Probabilistic Alternating-Time mu-Calculus

Reasoning about strategic abilities is key to an AI system consisting of multiple agents with random behaviors. We propose a probabilistic extension of Alternating mu-Calculus (AMC), named PAMC, for reasoning about strategic abilities of agents in stochastic multi-agent systems. PAMC subsumes existing logics AMC and P mu TL. The usefulness of PAMC is exemplified by applications in genetic regulatory networks. We show that, for PAMC, the model checking problem is in UP boolean AND co-UP, and the satisfiability problem is EXPTIME complete, both of which are the same as those for AMC. Moreover, PAMC admits the small model property. We implement the satisfiability checking procedure in a tool PAMCSolver.