Abstract
We consider an extension of modal logic with an operator for constructing inflationary fixed points, just as the modal μ-calculus extends basic modal logic with an operator for least fixed points. Least and inflationary fixed point operators have been studied and compared in other contexts, particularly in finite model theory, where it is known that the logics IFP and LFP that result from adding such fixed point operators to first order logic have equal expressive power. As we show, the situation in modal logic is quite different, as the modal iteration calculus (MIC) we introduce has much greater expressive power than the μ-calculus. Greater expressive power comes at a cost: the calculus is algorithmically much less manageable.
Research supported by EPSRC grants GR/L69596 and GR/N23028.
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References
A. Arnold and D. Niwinski. Rudiments of μ-calculus, North-Holland, 2001.
J. van Benthem. Modal Logic and Classical Logic. Bibliopolis, Napoli, 1983.
J. Büchi. Weak second-order arithmetic and finite automata. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 6:66–92, 1960.
A. Dawar. Feasible Computation Through Model Theory. PhD thesis, University of Pennsylvania, 1993.
A. Dawar, E. Grädel, and S. Kreutzer. Inflationary Fixed Points in Modal Logics. See “http://www.cl.cam.ac.uk/users/ad260/pubs.html” or “http://www-mgi.informatik.rwth-aachen.de/Publications/”.
A. Dawar and Y. Gurevich. Fixed point logics. In preparation.
A. Dawar and L. Hella. The expressive power of finitely many generalized quantifiers. Information and Computation, 123:172–184, 1995.
A. Dawar, S. Lindell, and S. Weinstein. Infinitary logic and inductive definability over finite structures. Information and Computation, 119:160–175, 1994.
S. Dziembowski. Bounded-variable fixpoint queries are PSPACE-complete. In 10th Annual Conference on Computer Science Logic CSL 96. Selected papers, volume 1258 of Lecture Notes in Computer Science, pages 89–105. Springer, 1996.
H.-D. Ebbinghaus and J. Flum. Finite Model Theory. Springer, 2nd edition, 1999.
E. Grädel and M. Otto. On logics with two variables. Theoretical Computer Science, 224:73–113, 1999.
Y. Gurevich and S. Shelah. Fixed-point extensions of first-order logic. Annals of Pure and Applied Logic, 32:265–280, 1986.
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, number 1119 in Lecture Notes in Computer Science, pages 263–277. Springer-Verlag, 1996.
M. Jurdzinski. Deciding the winner in parity games is in UP ∩ Co-UP. Information Processing Letters, 68:119–124, 1998.
M. Otto. Bisimulation-invariant Ptime and higher-dimensional mu-calculus. The-oretical Computer Science, 224:237–265, 1999.
J. Shallit and Y. Breitbart. Automaticity I: Properties of a measure of descriptional complexity. Journal of Computer and System Sciences, 53:10–25, 1996.
M. Vardi. On the complexity of bounded-variable queries. In Proc. 14th ACM Symp. on Principles of Database Systems, pages 266–267, 1995.
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Dawar, A., Grädel, E., Kreutzer, S. (2001). Inflationary Fixed Points in Modal Logic. In: Fribourg, L. (eds) Computer Science Logic. CSL 2001. Lecture Notes in Computer Science, vol 2142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44802-0_20
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DOI: https://doi.org/10.1007/3-540-44802-0_20
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