Equivalence Checking of Dissimilar Circuits II

Transcription

1 Equvalence Checkng of Dssmlar Crcuts II Cadence Berkeley Labs 1995 Unversty Ave.,Sute 460, Berkeley, Calforna,94704 phone: (510) , fax: (510) CDNL-TR August 2004 Eugene Goldberg (Cadence Berkeley Labs), Abstract In ths report we revst the theory ntroduced n [2]. We formulate t n terms of correlaton functons so showng that the ntroducton of flterng functons s not necessary. We also descrbe an algorthm of equvalence checkng for crcuts wth a known specfcaton that s based on computaton of correlaton functons only (no flterng functons are computed). 1. Introducton Ths report s smply a rewrte of paper [2] that presented some results on complexty of equvalence checkng of crcuts wth a common specfcaton. The results of [2] were formulated n terms of correlaton and flterng functons. In ths report, we show that the ntroducton of flterng functons s not necessary and the theory of [2] can be formulated n terms of correlaton functons only. Ths also apples to publcatons [3] and [4] n whch the ntroducton of flterng functons can be avoded ether. (Papers [2] and [4] are avalable onlne from the web page ) In the report we follow the structure of paper [2] wth the followng exceptons. To reduce the repetton of text we omtted materal presented n Sectons 1, 5 and 6 of [2]. The major changes are made n secton 4 whch presents the man result. In partcular, we added subsecton 4.4 n whch we explan why flterng functons can be dropped. Fnally we added a secton descrbng an algorthm for equvalence checkng of crcuts N 1, N 2 wth a known common specfcaton S (Secton 5). Ths algorthm s dentcal to that of [4] wth the followng two exceptons. We formulate ths algorthm n terms of correlaton functons only (flterng functons are not used). We emphasze the fact that for ths algorthm the specfcaton S does not have to be represented explctly. The algorthm only needs to know the parttonng of N 1 nto subcrcuts that are mplementatons of blocks of S. 2. Common Specfcaton of Boolean Crcuts In ths secton, we ntroduce the noton of a common specfcaton of Boolean crcuts. Let S be a combnatonal crcut of mult-valued blocks (further referred to as a specfcaton) specfed by a drected acyclc graph H. The sources and snks of H correspond to prmary nputs and outputs of S. Each non-source node of H corresponds to a mult-valued block computng a mult-valued functon of mult-valued arguments. Each node of n of H s assocated wth a mult-valued varable A. If n s a source of H, then the correspondng varable specfes values taken by the correspondng prmary nput of S. If n s a non-source node of S then the correspondng varable descrbes the values taken by the output of the block specfed by n. If n s a source (respectvely a snk), then the correspondng varable s called a prmary nput varable (respectvely prmary output varable). We wll use the notaton C=G(A,B) to ndcate that a) the output of a block G s assocated wth a varable C; b) the functon computed by the block G s G(A,B); c) only two nodes of H are connected to the node n n H and these nodes are assocated wth varables A and B. Denote by D(A) the doman of the varable A assocated wth a node of H. The value of D(A) s called the multplcty of A. If the multplcty of every varable A of S s equal to 2 then S s a Boolean crcut.

2 Now we descrbe how a Boolean crcut N can be produced from a specfcaton S by encodng the mult-valued varables. Let D(A)={a 1,,a t } be the doman of a varable A of S. Denote by q(a) a Boolean encodng of the values of D(A) that s a mappng q:d(a) {0,1} m. Denote by length(q(a)) the number of bts n q that s the value of m. The value of q(a ), a D(A) s called the code of a. Gven an encodng q of length m of a varable A assocated wth a block of S, denote by v(a) the set of m codng Boolean varables. In the followng exposton we make the assumptons below. Assumpton 1. Each gate of a Boolean crcut and each block of a specfcaton has two nputs and one output. Assumpton 2. The multplcty of each prmary nput (or output) varable of a specfcaton s a power of 2. Assumpton 3. If A s a prmary nput (or output) varable of a specfcaton, then length(q(a))=log 2 ( D(A) ) Assumpton 4. If a 1 and a 2 are values of a varable A of a specfcaton and a 1 a 2, then q(a 1 ) q(a 2 ). Assumpton 5. If A and B are two dfferent varables of a specfcaton, then v(a) v(b) =. Remark 1. From Assumpton 2, Assumpton 3, and Assumpton 4 t follows that f A s a prmary nput (or output) varable, a mappng q:d(a) {0,1} m s bjectve. In partcular, any assgnment to the varables of v(a) s a code of some value a D(A). Defnton 1. Gven a Boolean crcut I, denote by Inp(I) (respectvely Out(I)) the set of varables assocated wth prmary nputs (respectvely prmary outputs) of I. Defnton 2. Let X 1 and X 2 be sets of Boolean varables and X 2 X 1. Let y be an assgnment to the varables of X 1. Denote by proj(y,x 2 ) the projecton of y on X 2.e. the part of y that conssts of the assgnments to the varables of X 2. Defnton 3. Let C=G(A,B) be a block of specfcaton S. Let q(a),q(b),q(c) be encodngs of varables A,B, and C respectvely. A Boolean crcut I s sad to mplement the block G f the followng three condtons hold: 1) The set Inp(I) s a subset of v(a) v(b). 2) The set Out(I) s equal to v(c). 3) If the set of values assgned to v(a) and v(b) form codes q(a) and q(b) respectvely where a D(A), b D(B), then I(z )=q(c). Here z s the projecton of the assgnment z=(q(a),q(b)) on Inp(I), I(z ) s the value taken by I at z, and c=g(a,b). Remark 2. The reason why Inp(I) may not nclude all the varables of v(a) and/or v(b) s that the functon G(A,B) may not dstngush some values of A or B. (G(A,B) does not dstngush, say, values a 1,a 2 D(A), f for any b D(B), G(a 1,b)=G(a 2,b).) So to mplement G(A,B) the crcut I may need only a subset of varables of v(a) v(b). Ths sad, for the sake of smplcty, we wll wrte I(q(a),q(b)) meanng I(q (a),q (b)), q (a)= proj(q(a),inp(i)) and q (b)=proj(q(b),inp(i)). Defnton 4. Let S be a mult-valued crcut. A Boolean crcut N s sad to mplement the specfcaton S, f t s bult accordng to the followng two rules. 1) Each block G of S s replaced wth an mplementaton I of G. 2) Let the output of block G 1 (specfed by varable R) be connected to an nput of block G 2 (specfed by the same varable R) n S. Then the outputs of the crcut I 1 mplementng G 1 are properly connected to nputs of crcut I 2 mplementng G 2. Namely, the prmary output of I 1 specfed by a Boolean varable x v(r) s connected to the nput of I 2 specfed by the same varable of v(r) f x Inp(I 2 ). In Fg. 1a a specfcaton of three blocks s shown. The functonalty of two dfferent mplementatons of the block C=G 1 (A,B) (Fg. 1b) are shown n Fg. 1c and 1d. Here D(A)={a 0,a 1 }, D(B)={b 0,b 1,b 2,b 3 } and D(C)={c 0,c 1,c 2 }. Snce A and B are prmary nput varables, they are encoded wth a mnmum length encodng and q 1 (A)=q 2 (A) and q 1 (B)=q 2 (B) where q 1 (a 0 )=0, q 1 (a 1 )=1, q 1 (b 0 )=00, q 1 (b 1 )=01, q 1 (b 2 )=10, q 1 (b 3 )=11. Fnally, the encodngs q 1 (C) and q 2 (C) are q 1 (c 0 )=00, q 1 (c 1 )=10, q 1 (c 2 )=01 and q 2 (c 0 )=100, q 2 (c 1 )=010, q 2 (c 2 )=001. Remark 3. Let N be an mplementaton of a specfcaton S. Let p be the largest number of gates used n an mplementaton of a mult-valued block of S n N. We wll say that S s a specfcaton of granularty p for N. C G 3 A B F E K G 1 G 2 (a) I 1 (q 1 (A),q 1 (B)) q 1(A) q 1(B) q 1(C) (c) C=G 1 (A,B) A B C a 0 b 0 c 0 a 0 b 1 c 1 a 0 b 2 c 1 a 0 b 3 c 0 a 1 b 0 c 1 a 1 b 1 c 2 a 1 b 2 c 2 a 1 b 3 c 0 q 2(A) q 2(B) q 2(C) Fgure 1. A specfcaton and the functonalty of two mplementatons of a block Defnton 5. The topologcal level of a block G n a specfcaton S s the length of the longest path from a prmary nput of S to G. (The length of a path s measured n the number of blocks on t. The topologcal level of a prmary nput s assumed to be 0.) Denote by level(g) the topologcal level of G n S. (d) (b) I 2 (q 2 (A),q 2 (B))

3 Let N be an mplementaton of a specfcaton S. From Assumpton 4 t follows that for any value assgnment h to the nput varables of N there s a unque set of values (x 1,,x k ), where x D(X ) such that h=(q(x 1 ),,q(x k )). That s there s oneto-one correspondence between assgnments to prmary nputs of S and N. The same apples to prmary outputs of S and N. Defnton 6. Let N be an mplementaton of S. Gven a Boolean vector y of assgnments to the prmary nputs of N, the correspondng vector Y=(x 1,..,x k ), x D(X ) such that y=(q(x 1 ),,q(x k )) s called the pre-mage of y. Proposton 1. Let N be a crcut mplementng specfcaton S. Let I(G) be the mplementaton of a block C=G(A,B) of S n N. Let y be a value assgnment to the prmary nput varables of N and Y be the pre-mage of y. Then the values of prmary outputs of I(G) form the code q(c) where c s the value taken by the output of G when the nputs of S take the values specfed by Y. Proof. The proposton can be proven by nducton n topologcal levels of varables of the specfcaton S. Accordng to Remark 1, the proposton holds for the varables of topologcal level 0 (prmary nput varables of S). Let C=G(A,B) be a block of S and level(g)=n, n>0. Let I(G) be the mplementaton of G n N. By the nducton hypothess, values taken by the varables of v(a) and v(b) n N under the nput assgnment y should be q(a) and q(b) respectvely. Here a and b are values of varables A and B under the nput assgnment Y. Then from Defnton 3 t follows that the outputs of I(G) take the values of q(c) where c=g(a,b). Proposton 2. Let N 1, N 2 be crcuts mplementng a specfcaton S. Let each prmary nput (or output) varable X of S have the same encodng n N 1. Then Boolean crcuts N 1 are functonally equvalent. Proof. Let y be an arbtrary assgnment to nput varables of N 1. Snce the encodngs of prmary nput varables of S n N 1 are the same, then the pre-mage Y of y for N 1 s the same. Let C be a prmary output varable of S assocated wth a block G. From Proposton 1 t follows that the values taken by the mplementatons I 1 (G) and I 2 (G) of G n N 1 are equal to q 1 (c) and q 2 (c) respectvely. Here c s the value taken by the output of G under nput assgnment Y and q 1 and q 2 are encodngs of the prmary output varable C n N 1. Snce C has the same encodng n N 1, then q 1 (c) = q 2 (c). Defnton 7. Let N 1, N 2 be two functonally equvalent Boolean crcuts. Let N 1, N 2 mplement a specfcaton S so that for every prmary nput (output) varable X encodngs q 1 (X) and q 2 (X) (used when producng N 1 respectvely) are dentcal. Then N s called a common specfcaton (CS) of N 1. Defnton 8. Let S be a CS of N 1. Let p 1 (respectvely p 2 ) be the granularty of S wth respect to N 1 (respectvely N 2 ). Then we wll say that S s a CS of N 1 of granularty p = max(p 1,p 2 ). Defnton 9. Gven two functonally equvalent Boolean crcuts N 1, N 2, S s called the fnest common specfcaton f t has the smallest granularty p among all the CSs of N Equvalence Checkng as SAT Snce n ths report we formulate the complexty of equvalence checkng n terms of resoluton proofs, we recall a common way of reducng equvalence checkng to the satsfablty problem. Defnton 10. A dsjuncton of lterals of Boolean varables not contanng two lterals of the same varable s called a clause. A conjuncton of clauses s called a conjunctve normal form (CNF). Defnton 11. Gven a CNF F, the satsfablty problem (SAT) s to fnd a value assgnment to the varables of F for whch F evaluates to 1 (also called a satsfyng assgnment) or to prove that such an assgnment does not exst. A clause K of F s sad to be satsfed by a value assgnment y f K(y)=1. The standard converson of an equvalence checkng problem nto an nstance of SAT s performed n two steps. Let N 1 be Boolean crcuts to be checked for equvalence. At the frst step of ths converson, a crcut M called a mter [1] s formed from N 1. The mter M s obtaned by 1) dentfyng the correspondng prmary nputs of N 1 ; 2) XORng each par of correspondng prmary outputs of N 1 ; 3) ORng the outputs of the added XOR gates. So the mter of N 1 evaluates to 1 f and only f for some nput assgnment a prmary output of N 1 and the correspondng output of N 2 evaluate to dfferent values. Therefore, the problem of checkng the equvalence of N 1 s equvalent to testng the satsfablty of the mter of N 1. At the second step of converson, the satsfablty of the mter s reduced to that of a CNF formula F. Ths formula s a conjuncton of CNF formulas F 1,..,F n specfyng the functonalty of the gates of M and a one-lteral clause that s satsfed only f the output of M s set to 1. The CNF F specfes the -th gate g of M. Any assgnment to the varables of F that s nconsstent wth the functonalty of g falsfes a clause of F (and vce versa, a consstent assgnment satsfes all the clauses of F.) For nstance, the AND gate y=x 1 x 2 s specfed by the followng three clauses ~x 1 ~x 2 y, x 1 ~y, x 2 ~y. 4. Equvalence Checkng n General Resoluton In ths secton, we prove some results about the complexty of equvalence checkng of crcuts wth a CS of granularty p. The man dea of the proof s that f S s a CS of N 1, then ther equvalence checkng reduces to computng the correlaton functon for encodngs q 1 (C),q 2 (C) of each varable C of S. The two man propertes of correlaton functons are that They can be bult based only on the nformaton about the topology of S and about assgnment of gates of N 1 to blocks of S. The correlaton functon for encodngs q 1 (C),q 2 (C) of the varable C specfyng the output of a block G(A,B) can be computed locally from the correlaton functons of encodngs of varables A and B and CNFs specfyng mplementatons I 1 (G) and I 2 (G). So these functons can be computed n topologcal order startng wth nputs and proceedng to outputs.

4 4.1 Class M(p) and general resoluton Defnton 12. Gven a constant p, a CNF formula F s a member of the class M(p) f and only f t satsfes the followng two condtons. F s the CNF formula (obtaned by the procedure descrbed n Secton 3) specfyng the mter of a par of functonally equvalent crcuts N 1. N 1 has a CS of granularty p. Defnton 13. Let K and K be clauses havng opposte lterals of a varable (say varable x) and there s only one such varable. The resolvent of K, K n varable x s the clause that contans all the lterals of K and K but the postve (.e. lteral x) and negatve (.e. lteral ~x) lterals of x. The operaton of producng the resolvent of K and K s called resoluton. Defnton 14. General resoluton s a proof system of propostonal logc that has only one nference rule. Ths rule s to resolve two exstng clauses to produce a new one. Gven a CNF formula F, a proof L(F) of unsatsfablty of F n the general resoluton system conssts of a sequence of resolutons resultng n the dervaton of an empty clause (.e. a clause wthout lterals). General resoluton s complete, whch means that gven an unsatsfable formula F there s always a proof L(F) that derves an empty clause. Defnton 15. Let F be a set of clauses. Denote by supp(f) the set of varables whose lterals occur n clauses of F. The followng Proposton 3 and also Propostons 4,5,6 proved n subsectons 4.2, 4.3 are used n the proof of Proposton 7 that s the man result of ths paper. Proposton 3. Let F be a set of clauses that mples a clause K. Then there s a sequence of resolutons of at most 3 supp(f) steps that results n the dervaton of a clause that mples K. Proof. Denote by F the formula that s obtaned from F by makng the assgnments that set the lterals of K to 0 (and removng the satsfed clauses and the lterals set to 0). It s not hard to see that F s unsatsfable snce t mples an empty clause. So there s a resoluton proof L(F ) that results n deducng an empty clause. Then by replacng each clause of F nvolved n L(F ) wth ts parent clause from F we get a sequence of resolutons resultng n deducng a clause that mples K. The number of resolvents n L(F ) cannot be more than 3 supp(f ) (.e. the total number of clauses of supp(f ) varables) and so t cannot be more than 3 supp(f). 4.2 Correlaton functons Defnton 16. Let S be a CS of crcuts N 1 and C be a varable of S. A functon Cf (v 1 (C),v 2 (C)) s called a correlaton functon for encodngs q 1 and q 2 of the values of C (used when producng N 1 ) f : supp(cf ) v 1 (C) v 2 (C). Cf(z 1, z 2 )=1 for any assgnment z 1 to v 1 (C) and z 2 to v 2 (C) such that z 1 =q 1 (c) and z 2 =q 2 (c) where c D(C). Otherwse Cf(z 1, z 2 )=0. From now on we wll say that Cf(v 1 (C),v 2 (C)) s the correlaton functon for the varable C meanng that t s the correlaton functon for encodngs q 1 (C),q 2 (C) of the varable C. Remark 4. If C s a prmary nput varable of S, then Cf(v 1 (C),v 2 (C)) Eq(v 1 (C),v 2 (C)). The functon Eq(z 1, z 2 ) s equal to 1 ff z 1 =q 1 (c), z 2 = q 2 (c), where c D(C) and q 1 (c) = q 2 (c). Otherwse, Eq(v 1 (C),v 2 (C)) s equal to 0. Indeed, as t follows from Remark 1, sets v 1 (C) and v 2 (C) have the same number of codng varables and any assgnment to v 1 (C) or v 2 (C) s the code of a value c D(C). Besdes, from the defnton of CS t follows that q 1 (C)=q 2 (C). So every permssble assgnment (x,y) to the varables of v 1 (C),v 2 (C) can be represented as (q 1 (c),q 2 (c)), were c D(C) and q 1 (c) = q 2 (c). Proposton 4. Let S be a CS of crcuts N 1. Let C=G(A,B) be a block of S. Let F be the CNF formula specfyng the mter of N 1 bult as descrbed n Secton 3. Let F(I 1 (G)) and F(I 2 (G)) be the part of F specfyng the mplementaton I 1 (G) and I 2 (G) of G n N 1 respectvely. Then P mples Cf(v 1 (C),v 2 (C)). Here P = Correlaton Implementaton and, Correlaton = Cf(v 1 (A),v 2 (A)) Cf(v 1 (B),v 2 (B)), Implementaton = F(I 1 (G)) F(I 2 (G)). Proof. To prove that P mples Cf(v 1 (C),v 2 (C)) one needs to show that any assgnment that sets P to 1 also sets Cf(v 1 (C),v 2 (C)) to 1. It s not hard to see that the support of all the functons of the expresson P Cf(v 1 (C),v 2 (C)) s a subset of supp(f(i 1 (G)) supp(f(i 2 (G)). Let h=(x 1, x 2, y 1, y 2, z 1, z 2 ) be an assgnment that sets P to 1 where x 1, x 2, y 1, y 2, z 1, z 2 are assgnments to v 1 (A), v 2 (A), v 1 (B), v 2 (B), v 1 (C), v 2 (C) respectvely. Then h has to set to 1 all the functons the conjuncton of whch forms P. Snce h has to set the functon Correlaton to 1, then x 1 =q 1 (a), x 2 =q 2 (a) where a D(A) and y 1 =q 1 (b), y 2 =q 2 (b), where b D(B). So h=(q 1 (a),q 2 (a), q 1 (b),q 2 (b), z 1, z 2 ). Snce h sets the functon Implementaton to 1, then z 1 has to be equal to q 1 (c), c=g(a,b) and z 2 has to be equal to q 2 (c). So h s equal to (q 1 (a),q 2 (a),q 1 (b),q 2 (b),q 1 (c),q 2 (c)) and hence t sets the correlaton functon Cf(v 1 (C),v 2 (C)) to Complexty of formulas from M(p) Proposton 5. Let A,B,C be Boolean functons and A B C. Then for any functon A such that A A, t s true that A B C. Proof. Let x be an assgnment that sets A B to 1. Then A (x)=1 and B(x)=1. Snce A A, then A(x)=1. Then A(x) B(x) = 1 and so C(x)=1. Proposton 6. Let X 1 and X 2 be sets of Boolean varables, F(X 1,X 2 ) and H(X 2 ) be CNF formulas and F mply H. Then n at most 3 supp(f) resoluton steps one can derve a CNF formula H that mples H(X 2 ) such that supp(h ) supp(h). Proof. Let K be a clause of H. From Proposton 3 t follows that n at most 3 supp(f) steps one can derve a clause K that mples K. Snce K K, then supp(k ) supp(k). So n at most H 3 supp(f) steps, where H s the number of clauses n H, one can derve a CNF H mplyng H such that supp(h ) supp(h). (The fact that H mples H follows from Proposton 5.) However, f

5 one does not produce the same resolvent twce, the total number of resoluton steps when dervng H cannot be more than 3 supp(f) (because t s the total number of clauses of supp(f) varables). Proposton 7. Let F be a formula of M(p) specfyng the mter of crcuts N 1 obtaned from a CS S of granularty p. The unsatsfablty of F can be proven by a resoluton proof of no more than d n 3 6p resoluton steps where n s the number of blocks n S and d s a constant. Proof. From Proposton 4 t follows that one can deduce correlaton functons for all the varables of S startng wth blocks of topologcal level 1 and proceedng n topologcal order. Indeed, let C=G(A,B) be a block of topologcal level 1. Then A and B are prmary nput varables and the correlaton functons for them are equal to Eq(v 1 (A),v 2 (A)) and Eq(v 1 (B),v 2 (B)) respectvely (see Remark 4). The correlaton functon Cf(v 1 (C),v 2 (C)) s mpled by P =F(I 1 (G)) F(I 2 (G)) Eq(v 1 (A),v 2 (A)) Eq(v 1 (B),v 2 (B)). So a functon mplyng Cf(v 1 (C),v 2 (C)) can be derved from P by resoluton. From Proposton 5 t follows that to apply Proposton 4, nstead of the functon Cf(v 1 (C),v 2 (C)), one can use any functon mplyng t. After correlaton functons are computed for all the varables of level 1, the same procedure can be appled to varables of topologcal level 2 and so on. If S conssts of n blocks, then n n steps one can deduce correlaton functons for the prmary output varables of S. At each step the correlaton functon s computed for a varable C=G(A,B) of S. The complexty of ths step s no more than 3 6p. Indeed, the support of all functons mentoned n Proposton 4 needed for computng Cf(v 1 (C),v 2 (C)) s a subset of A=supp(F(I 1 (G))) supp(f(i 2 (G))). The total number of gates n I 1 (G) and I 2 (G) s bounded by 2p, each gate havng 2 nputs and 1 output. So the total number of varables n A cannot be more than 6p. Then from Proposton 6 t follows that n at most 3 6p steps one can deduce CNFs mplyng Cf(v 1 (C),v 2 (C)). Then the total number of resoluton steps one needs to deduce functons mplyng the correlaton functons for the prmary output varables of S s bounded by n 3 6p. Now we show that from the correlaton functons for prmary output varables of S, one can deduce an empty clause n the number of resoluton steps lnear n n p. Let C be a prmary output varable specfyng the output of a block G of N. Let I 1 (G) and I 2 (G) be the mplementatons of G n N 1 respectvely. Let D(C) =2 k (By Assumpton 2 the multplcty of C s a power of 2.) Then length(q 1 (C))= length(q 2 (C))=k. (By Assumpton 3, values of S are encoded by a mnmal length encodng.) Now we show that there s always a correlaton functon Cf(v 1 (C),v 2 (C)) that mples the CNF consstng of k pars of two lteral clauses specfyng the equvalence of correspondng outputs of I 1 (G) and I 2 (G). Let f 1 and f 2 be two Boolean varables of v 1 (C) and v 2 (C) respectvely that specfy correspondng outputs of N 1. Snce S s a CS of N 1, then q 1 (C)=q 2 (C). So any assgnment q 1 (c),q 2 (c) to v 1 (C) and v 2 (C) that satsfes Cf(v 1 (C),v 2 (C)) also satsfes clauses K =f 1 ~f 2 and K =~f 1 f 2. So K and K are mpled by Cf(v 1 (C),v 2 (C)) and so clauses mplyng them can be deduced by the procedure descrbed n the proof of Proposton 4. (The resoluton steps one needs to deduce equvalence clauses are already counted n the expresson n 3 6p ) Usng each par of equvalence clauses K and K (or clauses mplyng them) and the clauses specfyng the gate g=xor(f 1,f 2 ) of the mter, one can deduce a sngle lteral clause ~g. Ths clause requres settng the output of ths XOR gate to 0. Each such a clause can be deduced n the number of resolutons bounded by a constant and the total number of such clauses cannot be more than n p. Fnally, from these unt clauses and the clauses specfyng the fnal OR gate of the mter, the empty clause can be deduced n the number of resolutons bounded by n p. So the empty clause s deduced n no more than n 3 6p + d n p steps where d s a constant. Fnally, one can pck a constant d such n 3 6p + d n p d n 3 6p Remark 5. The essence of the resoluton proof descrbed n Proposton 7 s to compute correlaton functons nductvely movng from nputs to outputs. It s not hard to see that ths computaton s not restrcted to general resoluton. Indeed, all the terms of the expresson P=Cf(v 1 (A), v 2 (A)) Cf(v 1 (B), v 2 (B)) F(I 1 (G)) F(I 2 (G)) are just functons and so can be represented n any possble way (.e. not only as CNF formulas). Besdes, Proposton 7 can be proven n terms of exstental quantfcaton ntroduced by Defnton 17. Indeed, from Proposton 8 below t follows that Cf(v 1 (C),v 2 (C)) (or a functon mplyng t) can be obtaned from P by exstentally quantfyng away all the varables except those of v 1 (C) v 2 (C). Exstental quantfcaton of a functon can be done n many ways, for example, by usng BDDs. So, summarzng, Proposton 7 can be formulated and proven n terms of functons and exstental quantfcaton.e. ndependently of a proof system. Defnton 17. Let f be a Boolean functon. We wll say that functon f* s obtaned from f by exstentally quantfyng away Boolean varable x f f* = f(,x=0, ) f(,x=1,.). Proposton 8. Let X 1 and X 2 be two dsjont sets of Boolean varables. Let F(X 1,X 2 ) and H(X 2 ) be two Boolean functons and F mply H. Let F*(X 1 \ {x}, X 2 ) be obtaned from F(X 1,X 2 ) by exstentally quantfyng away the varable x. Then F*(X 1 \ {x},x 2 ) also mples H(X 2 ). Proof. Denote by X 1 the set X 1 \{x}. Let (z, z 1,z 2 ) be a boolean vector representng an assgnment to the varables of X 1 X 2. Here z s a Boolean value assgned to the varable x and z 1, z 2 are Boolean vectors representng assgnments to the varables of X 1 and X 2 respectvely. Suppose that F*(z 1, z 2 ) = 1. Accordng to Defnton 17, F*(X 1,X 2 ) = F(0, X 1,X 2 ) F(1, X 1,X 2 ) and so ether F(0, z 1, z 2 ) or F(0, z 1, z 2 ) has to be equal to 1. Snce F(x,X 1,X 2 ) mples H(X 2 ) then H(z 2 ) = 1. So from F*=1 t follows that H=1. Hence F* mples H. Remark 6. In Proposton 7 we gve a worst case estmate for the complexty of correlaton functon computaton. In practce, ths complexty can be much lower. In a sense, the best way to nterpret the theory developed n ths secton s that the complexty of equvalence checkng of crcuts N 1 wth a CS S s lnear n the number of blocks n S. Remark 7. In ths report, for the sake of clarty, we assumed that every block of a specfcaton has two nputs and one output (Assumpton 1). However, one can easly extend Proposton 7 to the case of a specfcaton S where a block may have an arbtrary (but fnte) number of nputs. (We stll assume that every gate of crcuts N 1 mplementng S have two nputs and one output). Indeed, let G be a block of S wth n nputs and let C, A 1,.., A n be varables assocated wth ts output

6 and n nputs respectvely. Then one can prove (n the same manner as n Proposton 4) that correlaton functon Cf(v 1 (C),v 2 (C)) s mpled by the expresson P = Correlaton Implementaton. Here Implementaton= F(I 1 (G)) F(I 2 (G)) s the same as n Proposton 4 and Correlaton = Cf(v 1 (A 1 ), v 2 (A 1 )) Cf(v 1 (A n ), v 2 (A n )). So to compute the correlaton functon for the output of an n-nput block one needs to compute n correlaton functons correspondng to n nput varables. Other than that, the proof of Proposton 7 does not change. 4.4 A few words about flterng functons In subsecton 4.3, we reproduced the result of [2] wthout ntroducng flterng functons. To make thngs even more clear, n ths subsecton we gve an nformal explanaton of why flterng functons can be dropped. We also explan under what crcumstances flterng functons mght come useful. Here s the defnton of flterng functons from [2]. Defnton 18. Let N be an mplementaton of a specfcaton S. Let C be a varable of S assocated wth the output of a block G. A functon Ff s called a flterng functon f: supp(ff ) v(c). If an assgnment z to the varables of v(c) s a code q(c), c D(C), then Ff(z)=1. Otherwse, Ff(z)=0. Let N 1 be crcuts wth a CS S. Let v 1 (C) and v 2 (C) be the codng varables of the varable C of S correspondng to the mplementatons I 1 (G) and I 2 (G) n N 1 respectvely. From Defnton 18 of flterng functons and Defnton 16 of correlaton functons t follows that Ff(v 1 (C)) Ff(v 2 (C)) Cf(v 1 (C),v 2 (C)) = Cf(v 1 (C),v 2 (C)). On the other hand, n Proposton 7 of [2] (used to prove the man result.e. Proposton 8) flterng functons Ff(v 1 (C)) Ff(v 2 (C)) appear only n conjuncton wth the correlaton functon Cf(v 1 (C),v 2 (C)). So flterng functons can be removed from the proof. The reason why flterng functons appeared n [2] was that orgnally the defnton of correlaton functons used n the manuscrpt was as follows. Defnton 19. Let S be a CS of crcuts N 1 and C be a varable of S. A functon Cf s called a correlaton functon for encodngs q 1 and q 2 of the values of C (used when obtanng N 1 ) f : supp(cf ) v 1 (C) v 2 (C). Cf(z 1, z 2 )=0 for any assgnment z 1 to v 1 (C) and z 2 to v 2 (C) such that z 1 =q 1 (c) and z 2 =q 2 (c*) where c,c* D(C) and c c*. It s not hard to see that Defnton 19, only partally defnes Cf. Defnton 19 can be vewed s a relaxaton of Defnton 16, meanng that the correlaton functon specfed by the latter s an mplementaton of the correlaton functon specfed by the former. If correlaton functons are specfed by Defnton 19, then to prove Proposton 7 one needs flterng functons. (Because now Ff(v 1 (C)) Ff(v 2 (C)) Cf(v 1 (C),v 2 (C)), n general, s not equvalent to Cf(v 1 (C),v 2 (C)).) Later, the defnton of correlaton functons was changed to the one used n [2] whch made flterng functons redundant. Defnton 16 s preferable from a theoretcal pont of vew because t reduces the number of objects employed n our theory. However, n practcal mplementatons of the algorthm of equvalence checkng descrbed n Secton 5, the use of flterng and correlaton functons specfed by Defnton 18 and Defnton 19 respectvely, nstead of correlaton functons specfed by Defnton 16, may make sense. The reason s that Defnton 16 mxes up two unary and one bnary relaton specfed over the set of output assgnments of subcrcuts I 1 (G) and I 2 (G). On the other hand, n practcal applcatons one may want to compute them separately. (The unary relatons are specfed by the flterng functons Ff(v 1 (C)) and Ff(v 2 (C)) that sngle out output assgnments of I 1 (G) and I 2 (G) that are codes of C. The bnary relaton s gven by the correlaton functon Cf(v 1 (C),v 2 (C)) specfed by Defnton 19. ) 5. Algorthm of Equvalence Checkng wth a Known Specfcaton In ths secton we descrbe an algorthm for equvalence checkng of crcuts wth a known specfcaton. Ths algorthm s dentcal to the one ntroduced n [4] wth a few exceptons. Frst, we formulate ths algorthm n terms of correlaton functons only (omttng flterng functons). Second, we emphasze the fact that ths equvalence checkng procedure needs only an mplct representaton of a CS of crcuts N 1, N 2. Ths representaton s gven as a parttonng of N 1, N 2 nto subcrcuts. In Secton 4 we consdered equvalence checkng n general resoluton that s a non-determnstc proof system. Ths means that ths proof s guded by an oracle that ponts to the next par of clauses to be resolved. Now we summarze the results of Secton 4 n a determnstc procedure of equvalence checkng of crcuts N 1 wth a CS S of granularty p. The dea s that f a CS S of N 1 s known, then S tself can be vewed as an oracle. Ths oracle s powerful enough to make equvalence checkng of N 1 effcent. (However, f S s unknown t s unlkely that there s an effcent algorthm for equvalence checkng of N 1 even f there exsts a CS of N 1 of small granularty.) For the sake of smplcty, we wll assume that all the prmary nput and output varables of S are bnary. (A more general case mpled by Assumpton 2 and Assumpton 3 s not much dfferent but makes explanaton more wordy.) We wll also assume that N 1 have only one prmary output. Besdes, we gve the descrpton of the algorthm that s ndependent of the proof system (see Remark 5). Note that n the proof of Proposton 7 we never used an explct representaton of blocks of S. We only needed to know how gates of N 1 are assgned to subcrcuts that are mplementatons of blocks of S. So n the algorthm descrpton shown n Fgure 2, a k-block specfcaton S of N 1 s represented mplctly as a parttonng of these two crcuts nto k subcrcuts N 1 1,..,N k 1, N 1 2,..,N k 2. We assume that N 1 and N 2 are mplementatons of the same block of specfcaton S. We also

7 assume that subcrcuts are numbered n the topologcal order of blocks n S. That s f > j, then the topologcal level of the block mplemented by N 1 s greater or equal to the topologcal level of the block mplemented by N 1 j. /* Part(N 1 )= {N 1 1,..,N 1 k }, Part(N 2 )= {N 2 1,.. k } */ check_for_equvalence(n 1, N 2, Part(N 1 ),Part(N 2 )) { /* check that specfcaton s correct topolgcally */ f (check_parttonns(part(n 1 ),Part(N 2 )) == ncorrect ) return( unsolved ); /* compute correlaton functons */ for (=1; <= k ; ++) {Correlaton = comp_np_corr_func(n 1 ); Cf(N 1, N 2 ) = comp_out_corr_func(n 1, Correlaton); } /* check the correlaton functon of the last par of subcrcuts */ f (Cf(N 1 k, N 2 k ) mples equvalence_ functon) return( equvalent ); else return( unsolved ); } Fgure 2. Pseudocode of equvalence checkng algorthm The pseudocode of our algorthm for equvalence checkng s gven n Fgure 2. The procedure check_parttons checks that specfcaton S represented by Part(N 1 ),Part(N 2 ) s correct topologcally. Namely, t checks that f outputs of subcrcut N 1 are (not) connected to nputs of subcrcut N j 1, then outputs of subcrcut N 2 should (not) be connected to nputs of N j 2. If ths s not true, the check_parttons procedure returns result ncorrect. In the man loop we compute the correlaton functons Cf(N 1,N 2 ) n topologcal order. (Note that n Fgure 2 we denote correlaton functon dfferently to emphasze the fact that specfcaton S s represented mplctly. Here Cf(N 1,N 2 ) denotes what we prevously denoted as Cf(v 1 (C),v 2 (C)) where C s the varable assocated wth the output of the block of S mplemented by N 1 and N 2.). Before computng Cf(N 1,N 2 ) the procedure comp_np_corr_func forms the expresson Correlaton. Ths expresson s a conjuncton of the correlaton functons correspondng to subcrcuts whose outputs are connected to nputs of N 1 and N 2. the correlaton functons correspondng to the prmary nputs of N 1 (f any) that are n the fann of N 1 or N 2. If, for example, nputs of N 1 are connected only to outputs of subcrcuts N j 1 and N m 1 (and so nputs of N 2 are connected only to j outputs of N 2 and N m 2 ), then Correlaton = Cf(N j 1,N j 2 ) Cf(N m 1,N m 2 ). On the other hand, f an nput x 1 of N 1 s a prmary nput of N 1 (and so the correspondng nput x 2 of N 2 s a prmary nput of N 2 ), then n the conjuncton of terms specfyng Correlaton there s term Eq(x 1,x 2 ) descrbng the equvalence of x 1 and x 2. The functon comp_out_corr_func computes the correlaton functon Cf(N 1,N 2 ) by exstentally quantfyng the functon P=Implementaton Correlaton. The functon Implementaton = F(N 1 ) F(N 2 ) descrbes consstent assgnments to the varables of N 1 and N 2. The functon Cf(N 1,N 2 ) s obtaned from P by exstentally quantfyng away all the varables of N 1 except the ones correspondng to outputs of N 1 and N 2. Fnally, the algorthm checks f the correlaton functon of subcrcuts N k 1 and N k 2 (whose prmary outputs are prmary outputs of N 1 ) mples the equvalence functon. If yes, then N 1 are equvalent. Otherwse, the algorthm returns the unsolved answer. The complexty of the algorthm shown n Fgure 2 s the same as n general resoluton.e. d n 3 6p where d s a constant. That s for the class of formulas M(p) wth the fxed value of p, the complexty of ths algorthm s lnear n crcut sze. 6. Conclusons In ths report, we prove the results of paper [2] wthout usng the noton of flterng functons. Ths allows us to smplfy the formulaton of our theory for equvalence checkng of crcuts wth a common specfcaton. Besdes, we gve a modfed descrpton of the algorthm for equvalence checkng of crcuts wth a known specfcaton. In ths descrpton we use only correlaton functons (omttng fterng functons) and emphasze the fact that CS s represented mplctly. 7. References [1] Brand, D., Verfcaton of large syntheszed desgns. Proceedngs of ICCAD-1993,pp [2] E.Goldberg, Y.Novkov. Equvalence Checkng of Dssmlar Crcuts. Internatonal Workshop on Logc and Synthess, May 28-30,2003,USA. [3] E.Goldberg, What Sat-solvers can and cannot do. pp n Advanced Formal Verfcaton, edted by Rolf Drechsler,2004, Kluwer Academc Publshers. [4] E.Goldberg, Y.Novkov. On complexty of equvalence checkng. Techncal Report, CDNL-TR , August, 2003.