Canonical completeness of infinitary μ

Abstract

This paper presents a new model construction for a natural cut-free infinitary version K + ω (µ) of the propositional modal µ-calculus. Based on that the completeness of K + ω (µ) and the related system K ω (µ) can be established directly-no detour, for example through automata theory, is needed. As a side result we also obtain a finite, cut-free sound and complete system for the propositional modal µ-calculus.

Circular proofs for the modal mu‐calculus

“…We plan to further study the connection between circular proofs, Kozen's proof system and the infinitary calculus of [2]. Of particular interest are translations between the three calculi and potential cut-elimination algorithms for circular proofs.…”

“…It remains an open question, however, whether there exists a finitary cut-free proof system for the mu-calculus. A notable approach is [2] where a sound and complete semi-finite cut-free proof system is given by utilising an infinitary sequent calculus.…”

On the Proof Theory of the Modal mu-Calculus

“…That is we consider soundness and completeness with respect to a standard notion of validity, see for instance [2,12,13,16]. …”

“…The infinitary calculus T ω µ+ is introduced in [12]. This deductive system provides a cut-free, sound and complete axiomatization for the modal µ-calculus.…”

“…However, note that it makes crucial use of a cut rule. Jäger, Kretz and Studer [12] introduce the cut-free system T ω µ+ Presented by Heinrich Wansing; Received August 14, 2007 which is also based on the ω-rule. Completeness of T ω µ+ is established by a canonical counter-model construction.…”