Completeness for Flat Modal Fixpoint Logics

Luigi Santocanale (1) , Yde Venema (2)

1 LIF - Laboratoire d'informatique Fondamentale de Marseille - UMR 6166
2 ILLC - Institute for Logic, Language and Computation

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.

Mots clés: fixpoint logic; modal logic; axiomatization; completeness; least fixpoint; modal algebra; representation theorem

Domaines: Logique en informatique [cs.LO]; Logique [math.LO]

Luigi Santocanale, Yde Venema. Completeness for Flat Modal Fixpoint Logics. Annals of Pure and Applied Logic, 2010, 162 (1), pp.55-82. ⟨10.1016/j.apal.2010.07.003⟩. ⟨hal-00346782⟩