Alternating-time temporal logic (ATL) and its extensions, including the alternating-time $\mu$-calculus (AMC), serve the specification of the strategic abilities of coalitions of agents in concurrent game structures. The key ingredient of the logic are path quantifiers specifying that some coalition of agents has a joint strategy to enforce a given goal. This basic setup has been extended to let some of the agents (revocably) commit to using certain named strategies, as in ATL with explicit strategies (ATLES). In the present work, we extend ATLES with fixpoint operators and strategy disjunction, arriving at the alternating-time $\mu$-calculus with disjunctive explicit strategies (AMCDES), which allows for a more flexible formulation of temporal properties (e.g. fairness) and, through strategy disjunction, a form of controlled nondeterminism in commitments. Our main result is an ExpTime upper bound for satisfiability checking (which is thus ExpTime-complete). We also prove upper bounds QP (quasipolynomial time) and NP $\cap$ coNP for model checking under fixed interpretations of explicit strategies, and NP under open interpretation. Our key technical tool is a treatment of the AMCDES within the generic framework of coalgebraic logic, which in particular reduces the analysis of most reasoning tasks to the treatment of a very simple one-step logic featuring only propositional operators and next-step operators without nesting; we give a new model construction principle for this one-step logic that relies on a set-valued variant of first-order resolution.

本文将ATLES与fixpoint算子、策略分离扩展为具有非确定性控制的AMCDES，其可以灵活地表述时间性质，主要结果是ExpTime上限可满足检测（为ExpTime完全问题），并且Model Checking的上限是QP和NP∪coNP或NP，取决于具体策略的解释。

具有析取显式策略的交替时间μ演算