Abstract
Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are “reasonably complete” even in the presence of free function symbols ranging into a background theory sort. In this paper we consider the case when all variables occurring below such function symbols are quantified over a finite subset of their domains. We present a non-naive decision procedure for background theories extended this way on top of black-box decision procedures for the EA-fragment of the background theory. In its core, it employs a model-guided instantiation strategy for obtaining pure background formulas that are equi-satisfiable with the original formula. Unlike traditional finite model finders, it avoids exhaustive instantiation and, hence, is expected to scale better with the size of the domains. Our main results in this paper are a correctness proof and first experimental results.
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References
Althaus, E., Kruglov, E., Weidenbach, C.: Superposition modulo linear arithmetic SUP(LA). In: Ghilardi, S., Sebastiani, R. (eds.) FroCoS 2009. LNCS (LNAI), vol. 5749, pp. 84–99. Springer, Heidelberg (2009)
Bachmair, L., Ganzinger, H.: Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation 4(3), 217–247 (1994)
Bachmair, L., Ganzinger, H., Waldmann, U.: Refutational theorem proving for hierarchic first-order theories. Appl. Algebra Eng. Commun. Comput 5, 193–212 (1994)
Baumgartner, P., Fuchs, A., de Nivelle, H., Tinelli, C.: Computing finite models by reduction to function-free clause logic. Journal of Applied Logic 7(1), 58–74 (2009)
Baumgartner, P., Tinelli, C.: Model evolution with equality modulo built-in theories. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 85–100. Springer, Heidelberg (2011)
Baumgartner, P., Waldmann, U.: Hierarchic superposition: Completeness without compactness. In: Kosta, M., Sturm, T. (eds.) MACIS (2013)
Baumgartner, P., Waldmann, U.: Hierarchic superposition with weak abstraction. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 39–57. Springer, Heidelberg (2013)
Claessen, K., Sörensson, N.: New techniques that improve MACE-style finite model building. In: Baumgartner, P., Fermüller, C.G. (eds.) CADE-19 Workshop: Model Computation – Principles, Algorithms, Applications (2003)
Ganzinger, H., Korovin, K.: Theory instantiation. In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS (LNAI), vol. 4246, pp. 497–511. Springer, Heidelberg (2006)
Ge, Y., Barrett, C.W., 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)
Ge, Y., de Moura, L.: Complete instantiation for quantified formulas in satisfiabiliby modulo theories. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 306–320. Springer, Heidelberg (2009)
Halpern, J.: Presburger Arithmetic With Unary Predicates is \(\Pi_1^1\)-Complete. Journal of Symbolic Logic 56(2), 637–642 (1991)
Korovin, K., Voronkov, A.: Integrating linear arithmetic into superposition calculus. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 223–237. Springer, Heidelberg (2007)
Kruglov, E., Weidenbach, C.: Superposition decides the first-order logic fragment over ground theories. In: Mathematics in Computer Science, pp. 1–30 (2012)
McCune, W.: Mace4 reference manual and guide. Technical Report ANL/MCS-TM-264, Argonne National Laboratory (2003)
de Moura, L., Bjørner, N.S.: Efficient E-matching for SMT solvers. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 183–198. Springer, Heidelberg (2007)
de Moura, L., Bjørner, N.S.: Z3: An efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)
Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving SAT and SAT modulo theories: from an abstract Davis-Putnam-Logemann-Loveland Procedure to DPLL(T). Journal of the ACM 53(6), 937–977 (2006)
Reynolds, A., Tinelli, C., Goel, A., Krstić, S.: Finite model finding in SMT. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 640–655. Springer, Heidelberg (2013)
Reynolds, A., Tinelli, C., Goel, A., Krstić, S., Deters, M., Barrett, C.: Quantifier instantiation techniques for finite model finding in SMT. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 377–391. Springer, Heidelberg (2013)
Rümmer, P.: A constraint sequent calculus for first-order logic with linear integer arithmetic. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 274–289. Springer, Heidelberg (2008)
Slaney, J.: Finder (finite domain enumerator): Notes and guide. Technical Report TR-ARP-1/92, Australian National University, Automated Reasoning Project, Canberra (1992)
Zhang, J., Zhang, H.: SEM: a system for enumerating models. In: Mellish, C. (ed.) IJCAI 1995. Morgan Kaufmann (1995)
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Baumgartner, P., Bax, J., Waldmann, U. (2014). Finite Quantification in Hierarchic Theorem Proving. In: Demri, S., Kapur, D., Weidenbach, C. (eds) Automated Reasoning. IJCAR 2014. Lecture Notes in Computer Science(), vol 8562. Springer, Cham. https://doi.org/10.1007/978-3-319-08587-6_11
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DOI: https://doi.org/10.1007/978-3-319-08587-6_11
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