Abstract
We study the decision problem for the language DGRA (directed graphs with reachability and acyclicity), a quantifier-free fragment of graph theory involving the notions of reachability and acyclicity.
We prove that the language DGRA is decidable, and that its decidability problem is NP-complete. We do so by showing that the language enjoys a small model property: If a formula is satisfiable, then it has a model whose cardinality is polynomial in the size of the formula.
Moreover, we show how the small model property can be used in order to devise a tableau-based decision procedure for DGRA.
The second author was in part supported by grants NSF ITR CCR-0113611 and NSF CCR-0098114.
This is a preview of subscription content, log in via an institution to check access.
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
Baader, F.: Augmenting concept languages by transitive closure of roles: An alternative to terminological cycles. In: Mylopoulos, J., Reiter, R. (eds.) International Joint Conference on Artificial Intelligence, pp. 446–451 (1991)
Benedikt, M., Reps, T.W., Sagiv, S.: A decidable logic for describing linked data structures. In: Swierstra, S.D. (ed.) ESOP 1999. LNCS, vol. 1576, pp. 2–19. Springer, Heidelberg (1999)
Burstall, R.M.: Some techniques for proving correctness of programs which alter data structures. Machine Intelligence 7, 23–50 (1972)
Cantone, D., Cincotti, G.: The decision problem in graph theory with reachability related constructs. In: Baumgartner, P., Zhang, H., (eds.), First-Order Theorem Proving, Technical Report 5/2000, pp. 68–90. Universität Koblenz-Landau (2000)
Cantone, D., Cutello, V.: A decidable fragment of the elementary theory of relations and some applications. In: International Symposium on Symbolic and Algebraic Computation, pp. 24–29 (1990)
Cantone, D., Zarba, C.G.: A tableau-based decision procedure for a fragment of graph theory involving reachability and acyclicity. Technical report, Department of Computer Science, University of New Mexico (2005)
Gallier, J.H.: Logic for Computer Science: Foundations of Automatic Theorem Proving. Harper & Row, New York (1986)
Kuncak, V., Rinard, M.: Generalized records and spatial conjunction in role logic. In: Giacobazzi, R. (ed.) SAS 2004. LNCS, vol. 3148, pp. 361–376. Springer, Heidelberg (2004)
Moser, L.E.: A decision procedure for unquantified formulas of graph theory. In: Lusk, E.R., Overbeek, R. (eds.) CADE 1988. LNCS, vol. 310, pp. 344–357. Springer, Heidelberg (1988)
Ranise, S., Zarba, C.G.: A decidable logic for pointer programs manipulating linked lists. Unpublished (2005)
Rensink, A.: Canonical graph shapes. In: Schmidt, D. (ed.) ESOP 2004. LNCS, vol. 2986, pp. 401–415. Springer, Heidelberg (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cantone, D., Zarba, C.G. (2005). A Tableau-Based Decision Procedure for a Fragment of Graph Theory Involving Reachability and Acyclicity. In: Beckert, B. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2005. Lecture Notes in Computer Science(), vol 3702. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11554554_9
Download citation
DOI: https://doi.org/10.1007/11554554_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28931-9
Online ISBN: 978-3-540-31822-4
eBook Packages: Computer ScienceComputer Science (R0)