Alternating-time Temporal Logics with Irrevocable Strategies Thomas ˚ Agotnes Dept. of Computer Engineering Bergen University College, Bergen, Norway tag@hib.no Valentin Goranko School of Mathematics Univ. of the Witwatersrand Johannesburg, South Africa goranko@maths.wits.ac.za Wojciech Jamroga Department of Informatics Clausthal Univ. of Technology Clausthal, Germany wjamroga@in.tu-clausthal.de Abstract In Alternating-time Temporal Logic (atl), one can express statements about the strate- gic ability of an agent (or a coalition of agents) to achieve a goal φ such as: “agent i can choose a strategy such that, if i fol- lows this strategy then, no matter what other agents do, φ will always be true”. How- ever, strategies in atl are revocable in the sense that in the evaluation of the goal φ the agent i is no longer restricted by the strategy she has chosen in order to reach the state where the goal is evaluated. In this paper we consider alternative variants of atl where strategies, on the contrary, are irrevocable. The difference between revocable and irrevo- cable strategies shows up when we consider the ability to achieve a goal which, again, involves (nested) strategic ability. Further- more, unlike in the standard semantics of atl, memory plays an essential role in the semantics based on irrevocable strategies. 1 Introduction Logics for game-like scenarios have received much in- terest recently [1, 5, 6, 8], for example as a part of the foundations of multi-agent systems. Alternating-time Temporal Logic atl [1] is probably the most popu- lar logic of this kind now. Atl is an extension of the Computational Tree Logic ctl [2], one of the most successful temporal logics in computer science. The main semantic assumption behind atl is that a sys- tem at a given time is in one of several possible states, and that the next state of the system is determined by the current state and the actions chosen by each of κ agents present in the system. Atl involves strate- gic quantifiers (called cooperation modalities ), such as 〈〈C 〉〉X and 〈〈C 〉〉G where C is a set of agents. For- mula 〈〈C 〉〉X φ is intended to mean that coalition C can achieve φ in the neX t state of the system, or, in more detail, that the agents in C can choose their strategies so that, if they use these strategies then φ will be true in the next state – no matter what the agents outside C do. Similarly, 〈〈C 〉〉Gφ means that C can force φ to be true in all future states (Globally). An interesting feature of atl is that strategies in the logic are revocable, in the sense that in the evaluation of the goal φ an agent i ∈ C is no longer restricted by the strategy she has chosen. That is, if φ includes a nested cooperation modality for a coalition including i , then i is again free to choose any strategy to demon- strate the truth of φ. This is very much in agreement with the semantics of ctl path quantifiers, where it is natural to express facts like “there is a path, such that the system can always deviate from the path to another path which satisfies φ”(EGEφ). On a more general level, this reflects the way in which quantifiers are treated in classical mathematical logic: in the for- mula ∃x (x =1 ∧∃xx = 0), the second occurrence of ∃x supersedes the first one in its scope of binding, and in consequence the formula is true in any sensible arithmetics. The semantics of the strategic quantifiers in atl works similarly, and there are many scenarios, where one would like to talk about strategies and abil- ities exactly in this way, too. However, it somehow contradicts the usual game-theoretical view of a strat- egy as a conditional plan that completely specifies the agent’s future behavior. In this paper, we focus on the latter view, and consider a variant of atl, in which strategies are irrevocable – they are chosen once and forever. We begin by recalling the language and semantics of atl, and an informal discussion on possible semantics of strategic quantifiers. Then, we present and study our alternative semantics of cooperation modalities in a formal way. Validity, satisfiability, and model check- ing problems are discussed in the subsequent sections. Irrevocable strategies are also used in the context of