Completeness for Flat Modal Fixpoint Logics

Abstract

Given a set Γ of modal formulas of the form γ(x, p), where x occurs positively in γ, the language $\mathcal{L}_\sharp({\it \Gamma})$ is obtained by adding to the language of polymodal logic K connectives $\sharp_\gamma$, γε Γ. Each term $\sharp_\gamma$ is meant to be interpreted as the parametrized least fixed point of the functional interpretation of the term γ(x). Given such a Γ, we construct an axiom system ${\bf K}_\sharp(\Gamma)$ which is sound and complete w.r.t. the concrete interpretation of the language $\mathcal{L}_\sharp({\it \Gamma})$ on Kripke frames. If Γ is finite, then ${\bf K}_\sharp(\Gamma)$ is a finite set of axioms and inference rules.