Symbolic Algorithms for In nite-State Games ?

A procedure for the analysis of state spaces is called sym bolic 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 in nite state spaces provided termination is guaranteed We present symbolic procedures and corresponding termination criteria for the so lution of in nite state games which occur in the control and modular veri cation of in nite state systems To characterize the termination of symbolic procedures for solving in nite state games we classify these game structures into four increasingly restrictive categories Class consists of in nite state structures for which all safety and reachability games can be solved Class consists of in nite state structures for which all regular games can be solved Class consists of in nite state structures for which all nested pos itive boolean combinations of regular games can be solved Class consists of in nite state structures for 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 in nite 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 in nite state games with arbitrary regular objectives and prove termination over all class structures This settles in particular the symbolic controller synthesis problem for rectangular hybrid systems

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