Generalising Automaticity to Modal Properties of Finite Structures
We introduce a complexity measure of modal properties of finite structures which generalises the automaticity of languages. It is based on graph-automata like devices called labelling systems. We define a measure of the size of a structure that we call rank, and show that any modal property of structures can be approximated up to any fixed rank n by a labelling system. The function that takes n to the size of the smallest labelling system doing this is called the labelling index of the property. We demonstrate that this is a useful and fine-grained measure of complexity and show that it is especially well suited to characterise the expressive power of modal fixed-point logics. From this we derive several separation results of modal and non-modal fixed-point logics, some of which are already known whereas others are new.
Unable to display preview. Download preview PDF.
- 1.A. Arnold and D. Niwiński. Rudiments of μ-calculus. North Holland, 2001.Google Scholar
- 2.P. Blackburn, M. de Rijke, and Y. Venema. Modal Logic. Cambridge University Press, 2001.Google Scholar
- 4.F. Gécseg and M. Steinby. Tree languages. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 3, pages 1–68. Springer, 1997.Google Scholar
- 5.D. Janin and I. Walukiewicz. On the expressive completeness of the propositional mu-calculus with respect to monadic second order logic. In Proceedings of 7th International Conference on Concurrency Theory CONCUR’ 96, volume 1119 of Lecture Notes in Computer Science. Springer-Verlag, 1996.Google Scholar
- 8.C. Stirling. Modal and Temporal Properties of Processes. Springer, 2001.Google Scholar
- 9.W. Thomas. On logics, tilings, and automata. In J. Leach et al., editor, Automata, Languages, and Programming, Lecture Notes in Computer Science Nr. 510, pages 441–453. Springer-Verlag, 1991.Google Scholar
- 10.W. Thomas. Finite-state recognizability and logic: from words to graphs. In 13th World Computer Congress 94, volume 1, pages 499–506. Elsevier Science, 1994.Google Scholar
- 11.W. Thomas. Languages, automata and logic. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 3, pages 389–455. Springer, 1997.Google Scholar