Establishing Mutual Beliefs by Joint Attention: Towards a Formal Model of Public Events

Establishing Mutual Beliefs by Joint Attention: Towards a Formal Model of Public Events Emiliano Lorini (emiliano.lorini@istc.cnr.it) Institute of Cognitive Sciences and Technologies, Viale Marx 15 00137, Roma, ITALY Luca Tummolini (luca.tummolini@istc.cnr.it) Institute of Cognitive Sciences and Technologies, Viale Marx 15 00137, Roma, ITALY Andreas Herzig (herzig@irit.fr) Institut de Recherche en Informatique de Toulouse, 118 Route de Narbonne F-3106, Toulouse, Cedex 4, FRANCE Abstract While the role of mutual beliefs in coordination and collaboration has been extensively acknowledged, the cognitive processes supporting their establishment are left unexplained or simply assumed. Notions like “public event” or “public announcement” usually refer to events or speech acts that create such mutual information states. The goal of this paper is to provide a formal model of the conditions under which mutual beliefs can be established. Agents should be able to perceive and reason about each other epistemic activities in a shared world. To express such reasoning a simple version of propositional dynamic logic with converse operator (CPDL) is adopted. Keywords: Mutual Belief Knowledge; Joint Attention. achievement; Common Introduction The notion of common or mutual belief is a widespread interpretive concept shared by many diverse disciplines 1 . Since the seminal work of Lewis (1969), it has been widely adopted as a crucial notion to explain coordination in a variety of social settings from discourse understanding and definite reference (Clark & Marshall 1981), to strategic reasoning in game theory (Bacharach 1992; Geanakoplos 1992), to collaborative and group activity in AI (Grosz & Kraus 1996). To act effectively in these situations, it is not enough for a group of agents that they all believe something; they should also have attitudes towards the mental states of their peers. Consequently, the problem of the genesis of common belief is of fundamental importance for modeling social interaction between cognitive agents. Such genesis is often considered as related either to public events or to public announcements. In the former situation a common belief is a consequence of an event whose occurrence is so evident (viz. public) that agents cannot but recognize it as when, during a soccer match, players mutually believe that they are playing soccer. In the latter, common belief is the product of a special event that is a communication process as when the referee publicly announces that one player is expelled. From there on each player believes that each other player believes and so on… that one of them has been expelled. Intuitively, an event is considered “public” as long as its occurrence is epistemically accessible by everybody such that it becomes common knowledge between them. Such a definition is usually given for granted but can be explicitly stated as: Public(e) ↔ (Happens(e) → CB (Happens(e))). However what are the “intuitive” conditions that make an event to be qualified as public? What are the reasons to believe that an occurring event is commonly believed? To achieve a common belief, agents need to be aware of each other current epistemic activities (attending to, looking at) both at the event itself and at each other epistemic activities. Looking at each other (i.e. by eye contact) while accessing the event provides reasons to accept the mutual information state. Such condition is usually described as joint attentional state (see for instance Tomasello 1999). Moving beyond traditional approaches to public announcements mostly focused on belief update at the group level (Baltag et al. 2003), in this paper first we introduce a propositional dynamic logic to reason about epistemic and pragmatic actions and beliefs. Then we advance a logic for perception and mutual perception that let the agents infer from the fact that they are jointly attending at something (i.e. by mutually checking whether something is true in the world) that a certain proposition is mutually believed by them. After discussing our model we conclude and point to future work. Language While knowledge is generally considered as justified true belief in this paper we will adopt only the weaker belief mental attitude. Let denumerable sets AGT = {1,…,n} of agents, Π of propositional symbols {p, q, r,…} and ACT of atomic pragmatic actions {a, b, c,…} be given. The language L is the smallest superset of Π such that: if ϕ, ψ ∈ L, i ∈ AGT, α∈ACT’ then ¬ϕ, ϕ ∨ ψ, Bel i ϕ, i ϕ, i ψ, i ψ, i ψ ∈L where ACT’ is the smallest superset of ACT such that if ϕ ∈ L and α 1 , α 2 α∈ACT’ then