Abstract
We introduce a new decidable logic for reasoning about infinite arrays of integers. The logic is in the \(\exists^* \forall^*\) first-order fragment and allows (1) Presburger constraints on existentially quantified variables, (2) difference constraints as well as periodicity constraints on universally quantified indices, and (3) difference constraints on values. In particular, using our logic, one can express constraints on consecutive elements of arrays (e.g., ∀ i . 0 ≤ i < n →a[i + 1] = a[i] − 1) as well as periodic facts (e.g., ∀ i . i ≡ 2 0 →a[i] = 0). The decision procedure follows the automata-theoretic approach: we translate formulae into a special class of Büchi counter automata such that any model of a formula corresponds to an accepting run of an automaton, and vice versa. The emptiness problem for this class of counter automata is shown to be decidable as a consequence of earlier results on counter automata with a flat control structure and transitions based on difference constraints.
The work was supported by the French Ministry of Research (RNTL project AVERILES), the Czech Grant Agency (projects 102/07/0322, 102/05/H050), the Czech-French Barrande project 2-06-27, and the Czech Ministry of Education by project MSM 0021630528.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Armando, A., Ranise, S., Rusinowitch, M.: Uniform Derivation of Decision Procedures by Superposition. In: Fribourg, L. (ed.) CSL 2001. LNCS, vol. 2142, p. 2001. Springer, Heidelberg (2001)
Arons, T., Pnueli, A., Ruah, S., Xu, J., Zuck, L.: Parameterized Verification with Automatically Computed Inductive Assertions. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, Springer, Heidelberg (2001)
Bouajjani, A., Jurski, Y., Sighireanu, M.: A Generic Framework for Reasoning About Dynamic Networks of Infinite-State Processes. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, Springer, Heidelberg (2007)
Bozga, M., Iosif, R., Lakhnech, Y.: Flat Parametric Counter Automata. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, Springer, Heidelberg (2006)
Bradley, A.R., Manna, Z., Sipma, H.B.: What ’s Decidable About Arrays? In: Emerson, E.A., Namjoshi, K.S. (eds.) VMCAI 2006. LNCS, vol. 3855, Springer, Heidelberg (2005)
Comon, H., Jurski, Y.: Multiple Counters Automata, Safety Analysis and Presburger Arithmetic. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, Springer, Heidelberg (1998)
Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Decision Procedures for Extensions of the Theory of Arrays. Annals of Mathematics and Artificial Intelligence 50 (2007)
Habermehl, P., Iosif, R., Vojnar, T.: What else is decidable about integer arrays? Technical Report TR-2007-8, Verimag (2007)
Jaffar, J.: Presburger Arithmetic with Array Segments. Inform. Proc. Letters 12 (1981)
King, J.: A Program Verifier. PhD thesis, Carnegie Mellon University (1969)
Mateti, P.: A Decision Procedure for the Correctness of a Class of Programs. Journal of the ACM 28(2) (1980)
McCarthy, J.: Towards a Mathematical Science of Computation. In: IFIP Congress (1962)
Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice-Hall, Inc., Englewood Cliffs (1967)
Nivat, M., Perrin, D.: Ensembles reconnaissables de mots biinfinis. Canad. J. Math. 38, 513–537 (1986)
Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In: Comptes Rendus du Premier Congrès des Mathématiciens des Pays Slaves, Warsaw, Poland, pp. 92–101 (1929)
Stump, A., Barrett, C.W., Dill, D.L., Levitt, J.R.: A Decision Procedure for an Extensional Theory of Arrays. In: Proc. of LICS 2001 (2001)
Suzuki, N., Jefferson, D.: Verification Decidability of Presburger Array Programs. Journal of the ACM 27(1) (1980)
Thomas, W.: Automata on Infinite Objects. In: Handbook of Theoretical Computer Science. Formal Models and Semantics, vol. B, Elsevier, Amsterdam (1990)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Habermehl, P., Iosif, R., Vojnar, T. (2008). What Else Is Decidable about Integer Arrays?. In: Amadio, R. (eds) Foundations of Software Science and Computational Structures. FoSSaCS 2008. Lecture Notes in Computer Science, vol 4962. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78499-9_33
Download citation
DOI: https://doi.org/10.1007/978-3-540-78499-9_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78497-5
Online ISBN: 978-3-540-78499-9
eBook Packages: Computer ScienceComputer Science (R0)