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Asymptotic probabilities of languages with generalized quantifiers

Guy Fayolle 1 Stéphane Grumbach 2 Christophe Tollu 3
2 VERSO - Databases
Inria Paris-Rocquencourt, CNRS - Centre National de la Recherche Scientifique : UMR8629
Abstract : We study the impact of adding certain families of generalized quantifiers to first-order logic (FO) on the asymptotic behavior of sentences. All our results are stated and proved for languages disallowing free variables in the scope of generalized quantifiers.for a class K of finite structures closed under isomorphism, the quantifier Qk is said to be strongly monotonic, sm, if membership in the class is preserved under a loose form of extensions. Our first theorem (0/1 law for FO with any set of sm quantifiers) subsumes a previous criterion for proving that almost no graphs satisfy a given property. We also establish a 0/1 law for FO with Hartig quantifiers (equicardinality quantifiers) and a limit law for a fragment of FO with Rescher quantifiers (expressing inequalities of cardinalities). The proofs of these last two results combine standard combinatorial enumerations with more sophisticated techniques from complex analysis. We also prove that the 0/1 law fails for the extension of FO with Hartig quantifiers if the above syntactic restriction is relaxed. We therefore get the best upper bound for the existence of a 0/1 law for FO with Hartigg quantifiers.
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Submitted on : Monday, May 29, 2006 - 5:16:43 PM
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Guy Fayolle, Stéphane Grumbach, Christophe Tollu. Asymptotic probabilities of languages with generalized quantifiers. Eighth Annual IEEE Symposium on Logic in Computer Science, Jun 1993, Montréal / Canada, pp.199-207, ⟨10.1109/LICS.1993.287587⟩. ⟨inria-00077192⟩

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