Modular Games for Coalgebraic Fixed Point Logics
Electronical Notes in Theoretical Computer Science
We build on existing work on finitary modular coalgebraic logics [3, 4] , which we extend with general fixed points, including CTL-and PDL-like fixed points, and modular evaluation games. These results are generalisations of their correspondents in the modal μ-calculus, as described e.g. in  . Inspired by recent work of Venema , we provide our logics with evaluation games that come equipped with a modular way of building the game boards. We also study a specific class of modular
... aic logics that allow for the introduction of an implicit negation operator. Open access under CC BY-NC-ND license. Coalgebraic fixed point logics were first considered in the work of Venema  , where a finitary version of the coalgebraic logic of Moss  was used as the underlying modal language. Our motivation for considering fixed point logics over different modal languages is rooted in our interest in verification techniques for systems modelled as coalgebras. In this setting, the logics obtained through the modular techniques described in  appear to be better suited as specification logics. The syntax of modular coalgebraic logics is based on the notion of syntax constructor , while their semantics uses a notion of one-step semantics for a syntax constructor  , which generalises the predicate liftings of  . The logics obtained from syntax constructors are originally boolean, but in order to ensure that fixed points have a well-defined semantics, we leave out negation from these languages. However, for the specific class of syntax constructors which are closed under duals (that is, for each modality they specify, a semantically dual modality is also specified), a safe negation becomes definable in the language, and thus the expressivity of the logic stays as before. For this class of syntax constructors, we also introduce a general way of defining CTL-and PDL-like fixed points, and illustrate their applicability via examples. For instance, we obtain the fixed points of Dynamic Epistemic Logic  via the coalgebraic semantics for this logic described in  . In standard model checking terminology, these fixed points are referred to as 'alternation-free', and enjoy a linear-time model checking algorithm based on parity games  . The results concerning the implicit negation and the alternation-free fragments of our logics make use of the notion of an S-modality (of some finite arity), with S a syntax constructor with an associated one-step semantics. This notion also allows us to relate logics induced by sets of polyadic predicate liftings, as considered in  , with logics induced by syntax constructors. As a result, we obtain a way to add fixed points to logics of the former type. In  , deciding about the satisfaction of formulae by states of a coalgebra is achieved through deciding the winner of so-called evaluation games. These are parity games that generalize those for the modal μ-calculus [15, 7, 10, 19, 20, 22] , by replacing the usual single moves of either the verifier or the refuter in positions that correspond to modal formulae by two consecutive moves: a move of the verifier, who has to exhibit a relation between sub-formulae of the original formula and states of the coalgebra, that witnesses the satisfaction of the given modal formula by a state of the coalgebra, and a move of the refuter, who has to choose an element of this relation. These two consecutive moves are, in turn, inspired by similar moves in the bisimulation game of Baltag . We introduce a variant of the evaluation games of  tailored to our fixed point logics, and prove their adequacy w.r.t. the standard coalgebraic semantics. The only difference w.r.t.  is in the moves corresponding to modal positions, where the one-step semantics for the syntax constructor defining the underlying modal language is used to define the valid moves. The distinctive feature of our games, however, is that they come equipped with one-step games. These adequately replace the two consecutive moves, of the verifier followed by the refuter, in modal positions, by an equivalent sub-game played between the verifier and the refuter.