More Precise Partition Abstractions

We define, for any partition of a state space and for formulas of the modal μ-calculus, two variants of precision for abstractions that have the partition set as state space. These variants are defined via satisfaction parity games in which the Refuter can replace a concrete state with any state in the same partition before, respectively after, a quantifier move. These games are independent of the kind of abstraction. Our first variant makes the abstraction games of de Alfaro et al. model-independent, captures the definition of precision given by Shoham & Grumberg, and corresponds to generalized Kripke modal transition systems. Our second variant is then shown to render more precise abstractions in the form of μ-automata without fairness. We discuss the relative tradeoffs of both variants in terms of the size of abstractions, the perceived cost of their synthesis via theorem provers, and the preservation of equations that are valid over concrete models. Finally, we sketch a combination of both abstraction methods.