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Outline

Title
Abstract
Figures
Introduction
Experimental Results and Comparisons
Results for NP-Complete Problems
Results for Chess Problems
Results for Exptime and Pspace Problems
Conclusions
References

Applying Modern SAT-solvers to Solving Hard Problems

Profile image of Artur NiewiadomskiArtur Niewiadomski

Fundamenta Informaticae

https://doi.org/10.3233/FI-2019-1788
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22 pages

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Abstract

We present nine SAT-solvers and compare their efficiency for several decision and combinatorial problems: three classical NP-complete problems of the graph theory, bounded Post correspondence problem (BPCP), extended string correction problem (ESCP), two popular chess problems, PSPACE-complete verification of UML systems, and the Towers of Hanoi (ToH) of exponential solutions. In addition to several known reductions to SAT for the problems of graph k-colouring, vertex k-cover, Hamiltonian path, and verification of UML systems, we also define new original reductions for the N-queens problem, the knight's tour problem, and ToH, SCP, and BPCP. Our extensive experimental results allow for drawing quite interesting conclusions on efficiency and applicability of SAT-solvers to different problems: they behave quite efficiently for NP-complete and harder problems but they are by far inferior to tailored algorithms for specific problems of lower complexity.

Figures (13)
Though it has been more than half a century since the DPLL algorithm was published, its core idea remains the basis of modern SAT-solvers. Of course, this by no means indicates that the progress made in the last fifty years has been insignificant. Most importantly, the satisfiability problem has matured, with its efficient solving having found specific applications in many production processes, such as the design and formal verification of integrated circuits (including Intel Core CPUs [19]), automated software ver- ification (used by Microsoft, among others [11]), or managing dependencies between optional software components (i.e. plug-ins for the Java Eclipse development environment [24]), and many more. This further bolsters the efforts towards improving the efficiency of SAT-solvers, as well as reinforces general interest in the topic, an example of which are International SAT Competitions, held mostly annually since 2002 and open to all interested developers. Crucially, the projects taking part in the competition are publicly available with full source code and are among the leading SAT-solvers today. Table 1. SAT-solvers selected for comparison Modern SAT-solvers can be classified in two main groups, depending on the algorithm used. The first approach constitutes an evolution of the original DPLL algorithm and is referred to as CDCL (Conflict- Driven Clause Learning). It augments the classic DPLL backtracking scheme with mechanisms such as non-chronological backjumping, clause learning, periodic restarts and various heuristics, both deter- ministic and non-deterministic. Currently, CDCL algorithms are represented in several state-of-the-art Driven Clause Learning). It augments the classic DPLL backtracking scheme with mechanisms such
Though it has been more than half a century since the DPLL algorithm was published, its core idea remains the basis of modern SAT-solvers. Of course, this by no means indicates that the progress made in the last fifty years has been insignificant. Most importantly, the satisfiability problem has matured, with its efficient solving having found specific applications in many production processes, such as the design and formal verification of integrated circuits (including Intel Core CPUs [19]), automated software ver- ification (used by Microsoft, among others [11]), or managing dependencies between optional software components (i.e. plug-ins for the Java Eclipse development environment [24]), and many more. This further bolsters the efforts towards improving the efficiency of SAT-solvers, as well as reinforces general interest in the topic, an example of which are International SAT Competitions, held mostly annually since 2002 and open to all interested developers. Crucially, the projects taking part in the competition are publicly available with full source code and are among the leading SAT-solvers today. Table 1. SAT-solvers selected for comparison Modern SAT-solvers can be classified in two main groups, depending on the algorithm used. The first approach constitutes an evolution of the original DPLL algorithm and is referred to as CDCL (Conflict- Driven Clause Learning). It augments the classic DPLL backtracking scheme with mechanisms such as non-chronological backjumping, clause learning, periodic restarts and various heuristics, both deter- ministic and non-deterministic. Currently, CDCL algorithms are represented in several state-of-the-art Driven Clause Learning). It augments the classic DPLL backtracking scheme with mechanisms such
Figure 1. An example instance of PCP, forn = 4 and © = {a,b}, W = (bab, bba, bab, b), V = (ab, a, aa, bb). The solution (4, 4,2, 4, 1) corresponds to the word | = U = bbbbabbab. solution length &, PCP is undecidable, as shown by Post in [34]. Thus, we deal with a bounded version of PCP (BPCP, for short), where k is bounded. BPCP is in NP [14] and was also studied in different contexts, for example, using DNA-based bio-computations [20]. An instance of PCP can be visualised using a list of tiles similar to domino bricks where the 7-th tile contains the word w, on the top, and the word v; at the bottom. Each tile has assigned an id equal to t he position of the tile in the input list and we assume that at least k copies of each tile is at our disposal. Thus, a solution is the sequence of tile’s zds, and a single element of the sequence is also called a solution step. An example instance and a solution is depicted in Fig. 1.
Figure 1. An example instance of PCP, forn = 4 and © = {a,b}, W = (bab, bba, bab, b), V = (ab, a, aa, bb). The solution (4, 4,2, 4, 1) corresponds to the word | = U = bbbbabbab. solution length &, PCP is undecidable, as shown by Post in [34]. Thus, we deal with a bounded version of PCP (BPCP, for short), where k is bounded. BPCP is in NP [14] and was also studied in different contexts, for example, using DNA-based bio-computations [20]. An instance of PCP can be visualised using a list of tiles similar to domino bricks where the 7-th tile contains the word w, on the top, and the word v; at the bottom. Each tile has assigned an id equal to t he position of the tile in the input list and we assume that at least k copies of each tile is at our disposal. Thus, a solution is the sequence of tile’s zds, and a single element of the sequence is also called a solution step. An example instance and a solution is depicted in Fig. 1.
Figure 2. An example of the problem with inputs: A = abcaab, B = baca, and the parameter k = 3. symbols switch their positions. Deletion is the removal of an individual instance of a symbol from a string, therefore shortening its length by 1. The deleted symbols are replaced with the special value ¢ denoting the empty character. Note that we have n > m as otherwise the problem is unsolvable without an insert operator. Consider an example presented in Fig. 2. A transformation from A = abcaab to
Figure 2. An example of the problem with inputs: A = abcaab, B = baca, and the parameter k = 3. symbols switch their positions. Deletion is the removal of an individual instance of a symbol from a string, therefore shortening its length by 1. The deleted symbols are replaced with the special value ¢ denoting the empty character. Note that we have n > m as otherwise the problem is unsolvable without an insert operator. Consider an example presented in Fig. 2. A transformation from A = abcaab to
In this section we discuss experimental results and compare the efficiency of the SAT-solvers applied. The test platform was a desktop PC with a quad-core Intel Core 15-3350 CPU operating at 3.1 Ghz, with- out HyperThreading support. The system had 16 GB of RAM and was running Windows 7 Professional. For time measurement, built-in statistics were used for every SAT-solver. Given the computational com- plexity of SAT, not all test instances were expected to be processed in a reasonable amount of time. It is of concern especially in the case of older, classic SAT-solvers, i.e., zChaff, as well as unsatisfiable instances of selected problems. The timeout was set at 60 minutes. Figure 3. Average results for the graph k-colouring problem, grouped by input size
In this section we discuss experimental results and compare the efficiency of the SAT-solvers applied. The test platform was a desktop PC with a quad-core Intel Core 15-3350 CPU operating at 3.1 Ghz, with- out HyperThreading support. The system had 16 GB of RAM and was running Windows 7 Professional. For time measurement, built-in statistics were used for every SAT-solver. Given the computational com- plexity of SAT, not all test instances were expected to be processed in a reasonable amount of time. It is of concern especially in the case of older, classic SAT-solvers, i.e., zChaff, as well as unsatisfiable instances of selected problems. The timeout was set at 60 minutes. Figure 3. Average results for the graph k-colouring problem, grouped by input size
Figure 4. Average results for the vertex k-cover problem, grouped by input size
Figure 4. Average results for the vertex k-cover problem, grouped by input size
Figure 5. Average results for the Hamiltonian path problem, grouped by input size
Figure 5. Average results for the Hamiltonian path problem, grouped by input size
Figure 6. Results for bounded Post correspondence problem, grouped by input size
Figure 6. Results for bounded Post correspondence problem, grouped by input size
Figure 7. Results for the string correction problem, grouped by input size
Figure 7. Results for the string correction problem, grouped by input size
Figure 9. Results for the knight’s tour problem, grouped by input size Figure 8. Results for the N-queens problem, grouped by input size
Figure 9. Results for the knight’s tour problem, grouped by input size Figure 8. Results for the N-queens problem, grouped by input size
Figure 11. Results for model checking of UML systems Figure 10. Results for the Towers of Hanoi problem Finally, in Fig. 11 we present the results of GRC benchmark for 20 tracks, scaled w.r.t. the length of the event queues (3, 5, 8, and 10), where the obtained formula is satisfiable (at depth 18) and the integers are encoded over 10 propositional variables. It is easy to observe that modern solvers, especially Glucose and Clasp, handle the benchmarks almost effortlessly, taking advantage of the system symmetry. For older SAT-solvers, running time does increase with the size of the input, but nowhere near as significantly as in previous examples. This is also evidenced by the fact even zChaff was able to handle all instances
Figure 11. Results for model checking of UML systems Figure 10. Results for the Towers of Hanoi problem Finally, in Fig. 11 we present the results of GRC benchmark for 20 tracks, scaled w.r.t. the length of the event queues (3, 5, 8, and 10), where the obtained formula is satisfiable (at depth 18) and the integers are encoded over 10 propositional variables. It is easy to observe that modern solvers, especially Glucose and Clasp, handle the benchmarks almost effortlessly, taking advantage of the system symmetry. For older SAT-solvers, running time does increase with the size of the input, but nowhere near as significantly as in previous examples. This is also evidenced by the fact even zChaff was able to handle all instances
Table 2. Total time consumed by the solvers for satisfiable and unsatisfiable benchmarks
Table 2. Total time consumed by the solvers for satisfiable and unsatisfiable benchmarks

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References (42)

  1. P. A. Abdulla, P. Bjesse, and N. Eén. Symbolic reachability analysis based on SAT-solvers. In Proc. of the 6th Int. Conf. on Tools and Algorithms for the Construction and Analysis of Systems (TACAS'00), volume 1785 of LNCS, pages 411-425. Springer-Verlag, 2000.
  2. A. Armando and L. Compagna. An optimized intruder model for SAT-based model-checking of security protocols. In Electronic Notes in Theoretical Computer Science, volume 125, pages 91-108. Elsevier Science Publishers, March 2005.
  3. G. Audemard and L. Simon. Glucose and syrup in the SAT race 2015. SAT Race, 2015.
  4. J. Bell and B. Stevens. A survey of known results and research areas for n-queens. Discrete Mathematics, 309(1):1 -31, 2009.
  5. A. Biere. Lingeling and friends entering the SAT challenge 2012. Proceedings of SAT Challenge, pages 33-34, 2012.
  6. A. Biere et al. Symbolic model checking using SAT procedures instead of BDDs. In In Proc. of the ACM/IEEE Design Automation Conference (DAC), pages 317-320, 1999.
  7. E. Clarke, A. Biere, R. Raimi, and Y. Zhu. Bounded model checking using satisfiability solving. Formal Methods in System Design, 19(1):7-34, 2001.
  8. F. J. Damerau. A technique for computer detection and correction of spelling errors. Commun. ACM, 7(3):171--176, 1964.
  9. M. Davis, G. Logemann, and D. Loveland. A machine program for theorem-proving. Commun. ACM, 5(7):394-397, July 1962.
  10. M. Davis and H. Putnam. A computing procedure for quantification theory. J. ACM, 7(3):201-215, July 1960.
  11. L. de Moura and N. Bjørner. Bugs, Moles and Skeletons: Symbolic Reasoning for Software Development, pages 400-411. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010.
  12. L. M. de Moura and N. Bjørner. Z3: An efficient SMT solver. In Proc. of TACAS'08, volume 4963 of LNCS, pages 337-340. Springer-Verlag, 2008.
  13. G. De Smet et al. Optaplanner user guide. http://www. optaplanner. org., 2017.
  14. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP- Completeness. W. H. Freeman, 1979.
  15. M. Gebser et al. clasp : A conflict-driven answer set solver. In LPNMR, 2007.
  16. I. P. Gent, Ch. Jefferson, and P. Nightingale. Complexity of n-queens completion. J. Artif. Int. Res., 59(1):815-848, May 2017.
  17. Y. Hamadi, S. Jabbour, and L. Sais. ManySAT: a Parallel SAT Solver. JSAT, 6(4):245-262, 2009.
  18. M. Kacprzak and W. Penczek. Unbounded model checking for alternating-time temporal logic. In 3rd International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS 2004), 19-23 August 2004, New York, NY, USA, pages 646-653, 2004.
  19. R. Kaivola et al. Replacing Testing with Formal Verification in Intel Core I7 Processor Execution Engine Validation. In Proceedings of the 21st International Conference on Computer Aided Verification, CAV '09, pages 414-429, Berlin, Heidelberg, 2009. Springer-Verlag.
  20. Lila Kari, Greg Gloor, and Sheng Yu. Using DNA to solve the bounded post correspondence problem. Theor. Comput. Sci., 231(2):193-203, 2000.
  21. R. Karp. Reducibility among combinatorial problems. In Complexity of Computer Computations, pages 85-103. Plenum Press, 1972.
  22. H. Kautz and B. Selman. Planning as satisfiability. In In ECAI 92: Proceedings of the 10th European conference on Artificial intelligence, pages 359-363, 1992.
  23. M. Knapik and W. Penczek. Bounded model checking for parametric timed automata. Trans. Petri Nets and Other Models of Concurrency, 5:141-159, 2012.
  24. D. Le Berre and P. Rapicault. Dependency management for the eclipse ecosystem: Eclipse p2, metadata and resolution. In Proceedings of the 1st International Workshop on Open Component Ecosystems, IWOCE '09, pages 21-30, New York, NY, USA, 2009. ACM.
  25. V. I. Levenshtein. Binary codes capable of correcting deletions, insertions, and reversals. Soviet Physics Doklady, 10(8):707--710, 1966.
  26. Shun-Shii Lin and Chung-Liang Wei. Optimal algorithms for constructing knight's tours on arbitrary n x m chessboards. Discrete Applied Mathematics, 146(3):219 -232, 2005.
  27. Y. Mahajan, Z. Fu, and S. Malik. Zchaff2004: An efficient sat solver. In Int. Conf. on Theory and Applications of Satisfiability Testing, pages 360-375. Springer, 2004.
  28. R. Martins and I. Lynce. Effective cnf encodings for the towers of hanoi. In International Conference on Logic for Programming Artificial Intelligence and Reasoning, 2008.
  29. A. Męski, W. Penczek, M. Szreter, B. Woźna-Szcześniak, and A. Zbrzezny. BDD-versus SAT-based bounded model checking for the existential fragment of linear temporal logic with knowledge: algorithms and their performance. Autonomous Agents and Multi-Agent Systems, 28(4):558-604, 2014.
  30. A. Niewiadomski. Automatyczna weryfikacja systemow specyfikowanych w UML (in Polish). PhD thesis, Polish Academy of Science, ICS, January 2011.
  31. A. Niewiadomski et al. Towards automatic composition of web services: SAT-based concretisation of abstract scenarios. Fundam. Inform., 120(2):181-203, 2012.
  32. A. Niewiadomski, W. Penczek, and M. Szreter. A new approach to model checking of UML state machines. Fundamenta Informaticae, 93(1-3):289-303, 2009.
  33. I. Parberry. An efficient algorithm for the knight's tour problem. Discrete Appl. Math., 73(3):251-260, 1997.
  34. Emil L Post. A variant of a recursively unsolvable problem. Bulletin of the American Mathematical Society, 52(4):264-268, 1946.
  35. B. Selman, H. Kautz, and B. Cohen. Local search strategies for satisfiability testing. In Dimacs Series in Discrete Mathematics and Theoretical Computer Science, pages 521-532, 1995.
  36. B. Selman, H. Levesque, and D. Mitchell. A new method for solving hard satisfiability problems. In Pro- ceedings of the Tenth National Conference on Artificial Intelligence, AAAI'92, pages 440-446. AAAI Press, 1992.
  37. N. Sorensson and N. Een. Minisat v1. 13 -a SAT solver with conflict-clause minimization. SAT, 2005(53):1- 2, 2005.
  38. H. Stamm-Wilbrandt. Programming in propositional logic or reductions: Back to the roots (satisfiability), 1993.
  39. OMG UML. Unified modeling language. Infrastructure Specification,, 2(1), 2007.
  40. R. A. Wagner and M. J. Fischer. The string-to-string correction problem. Journal of the ACM, 21(1):168- -173, 1974.
  41. B. Woźna, A. Zbrzezny, and W. Penczek. Checking reachability properties for Timed Automata via SAT. Fundamenta Informaticae, 55(2):223-241, 2003.
  42. B. Woźna-Szcześniak. SAT-based bounded model checking for weighted deontic interpreted systems. Fun- dam. Inform., 143(1-2):173-205, 2016.

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