Quantifying Over Coalitions in Epistemic Logic Thomas ◦ Agotnes Department of Computer Engineering, Bergen University College PB. 7030, N-5020 Bergen, Norway tag@hib.no Wiebe van der Hoek Department of Computer Science, University of Liverpool Liverpool L69 7ZF, UK wiebe@csc.liv.ac.uk Michael Wooldridge Department of Computer Science, University of Liverpool Liverpool L69 7ZF, UK mjw@liv.ac.uk ABSTRACT Some natural epistemic properties which may arise in applications can only be expressed in standard epistemic logic by formulae which are exponentially long in the number of agents in the system. An example is the property “at least m agents know that at most n agents know ϕ”. We present Epistemic Logic with Quantifica- tion over Coalitions (ELQC), where the standard common knowl- edge operator has been replaced allowing expressions of the form 〈P 〉C ϕ and [P ]C ϕ where P is a coalition predicate, meaning that there is a coalition satisfying P which have common knowledge of ϕ and that all coalitions satisfying P have common knowledge of ϕ, respectively; and similarly for distributed knowledge and everybody-knows. While the language is no more expressive than standard epistemic logic, it is exponentially more succinct. We give a sound and complete axiomatisation for ELQC, and characterise the complexity of its model checking problem. Categories and Subject Descriptors I.2.11 [Distributed Artificial Intelligence]: Multiagent Systems; I.2.4 [Knowledge representation formalisms and methods] General Terms Theory Keywords epistemic logic, expressivity, succinctness, model checking, com- plexity 1. INTRODUCTION Epistemic logic has proved to be a highly influential formalism for expressing properties of multi-agent and distributed systems [3, 8]. Central to the success of epistemic logic has been the concept of group knowledge, in the form of, e.g., common and distributed knowledge. However, conventional epistemic logics provide only very simple mechanisms for expressing group properties of knowl- edge, and specifically, some natural notions of group knowledge cannot be succinctly expressed within conventional epistemic logic. Consider the following epistemic property that one might wish to express of a system: Cite as: Quantifying Over Coalitions in Epistemic Logic, Thomas Ågotnes, Wiebe van der Hoek and Michael Wooldridge, Proc. of 7th Int. Conf. on Autonomous Agents and Multiagent Systems (AAMAS 2008), Padgham, Parkes, Müller and Parsons (eds.), May, 12-16., 2008, Estoril, Portugal, pp. 665-672. Copyright c  2008, International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org). All rights reserved. At least two agents know that at most three agents know ϕ, from an overall set of agents {1, 2, 3, 4}. The obvious way to express this fact in the conventional S5 C ,D n logic of knowledge is as follows: E {1,2} ψ ∨ E {1,3} ψ ∨ E {1,4} ψ ∨ E {2,3} ψ ∨ E {2,4} ψ ∨ E {3,4} ψ ∨ E {1,2,3} ψ ∨ E {1,2,4} ψ ∨ E {1,3,4} ψ ∨ E {2,3,4} ψ ∨ E {1,2,3,4} ψ where ψ is: (¬K1ϕ ∨¬K2ϕ ∨¬K3ϕ ∨¬K4ϕ) However, the construction has a very obvious disadvantage: the formula is (very) big. Can we do any better in S5 C ,D n ? As we will see later, the answer is no. In fact, to express the fact that at least m out of n agents know ϕ using S5 C ,D n will require a formula that is exponential in n . Since such formulae are clearly unrealistic for any practical purposes, this seems to imply that we cannot use log- ics like S5 C ,D n if we are interested in properties such as that above. The obvious answer (cf. the discussion in [1]) is to add an appara- tus for quantifying over coalitions to S5 C ,D n ; one might imagine a formula something like the following, expressing the above prop- erty: ∃G :(|G|≥ 2) ∧ EG ψ However, adding this kind of quantification into S5 C ,D n in a naive way will rapidly lead to very high complexity (possibly undecid- ability). So, can we add quantification over coalitions to S5 C ,D n in such a way that we are able to succinctly express properties such as the one above, without the logic becoming too complex to manage computationally? This paper addresses this issue. Our approach is inspired by [1], in which a similar issue was considered in the context of logics of strategic ability. We develop a logic ELQC (“Epistemic Logic with Quantification over Coalitions”), in which for example the C operator of S5 C ,D n is replaced by operators 〈P 〉C and [P ]C , where P is a coalition predicate. The idea is that 〈P 〉C ϕ means “there exists a coalition G satisfying property P such that ϕ is common knowledge in G”, while [P ]C ϕ means “it is common knowledge in every coalition G satisfying property P that ϕ”. The paper is organised as follows. In the next section we intro- duce the language of coalition predicates. Then we present ELQC by defining its syntax and semantics, showing the definition of some general epistemic properties in the language, and discussing examples of logical validities. In Section 4 we discuss the ex- pressivity of ELQC and show that while it is equally expressive 665