Reasoning about Resource-Sensitive Multi-Agents

Abstract

Aims. Formalizing common knowledge reasoning in multi-agent systems is of growing importance in Computer Science, Artificial Intelligence, Economics, Philosophy and Psychology. Obtaining a concrete logical foundation for common knowledge reasoning plays an important role for formal treatment and verification ofmulti-agent systems. For this reason, formalizing common knowledge reasoning is also a traditional issue for multi-agent epistemic logics (Fagin et al., 1995; Halpern & Moses, 1992; Lismont & Mongin, 1994; Meyer & van der Hoek, 1995). The aim of this paper is to formalize more fine-grained common knowledge reasoning by a new logical foundation based on Girard’s linear logics. Common knowledge. The notion of common knowledge was probably first introduced by Lewis (Lewis, 1969). This notion is briefly explained below. Let A be a fixed set of agents and α be an idea. Suppose that α belongs to the common knowledge of A, and i and j are some members of A. Then, we have the facts “both i and j know α”, “i knows that j knows α” and “j knows that i knows α”. Moreover, we also have the facts “i knows that j knows that i knows α”, and so on. Then, these nesting structures develop an infinite hierarchy as a result. Iterative interpretation. Suppose that the underlying multi-agent logic has the knowledge operators ♥1,♥2, ..., ♥n, in which a formula ♥iα means “the agent i knows α.” The common knowledge of a formula α is defined below. For any m ≥ 0, an expression Km means the set {♥i1♥i2 · · · ♥im | each ♥it is one o f ♥1, ...,♥n and it = it+1 f or all t = 1, ...,m− 1}. When m = 0, ♥i1♥i2 · · · ♥im is interpreted as the null symbol. The common knowledge ♥cα of α is defined by using an infinitary conjunction ∧ as the so-called iterative interpretation of common knowledge:♥cα := ∧ {♥α | ♥ ∈ ⋃