Abstract We study pure coordination games where in every outcome, all players have identical payoffs, ‘win’ or ‘lose’. We identify and discuss a range of ‘purely rational principles’ guiding the reasoning of rational players in such games and compare the classes of coordination games that can be solved by such players with no preplay communication or conventions. We observe that it is highly nontrivial to delineate a boundary between purely rational principles and other decision methods, such as conventions, for solving such coordination games.

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].

We develop a game-theoretic semantics (GTS) for the fragment ATL + of the Alternating-time Temporal Logic ATL * , essentially extending a recently introduced GTS for ATL. We show that the new game-theoretic semantics is equivalent to the standard compositional semantics of ATL + (with perfect-recall strategies). Based on the new semantics, we provide an analysis of the memory and time resources needed for model checking ATL + and show that strategies of the verifier that use only a very limited amount of memory suffice. Furthermore, using the GTS we provide a new algorithm for model checking ATL + and identify a natural hierarchy of tractable fragments of ATL + that extend ATL.

In this paper we analyze k-ary inclusion-exclusion logic, INEX[k], which is obtained by extending first order logic with k-ary inclusion and exclusion atoms. We show that every formula of INEX[k] can be expressed with a formula of k-ary existential second order logic, ESO[k]. Conversely, every formula of ESO[k] with at most k-ary free relation variables can be expressed with a formula of INEX [k]. From this it follows that, on the level of sentences, INEX[k] captures the expressive power of ESO [k].We also introduce several useful operators that can be expressed in INEX [k]. In particular, we define inclusion and exclusion quantifiers and so-called term value preserving disjunction which is essential for the proofs of the main results in this paper. Furthermore, we present a novel method of relativization for team semantics and analyze the duality of inclusion and exclusion atoms.Keywords: inclusion logic, exclusion logic, dependence logic, team semantics, IF-logic, existential second order logic, expressive power. * This paper is an extended version of [18] with additional technical details.Next we generalize these notations for tuples of variables. Let s be an assignment, x := x 1 . . . x k a tuple of variables and a := (a 1 , . . . , a k ) ∈ M k . We use the notation s[ a/ x ] := s[a 1 /x 1 , . . . , a k /x k ]. For a team X, a set A ⊆ M k and a function F : X → P(M k ) we writeLet M be an L-model, s an assignment and t ∈ T L s.t. Vr(t) ⊆ dom(s). The interpretation of t with respect to M and s, t M s , is denoted simply by s(t). For a team X and t ∈ T L s.t. Vr(t) ⊆ dom(X) we write X(t) := {s(t) | s ∈ X}. Let t := t 1 . . . t k be a tuple of L-terms and let X be a team s.t. Vr( t ) ⊆ dom(X). We writeNote that s( t ) is a vector in M and X( t ) is a k-ary relation in M. We use the notation P * (A) for the power set of A excluding the empty set (that is P * (A) := P(A) \ {∅}). We are now ready to define team semantics for FO. Definition 2.2. Let M be an L-model, ϕ ∈ FO L and X a team such that Fr(ϕ) ⊆ dom(X). We define the truth of ϕ in M and X, denoted by M X ϕ:• M X t 1 = t 2 iff s(t 1 ) = s(t 2 ) for all s ∈ X.Definition 2.4. If t 1 , t 2 are k-tuples of L-terms, t 1 ⊆ t 2 is a k-ary inclusion atom. We define Fr( t 1 ⊆ t 2 ) = Vr( t 1 ) ∪ Vr( t 2 ). The language INC L is defined by adding the following condition to the definition of FO L (Definition 2.1).• If t 1 , t 2 are tuples of L-terms of the same length, then t 1 ⊆ t 2 ∈ S.Note that we do not allow negation to appear in front of inclusion atoms. For literals, connectives and quantifiers we use the same semantics as for FO with team semantics. Inclusion atoms have the following truth condition:Definition 2.5. Let M be a model and X a team s.t. Vr( t 1 t 2 ) ⊆ dom(X). We define the truth of t 1 ⊆ t 2 in the model M and the team X:This truth condition can be written equivalently as follows:Example 2.1. Let t 1 , . . . , t m be k-tuples of L-terms and x a k-tuple of fresh variables. Now the following holds for all nonempty teams X:In particular, for t ∈ T...

We analyze from a global point of view the expressive resources of IF logic that do not stem from Henkin (partially-ordered) quantification. When one restricts attention to regular IF sentences, this amounts to the study of the fragment of IF logic which is individuated by the game-theoretical property of Action Recall. We prove that the fragment of Action Recall can express all existential second-order (ESO) properties. This can be accomplished already by the prenex fragment of Action Recall, whose only second-order source of expressiveness are the so-called signalling patterns. The proof shows that a complete set of Henkin prefixes is explicitly definable in the fragment of Action Recall. In the more general case, in which also irregular IF sentences are allowed, we show that full ESO expressive power can be achieved using neither Henkin nor signalling patterns.

We consider pure win-lose coordination games where the representation of the game structure has additional features that are commonly known to the players, such as colouring, naming, or ordering of the available choices or of the players. We study how the information provided by such enriched representations affects the solvability of these games by means of principles of rational reasoning in coordination scenarios with no prior communication or conventions.

We introduce a new game-theoretic semantics (GTS) for the modal mu-calculus. Our so-called bounded GTS replaces parity games with alternative evaluation games where only finite paths arise; infinite paths are not needed even when the considered transition system is infinite. The novel games offer alternative approaches to various constructions in the framework of the mu-calculus. For example, they have already been successfully used as a basis for an approach leading to a natural formula size game for the logic. While our main focus is introducing the new GTS, we also consider some applications to demonstrate its uses. For example, we consider a natural model transformation procedure that reduces model checking games to checking a single, fixed formula in the constructed models, and we also use the GTS to identify new alternative variants of the mu-calculus with PTime model checking.

We study pure coordination games where in every outcome, all players have identical payoffs, 'win' or 'lose'. We identify and discuss a range of 'purely rational principles' guiding the reasoning of rational players in such games and analyse which classes of coordination games can be solved by such players with no preplay communication or conventions. We observe that it is highly nontrivial to delineate a boundary between purely rational principles and other decision methods, such as conventions, for solving such coordination games.