The Expressive Power of Epistemic $μ$-Calculus

While the μ-calculus notoriously subsumes Alternating-time Temporal Logic (ATL), we show that the epistemic μ-calculus does not subsume ATL with imperfect information (ATLi), for the synchronous perfect-recall semantics. To prove this we first establish that jumping parity tree automata (JTA), a recently introduced extension of alternating parity tree automata, are expressively equivalent to the epistemic μ-calculus, and this for any knowledge semantics. Using this result we also show that, for bounded-memory semantics, the epistemic μ-calculus is not more expressive than the standard μ-calculus, and that its satisfiability problem is Exptime-complete. 1998 ACM Subject Classification F.4.1 Mathematical Logic, F.4.3 Formal Languages