The Constructive $\mu$-calculus: Game Semantics and Non-Wellfounded Proof Systems

We study a variant of the modal $\mu$-calculus based on the constructive modal logic $\mathsf{C}\mathsf{K}$. We define game semantics for the constructive $\mu$-calculus and prove its equivalence to the birelational Kripke semantics. We then use the game semantics to prove the soundness and completeness of a fully-labeled non-wellfounded proof system for it. At last, we briefly describe how to adapt the game semantics and proof system to the $\mu$-calculus over other non-classical modal logics.