 NETTRA-G3, is generally faster than the procedure based on SOGPC, i.e., 
 
57 
 
 
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59 
 
 NETTRA-GU, because the latter uses possibly-connectable conditions which 
 requires relatively long computation time. This observation agrees with 
 the result shown in Table U.3.2, where the total computation time for 20 
 functions are 10.19 seconds and 17.82 seconds by NETTRA-G3 and NETTRA-GU, 
 respectively. 
 
 As an example, Figure k. 3.1(a) shows a network of 25 NOR gates and 
 100 connections corresponding to the 3-level initial network for the 
 function z = (4 9 F36 3 CD) l6 . This network is transformed by program NETTRA-G3 
 to a network of 13 gates and U5 connections as shown in Figure U. 3.1(b). 
 Program NETTRA-GU, however, could only obtain a network shown in Figure 
 ^.3. 1(c) that consists of Hi gates and 52 connections. Figure k. 3.1(d) 
 shows the network obtained by applying NETTRA-G^ on the network of Figure 
 h. 3.1(b). By comparing these two networks, it is clear that NETTRA-Gl* 
 could remove gate 6 by substituting three connections from gates 2fc, Hi 
 and 15 to gates 8, 18 and 25, respectively, for the connections originally 
 from gate 6. Obviously, this type of transductions cannot be done by 
 procedures based on gate merging. 
 
x l - 
 
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 Co 
 
 Xi 
 
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 ->: 
 
 (a) Initial three-level network for function z = (49F363CD) 1 ^ 
 
 Figure k.3.1 An Example by NETTRA.-G3 and NETTRA.-G4. 
 
6l 
 
 (b) Network derived from (a) by program NETTRA-G3, 
 
 Figure if. 3.1 An Example by NETTRA-G3 and NETTRA-G4 (continued) 
 
62 
 
 16 
 
 x l-[ 
 X 2- 
 
 l£ 
 
 21 
 
 X l- 
 x 5 
 
 10 
 
 x u- 
 
 12 
 
 25 
 
 *2-r 
 
 x 3 - 
 x 5 -L 
 
 13 
 
 (c) Network derived from (a) by program NETTRA-G^. 
 
 Figure U.3.1 An Example by NETTRA-G3 and NETTRA-GU (continued) 
 
63 
 
 Gate 6 is removed by NETTRA-G4. The three heavy lines 
 represent the connections added by NETTRA-GU to replace 
 the connections from gate 6. 
 
 (d) Network derived from (b) by program NETTRA-G4. 
 
 Figure If. 3.1 An Example by NETTRA-G3 and NETTM-G^ (continued) 
 
6k 
 
 5. CONCLUSIONS 
 
 Two types of NOR-network transduction procedures are presented in 
 this report. The computational experiments show that they are quite effective 
 and efficient in reducing the number of gates in a given network. These 
 procedures, however, are aimed only at reducing the number of gates, and 
 therefore they may not be capable to remove other types of redundancies in 
 a given network. For example, the network in Figure k. 3.1(c) which is 
 derived by program NETTRA-GH, has an apparently redundant connection, i.e., 
 the connection from x to gate 2k. Since both immediate successors of gate 
 2k have an input x , the input x for gate 2k is redundant by the generalized 
 
 .... [8] 
 triangular condition 
 
 For the above reason, the procedures presented here should be 
 
 combined with other transduction procedures in order to implement a powerful 
 
 computer-aided logic design system. In such a system these procedures should 
 
 be applied in the early part of the entire procedure because of their 
 
 effectiveness in efficiently removing relatively simple redundant gates. 
 
65 
 
 REFERENCES 
 
 [ 1] Y. Kambayashi and S. Muroga, "Network transduction "based on permissible 
 functions (^General principles of NOR network transduction NETTRA 
 programs) " to appear as a report, Dept. of Computer Science, University 
 of Illinois. ' J 
 
 1 2] ij 1 " Culline ^> H ' C ' Lai > «id Y - Kambayashi, "Pruning procedures for 
 NOR networks using permissible functions (Principles of NOR network 
 transduction programs NETTRA-PG1, NETTRA-P1 , and NETTRA-P2 ) " Report 
 No. UIUCDCS-R-7^690, Dept. of Computer Science, University 'of Illinois 
 November 1974. 
 
 [3] H. C. Lai and J. N. Culliney, "Program manual: NOR network pruning 
 
 procedures using permissible functions (Reference manual of NOR network 
 transduction programs NETTRA-PG1, NETTRA-P1, and NETTRA-P2) " Report 
 No. UIUCDCS-R- T l*-686, Dept. of Computer Science, University 'of Illinois, 
 November 1974. 
 
 [ h] Y. Kambayashi and J. N. Culliney, "NOR network transduction procedures 
 based on connectable and disconnect able conditions (Principles of NOR 
 network transduction programs NETTRA-G1 and NETTRA-G2) ," to appear as a 
 report, Dept. of Computer Science, University of Illinois. 
 
 I 5] J. N. Culliney, "Program manual: NOR network transduction based on 
 connectable and disconnect able conditions (Reference manual of NOR 
 network transduction programs NETTRA-G1 and NETTRA-G2)," Report No 
 UIUCDCS-R-74-698, Dept. of Computer Science, University of Illinois 
 February 1975. 
 
 [ 6] D. T. Ellis, "A synthesis of combinational logic with NAND and NOR 
 
 elements, IEEE Trans, on Electronic Computers, Vol. EC-14, pp. 701-705, 
 
 [ 7] H. Lee and E. S. Davidson, "A transform for NAND network design," IEEE 
 Trans, on Computers, Vol. C-21, pp. 12-20, 1970. 
 
 [ 8] H. C. Lai, T. Nakagawa, and S. Muroga, "Redundancy check technique for 
 designing optimal networks by branch-and-bound method," International 
 Journal of Computer and Information Sciences, Vol. 3, pp. 251-271, 197U. 
 
 [ 9] ^J* 8 *!*'"? ^ d H ' C * Lai ' " A branch-and-bound algorithm for optimal 
 NOR networks (The algorithm description)," Report No. U38, Dept. of 
 Computer Science, University of Illinois, April 1971. 
 
 [10] T Nakagawa and H. C. Lai, "Reference manual of FORTRAN programs 
 ILLOD-(NOR-B) for optimal NOR networks," Report No. U88, Dept of 
 Computer Science, University of Illinois, December 1971' 
 
 [11] H C. Lai, "Program manual: NOR network transduction by generalized 
 gate merging and substitution (Reference manual of NOR network 
 transduction programs NETTRA-G3 and NETTRA-GM ," Report No. UIUCDCS-R- 
 rW14, Dept. of Computer Science, University of Illinois, April 1975 
 
BIBLIOGRAPHIC DATA 
 SHEET 
 
 4. Title and Subtitle 
 
 1. Report No. 
 
 UIUCDCS-R-75-728 
 
 NOR NETWORK TRANSDUCTION BY GENERALIZED GATE MERGING AND 
 SUBSTITUTION PROCEDURES (Principles of NOR Network 
 Transduction Programs NETTRA-G3 and NETTRA-GU) 
 
 7. Author(s) 
 
 3. Recipient's Accession N. 
 
 5. Report Date 
 
 June 1975 
 
 H. C. Lai. Y. Kambayashi 
 
 9. Performing Organization Name and Address 
 
 Department of Computer Science 
 
 University of Illinois at Urban a- Champaign 
 
 Urbana, Illinois 6l801 
 
 12. Sponsoring Organization Name and Address 
 
 15. Supplementary Notes 
 
 8. Performing Organization Rept. 
 No - UIUCDCS-R-75-728 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 13. Type of Report & Period 
 Covered 
 
 14. 
 
 16. Abstracts 
 
 Several NOR (or NAND) network transduction procedures aiming at reducing 
 the number of gates on a given network are introduced. These procedures are 
 generalizations, based on compatible sets of permissible functions, of simple 
 gate merging and substitution procedures. The programs NETTRA-G3 and NETTRA-Glj- 
 based on generalized gate merging and substitution, respectively, have been 
 prepared and tested. The computational experiments show that these programs 
 are relatively effective and efficient in reducing the number of gates in a 
 network with many redundant gates. 
 
 17. Key Words and Document Analysis. 17a. Descriptors 
 
 Logic design, logic circuits, logical elements, programs (computers) 
 
 17b. Identifiers/Open-Ended Terms 
 
 Computer-aided design, permissible functions, network transformation, network 
 transduction, gate merging, gate substitution, NOR, NAND, CSPF. 
 
 17c. COSATI Field/Group 
 
 18. Availability Statement 
 
 Unlimited 
 
 FORM NTIS-35 (10-70) 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 
 71 
 
 22. Price 
 
 USCOMM-DC 4 0329-P7 1 
 
Jll 1 
 
 b *ft 
 
^ 
 
I