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Minimal predicates, fixed-points, and definability

  • Johan Van Benthem (a1) (a2)
Abstract
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.

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[1977] P. Aczel , An introduction to inductive definitions. Handbook of mathematical logic ( J. Barwise , editor). North-Holland, Amsterdam. 1977. pp. 739782.

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[1980] J. McCarthy , Circumscription–a form of nonmonotonic reasoning. Artificial Intelligence, vol. 13 (1980), pp. 2739.

[1990] M-A Papalaskari and S. Weinstein , Minimal consequence in sentential logic. Journal of Logic Programming, vol. 9 (1990). pp. 1931.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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