Deduction chains for common knowledge

Abstract

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.

Syntactic cut-elimination for common knowledge

“…This results in a system in which proofs have transfinite depth and in which common knowledge is the greatest fixpoint of the function described above. Although this system has been further studied in [12,10], no syntactic cut-elimination procedure has been found. Cut-elimination was proved only indirectly by showing completeness of the cut-free system.…”