Dynamic programs, or fixpoint iteration schemes, are useful for solving many problems on state spaces, including model checking on Kripke structures ("verification"), computing shortest paths on weighted graphs ("optimization"), computing the value of games played on game graphs ("control"). For Kripke structures, a rich fixpoint theory is available in the form of the -calculus. Yet few connections have been made between different interpretations of fixpoint algorithms. We study the question of when a particular fixpoint iteration scheme ' for verifying an !-regular property on a Kripke structure can be used also for solving a two-player game on a game graph with winning objective. We provide a sufficient and necessary criterion for the answer to be a rmative in the form of an extremal-model theorem for games: under a game interpretation, the dynamic program' solves the game with objective if and only if both (1) under an existential interpretation on Kripke structures,' is equivalent to 9, and (2) under a universal interpretation on Kripke structures,' is equivalent to 8. In other words,' is correct on all two-player game graphs i it is correct on all extremal game graphs, where one or the other player has no choice of moves. The theorem generalizes to quantitative interpretations, where it connects two-player games with costs to weighted graphs. While the standard translations from !-regular properties to the-calculus violate (1) or (2), we give a translation that satisfies both conditions. Our construction, therefore, yields fixpoint iteration schemes that can be uniformly applied on Kripke structures, weighted graphs, game graphs, and game graphs with costs, in order to meet or optimize a given !-regular objective.

Abstract. The complementation problem for nondeterministic automata on infinite words has numerous applications in formal verification. In particular, the language-containment problem, to which many verification problems are reduced, involves complementation. Traditional optimal complementation constructions are quite complicated and have not been implemented. Recently, we have developed an analysis techniques for runs of co-Büchi and generalized co-Büchi automata and used the analysis to describe simpler optimal complementation constructions for Büchi and generalized Büchi automata. In this work, we extend the analysis technique to Rabin and Streett automata, and use the analysis to describe novel and simple complementation constructions for them. 1