An Efficient Divide and Conquer Algorithm for Exact Hazard Free Logic Minimization

Transcription

1 An Efficient Divide nd Conquer Algorithm for Exct Hzrd Free Logic Minimiztion J.W.J.M. Rutten, M.R.C.M. Berkelr, C.A.J. vn Eijk, M.A.J. Kolsteren Eindhoven University of Technology Informtion nd Communiction Systems Group e mil: jeroen@ics.ele.tue.nl Abstrct In this pper we introduce the first divide nd conquer lgorithm tht is cpble of exct hzrd free logic minimiztion in constructive wy. We compre our lgorithm with the method of Dill/Nowick, which ws the only known method for exct hzrd free minimiztion. We show tht our lgorithm is much fster thn the method proposed by Dill/Nowick by voiding significnt prt of the serch spce. We rgue tht the proposed lgorithm is promising frmework for the development of efficient heuristic lgorithms. Introduction One of the bottlenecks of synchronous design is the logic synthesis step. During this step it is often required tht the logic expressions to be generted re free of hzrds for given set of trnsitions under the unbounded wire dely model []. This is e.g. the cse during the synthesis of Asynchronous Burst Mode Mchines. It hs been shown tht these mchines cn successfully be used to implement lrge synchronous designs [4]. Until recently hzrd free logic expressions hd to be designed by hnd, since no lgorithm ws known to generte them. The first lgorithm cpble of generting hzrd free logic expressions ws proposed by Dill/Nowick []. This lgorithm is cpble of generting n exct pl bsed solution. It uses n exct two level minimizer, like espresso [5], to generte ll prime implicnts. A filter is pplied to ll these prime implicnts to render them dynmic hzrd free (dhf). A miniml selection is mde of the fittest implicnts to generte n exct miniml solution in terms of the number of prime implicnts. Since there cn be n exponentil number of prime implicnts, this method is only pplicble to reltively smll exmples. In this pper we propose new exct divide nd conquer lgorithm, similr to the one used in espresso, tht is cpble of generting hzrd free logic expressions in constructive wy. Tht is, we directly generte hzrd free logic without the need of posteriori fixing. Not only is this very importnt from theoreticl point of view, since we believe it is the first in its kind, it is lso importnt from prcticl point of view since by some modifictions the lgorithm is cpble of generting both exct nd heuristic solutions. We show tht this lgorithm cn void prt of the serch spce nd therefore is much fster thn Dill/Nowick s method for lrger exmples. In this pper we focus on single output functions. Single output minimiztion is n importnt opertion during multi level logic synthesis nd the described lgorithms cn be pplied in tht field. We lso outline the modifictions necessry to llow the described lgorithms to operte on multiple output functions; due to lck of spce detils re omitted. 2 Bckground A binry vlued input, binry vlued output function cn be described by f : B n B {*} where B {0, }. Here, str indictes the prt of the function belonging to the don t cre set. The offset of f is denoted s R. A cube S is non empty Crtesin product spce S S 2 S n, where S i B. A minterm is cube where S i. An implicnt is cube tht hs no intersection with the offset. A prime implicnt is n implicnt tht is not covered by ny other implicnt. Given function f, the gol of 2 level minimiztion is to find miniml selection of prime implicnts such tht the on set of this function is completely covered. Let function f be expressed over the input vribles x x n. f xi is clled cofctor of function f with respect to input vrible x i. It is defined s: f xi f (x,..., x i,..., x n ). The negtive cofctor, f xi is defined s: f xi f (x,..., x i 0,..., x n ). A function is sid to be unte in vrible x i if f xi f xi or f xi f xi. In the first cse the function is positive unte in x i, in the ltter cse, the function is sid to be negtive unte in x i. A function is unte if it is unte in ech vrible. 3 A new divide nd conquer technique In this section we introduce new divide nd conquer lgorithm, the threewy method, tht is cpble of clculting ll prime implicnts. We introduce this lgorithm becuse it cn be converted to generte the set of ll dhf prime implicnts,

2 which, to the best knowledge of the uthors, is not possible with e.g. the lgorithm used by espresso. Furthermore, we show tht this lgorithm is cpble of voiding significnt prt of the serch spce by only generting contributing dhf prime implicnts. We strt by introducing the problem of hzrd free logic minimiztion. 3. Hzrd free logic minimiztion Hzrd free logic is bsed on the dynmic hzrd free (dhf) implementtion of set of trnsitions. A trnsition is tuple defined s: I, O where I D n nd O D m. D is given by: D {0,,, }. A trnsition cube [] is cube notted s [A, B], where A nd B re minterms tht indicte the strting point nd end point of trnsition in the input spce. The trnsition cube is the smllest cube tht covers both A nd B. Hzrd free logic considers ech trnsition individully: ech trnsition is implemented free from hzrds. At the strt of ech trnsition, the logic implementing the behvior of the trnsition is ssumed to be stble. Hzrds cn be sttic or dynmic []. Sttic hzrds cn be voided by demnding tht the trnsition cube is covered by n implicnt in the solution. Such trnsition cube is clled required cube. Dynmic hzrds re tken cre off by so clled privileged cubes. A privileged cube is trnsition cube with n nnotted strting point. It is formlly denoted s: p (p c, p s ), where p c is clled the body of the privileged cube nd p s is clled the strting point. All implicnts in dhf solution re only llowed to intersect privileged cube when the strting point is lso included. Otherwise the intersection is sid to be illegl. An implicnt tht does not illeglly intersect ny privileged cube is clled dhf implicnt. A dhf prime implicnt is not covered by ny other dhf implicnt. A miniml dhf solution consists of miniml selection of dhf prime implicnts tht covers ll required cubes []. Exmple Consider circuit with 4 inputs nd output tht is supposed to implement the following trnsitions free from dynmic hzrds: {< 000,>, < 0, >, <,0>, <, >, < 0, >, < 00,0>, <00 0, >}. The order of the input vribles in these trnsitions is bcd. Furthermore we ssume tht the circuit implementing ll trnsitions is stble t the strt of ech trnsition. In figure Krnugh mp is depicted showing ll the trnsitions nd the on/offset tht cn be derived from this set of trnsitions. An empty entry corresponds with don t cre entry. If we use 2 level minimizer, like espresso, to generte miniml set of prime implicnts tht covers the depicted onset, the solution would be: 0, 00. This solution is hzrdous, which cn be shown by exmining trnsition < 0, >. In this trnsition, input vribles b nd d turn on. Suppose input vrible b turns on before input vrible d. d Figure : Krnugh mp for given set of trnsitions We cn observe tht cube 00 will turn off. Cube 0 will turn on, so we hve term tkeover tht cn give rise to sttic hzrd: the output might temporrily turn off. This hzrd cn be voided by demnding tht the solution covers the smllest cube tht covers both the strting point of the trnsition nd the minterm entered by turning b on. Since solution must cover this cube, it is clled required cube. Here, the required cube is: 00. Let us dd this required cube to the solution. Now let us exmine if we hve successfully removed ll hzrdous situtions for the given trnsition. Agin, we ssume tht input vrible b is the first to turn on. Becuse of the dded required cube, the output will remin t one. Now input vrible d switches on. Both the dded required cube nd cube 0 will turn off in tht cse. However the behvior of this cube might be observed fter the required cube itself hs turned off. In fct, the behvior of cube 0 going on might be observed fter the required cube hs turned off, due to our unbounded wire dely model. This might led to dynmic hzrd: fter the output hs turned off, glitch representing the behvior of cube 0 might be observed. This dynmic hzrd cn be voided by demnding tht during trnsition fter which the output will turn off, no cube cn temporrily turn on. A cube my turn on, or it my turn off, but not both. This constrint is met by tking into ccount set of privileged cubes. The body of privileged cube is equl to the trnsition cube tht covers the trnsition. A privileged cube lso hs n nnotted strting point, corresponding to the strting point of the trnsition. A set of implicnts is sid to be dhf if the intersection of ech implicnt with the body of privileged cube lso contins the strting point of tht privileged cube. If this is not the cse, the implicnt is sid to intersect privileged cube illeglly. This cn led to dynmic hzrd. In this exmple, the privileged cube necessry to void dynmic hzrd during trnsition < 0, > is ( p c = 0, p s = 000). Observe tht prime implicnt 0 intersects the body of this privileged cube, but does not cover the strting point. Therefore this prime is not dynmic hzrd free. For this exmple correct miniml hzrd free 0 c b

3 R bd d cd F d bcd bd bc d c b R R R R 0 0 bd cd bd cd d d bd d bc bcd bd Figure 2: Exmple of the threewy method solution consists of three dhf prime implicnts nmely: 0, 00, The threewy method In the threewy method for binry vlued functions, the set of ll prime implicnts is clculted by supplying the method the offset of function f. Serching the set of ll prime implicnts becomes equl to serching ll mximl cubes tht do not intersect the offset. If we consider vrible with respect to cube, we cn discern three cses: the vrible is 0, or (don t cre). The min ide behind the threewy method is to divide the originl problem of clculting ll prime implicnts into the three sub problems of clculting the set of prime implicnts in which selected vrible hs one of these vlues. Hence the nme threewy method. For exmple, one of the sub problems is to generte the set of prime implicnts for which the selected vrible is equl to. The bsic ide of the method is depicted in figure 2. The ide to prtition the problem of clculting the set of ll prime implicnts into three sub problems is not new, it ws lso used by Coudert to implement n efficient implicit 2 level minimizer [3]. In the exmple in figure 2, vrible is used to reduce the originl problem of finding ll prime implicnts into the three sub problems of finding the prime implicnts for which vrible hs one of the three possible vlues. These three vlues re represented by three edges. The set of primes for which is equl to 0 cn never intersect with tht prt of the offset where is equl to. Therefore in order to clculte the set of primes for which is 0, the zero edge for future references, we only need to clculte the set of mximl cubes tht hve no intersection with the negtive cofctored offset, since we bstrct from vrible. This is expressed s R. For the one edge, the offset becomes R. For the don t cre edge, the offset becomes R R. This is becuse for ny prime for which vrible is don t cre, n intersection with R or R implies n intersection with R itself nd vicevers. The new sub problems re reduced: the crdinlity of the support set is decresed. The originl problem cn be divided into new sub problems repetedly until the offset becomes empty or equl to the universe cube. In the first cse, the set of primes becomes equl to the universe cube, since this is the mximl cube tht does not intersect the empty offset. In the second cse the set of primes is empty. The selection of edges tken, i.e. the pth, determines the ssignment of the vribles of the prime. For the given exmple in figure 2, the offset is given by R bd d cd, the cofctors re R bd d nd R bd cd. By repeting the splitting process, the set of primes not intersecting R cn be clculted. This set turns out to be equl to Primes bd. Similrly the set of primes not intersecting R is equl to Primes d bc. Also it turns out tht the set of primes not intersecting R R is equl to Primes bcd. Combining these primes results in: Primes Primes Primes Primes = bd d bc bcd. Figure 3 gives generl overview of the proposed threewy method: R 0 R R R R Figure 3: Generl threewy model The primes generted by the zero or one edge cn be covered by the primes generted by the don t cre edge. Therefore single cube continment (SCC) filtering hs to be performed, to remove ll covered cubes, while merging the solutions of the sub problems into the solution of the originl problem. This is expressed s: Primes SCC( Primes Primes Primes )

4 We will now prove tht the proposed method genertes ll prime implicnts. In order to do so, we need the following lemm: Lemm Ech cube hs n unique pth in the serch tree. No two different cubes follow the sme pth in the serch tree since this would imply tht for ech input vrible they re ssigned the sme vlues. Theorem Ech cube generted by the threewy method is n implicnt. Proof: We only generte cube when the offset becomes empty. Certin pths exist in the serch tree fter which the offset becomes empty. By lemm ech pth corresponds to unique cube. Following pth is equl to clculting the cofctor of the offset with respect to this cube. Since this cofctor is empty, the cube must be n implicnt. Theorem 2 Ech implicnt generted by the threewy method is prime. Proof: Suppose prime implicnt d covers n implicnt c generted by the threewy method. Since d does not intersect the offset, the cofctor of the offset w.r.t. d is empty. Therefore, following the pth tht belongs to d leds us to lef node where the offset is empty. So d is creted. Implicnt c might lso be creted, but due to the SCC filter, it is removed. Theorem 3 The threewy method will generte ll prime implicnts. Proof: Suppose prime d is missing. Agin we cn follow its pth nd observe tht the offset gin will be empty. Therefore d will be generted. Theorem 4 If the offset is unte in vrible, then the threewy method cn be reduced to twowy method. Proof: The primes generted by Primes or Primes cn be covered completely by the primes of Primes. This hppens when the offset, nd therefore lso the onset is unte in certin vrible. If the offset is unte in vrible then R R or R R. In tht cse R R R or R R R. Therefore the primes generted by the zero (one ) edge will be covered completely by the primes generted by the don t cre edge. Therefore the threewy method becomes twowy method. We cn use this observtion to significntly reduce the serch spce by selecting unte vribles s soon s possible. 3.3 Generting ll dhf prime implicnts with the dhf threewy method Here, we propose modifictions tht will llow the threewy method to generte the set of ll dhf prime implicnts. This is ccomplished by keeping trck of the set of privileged cubes during the process of splitting the originl problem of clculting ll dhf primes. The problem of clculting the set of ll prime implicnts is trnsformed into one of clculting ll dhf prime implicnts. For the sub problems we now wnt to clculte the set of ll dhf primes for which the splitting vrible ssumes vlues zero, one or don t cre. Let us exmine the set of dhf prime implicnts for which vrible is equl to. Since vrible is equl to for ny prime clculted by the edge, ny privileged cube tht hs strting point for which is equl to 0, will be intersected illeglly when its body is intersected. To void this intersection we must dd the cofctored body, p c of ech of these privileged cubes to the offset of the edge. This wy, we cn gurntee tht the set of primes generted by the edge will not intersect these privileged cubes illeglly. However, the primes still might intersect those privileged cubes tht hve strting point for which vrible is equl to one or don t cre. Therefore, we must keep trck of these privileged cubes by pssing them on. Resuming, the offset nd set of privileged cubes tht re pssed on to the edge re expressed s: P p p s R R p c p s We represent the set of privileged cubes nd the Offset pssed on to the edge by dding subscript. We now consider the cse where the splitting vrible is equl to don t cre: we follow the don t cre edge. We cn not determine which subset of the set of privileged cubes will lwys be intersected illeglly. Therefore we must consider the set of ll co fctored privileged cubes. This is expressed s: P 2 P P The offset is expressed s: R 2 R R A subscript of 2 represents the sets tht re pssed on to the don t cre edge. Note tht P 2 is not equl to P 0 P. In figure 4 the generl model of the dhf threewy method is depicted. (R, P) 0 (R 0, P 0 ) (R 2, P 2 ) (R, P ) Figure 4: Generl model of the dhf threewy method Theorem 5 The set of implicnts generted by the dhf threewy method is dynmic hzrd free. Proof: By contrdiction. Suppose the method genertes n implicnt c tht illeglly intersects privileged cube p. We now strt to follow the pth belonging to c. After n edge, p cn still be vilble, in cofctored form, or it is removed during the process of clculting the set of privileged cubes tht must be pssed on. In the ltter cse, the cofctored form

5 of p s is empty in which cse the cofctored body of p, p c is dded to the offset. This would mke n illegl intersection of c with p impossible. Since c is supposed to intersect privileged cube p illeglly, p must remin in the trnsported set of privileged cubes. We cn pply this story in recursive wy until the offset becomes empty. Here c is being creted nd it is equl to the universe cube. However, since c is equl to the universe cube, it cnnot intersect p illeglly, becuse the strting point is included in the intersection. Therefore c is dhf implicnt. Theorem 6 The set of implicnts generted by the dhf threewy method is dhf prime. Proof: Suppose d is dhf implicnt tht covers dhf implicnt c generted by the threewy method. Now let s follow the pth belonging to d. The threewy method didn t generte d, mening tht when we trce d we end up t lef node where the offset is not empty. We ssume tht d does not intersect the originl offset. The offset, intersected by d, therefore is due to the body of t lest one privileged cube tht hs been dded to the offset. This implies tht d will t lest illeglly intersect one privileged cube. Therefore d cnnot be dhf prime implicnt. Theorem 7 The set of dhf primes generted by the dhf threewy method is complete. Proof: The threewy method genertes ll prime implicnts. The dhf threewy method genertes dhf prime implicnts. Suppose dhf prime c is missing from the set generted. If we follow the pth belonging to this prime we will discover tht t the node where c should hve been creted, the offset is not empty. Therefore c is not vlid: it intersects the originl offset or it intersects privileged cube illeglly. The set of dhf primes is complete. Agin we cn reduce the threewy method to t most twowy method if the offset is unte in t lest one vrible. However, in the dhf threewy method this constrint is not enough. The set of privileged cubes must lso be tken into ccount since these privileged cubes cn eventully contribute to the offset in the process of clculting the set of dhf primes for one of the sub problems. Therefore, we cn only skip the 0 () edge if P 2 is equl to P 0 (P ) nd R 2 R 0 (R 2 R ). 3.4 Generting contributing (useful) dhf prime implicnts The dhf threewy method is cpble of generting ll dhf prime implicnts. However, mny of these dhf primes will never contribute to vlid solution becuse they only cover the don t cre set or becuse they only prtilly cover some required cubes. The threewy method for binry functions cn be modified to void the genertion of these primes. This is possible becuse the solutions of sub problems do not interct to generte the solution of the originl problem, s is the cse in the unte recursive prdigm used in espresso. This mens tht if we know tht n edge will not generte set of contributing/useful primes, we cn just discrd tht edge. It will hve no influence on the other primes being generted. In order to detect edges tht will not generte contributing primes we lso must keep trck of the set of required cubes. The ide is to refrin from clculting the primes belonging to certin edge when the set of required cubes is empty. In figure 5 the modifictions necessry to the dhf threewy method re depicted. (R, P, Q) 0 (R 0, P 0, Q 0 ) (R 2, P 2, Q 2 ) (R, P, Q ) Figure 5: Generl model to generte contributing dhf primes Here Q is used to represent the set of required cubes. We lredy know how to clculte sets R k nd P k. Wht remins to be nswered is how to determine sets Q k. Let us strt by exmining the edge. We only wnt to propgte those required cubes in Q for which vrible lso is equl to. All other required cubes re only prtilly covered, or not covered t ll. So we cn express Q s: Q q q q A similr expression exists for Q 0. For the don t cre edge we must propgte ll, cofctored, required cubes. So we cn express Q 2 s: Q 2 Q Q 3.5 Multiple output functions In this section we will discuss in n outline how the described threewy method cn be modified to tke into ccount multiple output functions. In order to do so, we hve extended the threewy method to be ble to del with multiple vlued functions. All output vribles cn be combined into one multi vlued vrible with crdinlity equl to the number of outputs [5]. This multi vlued vrible cn be used to split the originl problem into three new sub problems, just like binry vrible. This is done by prtitioning this multi vlued vrible into two prtitions. For the 0 nd edge the offset simply become equl to the offset cofctored to the pproprite prtition. This, however, is not the cse for the don t cre edge. It turns out tht the offset for the don t cre edge is equl to the combined sets of both cofctors of the offset, where the originl multiple vlued vrible is split into two new multiple vlued vribles. Ech new multiple vlued vrible corresponds to one of

6 the prtitions of the originl multiple vlued vrible. Due to lck of spce we will not discuss the specific detils nor the modifictions necessry to modify this method to llow it to generte the set of ll dhf prime implicnts. It will be the subject of future pper. 4 Results nd Conclusions We hve implemented the described threewy method nd compred it with Dill/Nowick s method []. In order to do so, we used the two level minimizer espresso to generte the set of ll prime implicnts. These primes re then rendered free from hzrds by Dill/Nowick s hzrd removl lgorithm. In Tble the results re depicted. All benchmrk circuits in the first column re derived from well known Asynchronous Burst Mode Mchines nd represent the biggest single output exmples vilble. The number of inputs nd the number of cubes of ech circuit re shown in the second column. In the third column the number of primes generted by espresso nd the runtime in seconds re given. A dsh ( ) indictes tht the runtime ws too smll to mesure with much ccurcy (less thn 0. [s]). The fourth column shows the number of dhf prime implicnts generted by Dill/Nowick s lgorithm, which uses the results generted by espresso, nd the runtimes. The fifth column represents the results by our threewy method when it is used to generte ll dhf prime implicnts. The number of primes in the fifth column therefore must be equl to the number of primes in the fourth column. Agin, the runtimes re lso represented. The sixth column represents the number of primes nd the runtime by the threewy method when it only produces the set of contributing dhf prime implicnts. The lst column represents the smllest solution, in terms of dhf primes, possible. This solution ws obtined by solving unte covering problem, where the minimum number of dhf primes ws selected tht covers ll required cubes. All benchmrk runs were performed on HP9000/K260 system. Espresso nd the threewy method Tble : results prototype were both compiled with the sme compiler options. From tble it cn be observed tht the runtimes of the threewy method re comprble with the runtimes of espresso, when the times spend in the column hzrd filter re dded. However, the runtimes of the threewy method re significntly reduced when it is llowed to only generte the set of contributing prime implicnts. This is cused by the significnt reduction of the serch spce nd of the number of primes clculted. In this pper we hve presented wht we believe is the first constructive exct method tht is cpble of clculting the set of ll dhf prime implicnts. We hve shown tht this method cn esily be enhnced to generte only the set of useful primes. This significntly reduces the solution serch spce nd hence the runtimes necessry. The proposed method cn lso be extended to del with multiple output functions. To cope with lrger prcticl problem instnces, it is lso importnt to serch for efficient heuristic methods [2], especilly when deling with multiple output functions. In future pper we will show tht the threewy method lso provides good bsis for developing efficient heuristic methods. References [] Steven M. Nowick, Dvil L. Dill, Exct Two Level Minimiztion of Hzrd Free Logic with Multiple Input Chnges, ICCAD 92, pp [2] Michel Theobld, Steven M. Nowick, To Wu, Espresso HF: A Heuristic Hzrd Free Minimizer for Two Level Logic, DAC 96, pp [3] Olivier Coudert, Two level logic minimiztion: n overview, Integrtion the VLSI journl, Vol. 7 No. 2, 994, pp [4] K.Y. Yun, Synthesis of Asynchronous Controllers For Heterogeneous systems, Stnford University, 994, Ph.D. Thesis [5] R. Rudell, A Sngiovnni vincentelli, Multiple Vlued Minimiztion for PLA Optimiztion, IEEE Trnsctions on computer ided design, Vol. CAD 6, No. 5, September 987 Nme I/C Espresso Hzrd Filter dhf threewy Contrib Minimize sbuf send ctl.s.pl 6/50 8/ 0/ 0/ 4/ 3 isend bm.s.pl 9/9 0/ 0/ 0/ 0/ 3 pe send ifc.s.pl 8/59 7/ 7/ 7/ 2/ 6 isend.s.pl 8/67 27/ 27/ 27/ 7/ 4 p2.s.pl 3/59 40/ 40/ 40/ 20/ 4 pscsi.s.pl 4/4 73/0.0 79/ 79/0.40 7/ p.s.pl 20/ / /0. 527/.3 69/ cche ctrl.s.pl 2/ /.2 543/ /.40 95/ scsi.s.pl 8/92 958/ / /3.7 44/0.6 4