From Veriication to Control: Dynamic Programs for Omega-regular Objectives

Dynamic programs, or xpoint iteration schemes, are useful for solving many problems on state spaces, including model checking on Kripke structures (\veri cation"), computing shortest paths on weighted graphs (\optimization"), computing the value of games played on game graphs (\control"). For Kripke structures, a rich xpoint theory is available in the form of the -calculus. Yet few connections have been made between di erent interpretations of xpoint algorithms. We study the question of when a particular xpoint 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 su cient 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 satis es both conditions. Our construction, therefore, yields xpoint 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.