Heuristics for Thelen s Prime Implicant Method 1

Transcription

1 S C H E D A E I N F O R M A T I C A E VOLUME Heuristics for Thelen s Prime Implicnt Method 1 Jcek Biegnowski, Andrei Krtkevich Institute of Computer Engineering nd Electronics, ul. Podgórn 50, Zielon Gór e-mil: J.Biegnowski@iie.u.gor.pl, A.Krtkevich@iie.u.gor.pl Astrct. Thelen s lgorithm is n efficient method for genertion of the prime implicnts of Boolen function represented in CNF. In the pper new heuristics re presented, llowing to ccelerte the lgorithm. Eperimentl nlsis of their effects is performed. Kewords: Boolen functions, CNF, prime implicnts, serch tree, Thelen s method. 1. Introduction A lot of tsks relted to computer-ided logicl design require clcultion of prime implicnts of logicl function, which is often represented in conjunctive norml form (CNF) i.e., s product of sums. The most known of such tsks is two-level minimition of Boolen functions. Most methods of minimition (especill of ect minimition), oth clssicl nd modern, require clcultion of ll the prime implicnts, from which suset representing miniml DNF is then selected. For emple, the most widel used minimition progrm ESPRESSO uses such pproch [8]. It is worth mentioning, however, tht some new efficient methods hve een developed voiding to generte ll the prime implicnts [3, 7, 8]. On 1 The reserch is prtill supported Polish Stte Committee for Scientific Reserch (KBN) grnt No. 4T11C

2 126 the other hnd, new vrints of minimition methods requiring ll the prime implicnts re still eing developed [8, 10, 11]. And there re lot of other pplictions of method of prime implicnts genertion. For emple, clcultion of the complement of Boolen function (in DNF), or trnsformtion of Boolen eqution from CNF to DNF. And vice vers s fr s due to the Morgn s lws trnsformtion from DNF to CNF cn e performed trnsformtion from CNF to DNF. One more ppliction is detecting dedlocks nd trps in Petri net, which cn e performed solving logicl equtions [13, 14]. Generll, solutions of logicl eqution cn e esil otined from prime implicnts of its left prt, if the right prt is 1. Also there re tsks, which cn e solved clculting the shortest prime implicnt or prime implicnts stisfing certin conditions. In [5] severl of such logicl design tsks re discussed. Covering prolems, oth unte nd inte covering, cn e esil represented s logicl epressions in CNF nd re usull solved one of two pproches: BDD-sed [1] or rnch nd ound, for which the shortest prime implicnt would correspond to n optiml solution [2]. The sme is true for some grph prolems, such s decclistion of grphs [4]. Tsk of detecting dedlocks in FSM networks cn e reduced to tsk of generting suset of prime implicnts. The pproch discussed in this pper cn e pplied (directl or with some modifictions) to the whole rnge of mentioned prolems. For genertion of prime implicnts severl lgorithms re known. The method of Nelson [9], prol historicll first such method for CNF, is sed on strightforwrd multipling the disjunctions nd deleting the products tht susume other products. Such trnsformtion is ver time- nd memor-consuming. More efficient methods re known: n lgorithm sed on serch tree, proposed B. Thelen [12], nd recursive method descried in [8]. Comprison of those two methods is eond the scope of our pper; the pper is dedicted to heuristics llowing to ccelerte Thelen s method. Eecution time of this lgorithm depends remrkl on the order of cluses nd literls in the epression. Hence we m suppose tht some reordering of the epression will increse efficienc of the lgorithm. As fr s the serch tree in Thelen s method is reduced mens of certin rules (descried elow), it is difficult to evlute priori effects of different vrints of reordering. So it is resonle to use heuristic pproch nd to verif the heuristics sttisticll. Some of such heuristics re descried in [5, 6]. The rticle descries some new heuristics, their nlsis nd comprison with known heuristics. Eperiments re performed using the rndoml generted smples; the optiml comintion of the heuristics is formulted on the sis of eperimentl results.

3 Thelen s lgorithm Thelen s prime implicnt lgorithm is sed on the method of Nelson [9], who hs shown, tht ll the prime implicnts of Boolen function in conjunctive form cn e otined its trnsformtion into disjunctive form. Nelson s trnsformtion is performed strightforwrd multipling the disjunctions nd deleting the products tht susume other products. Such trnsformtion is ver time- nd memor-consuming, ecuse ll the intermedite products should e kept in memor, nd their numer grows eponentill. Thelen s lgorithm trnsforms CNF into DNF in much more efficient w. It requires liner memor for trnsformtion nd dditionl memor for clculted prime implicnts. The susuming products re not kept in memor. A serch tree is uilt, such tht ever level of it corresponds to cluse of the CNF, nd the outgoing rcs of node correspond to the literls of the disjunction. Conjunction of ll the literls corresponding to the rcs t the pth from the root of the tree to node is ssocited with the node. Lef nodes of the tree re the elementr conjunctions eing the prime implicnts of the epression or the implicnts susuming the prime implicnts clculted efore. A smple tree is shown in Fig. 1. The tree is serched in DFS order, nd severl pruning rules re used to minimie it. The rules re listed elow. R 1 An rc is pruned, if its predecessor node-conjunction contins the complement of the rc-literl. An rc is pruned, if nother non-epnded rc on higher level still eists which hs the sme rc-literl. A disjunction is discrded, if it contins literl which ppers lso in the predecessor node-conjunction. The rules ove re sed on the following lws of Boolen lger: = = = 0 0 = 0 0 = = (1) ( ) = (2) = 1 (3) 1 = 1 (4) 1 = (5)

4 128 Rules R 1 nd follow immeditel from (2) nd (3). Rule provides tht the implicnts ssocited with the lef nodes, if the re not prime, susume the implicnts clculted efore. Tht mens tht the first clculted implicnt is lws simple. An rc t level k with rc-literl, such tht there is non-epnded rc with the sme rc-literl t level l higher thn k, is pruned it. An implicnt otined epnding the mentioned rc would e t lest one literl shorter thn the implicnt which would e otined without ppling rule. As fr s the pth two times comes through literl (t the levels k nd l), ccording to (1), (2) the longest of those two implicnts susumes the shortest one. Hence the first clculted implicnt cnnot susume the implicnts clculted lter, ut it cn e susumed them. So, ppling rule llows to check whether n implicnt is simple immeditel fter its clcultion. It is enough to compre it with ll the implicnts clculted efore. Due to this propert the lgorithm is less memor-consuming, ecuse onl prime implicnts re kept. --- c c R 1 c c R 1 c c c c c c d d d c c cd Fig. 1. An emple of the tree for Boolen formul: ( c)( )( c)( d) H.-J. Mthon [5] hs proposed the dditionl fourth reduction rule, which reduces the serch tree up to 25%.

5 129 R 4 An rc j is pruned, if nother lred epnded rc k with the sme rc-literl eists on higher level v nd if rule ws not pplied in the sutree of rc k with respect to rc p on level v which leds to rc j. But using this rule complictes the lgorithm remrkl, ecuse dditionl informtion on ppling rule hs to e kept. Additionl reduction reduces proilit of ppering the non-prime implicnts t the lef nodes. But there is no gurntee tht such implicnts will not pper, nd still it is necessr to perform checking, the sme s in the cse of tree uilt with using onl 3 pruning rules. The net epression is n emple for which non-prime implicnts still pper even if ll 4 rules re used: ( )( )( ) (Fig. 3). 3. Heuristics for Thelen s method One of the possiilities of reducing the serch tree is sorting the disjunctions their sie in scending order. Heuristic 1 (Sort Length [5]). Choose disjunction D j with the smllest numer of literls. Effect of this heuristics cn e illustrted with complete serch tree (without rc pruning). Its sie (numer of nodes) cn e clculted ccording to the formul: n i V = 1 + L j (6) i=1 j=1 where L j is the numer of literls in cluse numer j. Let formul consist of 5 cluses, ech hving different numer of literls, from 2 to 6. If the re sorted from miml to miniml length, the complete serch tree will contin 1237 nodes; if sorted from miniml to miml the tree will contin onl 873 nodes. In the second cse it is 30% smller. So sorting of cluses influences the tree sie remrkl. Of course for the reduced serch trees reltion m differ. Now let us turn to the pruning rules. Note tht ever rule cn e implemented onl if the disjunction under considertion contins the sme vriles s the disjunctions corresponding to the predecessor nodes. Tht mens tht if the net disjunction considers the vriles which pper in the previous disjunctions, there re possiilities of reduction t tht level; nd there is no possiilit of reduction for the new vriles. So we m suppose tht

6 130 sorting of the cluses ccording to the vriles lso m led to the tree reduction. Here the similr effect is used s in the cse of sorting length: disjunctions contining mn repeting vriles llow to reduce the tree remrkl, nd if such reduction cn e performed not fr from the root, the tree will e growing slower. So the following heuristics reorder the disjunctions in such w tht miniml numer of new vriles ppers t ever net level of the serch tree. Heuristic 2 (Sort Literls). Choose disjunction D j with the smllest numer of literls tht do not pper in the disjunctions chosen efore. Heuristic 2 (Sort Vriles). Choose disjunction D j with the smllest numer of vriles tht do not pper in the disjunctions chosen efore. The onl difference etween these two vrints is tht heuristic 2 compres cluses ccording to literls nd heuristic 2 ccording to vriles. This mens tht i nd i re two different items for heuristic 2 ut not for heuristic 2. Averge results of these heuristics re ver similr, ut there re emples where the tree reduction differs lot. So it is possile to otin etter results selecting the most effective heuristic for ever emple. It will e the suject of the forthcoming work. The effect of heuristics 2 nd 2 is comprle to the effect of heuristic 1, ut we m s, tht it is more inteligent, which is confirmed sttisticl nlsis. The results of computer eperiments re presented in the Tle 1, nd s fr s heuristics 2 nd 2 give similr verge results, onl one of them is presented. Reordering literls in cluses lso ffects the serch tree, ecuse it chnges the order of generted prime implicnts nd m mke rule pplicle or not t certin levels of the tree. Two following heuristics reorder literls in cluses. The first of them llows quick clcultion of the shortest prime implicnt, nd the net heuristic reduces the serch tree when it is necessr to clculte ll prime implicnts. Heuristic 3 (Sorting literls[5]). Choose literl v i with the mimum frequenc in the non-epnded prt of the epression. Sorting the literls ccording to heuristic 3 leds to the situtions such tht rule will e more often pplicle for the rcs t the left side thn t the right side. Hence prol the first clculted prime implicnts will e the shortest. Now using the rnch-nd-ound method most of other rcs will e cut in severl steps. If ll the prime implicnts hve to e generted, the ordering should e different: in such cse it is etter to generte shortest implicnts lter (reverse order of literls given heuristic 3). Tht reduces proilit of

7 131 ) --- ) --- R 4 c d c d c d c d c d c d Fig. 2. An emple of the tree, in which effects of heuristic 4 nd rule R 4 re the sme ppering non-prime implicnts t the lef nodes. As fr s due to rule n implicnt cn susume onl the implicnts clculted efore, if the implicnts clculted lter re in most cses shorter thn those clculted erlier, then chnce of susuming is smll. The net heuristic is reversion of heuristic 3. Heuristic 4 (Reordering Literls). Choose literl v i with the minimum frequenc in the non-epnded prt of the epression. In mn cses (ut not lws) effects of rule R 4 nd heuristic 4 re ver similr. Rule R 4 prunes n rc, if t higher level there is non-epnded rc with the sme rc-literl (let it e ). Tht mens tht t the level k literl is not the lst literl in the cluse. Let literl e fter in the cluse. From the rc corresponding to the literl there is pth to the node under considertion t level l. If literl would e the lst in the cluse, insted of R 4 the rule would e pplicle with the sme effect (Fig. 2). We m lso stte tht if literl ppers t level k nd lso t lower level l (tht mens in cluses D k or D l (k > l)), then if does not pper in the cluses with numers greter thn k, fter ppling heuristic 4 in the cluse D k literl will pper efore nd will e pplicle insted of R 4. But if ppers in the cluses with numers greter thn k, such n effect will not lws occur. Here is emple of heuristic 4: After ppling the heuristic: ( )( c d) ( )( c d)

8 132 Such ordering of literls cuses tht the rc leding to non-prime implicnt, which in the first cse could e pruned onl ppling rule R 4, now will e pruned rule. Another emple: ( c)( )( c d)( c) In this cse heuristic 4 does not chnge ordering of the literls. Literls nd, ppering in cluse 2, pper in the net cluses with the sme frequenc, nd without ppling rule R 4 the lgorithm will generte nonprime implicnt. T. 1. Results of computer eperiments Bool P STD H1 H2 H4 H2 + H4 R4 form. T N T N % T N % T N % T N % T N % Avg: On the other hnd, it m hppen tht reordering of literls heuristic 4 llows pruning the rcs which would not e pruned rule R 4. It is possile ecuse the rules nd R 4 re not completel smmetricl. Rule R 4 hs dditionl condition which is sent in rule. This condition m lock ppling rule R 4. But if the literls cn e reordered in such w tht rule will e pplicle, then such rc will e pruned. Fig. 3 illustrtes such sitution. In the tree for epression ( )( )( ) (Fig. 3) rule R 4 cnnot e pplied ecuse the condition is not stisfied (in the left sutree rule ws pplied). Tht is wh in the right su-tree non-prime implicnt ppers. But heuristic 4 chnges the epression into the form: ( )( )( ). Now the non-prime implicnt does not pper, ecuse the rc leding to it is pruned rule (Fig. 3). Eperiments demonstrte tht oth heuristic 4 nd rule R 4 efficientl nd lmost t the sme etent reduce the numer of generted non-prime

9 133 ) ) Fig. 3. An emple of the tree, in which there re differences etween heuristic 4 nd rule R 4 implicnts. But rule R 4 is more difficult for implementtion nd increses necessr memor mount. So it seems tht ppling heuristic 4 is more resonle ecuse llows otining similr effect with less effort. Results of computer eperiments re summried in T. 1. For the tests the rndoml generted Boolen epressions were used. In the first column numer of vriles nd the numer of cluses of n epression re given (e.g. 2018). T denotes the tree sie (numer of nodes); P denotes the numer of prime implicnts; N denotes the numer of non-prime implicnts, eing the leves of the serch tree. A column % shows for ever heuristic the percentge of the tree sie in respect of the sie in the cse when no heuristic is used. The eperiments show tht it is est to sort disjunctions ccording to heuristic 2, nd literls in the disjunctions ccording to heuristic 4.

10 Conclusion nd further work The presented heuristics, ccording to the eperimentl results, llow to generte ll the prime implicnts of logicl epression represented in the conjunctive norml form more quickl, thn it cn e done using Thelen s method with the heuristics known efore. Besides of tht, the presented heuristics reduce remrkl the numer of the lef nodes in the serch tree corresponding to non-prime implicnts. A prospective direction of future work is evlution of efficienc of the proposed heuristics for solving prolems mentioned in Introduction, for which Thelen s lgorithm cn e pplied. Tht m require tking into ccount dditionl optimition prmeters nd modifiction of heuristics. One more direction is comprison etween Thelen s pproch nd the BDD-sed pproch to solving prolems such s covering prolems. 5. References [1] Brton R.K. et l.; VIS: A Sstem for Verifiction nd Snthesis, in: The Proceedings of the Conf. on Computer-Aided Verifiction, August 1996, Springer Verlg, 1102, pp [2] Coudert O., Mdre J.K.; New Ides for Solving Covering Prolems, Design Automtion Conference, 1995, pp [3] Coudert O., Mdre J.K., Frisse H.; A New Viewpoint on Two-Level Logic Minimition, Design Automtion Conference, 1993, pp [4] Krtkevich A.; On Algorithms for Decclistion of Oriented Grphs, in: Proceedings of the Interntionl Workshop DESDes 01, Zielon Gór, Polnd, 2001, pp [5] Mthon H.J.; Universl logic design lgorithm nd its ppliction the snthesis of two-level switching circuits, IEE Proceedings, 136,3, 1989, pp [6] Mthon H.J.; Algorithmic Design of Two-Level nd Multi-Level Switching Circuits, (in Germn), PhD thesis, ITIV, Univ. of Krlsruhe, [7] McGeer P.C. et l.; Espresso-Signture: A New Ect Minimier for Logic Functions, Design Automtion Conference, 1993, pp [8] De Micheli D.; Snthesis nd Optimition of Digitl Circuits, Stnford Univ., McGrw-Hill, Inc., 1994.

11 135 [9] Nelson R.; Simplest Norml Truth Functions, Journl of Smolic Logic, 20,2, 1955, pp [10] Rudell R., Sngiovnni-Vincentelli A.; Multiple-vlued Minimition for PLA Optimition, IEEE Trnsctions on CAD/ICAS, Sept. 1987, CAD-6, 5, 1987, pp [11] Rtsr B., Miniuk V.; The Set-theoreticl Modifiction of Boolen Functions Minim Covering Method, in: Proceedings of the Interntionl Conference TCSET 2004, Lviv Slvsko, Ukrine, 2004, pp [12] Thelen B.; Investigtions of lgorithms for computer-ided logic design of digitl circuits, (in Germn), PhD thesis, ITIV, Univ. of Krlsruhe, [13] Wȩgrn A., Wȩgrn M.; Smolic Verifiction of Concurrent Logic Controllers Mens Petri Nets, in: Proceedings of the Third Interntionl Conference CAD DD 99, Minsk, Belrus, 1999, pp [14] Wȩgrn A., Krtkevich A., Biegnowski J.; Detection of dedlocks nd trps in Petri nets mens of Thelen s prime implicnt method, AMCS, 14,1, 2004, pp Received Mrch 8, 2004