Reasoning about cooperation, actions and preferences

Lena Kurzen In this paper, a logic for reasoning about coalitional power is developed which explicitly represents agents' preferences and the actions by which the agents can achieve certain results. A complete axiomatization is given and its satisfiability problem is shown to be decidable and EXPTIME-hard. - Cooperation of agents plays a major role in many fields such as computer science, economics, politics, social sciences and philosophy. Agents can decide to cooperate and to form groups in order to share complementary resources or because as a group they can achieve something better than individually. When analyzing interactive situations involving multiple agents, we are interested in what results agents can achieveindividually or together as groups. There can be many ways how agents can achieve some result. They can differ significantly, e.g. with respect to their feasibility, costs or side-effects. Hence, it is not only relevant what results groups of agents can achieve but also how exactly they can do so. In other words, plans and actions also play a central role if we want to reason about cooperation in an explicit way. However, cooperative ability of agents expressed only in terms of results and actions that lead to these results does not tell us why a group of agents would actually decide to achieve a certain result. We also need to take into account the preferences based on which the agents decide what to do. Summarizing, we can say that in interactive situations, the following three questions are of interest: What results can groups of agents achieve? How can they achieve something? Why would they want to achieve a certain result? The above considerations show that coalitional power, actions/plans and preferences play a major role in interactive situations and are moreover tightly connected. Thus, we argue that a formal theory for reasoning about agents cooperative abilities in an explicit way should also take into account actions/plans of agents and their preferences. Modal logics have been used to develop formal models for reasoning about each of these aspectsmostly separately. Coalitional power has mainly been investigated within the frameworks of ATL (Alur et al. 1998), coalition logic (Pauly 2002a) and their extensions. Recently, there have been some attempts to develop logics for reasoning about coalitional power that also take into account either agents preferences or actions. One group of such logics looks at cooperation from the perspective of cooperative games (gotnes et al. 2007a). In a non-cooperative setting preferences and strategic abilities have been considered in van Otterloo et al. (2004). Another path that has been taken in order to make coalitional power more explicit is to combine cooperation logics with (fragments of) action logics (Sauro et al. 2006; Borgo 2007; Walther et al. 2007). In this paper, a logic for reasoning about cooperation, actions and preferences (CLA+P) is developed, which is obtained by combining the cooperation logic with actions CLA (Sauro et al. 2006) with a preference logic (van Benthem et al. 2005; van Benthem et al. 2007). Soundness and completeness are shown and the logics expressivity and computational complexity are investigated. The remainder of this paper is structured as follows. Section 2 gives an overview of CLA. In Sect. 3, CLA+P is developed, a complete axiomatization is given and its expressivity is discussed. Section 4 gives complexity results and Sect. 5 concludes this work. 2 Cooperation logic with actions In this section, we briefly present the CLA developed by Sauro et al. (2006), which will be extended in the next section by combining it with a preference logic. The idea of CLA is to make coalitional power explicit by expressing it in terms of the ability to perform actions instead of expressing it directly in terms of the ability to achieve certain outcomes. CLA is a modular modal logic, consisting of an environment module for reasoning about actions and their effects, and an agents module for reasoning about agents abilities to perform actions. By combining both modules, a framework is obtained in which cooperative ability can be made more explicit. The environment is modeled as a transition system whose edges are labeled with sets of atomic actions. Definition 1 (Environment Model (Sauro et al. 2006)) An environment model is a set-labelled transition system E = S, Ac, ()AAc, V . S is a set of states, Ac is a finite set of atomic actions, A S S and V is a propositional valuation. Each A is required to be serial. The intuition being s A t is that if in s all actions and only the actions in A are performed concurrently, then the next state can be t . Then a modal language is defined with modalities [], for being a propositional formula built from atomic actions. The intended meaning of [] is that every transition A such that A (using the satisfaction relation of propositional logic1) leads to a -state: Due to space restrictions, we cannot go into the underlying philosophy of actions but refer the reader to Broersen (2003) for a detailed discussion of action logics. The restriction to a finite set of actions is reasonable for modelling many concrete situations and also ensures that we have a finite axiomatization. An environment logic E is developed, which is sound and complete (Sauro et al. 2006). It contains seriality axioms and the K axiom for each modality [], for being consistent. The environment logic can then be used for reasoning about the effects of concurrent actions. Then an agents module is developed for reasoning about the ability of (groups of) agents to act. Each agent is assigned a set of atomic actions and a group is assigned the actions its members can perform. Definition 2 (Agents Model (Sauro et al. 2006)) An agents model is a triple Ag, Ac, act , where Ag is a set of agents, Ac is a set of atomic actions and act is a function act: Ag P( Ac) such that iAgact(i ) = Ac. For G Ag, define act(G) := iG act(i ). We are also interested in agents abilities to force more complex actions. A language is developed with expressions [G] , meaning that the group G can force that a concurrent action is performed that satisfies . This means that G can perform some set of atomic actions such that no matter what the other agents do, the resulting set of actions satisfies . [G] iff A act(G) : B act( Ag\G) : A B Then a cooperation logic for actions is developed, which is very much in the style of coalition logic (Pauly 2002a)the main difference being that it is concerned with the cooperative ability to force actions. Definition 3 (Coalition Logic for Actions (Sauro et al. 2006)) The coalition logic for actions A is defined to be the logic derived from the following set of axioms, with rule of inference modus ponens. 1 That is, A a iff a A, A iff A , and A iff A and A . (1) [G] , for all G Ag, (2) [G] [ Ag\G] , (3) [G] [G] if in proposition (...truncated)