Automata-Theoretic Decision of Timed Games
VMCAI '02 Revised Papers from the Third International Workshop on Verification, Model Checking, and Abstract Interpretation
Optimal Complexity Bounds for Positive LTL Games
CSL '02 Proceedings of the 16th International Workshop and 11th Annual Conference of the EACSL on Computer Science Logic
Dealing with Nondeterminism in Symbolic Control
HSCC '08 Proceedings of the 11th international workshop on Hybrid Systems: Computation and Control
Journal of Computer and System Sciences
CAV'05 Proceedings of the 17th international conference on Computer Aided Verification
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Abstract: Deciding infinite two-player games on finite graphs with the winning condition specified by a linear temporal logic (Ltl) formula, is known to be 2Exptime-complete. In this paper, we identify Ltl fragments of lower complexity. Solving Ltl games typically involves a doubly-exponential translation from Ltl formulas to deterministic omega-automata. First, we show that the longest distance (length of the longest simple path) of the generator is also an important parameter, by giving an O(d log n)-space procedure to solve a Büchi game on a graph with n vertices and longest distance d. Then, for the Ltl fragment with only eventualities and conjunctions, we provide a translation to deterministic generators of exponential size and linear longest distance, show both of these bounds to be optimal, and prove the corresponding games to be Pspace-complete. Introducing next modalities in this fragment, we provide a translation to deterministic generators still of exponential size but also with exponential longest distance, show both of these bounds to be optimal, and prove the corresponding games to be Exptime-complete. For the fragment resulting by further adding disjunctions, we provide a translation to deterministic generators of doubly-exponential size and exponential longest distance, show both of these bounds to be optimal, and prove the corresponding games to be Expspace. Finally, we show tightness of the double-exponential bound on the size as well as the longest distance for deterministic generators for Ltl even in the absence of next and until modalities.