Game-Theoretic Semantics for Alternating-Time Temporal Logic

Abstract

We introduce versions of game-theoretic semantics (GTS) for Alternating-Time Temporal Logic (ATL). In GTS, truth is defined in terms of existence of a winning strategy in a semantic evaluation game, and thus the game-theoretic perspective appears in the framework of ATL on two semantic levels: on the object level in the standard semantics of the strategic operators, and on the meta-level where game-theoretic logical semantics is applied to ATL. We unify these two perspectives into semantic evaluation games specially designed for ATL. The game-theoretic perspective enables us to identify new variants of the semantics of ATL based on limiting the time resources available to the verifier and falsifier in the semantic evaluation game. We introduce and analyse an unbounded and (ordinal) bounded GTS and prove these to be equivalent to the standard (Tarski-style) compositional semantics. We show that in these both versions of GTS, truth of ATL formulae can always be determined in finite time, i.e., without constructing infinite paths. We also introduce a non-equivalent finitely bounded semantics and argue that it is natural from both logical and game-theoretic perspectives. * This paper is a substantially extended and revised version of the conference paper [8].

Game-theoretic semantics for ATL+ with applications to model checking

“…We prove-as shown for ATL in [11]-that our GTS for ATL + ensures (even on infinite models) that it is always sufficient to construct finite paths only when formulae are evaluated. While the proof for ATL + is substantially more complex, the technical machinery we use is in many respects lighter, we claim, than the one used in [11].…”

“…Several new ideas and technical notions, such as the role of a seeker and the use of a truth function, will be introduced in order to enable the transition from ATL to ATL + in the GTS setting. Also, a connection of our GTS with Büchi games will be established; the connection applies trivially also to the games of [11]. Most importantly, we can directly use the new upgraded semantics in a model checking procedure for ATL + and the fragments ATL k .…”

“…The intuitions underlying the finitely bounded and unbounded GTS are similar. A reader unfamiliar with the concept of GTS may find it useful to consult, for example, [13] for GTS in general, or [11] for ATL-specific GTS. The particular GTS for ATL + presented here follows the general principles of GTS, the main original feature here being the treatment of strategic formulae A Φ.…”

“…The procedure in this phase is analogous to the step game, step(V, A, q), which was introduced for the GTS for ATL ([11]). …”

Bounded Game-Theoretic Semantics for Modal Mu-Calculus and Some Variants

“…In this section we define a bounded game-theoretic semantics for the µ-calculus, or bounded GTS. The semantics shares some features with a similar GTS for the Alternating-time temporal logic (ATL) defined in [8] (see also [10]).…”