Games Programs Play: Analyzing Multiplayer Programs

In this paper we present the first automatic technique for symbolically proving alternation-free μ-calculus properties of infinitestate, higher-order programs. In particular, we show how weak-recurrence solvability can be lifted from finite-state programs to higher-order recursive programs. Our strategy reduces the search for a proof of a winning strategy to nontermination of higher-order programs for safety games, termination for liveness games, and an iterated combination of the two— along with winning preconditions—for weak recurrence games. We can thus leverage existing reasoning techniques based on dependent refinement types to automatically generate abstractions. From the resulting types, one can construct the winning strategy. Our technique even allows one to solve games in which the state space consists of higher-order expressions and algebraic data-structures. We have implemented our technique in a prototype tool PGS, and discovered winning strategies for a variety of games, including safety games, liveness games, general recurrence games, and alternation-free μ-calculus verification games.