Abstract
Manna and Pnueli have extensively shown how a mixture of first-order logic (FOL) and discrete Linear time Temporal Logic (LTL) is sufficient to precisely state verification problems for the class of reactive systems. Theories in FOL model the (possibly infinite) data structures used by a reactive system while LTL specifies its (dynamic) behavior. In this paper, we derive undecidability and decidability results for both the satisfiability of (quantifier-free) formulae and the model-checking of safety properties by lifting combination methods for (non-disjoint) theories in FOL. The proofs of our decidability results suggest how decision procedures for the constraint satisfiability problem of theories in FOL and algorithms for checking the satisfiability of propositional LTL formulae can be integrated. This paves the way to employ efficient Satisfiability Modulo Theories solvers in the model-checking of infinite state systems. We illustrate our techniques on two examples.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bonacina, M.P., Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Decidability and undecidability results for Nelson-Oppen and rewrite-based decision procedures. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, Springer, Heidelberg (2006)
Burkart, O., Caucal, D., Moller, F., Steffen, B.: Verification of infinite state structures. In: Handbook of Process Algebras (2001)
Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT Press, Cambridge (2000)
de Moura, L., Rueß, H., Sorea, M.: Lazy theorem proving for bounded model checking over infinite domains. In: Voronkov, A. (ed.) Automated Deduction - CADE-18. LNCS (LNAI), vol. 2392, Springer, Heidelberg (2002)
Demri, S.: Linear-time temporal logics with Presburger constraints: An overview. Journal of Applied Non-Classical Logics 16(3-4) (2006)
Demri, S., Finkel, A., Goranko, V., van Drimmelen, G.: Towards a model-checker for counter systems. In: Graf, S., Zhang, W. (eds.) ATVA 2006. LNCS, vol. 4218, Springer, Heidelberg (2006)
Gabbay, D.M., Kurucz, A., Wolter, F., Zakharyaschev, M.: Many-Dimensional Modal Logics: Theory and Applications. North-Holland Publishing Co. (2003)
Ghilardi, S.: Model theoretic methods in combined constraint satisfiability. Journal of Automated Reasoning 33(3-4) (2004)
Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Combination methods for satisfiability and model-checking of infinite-state systems. Technical Report RI313-07, Università degli Studi di Milano (2007), Available at http://homes.dsi.unimi.it/~zucchell/publications/techreport/GhiNiRaZu-RI313-07.pdf
Graf, S., Saïdi, H.: Construction of abstract state graphs with PVS. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, Springer, Heidelberg (1997)
Maidl, M.: A unifying model checking approach for safety properties of parameterized systems. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, Springer, Heidelberg (2001)
Manna, Z., Pnueli, A.: Temporal Verification of Reactive Systems: Safety. Springer, Heidelberg (1995)
Minsky, M.L.: Recursive unsolvability of Post’s problem of “tag” and other topics in the theory of Turing machines. Annals of Mathematics 74(3) (1961)
Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Transaction on Programming Languages and Systems 1(2) (1979)
Plaisted, D.A.: A decision procedure for combination of propositional temporal logic and other specialized theories. Journal of Automated Reasoning 2(2) (1986)
Pnueli, A., Ruath, S., Zuck, L.D.: Automatic deductive verification with invisible invariants. In: Margaria, T., Yi, W. (eds.) ETAPS 2001 and TACAS 2001. LNCS, vol. 2031, Springer, Heidelberg (2001)
Sipma, H.B., Uribe, T.E., Manna, Z.: Deductive model checking. Formal Methods in System Design 15(1) (1999)
Sofronie-Stokkermans, V.: Interpolation in local theory extensions. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, Springer, Heidelberg (2006)
Tinelli, C., Harandi, M.T.: A new correctness proof of the Nelson-Oppen combination procedure. In: Proc. of FroCoS 1996 (1996)
Vardi, M.Y.: Verification of concurrent programs: the automata-theoretic framework. Annals of Pure and Applied Logic 51(1-2) (1991)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D. (2007). Combination Methods for Satisfiability and Model-Checking of Infinite-State Systems. In: Pfenning, F. (eds) Automated Deduction – CADE-21. CADE 2007. Lecture Notes in Computer Science(), vol 4603. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73595-3_25
Download citation
DOI: https://doi.org/10.1007/978-3-540-73595-3_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73594-6
Online ISBN: 978-3-540-73595-3
eBook Packages: Computer ScienceComputer Science (R0)