2010-10-01, English, Article, Journal or magazine article edition: Completeness for flat modal fixpoint logics. Santocanale, Luigi; Venema, Yde

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Physical Description
  • Magazine/​Journal article
Published
  • Elsevier B.V., 2010 Oct
Language
  • English

Edition details

Title
  • Completeness for flat modal fixpoint logics.
Appears In
  • Annals of Pure and Applied Logic, v.162, no.1, 2010 Oct, p.55(28) (ISSN: 0168-0072)
Author
  • Santocanale, Luigi
  • Venema, Yde
Published
  • Elsevier B.V., 2010 Oct
Physical Description
  • Magazine/​Journal article
Subjects
Target Audience
  • Academic
Summary
  • To link to full-text access for this article, visit this link: http:/​/​dx.doi.org/​10.1016/​j.apal.2010.07.003 Byline: Luigi Santocanale, Yde Venema Abstract: This paper exhibits a general and uniform method to prove axiomatic completeness for certain modal fixpoint logics. Given a set I of modal formulas of the form [gamma](x,p.sub.1,...,p.sub.n), where x occurs only positively in [gamma], we obtain the flat modal fixpoint language L.sub.a[macron](I) by adding to the language of polymodal logic a connective a[macron].sub.[gamma] for each [gamma][member of]I. The term a[macron].sub.[gamma]([phi].sub.1,...,[phi].sub.n) is meant to be interpreted as the least fixed point of the functional interpretation of the term [gamma](x,[phi].sub.1,...,[phi].sub.n). We consider the following problem: given I, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language L.sub.a[macron](I) on Kripke structures. We prove two results that solve this problem. First, let K.sub.a[macron](I) 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 a[macron].sub.[gamma]. Provided that each indexing formula [gamma] satisfies a certain syntactic criterion, we prove this axiom system to be complete. Second, addressing the general case, we prove the soundness and completeness of an extension K.sub.a[macron].sup.+(I) of K.sub.a[macron](I). This extension is obtained via an effective procedure that, given an indexing formula [gamma] as input, returns a finite set of axioms and derivation rules for a[macron].sub.[gamma], of size bounded by the length of [gamma]. Thus the axiom system K.sub.a[macron].sup.+(I) is finite whenever I is finite. Article History: Received 12 December 2009; Revised 9 April 2010; Accepted 4 June 2010 Article Note: (footnote) [star] The research leading to this paper was supported by the Van Gogh research project Modal Fixpoint Logics.
Terms of Use
  • COPYRIGHT 2010 Elsevier B.V.
Language
  • English
Database
  • Gale Academic OneFile
Identifier
  • 237620281

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