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Minimal predicates, fixed-points, and definability

Published online by Cambridge University Press:  12 March 2014

Johan Van Benthem
Johan Van Benthem*
Affiliation:
Institute for Logic, Language and Computation Illc, Universiteit Van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The NederlandsE-mail:, johan@science.uva.nlDepartment of Philosophy, Stanford University, Stanford. CA 94305-2155., USAE-mail:, johan@science.uva.nl
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Abstract

Minimal predicates P satisfying a given first-order description ϕ(P) occur widely in mathematical logic and computer science. We give an explicit first-order syntax for special first-order ‘PIA conditions’ ϕ(P) which guarantees unique existence of such minimal predicates. Our main technical result is a preservation theorem showing PIA-conditions to be expressively complete for all those first-order formulas that are preserved under a natural model-theoretic operation of ‘predicate intersection’. Next, we show how iterated predicate minimization on PIA-conditions yields a language MIN(FO) equal in expressive power to LFP(FO), first-order logic closed under smallest fixed-points for monotone operations. As a concrete illustration of these notions, we show how our sort of predicate minimization extends the usual frame correspondence theory of modal logic, leading to a proper hierarchy of modal axioms: first-order-definable, first-order fixed-point definable, and beyond.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 3 , September 2005, pp. 696 - 712
Copyright
Copyright © Association for Symbolic Logic 2005

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References

REFERENCES

[1977]Aczel, P., An introduction to inductive definitions. Handbook of mathematical logic (Barwise, J., editor). North-Holland, Amsterdam. 1977. pp. 739–782.CrossRefGoogle Scholar
[1999]Barwise, J. and van Benthem, J., Interpolation, preservation, and pebble games, this Journal, vol. 64 (1999), no. 2, pp. 881–903.Google Scholar
[1983]van Benthem, J., Modal logic and classical logic, Bibliopolis, Naples, 1983.Google Scholar
[1985]van Benthem, J., Logic programming. Lecture Notes, Philosophical Institute, Rijksuniversiteit Groningen, 1985.Google Scholar
[1996]van Benthem, J.. Exploring logical dynamics, CSLI Publications, Stanford, CA, 1996.Google Scholar
[2005]van Benthem, J.. Modal frame correspondence generalized, 2005. Preprint DARE 148523. Institute for Logic, Language and Computation, University of Amsterdam. Stiulia Logica. To appear.Google Scholar
[2002]Benton, R., A simple incomplete extension of T, Journal of Philosophical Logic, vol. 31 (2002), no. 6, pp. 527–541.CrossRefGoogle Scholar
[2001]Blackburn, P., de Ruke, M., and Venema, Y.. Modal logic, Cambridge University Press. Cambridge, 2001.CrossRefGoogle Scholar
[1973]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[1994]Doets, H.C., From logic to logic programming, MIT Press, Cambridge, MA. 1994.CrossRefGoogle Scholar
[1999]Ebbinghaus, H-D and Flum, J., Finite model theory, Springer-Verlag, Berlin. 1999.Google Scholar
[1992]Gabbay, D. and Ohlbach, H.J., Quantifier elimination in second-order predicate logic, South African Computer Journal, vol. 7 (1992), pp. 35–43.Google Scholar
[2003]Goranko, V. and Vakarelov, D.. Elementary canonical formulas I. Extending Sahlqvist's theorem. Department of Mathematics. Rand Afrikaans University, Johannesburg. 2003, and Faculty of Mathematics and Computer Science, Kliment Ohridski University, Sofia. Submitted for publication.Google Scholar
[2002]Bars, J.M. Le, The 0-1 law fails for frame satisfiability of propositional modal logic, Proceedings of Logic in Computer Science, 2002.Google Scholar
[1983]Mahr, B. and Makovsky, J., Characterizing specification languages which admit initial semantics, Proceedings of the 8th CAAP, Springer, Berlin, 1983.Google Scholar
[1971]Mal′cev, A.I., The metamathematics of algebraic systems. North-Holland, Amsterdam, 1971.Google Scholar
[1980]McCarthy, J., Circumscription–a form of nonmonotonic reasoning. Artificial Intelligence, vol. 13 (1980), pp. 27–39.CrossRefGoogle Scholar
[1974]Moschovakis, Y. N., Elementary induction on abstract structures, North-Holland, Amsterdam, 1974.Google Scholar
[1999]Nonnengart, A. and Szałas, A., A fixed-point approach to second-order quantifier elimination with applications to modal correspondence theory. Logic at work (Orlowska, E., editor), Physica-Verlag, Heidelberg, 1999, pp. 89–108.Google Scholar
[1990]Papalaskari, M-A and Weinstein, S., Minimal consequence in sentential logic. Journal of Logic Programming, vol. 9 (1990). pp. 19–31.CrossRefGoogle Scholar
[1999]Stirling, C., Bisimulation, modal logic and model checking games, Logic Journal of the Interest Group in Pure and Applied Logics, vol. 7 (1999), no. 1, pp. 103–124.Google Scholar
[1965]Weinstein, J. M., First-order properties preserved by direct products, Ph;D. thesis, University of Wisconsin. Madison, 1965.Google Scholar