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While the $\mu$-calculus notoriously subsumes Alternating-time Temporal Logic (ATL), we show that the epistemic $\mu$-calculus does not subsume ATL with imperfect information (ATL$_i$) 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 $\mu$-calculus, and this for any knowledge semantics. Using this result we also show that, for bounded-memory semantics, the epistemic $\mu$-calculus is not more expressive than the standard $\mu$-calculus, and that its satisfiability problem is EXPTIME-complete.

We show that the model-checking problem is decidable for a fragment of the epistemic \mu-calculus. The fragment allows free variables within the scope of epistemic modalities in a restricted form that avoids constructing formulas embodying any form of common knowledge. Our calculus subsumes known decidable fragments of epistemic CTL/LTL. Its modal variant can express winning strategies in two-player games with one player having imperfect information and non-observable objectives, and, with a suitable encoding, decidable instances of the model-checking problem for ATL with imperfect information and perfect recall can be encoded as instances of the model-checking problem for this epistemic \mu-calculus.

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