We consider an extension of Church's synthesis problem to ordinals by adding limit transitions to graph games. We consider game arenas where these limit transitions are defined using the sets of cofinal states. In a previous paper, we have shown that such games of ordinal length are determined and that the winner problem is PSPACE-complete, for a subclass of arenas where the length of plays is always smaller than $\omega^\omega$. However, the proof uses a rather involved reduction to classical Muller games, and the resulting strategies need infinite memory. We adapt the LAR reduction to prove the determinacy in the general case, and to generate strategies with finite memory, using a reduction to games where the limit transitions are defined by priorities. We provide an algorithm for computing the winning regions of both players in these games, with a complexity similar to parity games. Its analysis yields three results: determinacy without hypothesis on the length of the plays, existence of memoryless strategies, and membership of the winner problem in NP and co-NP.

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Schloss Dagstuhl - Leibniz Center of Informatics
R. Hariharan , M. Mukund , V. Vinay
Leibniz International Proceedings in Informatics
IARCS Annual Conference on the Foundations of Software Technology and Theoretical Computer Science
Networks and Optimization

Cristau, J, & Horn, F. (2008). Graph Games on Ordinals. In R Hariharan, M Mukund, & V Vinay (Eds.), Proceedings of the 28th IARCS Annual Conference on the Foundations of Software Technology and Theoretical Computer Science (FSTTCS\'08) (pp. 143–154). Schloss Dagstuhl - Leibniz Center of Informatics.