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

by   Gal Amram, et al.

ω-regular energy games, which are weighted two-player turn-based games with the quantitative objective to keep the energy levels non-negative, have been used in the context of verification and synthesis. The logic of modal μ-calculus, when applied over game graphs with ω-regular winning conditions, allows defining symbolic algorithms in the form of fixed-point formulas for computing the sets of winning states. In this paper, we introduce energy μ-calculus, a multi-valued extension of the μ-calculus that serves as a symbolic framework for solving ω-regular energy games. Energy μ-calculus enables the seamless reuse of existing, well-known symbolic μ-calculus algorithms for ω-regular games, to solve their corresponding energy augmented variants. We define the syntax and semantics of energy μ-calculus over symbolic representations of the game graphs, and show how to use it to solve the decision and the minimum credit problems for ω-regular energy games, for both bounded and unbounded energy level accumulations.



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