Cook, S. A. and D. G. Mitchell (1997): Finding Hard Instances of the Satisfiability Problem: A Survey. The DIMACS Workshop on Satisfiability Problems, American Mathematical Society.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Equivalence Classes in CNF; 4) Experimental Design and SAT; 5) Reports of Experiments; 6) Summary and Conclusions. 2 Background and Motivation Traditionally, the performance of SAT solvers has been evaluated experimentally either in terms of randomly generated instances of SAT problems, e.g. [24, 25], or structured instances, such as the instances from the DIMACS set [21] or the SATPLAN set [22] Merits of either approach are subject to on going critique and examination [15, 16, 26, 27, 28] The traditional way to report results of SAT solvers is the time to solve performance of single ....

S. Cook and D. Mitchell. Finding hard instances of the satisfiability problem: A survey, 1997. Available at http://dream.dai.ed.ac.uk/group/tw/sat/sat-survey3.ps.

....be mapped to SAT. Many problems in planning and scheduling can be represented using SAT; therefore solving SAT is a very attractive research area. However, it is known that SAT and 3 SAT are NP complete [9] The easiness hardness of solving SAT depends on a phenomena known as phase transition [5]. Problems before the phase transition are easy to solve and those after the phase transition are mostly unsatisfiable. Hard SAT problems exist around the phase transition region. A phase transition is defined by the ratio between the number of constraints, l, and the number of variables, n. For ....

....after the phase transition are mostly unsatisfiable. Hard SAT problems exist around the phase transition region. A phase transition is defined by the ratio between the number of constraints, l, and the number of variables, n. For 3 SAT, the phase transition was experimentally found to be 4. 3 [5]. 4 MBO for SAT In this section, the application of the MBO algorithm to the propositional satisfiability problem is introduced in two stages. First, the representation of a colony and the means of calculating the individuals fitness are presented in Section 4.1. Second, the MBO algorithm ....

S.A. Cook and D.C. Mitchell. Finding hard instances of the satisfiability problem: A survey. In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, editors, Satisfiability Problem: Theory and Applications. American Mathematical Society, 1997.

....A problem in SAT is considered to be an easy problem before its phase transition and becomes hard after its phase transition. The phase transition is defined as the ratio of the number of clauses over the number of literals. The phase transition of 3 SAT has been experimentally found to be 4. 3 [6]. 2.2 Swarm Intelligence Swarm intelligence is an emerging field of artificial intelligence inspired by the behavioral models of social insects such as ants, bees, wasps and termites [5] This approach utilizes simple and flexible agents that form a collective intelligence as a group. Swarm ....

S.A. Cook and D.C. Mitchell. Finding hard instances of the satisfiability problem: A survey. In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, editors, Satisfiability Problem: Theory and Applications. American Mathematical Society, 1997.

....the hardest instances to solve are concentrated in the sharp transition region. As well known, finding ways to generate hard instances for a problem is important both for understanding the complexity of the problem and for providing challenging benchmarks for experimental evaluation of algorithms [9]. So the finding of phase transition phenomena in computer science not only gives a new method to generate hard instances but also provides useful insights into the study of computational complexity from a new perspective. Although tremendous progress has been made in the study of phase ....

....to generate hard satisfiable instances. Besides practical importance, more interestingly, the problem of generating random hard satisfiable instances is related to some open problems in cryptography, e.g. computing a one way function, generating pseudo random numbers and private key cryptography [9, 13, 15]. 7 In fact, for constraint satisfaction and Boolean satisfiability problems, there is a natural strategy to generate instances that are guaranteed to have at least one satisfying assignment. The strategy is as follows [2] first generate a random truth assignment t, and then generate a certain ....

S. Cook and D. Mitchell. Finding Hard Instances of the Satisfiability Problem: A Survey, In: Satisfiability Problem: Theory and Applications. Du, Gu and Pardalos (Eds). DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 35, 1997.

....to particular behavior. p. 769) In computer science, the complexity of typical instances of NP complete problems has been investigated for decades. Highlights include Levin s theory of average case completeness [9] and studies of phase transitions in randomly generated combinatorial problems [10]. It remains open to show that some NP complete problem is hard on average, under a simple distribution, so long as P NP. However, worst case average case equivalence has been shown for several cryptographic problems, including one studied by Ajtai and Dwork [11] As for the cellular ....

S. A. Cook and D. Mitchell (1997), Finding hard instances of the satisfiability problem: a survey, DIMACS Series in Discrete Math and Theoretical Computer Science 35, pp. 1--17.

....and even make them worse. This was observed by others and is the focus of on going work by Preswitch, Kautz and Selman. We stress that the proposed flow may not be useful on SAT benchmarks that (i) are easy, or (ii) do not have symmetries. Many difficult SAT instances do not have symmetries [16]. On the other hand, many DIMACS benchmarks [21] have large numbers of symmetries, but are easy and can be solved faster than their symmetries can be found by existing methods. Our on going research seeks (i) faster symmetry detection, e.g. via incomplete algorithms, ii) finding [some] semantic ....

S. A. Cook and D. G. Mitchell, "Finding Hard Instances of the Satisfiability Problem: A Survey", In Satisfiability Problem: Theory and Applications, DIMACS Series in Discr. Math. and Theor. Comp. Sci, 25, pp. 1--17. Amer. Math. Soc., 1997.

....to a solution, they are fast and more suitable for large problems. Therefore, incomplete techniques become more attractive, especially with problems in planning which include thousands of variables [13] The easiness hardness of solving SAT depends on a phenomena known as phase transition [3]. Problems before the phase transition are easy to solve and those after the phase transition are mostly unsatisfiable. Hard SAT problems exist around the phase transition region. A phase transition is defined by the ratio between the number of clauses (constraints) l, and the number of literals ....

....transition are mostly unsatisfiable. Hard SAT problems exist around the phase transition region. A phase transition is defined by the ratio between the number of clauses (constraints) l, and the number of literals (variables) n. For 3 SAT, the phase transition was experimentally found to be 4. 3 [3]. 4 MBO for SAT In this section, the application of the MBO algorithm to the propositional satisfiability problem is introduced in three stages. First, the representation of a colony and the means of calculating the individuals fitness are presented in Section 4.1. Second, a description of the ....

S.A. Cook and D.C. Mitchell. Finding hard instances of the satisfiability problem: A survey. In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, editors, Satisfiability Problem: Theory and Applications. American Mathematical Society, 1997.

....j)I. For example, if we partition a given theory .4 into only two partitions (n = 2) sharing propositional symbols, the algorithm will take time O(21 fSAT( Assuming P NP, this is a significant improvement over a simple SAT procedure, for every that is small enough ( 9, and c 0. 582 [38, 14]) It is important to notice that both the MP procedure (Figure 2) and the LINEARPART SAT procedure (Figure 4) focus on structured problems and not random ones. In structured problems the labels of the links are small, leading to only a small overhead in space. Lemma 1 and Section 2.2 show that ....

....less those in the links, i.e. in Me L(i) Typically, having more partitions causes mi to become smaller. 3. n the number of partitions. Also, a simple analysis shows that given fixed values for l, d in Corollary 1, the maximal n that maintains l, d such that also n ln2 c m (c = 0. 582 [38, 14]) yields an optimal bound for LINEAR PART SAT In Section 2.2 we saw that the same parameters influence the number of derivations we can perform in MP: I(i)l influ ences the interpolant size and thus the proof length, and m influences the number of deductions resolutions we can perform. Thus, we ....

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S.A. Cook and D. G. Mitchell. Finding hard instances of the satisfiability problem: a survey. In Dimacs Series in Discrete Math. and Theoretical Comp. Sci., volume 35. AMS, 1997.

....not expect the proposed flow to give improvement on arbitrary SAT benchmarks. Many DIMACS benchmarks [12] have large numbers of symmetries, but can be solved so quickly that the symmetry detection overhead is not justified. On the other hand, many difficult SAT instances do not have symmetries [10]. Acknowledgements This work is funded by the DARPA MARCO Gigascale Silicon Research Center, an Agere Systems SRC Research fellowship and a DAC fellowship. 8. ....

S. A. Cook, D. G. Mitchell, "Finding Hard Instances of the Satisfiability Problem: A Survey", D I MACS Se r. D s c r. Ma h and Theo r. Comp Sc i , 1997.

....Equivalence Classes in CNF; 4) Experimental Design and SAT; 5) Reports of Experiments; 6) Summary and Conclusions. 2. BACKGROUND AND MOTIVATION Traditionally, the performance of SAT solvers has been evaluated experimentally either in terms of randomly generated instances of SAT problems, e.g. [24, 25], or structured instances, such as the instances from the DIMACS set [21] or the SATPLAN set [22] Merits of either approach are subject to on going critique and examination [15] 16] 26] 27] 28] The traditional way to report results of SAT solvers is the time to solve performance of ....

S. Cook and D. Mitchell. Finding hard instances of the satisfiability problem: A survey, 1997. Available at sat-survey3.ps.

....described in Table 1 but performed with GRASP [23] instead of CHAFF [6] our flow demonstrated speed ups 1.5 5 times even for the micro processor verification benchmarks. The proposed flow may not give improvement on arbitrary SAT benchmarks many difficult SAT instances do not have symmetries [8]. While many DIMACS benchmarks [10] have large numbers of symmetries, they are easy and can be solved faster than their symmetries can be detected. Acknowledgements. This work is funded by an Agere Systems SRC Research fellowship, a DAC fellowship and DARPA MARCO GSRC. 8. ....

S. A. Cook, D. G. Mitchell, "Finding Hard Instances of the Satisfiability Problem: A Survey", DIMACS Ser. Discr. Math. & Theor. Comp. Sci., `97.

....several optimizations to lpsat, including learning from constraint failure and randomized cuto#s. 1 Introduction Recent advances in satisfiability (SAT) solving technology have rendered large, previously intractable problems quickly solvable [ Crawford and Auton, 1993; Selman et al. 1996; Cook and Mitchell, 1997; Bayardo and Schrag, 1997; Li and Anbulagan, 1997; Gomes et al. 1998 ] SAT solving has become so successful that many other di#cult tasks are being compiled into propositional form to be solved as SAT problems. For example, SAT encoded solutions to graph coloring, planning, and circuit ....

....consistency of the runtimes on the harder problems; indeed, on log c five of the twenty Raw runs lasted longer than the longest Cuto# doubling run. 7 Related Work Limited space precludes a survey of propositional satisfiability algorithms and linear programming methods in this paper. See [ Cook and Mitchell, 1997 ] for a survey of satisfiability and [ Karlo#, 1991 ] for a survey of linear programming. Our work was inspired by the idea of compiling probabilistic planning problems to majsat [ Majercik and Littman, 1998 ] It seemed that if one could extend the SAT virtual machine to support ....

S. Cook and D. Mitchell. Finding hard instances of the satisfiability problem: A survey. Proceedings of the DIMACS Workshop on Satisfiability Problems, pages 11--13, 1997.

....optimizations to lpsat, including learning from constraint failure and randomized cuto#s. 1 Introduction Recent advances in satisfiability (SAT) solving technology have rendered large, previously intractable problems quickly solvable [ Crawford Auton, 1993; Selman, Kautz, Cohen, 1996; Cook Mitchell, 1997; Bayardo Schrag, 1997; Li Anbulagan, 1997; Gomes, Selman, Kautz, 1998 ] SAT solving has become so successful that many other di#cult tasks are being compiled into propositional form to be solved as SAT problems. For example, SAT encoded solutions to graph coloring, planning, and circuit ....

....consistency of the runtimes on the harder problems; indeed, on log c five of the twenty Raw runs lasted longer than the longest Cuto# doubling run. 7 Related Work Limited space precludes a survey of propositional satisfiability algorithms and linear programming methods in this paper. See [ Cook Mitchell, 1997 ] for a survey of satisfiability and [ Karlo#, 1991 ] for a survey of linear programming. Our work was inspired by the idea of compiling probabilistic planning problems to majsat [ Majercik Littman, 1998 ] It seemed that if one could extend the SAT virtual machine to support probabilistic ....

Cook, S., and Mitchell, D. 1997. Finding hard instances of the satisfiability problem: A survey. Proceedings of the DIMACS Workshop on Satisfiability Problems 11--13.

....optimizations to LPSAT, including learning from constraint failure and randomized cuto#s. 1 Introduction Recent advances in boolean satisfiability (SAT) solving technology have rendered large, previously intractable problems quickly solvable [ Crawford and Auton, 1993; Selman et al. 1996; Cook and Mitchell, 1997; Bayardo and Schrag, 1997; Li and Anbulagan, 1997; Gomes et al. 1998 ] SAT solving has become so successful that many other di#cult tasks are being compiled into propositional form to be solved as SAT problems. For example, SATencoded solutions to graph coloring, planning, and circuit ....

....the consistency of the runtimes on the harder problems; indeed, on m log c five of the twenty Raw runs lasted longer than the longest Cuto# doubling run. 9 Related Work Limited space precludes a survey of boolean satisfiability algorithms and linear programming methods in this paper. See [ Cook and Mitchell, 1997 ] for a survey of satisfiability and [ Karlo#, 1991 ] for a survey of linear programming. Our work was inspired by the idea of compiling probabilistic planning problems to majsat [ Majercik and Littman, 1998 ] It seemed that if one could extend the SAT virtual machine to support ....

S. Cook and D. Mitchell. Finding hard instances of the satisfiability problem: A survey. Proceedings of the DIMACS Workshop on Satisfiability Problems, pages 11--13, 1997.

....such as cha#, satire, and sato. We thank authors for the ready access and the exceptional ease of installation of these software packages. Traditionally, the performance of SAT algorithms has been evaluated experimentally either in terms of randomly generated instances of SAT problems, e.g. [28, 11], or structured instances, such as the instances from the DIMACS set [30] or the SATPLAN set [23] Merits of either approach are subject to on going critique and examination [10] 24] 26] in particular, and [21] 22] 25] in general. The papers [21] and [22] succinctly articulate the case for ....

S. Cook and D. Mitchell, Finding hard instances of the satisfiability problem: A survey, 1997. Available at http://dream.dai.ed.ac.uk/group/tw/sat/- sat-survey3.ps.

....problems are of moderate difficulty. This is due to the fact that the underconstrained problem are simply much larger after forward checking. Again note that this situation is different from that of random # cnf, where it is the overconstrained problems that are of moderate difficulty (Cook and Mitchell 1997). At the peak the balanced QWH problems are much harder than the filtered QCP problems, showing that we have achieved the goal of creating a better benchmark for testing incomplete solvers. Balancing can be added to the filtered QCP model; the resulting balanced QCP model are yet more difficult. ....

Cook, S.A. and Mitchell, D. (1997). Finding Hard Instances of the Satisfiability Problem: A Survey, in D. Du, J. Gu, and P. Pardalos, eds. The Satisfiability Problem. Vol. 35 of DIMACS Series in Discr. Math. and Theor. Comp. Sci., 1-17.

....the erratic behavior of the mean and the variance of the search cost. In fact, this phenomenon has led researchers studying the nature of computationally hard problems to use the median cost, instead of the mean, to characterize search di#culty, because the median is generally much more stable [9, 20, 27, 35, 46, 64]. More recently, the study of runtime distributions of search methods instead of just the moments and median has been shown to provide a better characterization of search methods and much useful information in the design of algorithms [18, 23, 21, 29, 36, 55] heavytails.tex; 4 10 1999; ....

Cook, S. and D. Mitchell: 1998, `Finding hard instances of the satisfiability problem: a survey'. In: Du, Gu, and Pardalos (eds.): Satisfiability Problem: Theory and Applications. Dimacs Series in Discrete Mathematics and Theoretical Computer Science, Vol. 35.

....of FORWARDMESSAGE PASSING (MP) on the computational e#ciency of resolutionbased inference, and identify some of the parameters of influence. Current measures for comparing automated deduction strategies are insu#cient for our purposes. Proof length (e.g. 110,52,112] also see the survey in [22]) is only marginally relevant. More relevant is comparing the sizes of search spaces induced by di#erent strategies (e.g. resolution of propositional Horn clauses [84] and contraction rules for FOL [12] These measures do not precisely address our needs, but we use them here, leaving the ....

....a given theory A into only two partitions (n = 2) sharing l propositional symbols, the algorithm will take time O(2 l # f SAT ( m 2 ) Assuming P #= NP , this is a significant improvement over a simple SAT procedure, for every l that is small enough (l #m 2 , and # # 0. 582 [92,22]) 5 Decomposing a Logical Theory The algorithms presented in previous sections assumed a given partitioning. In this section we address the critical problem of automatically decomposing a set of propositional or FOL clauses into a partitioned theory. Guided by the results of previous sections, we ....

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Stephen A. Cook and David G. Mitchell. Finding hard instances of the satisfiability problem: a survey. In Dimacs Series in Discrete Mathamatics and Theoretical Computer Science, volume 35, Providence, RI, USA, 1997. American Mathematical Society.

....that many instances of SAT, especially the random instances that were popular for experimental work a decade ago, are very easy. Indeed the model popular then is provably easy (on average) while random k SAT is (under certain conditions) provably hard for most popular algorithms [MSL92, Mit93, CM97] The main point of an experiment is to answer a question that we are unable (for whatever reason) to answer by deduction. Computational experiments typically consist of seeing what happens when a collection of programs is run on a collection of inputs. Although we may require experiment to ....

Stephen A. Cook and David G. Mitchell. Finding hard instances of the satisfiability problem: A survey, volume 35 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 1--17. American Mathematical Society, Association for Computing Machinery, 1997.

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Cook, S. A. and D. G. Mitchell (1997): Finding Hard Instances of the Satisfiability Problem: A Survey. The DIMACS Workshop on Satisfiability Problems, American Mathematical Society.

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Cook, S., Mitchell, D.G.: Finding hard instances of the satisfiability problem: A survey. In Du, Gu, Pardalos, eds.: Satisfiability Problem: Theory and Applications. Volume 35 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society (1997)

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S. A. Cook and D. G. Mitchell, "Finding hard instances of the satisfiability problem: A survey," in Satisfiability Problem: Theory and Applications, 1997, vol. 25, DIMACS Series in Discr. Math. and Theor. Comp. Sci, pp. 1--17.

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Cook, S., Mitchell, D.G.: Finding hard instances of the satisfiability problem: A survey. In Du, Gu, Pardalos, eds.: Satisfiability Problem: Theory and Applications. Volume 35 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society (1997)

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S. Cook and D. Mitchel, Finding hard instances of the satisfiability problem: A survey, in Satisfiability Problem: Theory and Applications, DIMACS series in Discrete Mathematics and Theoretical Computer Science 25, 1--17, American Mathematical Society, 1997.

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S.A. Cook and D.G. Mitchell, Finding Hard Instances of the Satisfiability Problem: A Survey, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 35, 1997, 1-17

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