Automatic Generation of Invariants for Circular Derivations in SUP(LA)

  • Arnaud Fietzke
  • Evgeny Kruglov
  • Christoph Weidenbach
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7180)

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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)CrossRefGoogle Scholar
  2. 2.
    Alur, R., Dill, D.L.: A theory of timed automata. Theor. Comput. Sci. 126(2), 183–235 (1994)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    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)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    Bozga, M., Iosif, R., Lakhnech, Y.: Flat parametric counter automata. Fundam. Inform. 91(2), 275–303 (2009)MathSciNetMATHGoogle Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    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)Google Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    Fietzke, A., Weidenbach, C.: Superposition as a decision procedure for timed automata. In: MACIS, pp. 52–62 (2011)Google Scholar
  14. 14.
    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)CrossRefGoogle Scholar
  15. 15.
    Halpern, J.Y.: Presburger arithmetic with unary predicates is \(\Pi_{1}^{1}\) complete. Journal of Symbolic Logic 56(2), 637–642 (1991)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Hendriks, M., Larsen, K.G.: Exact acceleration of real-time model checking. Electr. Notes Theor. Comput. Sci. 65(6) (2002)Google Scholar
  17. 17.
    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)CrossRefGoogle Scholar
  18. 18.
    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)CrossRefGoogle Scholar
  19. 19.
    Kruglov, E., Weidenbach, C.: SUP(T) decides the first-order logic fragment over ground theories. In: MACIS, pp. 126–148 (2011)Google Scholar
  20. 20.
    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)CrossRefGoogle Scholar
  21. 21.
    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)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Arnaud Fietzke
    • 1
    • 2
  • Evgeny Kruglov
    • 1
    • 2
  • Christoph Weidenbach
    • 1
    • 2
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.Computer ScienceSaarland UniversitySaarbrückenGermany

Personalised recommendations