Completeness for Flat Modal Fixpoint Logics

TL;DR

This paper develops a uniform method to prove completeness for a class of modal fixpoint logics, extending polymodal logic with least fixed point connectives interpreted on Kripke frames.

Contribution

It introduces a general approach to axiomatize and prove completeness for flat modal fixpoint logics with an effective procedure for finite axiomatizations.

Findings

Proves completeness of a logic with fixpoint connectives under syntactic criteria.

Provides an effective method to generate finite axiomatizations for these logics.

Establishes soundness and completeness of an extended logic for the general case.

Abstract

This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set \Gamma of modal formulas of the form \gamma(x, p1, . . ., pn), where x occurs only positively in \gamma, the language L\sharp (\Gamma) is obtained by adding to the language of polymodal logic a connective \sharp\_\gamma for each \gamma \epsilon. The term \sharp\_\gamma (\varphi1, . . ., \varphin) is meant to be interpreted as the least fixed point of the functional interpretation of the term \gamma(x, \varphi 1, . . ., \varphi n). We consider the following problem: given \Gamma, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language L\sharp (\Gamma) on Kripke frames. We prove two results that solve this problem. First, let K\sharp (\Gamma) 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 \sharp\_\gamma. Provided that each indexing formula \gamma 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+ (\Gamma) of K\_\sharp (\Gamma). 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 \sharp\_\gamma, of size bounded by the length of \gamma. Thus the axiom system K+ (\Gamma) is finite whenever \Gamma is finite.