Skip to main content
SpringerLink
Log in

Autarky Pruning in Propositional Model Elimination Reduces Failure Redundancy

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

Goal-sensitive resolution methods, such as Model Elimination, have been observed to have a higher degree of search redundancy than model-search methods. Therefore, resolution methods have not been seen in high-performance propositional satisfiability testers. A method to reduce search redundancy in goal-sensitive resolution methods is introduced. The idea at the heart of the method is to attempt to construct a refutation and a model simultaneously and incrementally, based on subsearch outcomes. The method exploits the concept of ‘autarky’, which can be informally described as a ‘self-sufficient’ model for some clauses, but which does not affect the remaining clauses of the formula. Incorporating this method into Model Elimination leads to an algorithm called Modoc. Modoc is shown, both analytically and experimentally, to be faster than Model Elimination by an exponential factor. Modoc, unlike Model Elimination, is able to find a model if it fails to find a refutation, essentially by combining autarkies. Unlike the pruning strategies of most refinements of resolution, autarky-related pruning does not prune any successful refutation; it only prunes attempts that ultimately will be unsuccessful; consequently, it will not force the underlying Modoc search to find an unnecessarily long refutation. To prove correctness and other properties, a game characterization of refutation search is introduced, which demonstrates some symmetries in the search for a refutation and the search for a model. Experimental data is presented on a variety of formula classes, comparing Modoc with Model Elimination and model-search algorithms. On random formulas, model-search methods are faster than Modoc, whereas Modoc is faster on structured formulas, including those derived from a circuit-testing application. Considerations for first-order refutation methods are discussed briefly.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson, R. and Bledsoe, W. W.: A linear format for resolution with merging and a new technique for establishing completeness, J. ACM 17(3) (1970), 525–534.

    Google Scholar 

  2. Astrachan, O. L. and Loveland, D. W.: The use of lemmas in the model elimination procedure, J. Automated Reasoning 19 (1997), 117–141.

    Google Scholar 

  3. Astrachan, O. L. and Stickel, M. E.: Caching and lemmaizing in model elimination theorem provers, in D. Kapur (ed.), Automated Deduction – CADE-11. Proceedings of 11th International Conference on Automated Deduction (Saratoga Springs, NY, USA, 15–18 June 1992), Springer-Verlag, Berlin, Germany, 1992, pp. 224–238.

    Google Scholar 

  4. Blair, C. E., Jeroslow, R. G. and Lowe, J. K.: Some results and experiments in programming techniques for propositional logic, Comput. Oper. Res. 13(5) (1986), 633–645.

    Google Scholar 

  5. Caferra, R.: A tableaux method for systematic simultaneous search for refutations and models using equational problems, J. Logic Comput. 3(1) (1993), 3–25.

    Google Scholar 

  6. Crawford, J. and Auton, L.: Experimental results on the cross-over point in satisfiability problems, in Proceedings of the Eleventh National Conference on Artificial Intelligence; AAAI-93 and IAAI-93 (Washington, DC, USA, 11–15 July 1993), AAAI Press, Menlo Park, CA, USA, 1993, pp. 21–27.

    Google Scholar 

  7. Dalal, M. and Etherington, D.: A hierarchy of tractable satisfiability problems, Inform. Process. Lett. 44 (1992), 173–180.

    Google Scholar 

  8. Davis, M., Logemann, G. and Loveland, D.: A machine program for theorem-proving, Comm. ACM 5 (1962), 394–397.

    Google Scholar 

  9. Davis, M. and Putnam, H.: A computing procedure for quantification theory, J. Assoc. Comput. Mach. 7 (1960), 201–215.

    Google Scholar 

  10. Dechter, R. and Rish, I.: Directional resolution: The Davis–Putnam procedure, revisited, in Proc. 4th Int'l Conf. on Principles of Knowledge Representation and Reasoning (KR'94), Morgan Kaufmann, San Francisco, 1994, pp. 134–145.

  11. Dubois, O., Andre, P., Boufkhad, Y. and Carlier, J.: SAT versus UNSAT, in D. S. Johnson and M. Trick (eds.), Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, DIMACS Series in Discrete Math. and Theoret. Comput. Sci., Amer. Math. Soc., 1995.

  12. Elkan, C.: Conspiracy numbers and caching for searching and/or trees and theorem-proving, in Eleventh Int'l Joint Conf. on Artificial Intelligence, Palo Alto, CA, Morgan Kaufmann, 1989, pp. 20–25.

  13. Fermuller, C. and Leitsch, A.: Model building by resolution, in E. Borger, G. Jager, H. Kleine Buning, S. Martini et al. (eds.), '92 (San Miniato, Italy, 28 Sept.–2 Oct. 1992), Springer-Verlag, Berlin, 1993, pp. 134–148.

    Google Scholar 

  14. Fleisig, S., Loveland, D. W., Smiley, A. K. and Yarmush, D. L.: An implementation of the model elimination proof procedure, JACM 21(1) (1974), 124–139.

    Google Scholar 

  15. Franco, J. and Paull, M.: Probabilistic analysis of the Davis–Putnam procedure for solving the satisfiability problem, Discrete Appl. Math. 5 (1983), 77–87.

    Google Scholar 

  16. Gent, I. P. and Walsh, T.: Towards an understanding of hill-climbing procedures for SAT, in Proceedings of the Eleventh National Conference on Artificial Intelligence; AAAI-93 and IAAI-93 (Washington, DC, USA, 11–15 July 1993), AAAI Press, Menlo Park, CA, USA, 1993, pp. 28–33.

    Google Scholar 

  17. Giunchiglia, F. and Sebastiani, R.: Building decision procedures for modal logics from propositional decision procedures – the case study of modal K, in 13th International Conference on Automated Deduction, Springer-Verlag, 1996, pp. 583–597.

  18. Giunchiglia, F. and Sebastiani, R.: A SAT-based decision procedure for ALC, in International Conference on Principles of Knowledge Representation and Reasoning, Cambridge, MA, USA, November 1996.

    Google Scholar 

  19. Harche, F., Hooker, J. N. and Thompson, G. L.: A computational study of satisfiability algorithms for propositional logic, ORSA J. Comput. 6(4) (1994), 423–435.

    Google Scholar 

  20. Jaumard, B., Stan, M. and Desrosiers, J.: Tabu search and a quadratic relaxation for the satisfiability problem, in D. S. Johnson and M. Trick (eds.), Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, DIMACS Series in Discrete Math. and Theoret. Comput. Sci., Amer. Math. Soc., 1995.

  21. Kowalski, R. and Kuehner, D.: Linear resolution with selection function, Artif. Intell. 2(2/3) (1971), 227–260.

    Google Scholar 

  22. Larrabee, T.: Test pattern generation using Boolean satisfiability, IEEE Trans. Comput.-Aided Design 11(1) (1992), 6–22.

    Google Scholar 

  23. Letz, R., Mayr, K. and Goller, C.: Controlled integration of the cut rule into connection tableau calculi, J. Automated Reasoning 13(3) (1994), 297–337.

    Google Scholar 

  24. Lewis, H. R.: Renaming a set of clauses as a Horn set, J. Assoc. Comput. Mach. 25(1) (1978), 134–135.

    Google Scholar 

  25. Loveland, D.W.:Mechanical theorem-proving by model elimination, J. Assoc. Comput. Mach. 15(2) (1968), 236–251.

    Google Scholar 

  26. Loveland, D. W.: A simplified format for the model elimination theorem-proving procedure, J. Assoc. Comput. Mach. 16(3) (1969), 349–363.

    Google Scholar 

  27. Loveland, D. W.: A unifying view of some linear Herbrand procedures, J. Assoc. Comput. Mach. 19(2) (1972), 366–384.

    Google Scholar 

  28. Loveland, D. W.: Automated Theorem Proving: A Logical Basis, North-Holland, Amsterdam, 1978.

    Google Scholar 

  29. Minker, J. and Zanon, G.: An extension to linear resolution with selection function, Inform. Process. Lett. 14(3) (1982), 191–194.

    Google Scholar 

  30. Mitchell, D., Selman, B. and Levesque, H.: Hard and easy distributions of SAT problems, in Proceedings of the Tenth National Conference on Artificial Intelligence (AAAI-92), San Jose, CA, 1992, pp. 459–465.

  31. Monien, B. and Speckenmeyer, E.: Solving satisfiability in less than 2in steps, Discrete Appl. Math. 10 (1985), 287–295.

    Google Scholar 

  32. Okushi, F.: Parallel cooperative propositional theorem proving, Ann. Math. Artificial Intelligence (1999), to appear.

  33. Plaisted, D. A.: Private communication, 1984.

  34. Plaisted, D. A.: The search efficiency of theorem proving strategies, in 12th International Conference on Automated Deduction, Springer-Verlag, 1994, pp. 57–71.

  35. Pretolani, D.: Efficiency and stability of hypergraph SAT algorithms, in D. S. Johnson and M. Trick (eds.), Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, DIMACS Series in Discrete Math. and Theoret. Comput. Sci., Amer. Math. Soc., 1995.

  36. Selman, B., Kautz, H. A. and Cohen, B.: Local search strategies for satisfiability testing, in D. S. Johnson and M. Trick (eds.), Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, DIMACS Series in Discrete Math. and Theoret. Comput. Sci., Amer. Math. Soc., 1995.

  37. Selman, B., Levesque, H. and Mitchell, D.: A new method for solving hard satisfiability problems, in Proceedings of the Tenth National Conference on Artificial Intelligence (AAAI-92), San Jose, CA, 1992, pp. 440–446.

  38. Shostak, R. E.: A Graph-Theoretic View of Resolution Theorem-Proving, Ph.D. thesis, Center for Research in Computing Technology, Harvard University, 1974. Also available from CSL, SRI International, Menlo Park, CA.

  39. Shostak, R. E.: Refutation graphs, Artif. Intell. 7 (1976), 51–64.

    Google Scholar 

  40. Slaney, J., Fujita, M. and Stickel, M.: Automated reasoning and exhaustive search: Quasigroup existence problems, Comput. Math. Appl. 29(2) (1995), 115–32.

    Google Scholar 

  41. Stickel, M. E.: Upside-down meta-interpretation of the model elimination theorem-proving procedure for deduction and abduction, J. Automated Reasoning 13(2) (1994), 189–210.

    Google Scholar 

  42. Urquhart, A.: Hard examples for resolution, J. Assoc. Comput. Mach. 34(1) (1987), 209–219.

    Google Scholar 

  43. Van Gelder, A. and Okushi, F.: A propositional theorem prover to solve planning and other problems, Ann. Math. Artificial Intelligence (1999), to appear.

  44. Van Gelder, A. and Okushi, F.: Lemma and cut strategies for propositional model elimination, Ann. Math. Artificial Intelligence (1999), to appear.

  45. Van Gelder, A. and Tsuji, Y. K.: Satisfiability testing with more reasoning and less guessing, in D. S. Johnson and M. Trick (eds.), Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Amer. Math. Soc., 1996.

  46. Zhang, H.: SATO: An efficient propositional prover, in 14th International Conference on Automated Deduction, 1997, pp. 272–275.

Download references

Author information

Authors and Affiliations

  1. University of California, Santa Cruz, U.S.A

    Allen Van Gelder

Authors
  1. Allen Van Gelder
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Van Gelder, A. Autarky Pruning in Propositional Model Elimination Reduces Failure Redundancy. Journal of Automated Reasoning 23, 137–193 (1999). https://doi.org/10.1023/A:1006143621319

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1006143621319

Search

Navigation