Abstract
Quantifier reasoning in Satisfiability Modulo Theories (SMT) is a long-standing challenge. The practical method employed in modern SMT solvers is to instantiate quantified formulas based on heuristics, which is not refutationally complete even for pure first-order logic. We present several decidable fragments of first order logic modulo theories. We show how to construct models for satisfiable formulas in these fragments. For richer undecidable fragments, we discuss conditions under which our procedure is refutationally complete. We also describe useful heuristics based on model checking for prioritizing or avoiding instantiations.
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References
Halpern, J.Y.: Presburger Arithmetic with unary predicates is \(\Pi_1^1\) Complete. Journal of Symbolic Logic 56, 637–642 (1991)
Deharbe, D., Ranise, S.: Satisfiability solving for software verification. International Journal on Software Tools Technology Transfer (2008) (to appear)
de Moura, L.M., Bjørner, N.: Efficient E-Matching for SMT Solvers. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 167–182. Springer, Heidelberg (2007)
Stickel, M.E.: Automated deduction by theory resolution. Journal of Automated Reasoning 1, 333–355 (1985)
Ge, Y., Barrett, C., Tinelli, C.: Solving quantified verification conditions using satisfiability modulo theories. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 167–182. Springer, Heidelberg (2007)
Flanagan, C., Joshi, R., Saxe, J.B.: Theorem proving using lazy proof explication. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 355–367. Springer, Heidelberg (2003)
Dutertre, B., de Moura, L.: A fast linear-arithmetic solver for DPLL(T). In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 81–94. Springer, Heidelberg (2006)
Detlefs, D., Nelson, G., Saxe, J.B.: Simplify: a theorem prover for program checking. J. ACM 52 (2005)
Barnett, M., Chang, B.Y.E., DeLine, R., Jacobs, B., Leino, K.R.M.: Boogie: A modular reusable verifier for object-oriented programs. In: de Boer, F.S., Bonsangue, M.M., Graf, S., de Roever, W.-P. (eds.) FMCO 2005. LNCS, vol. 4111, pp. 364–387. Springer, Heidelberg (2006)
Flanagan, C., Leino, K.R.M., Lillibridge, M., Nelson, G., Saxe, J.B., Stata, R.: Extended static checking for java. In: PLDI 2002. ACM, New York (2002)
Abadi, A., Rabinovich, A., Sagiv, M.: Decidable fragments of many-sorted logic. In: Dershowitz, N., Voronkov, A. (eds.) LPAR 2007. LNCS, vol. 4790, pp. 17–31. Springer, Heidelberg (2007)
Bradley, A.R., Manna, Z., Sipma, H.B.: What’s decidable about arrays? In: Emerson, E.A., Namjoshi, K.S. (eds.) VMCAI 2006. LNCS, vol. 3855, pp. 427–442. Springer, Heidelberg (2005)
Aiken, A.: Set Constraints: Results, Applications, and Future Directions. In: Second Workshop on the Principles and Practice of Constraint Programming (1994)
Ge, Y., de Moura, L.: Complete instantiation for quantified SMT formulas. Technical report, Microsoft Research (2009)
Barrett, C., Tinelli, C.: CVC3. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 298–302. Springer, Heidelberg (2007)
de Moura, L., Bjørner, N.: Z3: An Efficient SMT Solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)
Claessen, K.: Equinox, a new theorem prover for full first-order logic with equality. Presentation at Dagstuhl Seminar 05431 on Deduction and Applications (2005)
Habermehl, P., Iosif, R., Vojnar, T.: What else is decidable about integer arrays? In: Amadio, R. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 474–489. Springer, Heidelberg (2008)
Fontaine, P., Gribomont, E.P.: Decidability of invariant validation for paramaterized systems. In: Garavel, H., Hatcliff, J. (eds.) TACAS 2003. LNCS, vol. 2619, pp. 97–112. Springer, Heidelberg (2003)
Arons, T., Pnueli, A., Ruah, S., Xu, J., Zuck, L.D.: Parameterized verification with automatically computed inductive assertions. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, p. 221. Springer, Heidelberg (2001)
Ihlemann, C., Jacobs, S., Sofronie-Stokkermans, V.: On local reasoning in verification. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 265–281. Springer, Heidelberg (2008)
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Ge, Y., de Moura, L. (2009). Complete Instantiation for Quantified Formulas in Satisfiabiliby Modulo Theories. In: Bouajjani, A., Maler, O. (eds) Computer Aided Verification. CAV 2009. Lecture Notes in Computer Science, vol 5643. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02658-4_25
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DOI: https://doi.org/10.1007/978-3-642-02658-4_25
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