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DPLL: The Core of Modern Satisfiability Solvers

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Martin Davis on Computability, Computational Logic, and Mathematical Foundations

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 10))

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Abstract

Propositional theorem provers have become a core technology in a range of areas, such as in hardware and software verification, AI planning , and mathematical discovery. The theorem provers rely on fast Boolean satisfiabilty (SAT) solving procedures, whose roots can be traced back to the work by Martin Davis and colleagues in the late 1950s. We review the history of this work with recent advances and applications.

This paper is dedicated to Martin Davis, in recognition of his foundational contributions to the area of automated reasoning.

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Notes

  1. 1.

    The papers introducing DP [12] and DPLL [11] each have over 3000 citations listed in Google Scholar (May 2015), which is indicative of their tremendous influence over the years.

  2. 2.

    The Dunham, Fridshal, and Sward project might have begun in 1957.

  3. 3.

    Almost all of the papers cited in the introduction can be found in [46].

  4. 4.

    This project was also executed at IBM.

  5. 5.

    \(\overline{c}\) denotes \(not \ c\) here.

  6. 6.

    A complementary literal has the same variable name but opposite negation status.

  7. 7.

    An AE formula is a formula with all quantifiers leftmost with no existential quantifier to the left of a universal quantifier.

  8. 8.

    A tautological clause contains \(p \vee \lnot p\) for some variable p.

  9. 9.

    U.S. News and World Report: Best Grad Schools 2014.

References

  1. Audemard, G., & Simon, L. (2009, July). Predicting learnt clauses quality in modern SAT solvers. In 21st IJCAI (pp. 399–404), Pasadena, CA.

    Google Scholar 

  2. Beame, P., Kautz, H., & Sabharwal, A. (2004). Understanding and harnessing the potential of clause learning. JAIR, 22, 319–351.

    Google Scholar 

  3.  Biere, A. (2012). Plingeling: Solver description. In SAT Challenge 2012.

    Google Scholar 

  4. Biere, A., Heule, M., van Maaren, H., & Walsh, T. (Eds.) (2009). Handbook of satisfiability. IOS Press.

    Google Scholar 

  5. Bloom, B., Grove, D., Herta, B., Sabharwal, A., Samulowitz, H., & Saraswat, V. A. (2012). SatX10: A scalable plug&play parallel sat framework—(tool presentation). In 15th SAT (pp. 463–468).

    Google Scholar 

  6. Böhm, M., & Speckenmeyer, E. (1996). A fast parallel SAT-solver—efficient workload balancing. Annals of Mathematics and Artificial Intelligence, 17(3–4), 381–400.

    Article  Google Scholar 

  7. Cook, S. A. (1971, May). The complexity of theorem proving procedures. In Conference Record of 3rd STOC (pp. 151–158), Shaker Heights, OH.

    Google Scholar 

  8. Davis, M. (1983). A computer program for Presburger’s algorithm. In J. Siekmann and G. Wrightson (Eds.) Automated reasoning: Classical papers on computational logic (pp. 41–48). Springer.

    Google Scholar 

  9.  Davis, M. (2014). Private communication.

    Google Scholar 

  10. Davis, M. (2015). My life as a logician. In this volume. Springer.

    Google Scholar 

  11. Davis, M., Logemann, G., & Loveland, D. (1962) A machine program for theorem-proving. Communication of ACM, 5, 394–397. ISSN: 0001-0782.

    Google Scholar 

  12. Davis, M., & Putnam, H. (1960). A computing procedure for quantification theory. Journal of the Association for Computing Machinery, 7, 201–215. ISSN: 0004-5411.

    Google Scholar 

  13. Dunham, B., Fridshal, R., & Sward, G. (1959). A non-heuristic program for proving elementary logical theorems. In IFIP Congress (pp. 282–284).

    Google Scholar 

  14. Dunham, B., Fridshal, R., & Sward, G. (1959). A non-heuristic program for proving elementary logical theorems (abstract). CACM, 2, 19–20.

    Google Scholar 

  15. Eén, N., & Sörensson, N. (2005, June). MiniSat: A SAT solver with conflict-clause minimization. In 8th SAT, St. Andrews, U.K.

    Google Scholar 

  16. Frost, D., Rish, I., & Vila, L. (1997). Summarizing CSP hardness with continuous probability distributions. In Proceedings of the Fourteenth National Conference on Artificial Intelligence (AAAI-97) (pp. 327–334). New Providence, RI: AAAI Press.

    Google Scholar 

  17.  Gelernter, H. (1960). Realization of a geometry theorem proving machine. In Information processing (pp. 273–282). UNESCO, Paris: R. Oldenbourg, Munich: Butterworths, London.

    Google Scholar 

  18. Gelernter, H., Hansen, J. R.,  & Loveland, D. W. (1960). Empirical explorations of the geometry theorem machine. In Papers presented at the May 3–5, 1960, Western Joint IRE-AIEE-ACM computer conference (pp. 143–149). ACM.

    Google Scholar 

  19.  Gelfond, M. (2008). Answer sets. In F. van Harmelen, V. Lifschitz and B. Porter (Eds.), Handbook of knowledge representation. Foundations of artificial intelligence, Chapter 7 (Vol. 3, pp. 285–316). Elsevier.

    Google Scholar 

  20.  Gilmore, P. (1958). Private communication.

    Google Scholar 

  21. Gilmore, P. C. (1959). A program for the production of proofs of theorems derivable within the first order predicate calculus. In CACM (Vol. 2, pp. 19–19). ACM.

    Google Scholar 

  22. Gilmore, P. C. (1960). A proof method for quantification theory: Its justification and realization. IBM Journal of Research and Development, 4, 28–35. ISSN: 0018-8646.

    Google Scholar 

  23. Goldberg, E., & Novikov, Y. (2007). BerkMin: A fast and robust SAT-solver. Discrete Applied Mathematics, 155(12), 1549–1561.

    Article  Google Scholar 

  24. Gomes, C., Kautz, H., Sabharwal, A., & Selman, B. (2008). Satisfiability solvers. In F. Van Harmelen, V. Lifschitz and B. Porter (Eds.), Handbook of knowledge representation (pp. 89–134). Elsevier.

    Google Scholar 

  25. Gomes, C., Selman, B., & Crato, N. (1997). Heavy-tailed distributions in combinatorial search. In 3rd CP (pp. 121–135).

    Google Scholar 

  26. Gomes, C. P., Selman, B., & Kautz, H. (1998, July). Boosting combinatorial search through randomization. In 15th AAAI (pp. 431–437), Madison, WI.

    Google Scholar 

  27. Hamadi, Y., Jabbour, S., & Sais, L. (2009). Manysat: A parallel SAT solver. Journal on Satisfiability, 6(4), 245–262.

    Google Scholar 

  28. Hamadi, Y.,  & Wintersteiger, C. M. (2012). Seven challenges in parallel SAT solving. In 26th AAAI.

    Google Scholar 

  29. Hogg, T., & Williams, C. (1994). Expected gains from parallelizing constraint solving for hard problems. In Proceedings of the Twelfth National Conference on Artificial Intelligence (AAAI-94) (pp. 1310–1315). Seattle, WA: AAAI Press.

    Google Scholar 

  30. Jeroslow, R. G., & Wang, J. (1990). Solving propositional satisfiability problems. Annals of Mathematics and Artificial Intelligence, 1(1-4), 167–187.

    Google Scholar 

  31. Katsirelos, G., Sabharwal, A., Samulowitz, H., & Simon, L. (2013). Resolution and parallelizability: Barriers to the efficient parallelization of SAT solvers. In 27th AAAI.

    Google Scholar 

  32. Kautz, H., & Selman, B. (1996). Pushing the envelope: Planning, propositional logic, and stochastic search. In Proceedings of AAAI-96, Portland, OR.

    Google Scholar 

  33. Levin, L. A. (1973). Universal sequential search problems. Problems of Information Transmission, 9(3), 265–266. Originally in Russian.

    Google Scholar 

  34. Marques-Silva, J. P. (1999, September). The impact of branching heuristics in propositional satisfiability algorithms. In 9th Portuguese Conference on AI. LNCS (Vol. 1695, pp. 62–74).

    Google Scholar 

  35. Marques-Silva, J. P.,  & Sakallah, K. A. (1996, November). GRASP—a new search algorithm for satisfiability. In ICCAD (pp. 220–227), San Jose, CA.

    Google Scholar 

  36. Mitchell, D., & Levesque, H. (1996). Some pitfalls for experimenters with random SAT. Artificial Intelligence, 81(1–2), 111–125.

    Article  Google Scholar 

  37. Moskewicz, M. W., Madigan, C. F., Zhao, Y., Zhang, L., & Malik, S. (2001, June). Chaff: Engineering an efficient SAT solver. In 38th DAC (pp. 530–535), Las Vegas, NV.

    Google Scholar 

  38.  Nadel, A. (2002). Backtrack search algorithms for propositional logic satisfiability: Review and innovations. Master’s thesis, Hebrew University of Jerusalem.

    Google Scholar 

  39. Newell, A., Shaw, J. C., & Simon, H. (1957). Empirical explorations with the logic theory machine: A case study in heuristics. Western Joint Comp. Conf., 1, 49–73.

    Google Scholar 

  40. Nieuwenhuis, R., Oliveras, A., & Tinelli, C. (2006). Solving SAT and SAT modulo theories: From an abstract Davis-Putnam-Logemann-Loveland procedure to DPLL(T). Journal of Association for Computing Machinery, 53(6), 937–977.

    Article  Google Scholar 

  41. Palù, A. D., Dovier, A., Formisano, A., & Pontelli, E. (2015). CUD@SAT: SAT solving on GPUs. Journal of Experimental and Theoretical Artificial Intelligence, 27(3), 293–316.

    Article  Google Scholar 

  42. Pipatsrisawat, K., & Darwiche, A. (2006). RSat 1.03: SAT solver description. Technical Report D-152, Automated Reasoning Group, Computer Science Department, UCLA.

    Google Scholar 

  43. Pipatsrisawat, K., & Darwiche, A. (2011). On the power of clause-learning SAT solvers as resolution engines. AI Journal, 175(2), 512–525.

    Google Scholar 

  44. Selman, B., Kautz, H., & Cohen, B. (1996). Local search strategies for satisfiability testing. In D. S. Johnson and M. A. Trick (Eds.), Cliques, coloring, and satisfiability: The second DIMACS implementation challenge. DIMACS Series in Discrete Mathematics and Theoretical Computer Science (Vol. 26, pp. 521–532). American Mathematical Society.

    Google Scholar 

  45.  Selman, B., Levesque, H. J., & Mitchell, D. G. (1992, July). A new method for solving hard satisfiability problems. In 10th AAAI (pp. 440–446), San Jose, CA.

    Google Scholar 

  46. Siekmann, J., & Wrightson, G. (Eds.) (1983). Automation of Reasoning1: Classical Papers on Computational Logic 1957–1966. Springer Publishing Company Incorporated.

    Google Scholar 

  47. Stallman, R. M., & Sussman, G. J. (1977). Forward reasoning and dependency-directed backtracking in a system for computer-aided circuit analysis. AI Journal, 9, 135–196.

    Google Scholar 

  48. Szeider, S. (2005). Backdoor sets for DLL subsolvers. Journal of Automated Reasoning, 35(1–3).

    Google Scholar 

  49.  Wang, H. (1960, April). Proving theorems by pattern recognition—I. Communications of the ACM, 3(4), 220–234. ISSN: 0001-0782.

    Google Scholar 

  50.  Wang, H. (1960). Toward mechanical mathematics. IBM Journal of Research and Development, 4, 2–22. ISSN: 0018-8646.

    Google Scholar 

  51.  Wang, H. (1961). Proving theorems by pattern recognition—II. Bell System Technical Journal, 40(1), 1–41. ISSN: 1538-7305.

    Google Scholar 

  52. Williams, R., Gomes, C., & Selman, B. (2003). Backdoors to typical case complexity. In 18th IJCAI.

    Google Scholar 

  53. Zhang, H. (1997, July). SATO: An efficient propositional prover. In 14th CADE. LNCS (Vol. 1249, pp. 272–275), Townsville, Australia.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Duke University, Durham, USA

    Donald Loveland

  2. Allen Institute for AI, Seattle, USA

    Ashish Sabharwal

  3. Cornell University, Ithaca, USA

    Bart Selman

Authors
  1. Donald Loveland
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  2. Ashish Sabharwal
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  3. Bart Selman
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Corresponding author

Correspondence to Donald Loveland .

Editor information

Editors and Affiliations

  1. University of Trieste, Trieste, Italy

    Eugenio G. Omodeo

  2. University of Udine, Udine, Italy

    Alberto Policriti

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Loveland, D., Sabharwal, A., Selman, B. (2016). DPLL: The Core of Modern Satisfiability Solvers. In: Omodeo, E., Policriti, A. (eds) Martin Davis on Computability, Computational Logic, and Mathematical Foundations. Outstanding Contributions to Logic, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-41842-1_12

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