Abstract
First-order logic modulo the theory of integer arithmetic is the basis for reasoning in many areas, including deductive software verification and software model checking. While satisfiability checking for ground formulae in this logic is well understood, it is still an open question how the general case of quantified formulae can be handled in an efficient and systematic way. As a possible answer, we introduce a sequent calculus that combines ideas from free-variable constraint tableaux with the Omega quantifier elimination procedure. The calculus is complete for theorems of first-order logic (without functions, but with arbitrary uninterpreted predicates), can decide Presburger arithmetic, and is complete for a substantial fragment of the combination of both.
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References
Giese, M.: Incremental closure of free variable tableaux. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS, vol. 2083, pp. 545–560. Springer, Heidelberg (2001)
Pugh, W.: The Omega test: a fast and practical integer programming algorithm for dependence analysis. In: Proceedings, 1991 ACM/IEEE conference on Supercomputing, pp. 4–13. ACM, New York (1991)
Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In: Sprawozdanie z I Kongresu metematyków slowiańskich, Warszawa, Warsaw, Poland, vol. 1929, pp. 92–101, 395 (1930)
Rümmer, P.: Calculi for Program Incorrectness and Arithmetic. PhD thesis, Chalmers University of Technology (to appear, 2008)
Fitting, M.C.: First-Order Logic and Automated Theorem Proving, 2nd edn. Springer, New York (1996)
Dershowitz, N., Manna, Z.: Proving termination with multiset orderings. Commun. ACM 22, 465–476 (1979)
Baumgartner, P., Fuchs, A., Tinelli, C.: MELIA – model evolution with linear integer arithmetic constraints (to appear, 2008)
Rümmer, P.: A sequent calculus for integer arithmetic with counterexample generation. In: Beckert, B. (ed.) Proceedings, 4th International Verification Workshop. CEUR, vol. 259 (2007), http://ceur-ws.org/
Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986)
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, 937–977 (2006)
Dowek, G., Hardin, T., Kirchner, C.: Theorem proving modulo. Journal of Automated Reasoning 31, 33–72 (2003)
Platzer, A.: Differential dynamic logic for hybrid systems. Journal of Automated Reasoning 41, 143–189 (2008)
Stickel, M.E.: Automated deduction by theory resolution. Journal of Automated Reasoning 1, 333–355 (1985)
Bürckert, H.J.: A resolution principle for clauses with constraints. In: Stickel, M.E. (ed.) CADE 1990. LNCS, vol. 449, pp. 178–192. Springer, Heidelberg (1990)
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)
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Rümmer, P. (2008). A Constraint Sequent Calculus for First-Order Logic with Linear Integer Arithmetic. 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_20
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DOI: https://doi.org/10.1007/978-3-540-89439-1_20
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