Abstract
The hierarchic combination of linear arithmetic and first-order logic with free function symbols, FOL(LA), results in a strictly more expressive logic than its two parts. The SUP(LA) calculus can be turned into a decision procedure for interesting fragments of FOL(LA). For example, reachability problems for timed automata can be decided by SUP(LA) using an appropriate translation into FOL(LA). In this paper, we extend the SUP(LA) calculus with an additional inference rule, automatically generating inductive invariants from partial SUP(LA) derivations. The rule enables decidability of more expressive fragments, including reachability for timed automata with unbounded integer variables. We have implemented the rule in the SPASS(LA) theorem prover with promising results, showing that it can considerably speed up proof search and enable termination of saturation for practically relevant problems.
This work has been partly supported by the German Transregional Collaborative Research Center SFB/TR 14 AVACS.
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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, vol. 5749, pp. 84–99. Springer, Heidelberg (2009)
Alur, R., Dill, D.L.: A theory of timed automata. Theor. Comput. Sci. 126(2), 183–235 (1994)
Bachmair, L., Ganzinger, H., Waldmann, U.: Superposition with Simplification as a Decision Procedure for the Monadic Class with Equality. In: Mundici, D., Gottlob, G., Leitsch, A. (eds.) KGC 1993. LNCS, vol. 713, pp. 83–96. Springer, Heidelberg (1993)
Bachmair, L., Ganzinger, H., Waldmann, U.: Refutational theorem proving for hierarchic first-order theories. Applicable Algebra in Engineering, Communication and Computing, AAECC 5(3/4), 193–212 (1994)
Boigelot, B., Wolper, P.: Symbolic Verification with Periodic Sets. In: Dill, D.L. (ed.) CAV 1994. LNCS, vol. 818, pp. 55–67. Springer, Heidelberg (1994)
Bozga, M., Iosif, R., Konečný, F.: Fast Acceleration of Ultimately Periodic Relations. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 227–242. Springer, Heidelberg (2010)
Bozga, M., Iosif, R., Lakhnech, Y.: Flat parametric counter automata. Fundam. Inform. 91(2), 275–303 (2009)
Comon, H., Jurski, Y.: Multiple Counters Automata, Safety Analysis and Presburger Arithmetic. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, pp. 268–279. Springer, Heidelberg (1998)
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)
Fermüller, C.G., Leitsch, A., Hustadt, U., Tamet, T.: Resolution decision procedures. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. II, ch.25, pp. 1791–1849. Elsevier (2001)
Fietzke, A., Hermanns, H., Weidenbach, C.: Superposition-Based Analysis of First-Order Probabilistic Timed Automata. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17. LNCS, vol. 6397, pp. 302–316. Springer, Heidelberg (2010)
Fietzke, A., Kruglov, E., Weidenbach, C.: Automatic generation of inductive invariants by SUP(LA). Technical Report MPI-I-2012-RG1-002, Max-Planck-Institut für Informatik (2012)
Fietzke, A., Weidenbach, C.: Superposition as a decision procedure for timed automata. In: MACIS, pp. 52–62 (2011)
Finkel, A., Leroux, J.: How to Compose Presburger-Accelerations: Applications to Broadcast Protocols. In: Agrawal, M., Seth, A.K. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 145–156. Springer, Heidelberg (2002)
Halpern, J.Y.: Presburger arithmetic with unary predicates is \(\Pi_{1}^{1}\) complete. Journal of Symbolic Logic 56(2), 637–642 (1991)
Hendriks, M., Larsen, K.G.: Exact acceleration of real-time model checking. Electr. Notes Theor. Comput. Sci. 65(6) (2002)
Jacquemard, F., Rusinowitch, M., Vigneron, L.: Tree Automata with Equality Constraints Modulo Equational Theories. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 557–571. Springer, Heidelberg (2006)
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.: SUP(T) decides the first-order logic fragment over ground theories. In: MACIS, pp. 126–148 (2011)
Peltier, N.: A General Method for Using Schematizations in Automated Deduction. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 578–592. Springer, Heidelberg (2001)
Wolper, P., Boigelot, B.: Verifying Systems with Infinite but Regular State. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, pp. 88–97. Springer, Heidelberg (1998)
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Fietzke, A., Kruglov, E., Weidenbach, C. (2012). Automatic Generation of Invariants for Circular Derivations in SUP(LA). In: Bjørner, N., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2012. Lecture Notes in Computer Science, vol 7180. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28717-6_17
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