106 articles – 48 Notices  [english version]
 HAL : hal-00346782, version 1
 arXiv : 0812.2390
 Completeness for Flat Modal Fixpoint LogicsSantocanale L., Venema Y.http://hal.archives-ouvertes.fr/hal-00346782
Preprint, Working Paper, Document sans référence, etc.
 Informatique/Logique en informatique Mathématiques/Logique
Completeness for Flat Modal Fixpoint Logics
Luigi Santocanale () 1, Yde Venema () 2
 1 : Laboratoire d'informatique Fondamentale de Marseille (LIF) http://www.lif.univ-mrs.fr/ CNRS : UMR6166 – Université de la Méditerranée - Aix-Marseille II – Université de Provence - Aix-Marseille I CMI 39, Rue Joliot Curie 13453 MARSEILLE CEDEX 13 France 2 : Institute for Logic, Language and Computation (ILLC) http://www.illc.uva.nl/ Universiteit van Amsterdam Plantage Muidergracht 24, 1018 TV Amsterdam, Netherlands. Pays-Bas
This paper exhibits a general and uniform method to prove completeness for certain modal ﬁxpoint logics. Given a set Γ of modal formulas of the form γ(x, p1 , . . . , pn ), where x occurs only positively in γ, the language L♯ (Γ) is obtained by adding to the language of polymodal logic a connective ♯γ for each γ ∈ Γ. The term ♯γ (ϕ1 , . . . , ϕn ) is meant to be interpreted as the least ﬁxed point of the functional interpretation of the term γ(x, ϕ1 , . . . , ϕn ). We consider the following problem: given Γ, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language L♯ (Γ) on Kripke frames. We prove two results that solve this problem. First, let K♯ (Γ) be the logic obtained from the basic polymodal K by adding a Kozen-Park style ﬁxpoint axiom and a least ﬁxpoint rule, for each ﬁxpoint connective ♯γ . Provided that each indexing formula γ satisﬁes the syntactic criterion of being untied in x, we prove this axiom system to be complete. Second, addressing the general case, we prove the soundness and completeness of an extension K+ (Γ) of K♯ (Γ). This extension is obtained via an eﬀective procedure that, given an indexing formula γ as input, returns a ﬁnite set of axioms and derivation rules for ♯γ , of size bounded by the length of γ. Thus the axiom system K+ (Γ) is ﬁnite whenever Γ is ﬁnite.
Anglais
12/12/2008

fixpoint logic – modal logic – axiomatization – completeness – least fixpoint – modal algebra – representation theorem

Modal Fixpoint Logics (Van Gogh)
Egide

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 hal-00346782, version 1 http://hal.archives-ouvertes.fr/hal-00346782 oai:hal.archives-ouvertes.fr:hal-00346782 Contributeur : Luigi Santocanale <> Soumis le : Vendredi 12 Décembre 2008, 12:34:58 Dernière modification le : Mercredi 4 Mars 2009, 17:53:02