Luca de Alfaro, Thomas A. Henzinger, and Rupak Majumdar
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 infinite state spaces, provided termination is guaranteed. We present symbolic procedures, and corresponding termination criteria, for the solution of infinite-state games, which occur in the control and modular verification of infinite-state systems. To characterize the termination of symbolic procedures for solving infinite-state games, we classify these game structures into four increasingly restrictive categories:
Class 1 consists of infinite-state structures for which all safety and reachability games can be solved.
Class 2 consists of infinite-state structures for which all omega-regular games can be solved.
Class 3 consists of infinite-state structures for which all nested positive boolean combinations of omega-regular games can be solved.
Class 4 consists of infinite-state structures for which all nested boolean combinations of omega-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 infinite-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 infinite-state games with arbitrary omega-regular objectives, and prove termination over all class-2 structures. This settles, in particular, the symbolic controller synthesis problem for rectangular hybrid systems.
Proceedings of the 12th International Conference on Concurrency Theory (CONCUR), Lecture Notes in Computer Science 2154, Springer, 2001, pp. 536-550.