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Reducing Higher-Order Theorem Proving to a Sequence of SAT Problems

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Abstract

We describe a complete theorem proving procedure for higher-order logic that uses SAT-solving to do much of the heavy lifting. The theoretical basis for the procedure is a complete, cut-free, ground refutation calculus that incorporates a restriction on instantiations. The refined nature of the calculus makes it conceivable that one can search in the ground calculus itself, obtaining a complete procedure without resorting to meta-variables and a higher-order lifting lemma. Once one commits to searching in a ground calculus, a natural next step is to consider ground formulas as propositional literals and the rules of the calculus as propositional clauses relating the literals. With this view in mind, we describe a theorem proving procedure that primarily generates relevant formulas along with their corresponding propositional clauses. The procedure terminates when the set of propositional clauses is unsatisfiable. We prove soundness and completeness of the procedure. The procedure has been implemented in a new higher-order theorem prover, Satallax, which makes use of the SAT-solver MiniSat. We also describe the implementation and give several examples. Finally, we include experimental results of Satallax on the higher-order part of the TPTP library.

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References

  1. Andrews, P.B., Bishop, M., Brown, C.E.: System description: TPS: a theorem proving system for type theory. In: McAllester, D. (ed.) Proceedings of the 17th International Conference on Automated Deduction, Lecture Notes in Artificial Intelligence, vol. 1831, pp. 164–169. Springer-Verlag, Pittsburgh, PA (2000)

    Google Scholar 

  2. Andrews, P.B., Brown, C.E.: TPS: a hybrid automatic-interactive system for developing proofs. Journal of Applied Logic 4(4), 367–395 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Backes, J., Brown, C.E.: Analytic tableaux for higher-order logic with choice. In: Jürgen Giesl, R.H. (ed.) Proceedings of the 5th International Joint Conference Automated Reasoning, IJCAR 2010, LNCS/LNAI, vol. 6173, pp. 76–90. Springer, Edinburgh, 16–19 July 2010

    Google Scholar 

  4. Benzmüller, C., Otten, J., Raths, T.: Implementing and evaluating provers for first-order modal logics. In: Raedt, L.D., Bessière, C., Dubois, D., Doherty, P., Frasconi, P., Heintz, F., Lucas, P.J.F. (eds.) Frontiers in Artificial Intelligence and Applications, ECAI, vol. 242, pp. 163–168. IOS Press (2012)

  5. Benzmüller, C., Paulson, L., Theiss, F., Fietzke, A.: LEO-II—a cooperative automatic theorem prover for classical higher-order logic. In: 4th International Joint Conference on Automated Reasoning (IJCAR’08), LNCS (LNAI), vol. 5195, pp. 162–170. Springer (2008)

  6. Blanchette, J.C.: Automatic Proofs and Refutations for Higher-Order Logic. Ph.D. thesis, Department of Informatics, T.U. München (2012)

  7. Bledsoe, W.W., Feng, G.: Set-Var. J. Autom. Reason. 11, 293–314 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brown, C.E.: Solving for set variables in higher-order theorem proving. In: Voronkov, A. (ed.) Proceedings of the 18th International Conference on Automated Deduction, Lecture Notes in Artificial Intelligence, vol. 2392, pp. 408–422. Springer-Verlag, Copenhagen (2002)

    Google Scholar 

  9. Brown, C.E.: Reducing higher-order theorem proving to a sequence of SAT problems. In: Bjørner, N., Sofronie-Stockkermans, V. (eds.) CADE—the 23rd International Conference on Automated Deduction, LNCS/LNAI, vol. 6803, pp. 147–161. Springer (2011)

  10. Brown, C.E., Smolka, G.: Analytic tableaux for simple type theory and its first-order fragment. LMCS 6(2), 1-33 (2010)

    MathSciNet  Google Scholar 

  11. De Moura, L., Bjørner, N.: Satisfiability modulo theories: introduction and applications. Commun. ACM 54(9), 69–77 (2011). doi:10.1145/1995376.1995394

    Article  Google Scholar 

  12. Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) Theory and Applications of Satisfiability Testing, Lecture Notes in Computer Science, vol. 2919, pp. 333–336. Springer, Berlin/Heidelberg (2004)

    Chapter  Google Scholar 

  13. Korovin, K.: iProver—an instantiation-based theorem prover for first-order logic (system description). In: Armando, A., Baumgartner, P., Dowek G. (eds.) Proceedings of the 4th International Joint Conference on Automated Reasoning, (IJCAR 2008), Lecture Notes in Computer Science, vol. 5195, pp. 292–298. Springer (2008)

  14. Korovin, K., Sticksel, C.: iprover-eq: an instantiation-based theorem prover with equality. In: Jürgen Giesl, R.H. (ed.) Proceedings of the 5th International Joint Conference Automated Reasoning, IJCAR 2010, LNCS/LNAI, vol. 6173, pp. 196–201. Springer, Edinburgh, 16–19 July 2010

    Google Scholar 

  15. Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: a proof assistant for higher-order logic. In: LNCS, vol. 2283. Springer (2002)

  16. Paulson, L.C., Blanchette, J.C.: Three years of experience with Sledgehammer, a practical link between automatic and interactive theorem provers. In: Sutcliffe, G., Ternovska, E., Schulz, S. (eds.) IWIL-2010 (2010)

  17. Pfenning, F., Schürmann, C.: Algorithms for equality and unification in the presence of notational definitions. In: Altenkirch, T., Naraschewski, W., Reus, B. (eds.) TYPES 1998, Lecture Notes in Computer Science, vol. 1657, pp. 179–193. Springer (1999)

  18. Smullyan, R.M.: A unifying principle in quantification theory. Proc. Natl. Acad. Sci. U.S.A. 49, 828–832 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sutcliffe, G.: The TPTP problem library and associated infrastructure: the FOF and CNF Parts, v3.5.0. J. Autom. Reasoning 43(4), 337–362 (2009)

    Article  MATH  Google Scholar 

  20. Sutcliffe, G.: The 5th IJCAR Automated Theorem Proving System Competition—CASC-J5. AI Commun. 24(1), 75–89 (2011)

    MathSciNet  Google Scholar 

  21. Sutcliffe, G.: The CADE-23 automated theorem proving system competition—CASC-23. AI Commun. 25(1), 49–63 (2012)

    Google Scholar 

  22. Sutcliffe, G., Benzmüller, C.: Automated reasoning in higher-order logic using the TPTP THF infrastructure. J. Form. Reasoning 3(1), 1–27 (2010)

    MATH  Google Scholar 

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  1. Programming Systems Lab, Universität des Saarlandes, Campus E1 3, 66123, Saarbrücken, Germany

    Chad E. Brown

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  1. Chad E. Brown
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Brown, C.E. Reducing Higher-Order Theorem Proving to a Sequence of SAT Problems. J Autom Reasoning 51, 57–77 (2013). https://doi.org/10.1007/s10817-013-9283-8

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