Abstract
Many applications of automated deduction require reasoning modulo some form of integer arithmetic. Unfortunately, theory reasoning support for the integers in current theorem provers is sometimes too weak for practical purposes. In this paper we propose a novel calculus for a large fragment of first-order logic modulo Linear Integer Arithmetic (LIA) that overcomes several limitations of existing theory reasoning approaches. The new calculus — based on the Model Evolution calculus, a first-order logic version of the propositional DPLL procedure — supports restricted quantifiers, requires only a decision procedure for LIA-validity instead of a complete LIA-unification procedure, and is amenable to strong redundancy criteria. We present a basic version of the calculus and prove it sound and (refutationally) complete.
The work of the last two authors was partially supported by the National Science Foundation grant number 0237422.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bachmair, L., Ganzinger, H., Waldmann, U.: Refutational Theorem Proving for Hierachic First-Order Theories. Appl. Algebra Eng. Commun. Comput. 5, 193–212 (1994)
Baumgartner, P.: Theory Reasoning in Connection Calculi. LNCS (LNAI), vol. 1527. Springer, Heidelberg (1998)
Baumgartner, P.: Logical Engineering with Instance-Based Methods. In: Pfenning, F. (ed.) CADE 2007. LNCS, vol. 4603, pp. 404–409. Springer, Heidelberg (2007)
Baumgartner, P., Fuchs, A., de Nivelle, H., Tinelli, C.: Computing Finite Models by Reduction to Function-Free Clause Logic. Journal of Applied Logic (in Press, 2007)
Baumgartner, P., Fuchs, A., Tinelli, C.: Implementing the Model Evolution Calculus. International Journal of Artificial Intelligence Tools 15(1), 21–52 (2006)
Baumgartner, P., Fuchs, A., Tinelli, C.: – Model Evolution With Linear Integer Arithmetic Constraints. Technical Report, Department of Computer Science, The University of Iowa (2008), http://www.cs.uiowa.edu/~tinelli
Baumgartner, P., Tinelli, C.: The Model Evolution Calculus. In: Baader, F. (ed.) CADE 2003. LNCS (LNAI), vol. 2741, pp. 350–364. Springer, Heidelberg (2003)
Bürckert, H.J.: A Resolution Principle for Clauses with Constraints. In: Stickel, M.E. (ed.) CADE 1990. LNCS (LNAI), vol. 449, pp. 178–192. Springer, Heidelberg (1990)
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., Tinelli, C.: Solving Quantified Verification Conditions Using Satisfiability Modulo Theories. In: Pfenning, F. (ed.) CADE 2007. LNCS, vol. 4603. Springer, Heidelberg (2007)
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)
Antonio, J., Peréz, N.: Encoding and Solving Problems in Effectively Propositional Logic. PhD thesis, The University of Manchester (2007)
Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving SAT and SAT Modulo Theories: from an Abstract Davis-Putnam-Logemann-Loveland Procedure to DPLL(T). J. of the ACM 53(6), 937–977 (2006)
Nieuwenhuis, R., Rubio, A.: Paramodulation-Based Theorem Proving. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning. Elsevier/MIT press (2001)
Stickel, M.E.: Automated Deduction by Theory Resolution. J. of Aut. R. 1, 333–355 (1985)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Baumgartner, P., Fuchs, A., Tinelli, C. (2008). (LIA) - Model Evolution with Linear Integer Arithmetic Constraints. In: Cervesato, I., Veith, H., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2008. Lecture Notes in Computer Science(), vol 5330. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89439-1_19
Download citation
DOI: https://doi.org/10.1007/978-3-540-89439-1_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-89438-4
Online ISBN: 978-3-540-89439-1
eBook Packages: Computer ScienceComputer Science (R0)