Alternating-time Temporal Logic ATL Research Articles

Complexity of finite-variable fragments of propositional temporal and modal logics of computation

We prove that branching-time temporal logics CTL and CTL⁎ are polynomial-time embeddable into their single-variable fragments. It follows that satisfiability for CTL and CTL⁎, and therefore also for alternating-time temporal logics ATL and ATL⁎, in languages with one propositional variable is as algorithmically hard as satisfiability for the full logic: EXPTIME-complete for CTL and ATL, and 2EXPTIME-complete for CTL⁎ and ATL⁎. We discuss applicability of the technique used in the proofs to other modal and temporal logics, including propositional dynamic, epistemic, and temporal-epistemic logics, as well as many-dimensional and first-order modal and temporal logics.

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

We develop a game-theoretic semantics (GTS) for the fragment ATL+ of the alternating-time temporal logic ATL⁎, thereby extending the recently introduced GTS for ATL. We show that the 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 substantially extend ATL.

Alternating-time temporal logic ATL with finitely bounded semantics

We study a variant ATLFB of the alternating-time temporal logic ATL with a non-standard, ‘finitely bounded’ semantics (FBS). FBS was originally defined as a game-theoretic semantics where players must commit to time limits when attempting to verify eventuality (respectively, to falsify safety) formulae. It turns out that FBS has a natural corresponding compositional semantics that essentially evaluates formulae only on finite initial segments of paths and imposes uniform bounds on all plays for the fulfilment of eventualities. The resulting version ATLFB differs in some essential features from the standard ATL, as it no longer has the finite model property, though the two logics are equivalent on finite models.We develop two tableau systems for ATLFB. The first one deals with infinite sets of formulae and may run in a transfinite sequence of steps, whereas the second one deals only with finite sets of formulae in an extended language allowing explicit symbolic indication of time limits in formulae. We prove soundness and completeness of the infinitary tableau system and prove that it is equivalent to the finitary one. We also show that the finitary tableau system provides an exponential-time decision procedure for the satisfiability problem of ATLFB and thus establishes its EXPTIME-completeness. Furthermore, we present an infinitary axiomatization for ATLFB and prove its soundness and completeness.

Reasoning about Strategic Abilities

In alternating-time temporal logic ATL * , agents with perfect recall assign choices to sequences of states, i.e., to possible finite histories of the game. However, when a nested strategic modality is interpreted, the new strategy does not take into account the previous sequence of events. It is as if agents collect their observations in the nested game again from scratch, thus, effectively forgetting what they observed before. Intuitively, it does not fit the assumption of agents having perfect recall of the past. In this article, we investigate the alternative semantics for ATL * where the past is not forgotten in nested games. We show that the standard semantics of ATL * coincides with the “truly perfect recall” semantics for agents with perfect information and in case of so-called “objective” abilities under uncertainty. On the other hand, the two semantics differ significantly for the most popular (“subjective”) notion of ability under imperfect information. The same applies to the standard vs. “truly perfect recall” semantics of ATL * with persistent strategies. We compare the relevant variants of ATL * by looking at their expressive power, sets of validities, and tractability of model checking.

Optimal Tableau Method for Constructive Satisfiability Testing and Model Synthesis in the Alternating-Time Temporal Logic ATL +

We develop a sound, complete, and practically implementable tableau-based decision method for constructive satisfiability testing and model synthesis for the fragment ATL + of the full alternating-time temporal logic ALT * . The method extends in an essential way a previously developed tableau-based decision method for ATL and works in 2EXPTIME, which is the optimal worst-case complexity of the satisfiability problem for ATL + . We also discuss how suitable parameterizations and syntactic restrictions on the class of input ATL + formulas can reduce the complexity of the satisfiability problem.

State and path coalition effectivity models of concurrent multi-player games

We consider models of multi-player games where abilities of players and coalitions are defined in terms of sets of outcomes which they can effectively enforce. We extend the well studied state effectivity models of one-step games in two different ways. On the one hand, we develop multiple state effectivity functions associated with different long-term temporal operators. On the other hand, we define and study coalitional path effectivity models where the outcomes of strategic plays are infinite paths. For both extensions we obtain representation results with respect to concrete models arising from concurrent game structures. We also apply state and path coalitional effectivity models to provide alternative, arguably more natural and elegant semantics to the alternating-time temporal logic ATL*, and discuss their technical and conceptual advantages.

Alternating-time temporal logic with finite-memory strategies

Model-checking the alternating-time temporal logics ATL and ATL* with incomplete information is undecidable for perfect recall semantics. However, when restricting to memoryless strategies the model-checking problem becomes decidable. In this paper we consider two other types of semantics based on finite-memory strategies. One where the memory size allowed is bounded and one where the memory size is unbounded (but must be finite). This is motivated by the high complexity of model-checking with perfect recall semantics and the severe limitations of memoryless strategies. We show that both types of semantics introduced are different from perfect recall and memoryless semantics and next focus on the decidability and complexity of model-checking in both complete and incomplete information games for ATL/ATL*. In particular, we show that the complexity of model-checking with bounded-memory semantics is Delta_2p-complete for ATL and PSPACE-complete for ATL* in incomplete information games just as in the memoryless case. We also present a proof that ATL and ATL* model-checking is undecidable for n >= 3 players with finite-memory semantics in incomplete information games.

On timed alternating simulation for concurrent timed games

We address the problem of alternating simulation refinement for concurrent timed games (TG). We show that checking timed alternating simulation between TG is EXPTIME-complete, and provide a logical characterization of this preorder in terms of a meaningful fragment of a new logic, TAMTL*. TAMTL* is an action-based timed extension of standard alternating-time temporal logic ATL*, which allows to quantify over strategies where the designated coalition of players is not responsible for blocking time. While for full TAMTL*, model-checking TG is undecidable, we show that for its fragment TAMTL, corresponding to the timed version of ATL, the problem is instead decidable and in EXPTIME.

Tableau-based decision procedures for logics of strategic ability in multiagent systems

We develop an incremental tableau-based decision procedure for the alternating-time temporal logic ATL and some of its variants. While running within the theoretically established complexity upper bound, we believe that our tableaux are practically more efficient in the average case than other decision procedures for ATL known so far. Besides, the ease of its adaptation to variants of ATL demonstrates the flexibility of the proposed procedure.

What groups do, can do, and know they can do: an analysis in normal modal logics

We investigate a series of logics that allow to reason about agents' actions, abilities, and their knowledge about actions and abilities. These logics include Pauly's Coalition Logic CL, Alternating-time Temporal Logic ATL, the logic of ‘seeing-to-it-that' (STIT), and epistemic extensions thereof. While complete axiomatizations of CL and ATL exist, only the fragment of the STIT language without temporal operators and without groups has been axiomatized by Xu (called Ldm). We start by recalling a simplification of the Ldm that has been proposed in previous work, together with an alternative semantics in terms of standard Kripke models. We extend that semantics to groups via a principle of superadditivity, and give a sound and complete axiomatization that we call Ldm G. We then add a temporal ‘next' operator to Ldm G, and again give a sound and complete axiomatization. We show that Ldm G subsumes coalition logic CL. Finally, we extend these logics with standard S5 knowledge operators. This enables us to express that agents see to something under uncertainty about the present state or uncertainty about which action is being taken. We focus on the epistemic extension of X-Ldm G, noted E-X-Ldm G. In accordance with established terminology in the planning community, we call this extension of X-Ldm G the conformant X-Ldm G. The conformant X-Ldm G enables us to express that agents are able to perform a uniform strategy. We conclude that in that respect, our epistemic extension of X-Ldm G is better suited than epistemic extensions of ATL.

Model checking MASL specification of distributed real-time systems

We present model checking algorithms for MASL specification of distributed real-time systems. The proposed algorithms use symbolic model checking approach by analogy with model checking algorithms for branching-time temporal logic CTL and alternating-time temporal logic ATL. For the fixed environment case, the algorithm is polynomial-time in the specification length and sizes of the sets of system states and actions. For the dynamic environment case, the algorithm is polynomial-time in the model size, but it is exponential-time in the structure of environment specification.

Comparing Semantics of Logics eor Multi-Agent Systems

We draw parallels between several closely related logics that combine – in different proportions – elements of game theory, computation tree logics, and epistemic logics to reason about agents and their abilities. These are: the coalition game logics CL and ECL introduced by Pauly in 2000, the alternating-time temporal logic ATL developed by Alur, Henzinger and Kupferman between 1997 and 2002, and the alternating-time temporal epistemic logic ATEL by van der Hoek and Wooldridge (2002). In particular, we establish some subsumption and equivalence results for their semantics, as well as interpretation of the alternating-time temporal epistemic logic into ATL. The focus in this paper is on models: alternating transition systems, multi-player game models (alias concurrent game structures) and coalition effectivity models turn out to be intimately related, while alternating epistemic transition systems share much of their philosophical and formal apparatus. Our approach is constructive: we present ways to transform between different types of models and languages.

Alternating-time temporal logic

Temporal logic comes in two varieties: linear-time temporal logic assumes implicit universal quantification over all paths that are generated by the execution of a system; branching-time temporal logic allows explicit existential and universal quantification over all paths. We introduce a third, more general variety of temporal logic: alternating-time temporal logic offers selective quantification over those paths that are possible outcomes of games, such as the game in which the system and the environment alternate moves. While linear-time and branching-time logics are natural specification languages for closed systems, alternating-time logics are natural specification languages for open systems. For example, by preceding the temporal operator "eventually" with a selective path quantifier, we can specify that in the game between the system and the environment, the system has a strategy to reach a certain state. The problems of receptiveness, realizability, and controllability can be formulated as model-checking problems for alternating-time formulas. Depending on whether or not we admit arbitrary nesting of selective path quantifiers and temporal operators, we obtain the two alternating-time temporal logics ATL and ATL*.ATL and ATL* are interpreted over concurrent game structures . Every state transition of a concurrent game structure results from a choice of moves, one for each player. The players represent individual components and the environment of an open system. Concurrent game structures can capture various forms of synchronous composition for open systems, and if augmented with fairness constraints, also asynchronous composition. Over structures without fairness constraints, the model-checking complexity of ATL is linear in the size of the game structure and length of the formula, and the symbolic model-checking algorithm for CTL extends with few modifications to ATL. Over structures with weak-fairness constraints, ATL model checking requires the solution of 1-pair Rabin games, and can be done in polynomial time. Over structures with strong-fairness constraints, ATL model checking requires the solution of games with Boolean combinations of Büchi conditions, and can be done in PSPACE. In the case of ATL*, the model-checking problem is closely related to the synthesis problem for linear-time formulas, and requires doubly exponential time.

Alternating-time temporal logic

Temporal logic comes in two varieties: linear-time temporal logic assumes implicit universal quantification over all paths that are generated by the execution of a system; branching-time temporal logic allows explicit existential and universal quantification over all paths. We introduce a third, more general variety of temporal logic: alternating-time temporal logic offers selective quantification over those paths that are possible outcomes of games, such as the game in which the system and the environment alternate moves. While linear-time and branching-time logics are natural specification languages for closed systems, alternating-time logics are natural specification languages for open systems. For example, by preceding the temporal operator eventually with a selective path quantifier, we can specify that in the game between the system and the environment, the system has a strategy to reach a certain state. The problems of receptiveness, realizability, and controllability can be formulated as model-checking problems for alternating-time formulas. Depending on whether or not we admit arbitrary nesting of selective path quantifiers and temporal operators, we obtain the two alternating-time temporal logics ATL and ATLa.ATL and ATLa are interpreted over concurrent game structures. Every state transition of a concurrent game structure results from a choice of moves, one for each player. The players represent individual components and the environment of an open system. Concurrent game structures can capture various forms of synchronous composition for open systems, and if augmented with fairness constraints, also asynchronous composition. Over structures without fairness constraints, the model-checking complexity of ATL is linear in the size of the game structure and length of the formula, and the symbolic model-checking algorithm for CTL extends with few modifications to ATL. Over structures with weak-fairness constraints, ATL model checking requires the solution of 1-pair Rabin games, and can be done in polynomial time. Over structures with strong-fairness constraints, ATL model checking requires the solution of games with Boolean combinations of Buchi conditions, and can be done in PSPACE. In the case of ATLa, the model-checking problem is closely related to the synthesis problem for linear-time formulas, and requires doubly exponential time.