Abstract
We present modal logics for four classes of finite graphs: finite directed graphs, finite acyclic directed graphs, finite undirected graphs and finite loopless undirected graphs. For all these modal proof theories we discuss soundness and completeness results with respect to each of these classes of graphs. Moreover, we investigate whether some well-known properties of undirected graphs are modally definable or not: κ-colouring, planarity, connectivity and properties that a graph is Eulerian or Hamiltonian. Finally, we present an axiomatization for colouring and prove that it is sound and complete with respect to the class of finite κ-colourable graphs. One of most interesting feature of this approach is the use of the axioms of Dynamic Logic together with the Löb axiom to ensure acyclicity.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
P. Blackburn and W. Meyer-Viol. Linguistic, logic and finite trees. Bulletin of the IGPL, 2:2–29, 1994.
P. Blackburn, W. Meyer-Viol, and M. de Rijke. A Proof System for Finite Trees. Lecture Notes in Computer Science 1092. Springer-Verlag, Berlin, 1996. Computer Science Logic 95.
M. J. Fisher and R. F. Ladner. Prepositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18, 1979.
R. Goldblatt. Logics of Time and Computation. CSLI Lecture Notes 7. CSLI, Stanford, 1992.
H. Sahlqvist. Completeness and correspondence in first and second-order semantics for modal logic. In S. Kanger, editor, Proceedings of the third Scandinavian Logic Symposium. North-Holland, Amsterdam, 1975.
D. Vakarelov. Filtration Theorem for Dynamic Algebras with Test and Inverse Operators. Lecture Notes in Computer Science 148. Springer-Verlag, Berlin, 1983. Logics of Programs and their Applications.
J. van Benthem. Correspondence theory. In D. M. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, vol. 2. D. Reidel Publishing Company, Dordrecht, 1984.
J. van Benthem. Modal Logic and Classical Logic. Bibliopolis, Italy, 1985.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Kluwer Academic Publishers
About this chapter
Cite this chapter
Benevides, M.R.F. (2003). Modal Logics for Finite Graphs. In: de Queiroz, R.J.G.B. (eds) Logic for Concurrency and Synchronisation. Trends in Logic, vol 15. Springer, Dordrecht. https://doi.org/10.1007/0-306-48088-3_6
Download citation
DOI: https://doi.org/10.1007/0-306-48088-3_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-1270-9
Online ISBN: 978-0-306-48088-1
eBook Packages: Springer Book Archive