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$\omega$-regular energy games, which are weighted two-player turn-based gameswith the quantitative objective to keep the energy levels non-negative, havebeen used in the context of verification and synthesis. The logic of modal$\mu$-calculus, when applied over game graphs with $\omega$-regular winningconditions, allows defining symbolic algorithms in the form of fixed-pointformulas for computing the sets of winning states.  In this paper, we introduce energy $\mu$-calculus, a multi-valued extensionof the $\mu$-calculus that serves as a symbolic framework for solving$\omega$-regular energy games. Energy $\mu$-calculus enables the seamless reuseof existing, well-known symbolic $\mu$-calculus algorithms for $\omega$-regulargames, to solve their corresponding energy augmented variants. We define thesyntax and semantics of energy $\mu$-calculus over symbolic representations ofthe game graphs, and show how to use it to solve the decision and the minimumcredit problems for $\omega$-regular energy games, for both bounded andunbounded energy level accumulations.

Energy mu-Calculus: Symbolic Fixed-Point Algorithms for omega-Regular Energy Games