Thomas Ågotnes

Constructive knowledge: what agents can achieve under imperfect information

We propose a non-standard interpretation of Alternating-time Temporal Logic with imperfect information, for which no commonly accepted semantics has been proposed yet. Rather than changing the semantic structures, we generalize the usual interpretation of formulae in single states to sets of states. We also propose a new epistemic operator for “practical” or “constructive” knowledge, and we show that the new logic (which we call Constructive Strategic Logic) is strictly more expressive than most existing solutions, while it retains the same model checking complexity. Finally, we study properties of constructive knowledge and other operators in this non-standard semantics.

Group Announcement Logic

Two currently active strands of research on logics for multi-agent systems are dynamic epistemic logic, focusing on the epistemic consequences of actions, and logics of coalitional ability, focusing on what coalitions of agents can achieve by cooperating strategically. In this paper we bridge these topics by considering the question: “what can a coalition achieve by making public announcements?”. We propose an extension of public announcement logic with constructs of the form hGi’, where G is a group of agents, with the intuitive meaning that G can jointly execute a publicly observable action such that ’ will be true afterwards. Actions here are taken to be truthful public announcements, but turn out also to include sequences of such joint actions as well as protocols with alternating actions by dierent agents, in response to the actions of others. We also study in detail the dierence between ‘knowing how’ (knowing de re) and ‘knowing that’ (knowing de dicto) in our framework: both can elegantly be expressed in the single-agent case. We present several meta-logical properties of this Group Announcement Logic, including a sound and complete axiomatisation, expressivity and the complexity of model checking. The results are based on but greatly extend a part of [2].

Normative system games

We develop a model of normative systems in which agents are assumed to have multiple goals of increasing priority, and investigate the computational complexity and game theoretic properties of this model. In the underlying model of normative systems, we use Kripke structures to represent the possible transitions of a multiagent system. A normative system is then simply a subset of the Kripke structure, which contains the arcs that are forbidden by the normative system. We specify an agents goals as a hierarchy of formulae of Computation Tree Logic (CTL), a widely used logic for representing the properties of Kripke structures: the intuition is that goals further up the hierarchy are preferred by the agent over those that appear further down the hierarchy. Using this scheme, we define a model of ordinal utility, which in turn allows us to interpret our Kripke-based normative systems as games, in which agents must determine whether to comply with the normative system or not. We then characterise the computational complexity of a number of decision problems associated with these Kripke-based normative system games; for example, we show that the complexity of checking whether there exists a normative system which has the property of being a Nash implementation is NP-complete.

Quantified coalition logic

We add a limited but useful form of quantification to Coalition Logic, a popular formalism for reasoning about cooperation in game-like multi-agent systems. The basic constructs of Quantified Coalition Logic (QCL) allow us to express such properties as “every coalition satisfying property P can achieve φ” and “there exists a coalition C satisfying property P such that C can achieve φ”. We give an axiomatisation of QCL, and show that while it is no more expressive than Coalition Logic, it is nevertheless exponentially more succinct. The complexity of QCL model checking for symbolic and explicit state representations is shown to be no worse than that of Coalition Logic, and satisfiability for QCL is shown to be no worse than satisfiability for Coalition Logic. We illustrate the formalism by showing how to succinctly specify such social choice mechanisms as majority voting, which in Coalition Logic require specifications that are exponentially long in the number of agents.

Reasoning about coalitional games

We develop, investigate, and compare two logic-based knowledge representation formalisms for reasoning about coalitional games. The main constructs of Coalitional Game Logic (cgl) are expressions for representing the ability of coalitions, which may be combined with expressions for representing the preferences that agents have over outcomes. Modal Coalitional Game Logic (mcgl) is a normal modal logic, in which the main construct is a modality for expressing the preferences of groups of agents. For both frameworks, we give complete axiomatisations, and show how they can be used to characterise solution concepts for coalitional games. We show that, while cgl is more expressive than mcgl, the former can only be used to reason about coalitional games with finitely many outcomes, while mcgl can be used to reason also about games with infinitely many outcomes, and is in addition more succinct. We characterise the computational complexity of satisfiability for cgl, and give a tableaux-based decision procedure.

Action and knowledge in alternating-time temporal logic

Alternating-time temporal logic (ATL) is a branching time temporal logic in which statements about what coalitions of agents can achieve by strategic cooperation can be expressed. Alternating-time temporal epistemic logic (ATEL) extends ATL by adding knowledge modalities, with the usual possible worlds interpretation. This paper investigates how properties of agents’ actions can be expressed in ATL in general, and how properties of the interaction between action and knowledge can be expressed in ATEL in particular. One commonly discussed property is that an agent should know about all available actions, i.e., that the same actions should be available in indiscernible states. Van der Hoek and Wooldridge suggest a syntactic expression of this semantic property. This paper shows that this correspondence in fact does not hold. Furthermore, it is shown that the semantic property is not expressible in ATEL at all. In order to be able to express common and interesting properties of action in general and of the interaction between action and knowledge in particular, a generalization of the coalition modalities of ATL is proposed. The resulting logics, ATL-A and ATEL-A, have increased expressiveness without loosing ATL’s and ATEL’s tractability of model checking.

Robust normative systems and a logic of norm compliance

Although normative systems, or social laws, have proved to be a highly influential approach to coordination in multi-agent systems, the issue of compliance to such normative systems remains problematic. In all real systems, it is possible that some members of an agent population will not comply with the rules of a normative system, even if it is in their interests to do so. It is therefore important to consider the extent to which a normative system is robust, i.e., the extent to which it remains effective even if some agents do not comply with it. We formalise and investigate three different notions of robustness and related decision problems. We begin by considering sets of agents whose compliance is necessary and/or sufficient to guarantee the effectiveness of a normative system; we then consider quantitative approaches to robustness, where we try to identify the proportion of an agent population that must comply in order to ensure success; and finally, we consider a more general approach, where we characterise the compliance conditions required for success as a logical formula. We furthermore introduce a logic for specifying properties of norm compliance in general and norm robustness in particular.

On the logic of coalitional games

We develop a logic for representing and reasoning about coalitional games without transferable payoffs. Although a number of logics of cooperation have been proposed over the past decade (notably Coalition Logic [14] and Alternating-time Temporal Logic [1]), these logics focused primarily on the issue of strategic cooperative ability -- what states a coalition can effectively enforce - and have tended to ignore the essential issue of the preferences that agents have over such states; in addition, the connection between such logics and coalitional games, in the sense of cooperative game theory, is left implicit. The Coalitional Game Logic (CGL) that we develop in this paper differs from such previous logics in two important respects. First, CGL includes operators that make it directly possible to represent an agents preferences over outcomes. Second, we interpret formulae of CGL directly with respect to coalitional games without transferable payoff, thereby establishing an explicit link between formulae of the logic and properties of coalitional games. We show that these coalitional games cannot be seen directly as models for Coalition Logic. We give a complete axiomatization of CGL, prove that it is expressively complete with respect to coalitional games without transferable payoff, show that the satisfiability problem for the logic is NP-complete, and to illustrate its use, we show how the logic can be used to characterise axiomatically a number of well-known solution concepts for coalitional games, including for example non-emptiness of the core.

What will they say?—Public Announcement Games

Dynamic epistemic logic describes the possible information-changing actions available to individual agents, and their knowledge pre- and post conditions. For example, public announcement logic describes actions in the form of public, truthful announcements. However, little research so far has considered describing and analysing rational choice between such actions, i.e., predicting what rational self-interested agents actually will or should do. Since the outcome of information exchange ultimately depends on the actions chosen by all the agents in the system, and assuming that agents have preferences over such outcomes, this is a game theoretic scenario. This is, in our opinion, an interesting general research direction, combining logic and game theory in the study of rational information exchange. In this article we take some first steps in this direction: we consider the case where available actions are public announcements, and where each agent has a (typically epistemic) goal formula that she would like to become true. What will each agent announce? The truth of the goal formula also depends on the announcements made by other agents. We analyse such public announcement games.

The Dynamics of Syntactic Knowledge

The syntactic approach to epistemic logic avoids the logical omniscience problem by taking knowledge as primary rather than as defined in terms of possible worlds. In this study, we combine the syntactic approach with modal logic, using transition systems to model reasoning. We use two syntactic epistemic modalities: ‘knowing at least’ a set of formulae and ‘knowing at most’ a set of formulae. We are particularly interested in models restricting the set of formulae known by an agent at a point in time to be finite. The resulting systems are investigated from the point of view of axiomatization and complexity. We show how these logics can be used to formalise non-omniscient agents who know some inference rules, and study their relationship to other systems of syntactic epistemic logics,