From verification to control: dynamic programs for omega-regular objectives

Abstract

Dynamic programs, or fixpoint iteration schemes, are useful for solving many problems on state spaces. For Kripke structures, a rich fixpoint theory is available in the form of the /spl mu/-calculus, yet few connections have been made between different interpretations of fixpoint algorithms. We study the question of when a particular fixpoint iteration scheme /spl phi/ for verifying an /spl omega/-regular property /spl Psi/ on a Kripke structure can be used also for solving a two-player game on a game graph with winning objective /spl Psi/. We provide a sufficient and necessary criterion for the answer to be affirmative in the form of an extremal-model theorem for games: under a game interpretation, the dynamic program /spl phi/ solves the game with objective /spl Psi/ iff both (1) under an existential interpretation on Kripke structures, /spl phi/ is equivalent to /spl exist//spl Psi/, and (2) under a universal interpretation on Kripke structures, /spl phi/ is equivalent to /spl forall//spl Psi/. In other words, /spl phi/ is correct on all two-player game graphs iff 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 /spl omega/-regular properties to the /spl mu/-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 /spl omega/-regular objective.