## Finitary Winning in Omega-regular Games

*Krishnendu Chatterjee and
Thomas A. Henzinger*

Games on graphs with omega-regular objectives provide a model for the
control and synthesis of reactive systems. Every omega-regular
objective can be decomposed into a safety part and a liveness part.
The liveness part ensures that something good happens "eventually."
Two main strengths of the classical, infinite-limit formulation of
liveness are robustness (independence from the granularity of
transitions) and simplicity (abstraction of complicated time bounds).
However, the classical liveness formulation suffers from the drawback
that the time until something good happens may be unbounded. A
stronger formulation of liveness, so-called *finitary* liveness,
overcomes this drawback, while still retaining robustness and
simplicity. Finitary liveness requires that there exists an unknown,
fixed bound *b* such that something good happens within *b*
transitions. While for one-shot liveness (reachability) objectives,
classical and finitary liveness coincide, for repeated liveness
(Buechi) objectives, the finitary formulation is strictly stronger.
In this work we study games with finitary parity and Streett
(fairness) objectives. We prove the determinacy of these games,
present algorithms for solving these games, and characterize the
memory requirements of winning strategies. Our algorithms can be
used, for example, for synthesizing controllers that do not let the
response time of a system increase without bound.

*Proceedings of the
12th International Conference on Tools and Algorithms for the
Construction and Analysis of Systems*
(TACAS),
Lecture Notes in Computer Science 3920,
Springer, 2006, pp. 257-271.

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