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MINIMAL PREDICATES, FIXED-POINTS, AND DEFINABILITY (2004)
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@MISC{Benthem04minimalpredicates,,
author = {Johan van Benthem},
title = { MINIMAL PREDICATES, FIXED-POINTS, AND DEFINABILITY},
year = {2004}
}
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Abstract
Minimal predicates P satisfying a given first-order description (P) occur widely in mathematical logic and computer science. We give an explicit first-order syntax for special first-order 'PIA conditions' (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.
Keyphrases
minimal predicate first-order formula first-order logic predicate minimization first-order description usual frame correspondence theory modal frame correspondence theory main technical result language min mathematical logic pia-conditions yield standard proof rijke venema accessibility relation general definition suitable monadic second-order sentence special first-order computer science pia condition modal k4-axiom monotone operation expressive power modal logic unique existence qx px explicit first-order syntax ryz rxz first-order definability first-order fixed-point definable proper hierarchy modal axiom natural model-theoretic operation motivating example iterated predicate minimization predicate intersection rxy py first-order modal frame correspondence useful fashion straightforward case basic correspondence certain first-order description concrete illustration
