A General Approach to Multi-agent Minimal Knowledge

Abstract

We extend our general approach to characterizing informa- tion to multi-agent systems. In particular, we provide a formal descrip- tion of an agent’s knowledge containing exactly the information conveyed by some (honest)formula ϕ. Only knowing is important for dynamic agent systems in two ways. First of all,one wants to compare different states of knowledge of an agent and, secondly for agent a’s decisions it may be relevant that (he knows that)agent b does not know more than ϕ There are three ways to study the question whether a formula ϕ can be interpreted as minimal information. The first method is semantic and inspects ‘minimal’ models for ϕ (with respect to some order ≤ on states). The second one is syntactic and searches for stable expansions, minimal with respect to some language $\mathcal{L}^*$.The third method is a deductive test known as the disjunction property.We present a condition under which the three methods are equivalent. Then we show how to construct the order ≤ by collecting ‘layered or- ders’. We then focus on the multi-agent case and identify languages $\mathcal{L}^*$. for several orders ≤ and show how they yield different notions of hon- esty for different multi-modal systems.Finally some consequences of the different notions are discussed.

This author’s research partly took place during his sabbatical at CSLI, Stanford USA, which is gratefully acknowledged. He also thanks Michael Dunn (Indiana University Logic Group) and Arnis Vilks (Handelshochschule Leipzig) for kind invitations and useful comments.