Symbolic Algorithms for Innnite-state Games

A procedure for the analysis of state spaces is called symbolic if it manipulates not individual states, but sets of states that are represented by constraints. Such a procedure can be used for the analysis of innnite state spaces, provided termination is guaranteed. We present symbolic procedures, and corresponding termination criteria, for the solution of innnite-state games, which occur in the control and modular veriication of innnite-state systems. To characterize the termination of symbolic procedures for solving innnite-state games, we classify these game structures into four increasingly restrictive categories: 1. Class 1 consists of innnite-state structures for which all safety and reachability games can be solved. 2. Class 2 consists of innnite-state structures for which all !-regular games can be solved. 3. Class 3 consists of innnite-state structures for which all nested positive boolean combinations of !-regular games can be solved. 4. Class 4 consists of innnite-state structuresfor which all nested boolean combinations of !-regular games can be solved. We give a structural characterization for each class, using equivalence relations on the state spaces of games which range from game versions of trace equivalence to a game version of bisimilarity. We provide innnite-state examples for all four classes of games from control problems for hybrid systems. We conclude by presenting symbolic algorithms for the synthesis of winning strategies (\controller synthesis") for innnite-state games with arbitrary !-regular objectives, and prove termination over all class-2 structures. This settles, in particular, the symbolic controller synthesis problem for rectangular hybrid systems.