# Symbolic Algorithms for Infinite-State Games

## Abstract

*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:

- 1
Class 1 consists of infinite-state structures for which all safety and reachability games can be solved.

- 2
Class 2 consists of infinite-state structures for which all ω-regular games can be solved.

- 3
Class 3 consists of infinite-state structures for which all nested positive boolean combinations of ω-regular games can be solved.

- 4
Class 4 consists of infinite-state structures for which all nested boolean combinations of ω-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 infinitestate games with arbitrary ω-regular objectives, and prove termination over all class-2 structures. This settles, in particular, the symbolic controller synthesis problem for rectangular hybrid systems.

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