When Not Losing Is Better than Winning: Abstraction and Refinement for the Full µ-calculus

This work presents a novel game-based approach to abstraction-refinement for the full µ-calculus, interpreted over 3-valued semantics. A novel notion of non-losing strategy is introduced and exploited for refinement. Previous works on refinement in the context of 3-valued semantics require a direct algorithm for solving a 3-valued model checking game. This was necessary in order to have the information needed for refinement available on one game board. In contrast, while still considering a 3-valued model checking game, here we reduce the problem of solving the game to solving two 2-valued model checking (parity) games. In case the result is indefinite (don’t know), the corresponding non-losing strategies, when combined, hold all the information needed for refinement. This approach is beneficial since it can use any solver for 2-valued parity games. Thus, it can take advantage of newly developed such algorithms with improved complexity.