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.

Starting off from the infinitary system for common knowledge over multi-modal epistemic logic presented in [L. Alberucci, G. Jäger, About cut elimination for logics of common knowledge, Annals of Pure and Applied Logic 133 (2005) 73-99], we apply the finite model property to "finitize" this deductive system. The result is a cut-free, sound and complete sequent calculus for common knowledge.

Deduction chains represent a syntactic and in a certain sense constructive method for proving completeness of a formal system. Given a formula φ, the deduction chains of φ are built up by systematically decomposing φ into its subformulae. In the case where φ is a valid formula, the decomposition yields a (usually cut-free) proof of φ. If φ is not valid, the decomposition produces a countermodel for φ. In the current paper, we extend this technique to a semiformal system for the Logic of Common Knowledge. The presence of fixed point constructs in this logic leads to potentially infinite-length deduction chains of a non-valid formula, in which case fairness of decomposition requires special attention. An adequate order of decomposition also plays an important role in the reconstruction of the proof of a valid formula from the set of its deduction chains.