Exploring theKD45n Property of a Kripke Model after the Execution of an Action Sequence

The paper proposes a condition for preserving the KD45n property of a Kripke model when a sequence of update models is applied to it. The paper defines the notions of a primitive update model and a semi-reflexive KD45n (or sr-KD45n) Kripke model. It proves that updating a sr-KD45n Kripke model using a primitive update model results in a sr-KD45n Kripke model, i.e., a primitive update model preserves the properties of a sr-KD45n Kripke model. It shows that several update models for modeling well-known actions found in the literature are primitive. This result provides guarantees that can be useful in presence of multiple applications of actions in multi-agent system (e.g., multi-agent planning). Introduction and Motivations In a multi-agent action setting, agents need to not just reason about the properties of the world, but also about their knowledge and beliefs about the world and about other agents’ own knowledge and beliefs. Among the various formalizations of multi-agent actions and their impact on a physical and mental world, the action models introduced in (Baltag and Moss 2004; Baltag, Moss, and Solecki 1998) and later extended to update models in (van Benthem, van Eijck, and Kooi 2006; van Ditmarsch, van der Hoek, and Kooi 2007) are the most widely accepted. For example, these action models and update models1 have been employed in the study of epistemic planning problems in MAS (Bolander and Andersen 2011; Löwe, Pacuit, and Witzel 2011; van der Hoek and Wooldridge 2002; Baral et al. 2013). However, there is something fundamental missing in these formulations. Let us consider a variant of the coin example from (Baltag and Moss 2004): there are two agents, 1 and 2, and a coin in a box. The coin lies heads up, but the two agents are unaware of this fact. Let us assume that agent 1 alone learns that the coin lies heads up (e.g., by peeking into the box while 2 is looking away). Intuitively, in ∗The first two authors were partially supported by NSF-HRD 1345232. Copyright c © 2015, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. For the sake of simplicity, we will use the generic term “update model” to denote both “action models” and “update models.” Furthermore, we will frequently use the terminology “execution of an action” to indicate the result of applying an update model. the resulting state, we should be able to conclude that the following formulae are true (Kiφ and Biφ represent that i knows that φ is true (or i knows φ, for short) and i believes that φ is true (i believes φ); h denotes the coin being heads up): K1h (1 knows h); ¬K2h∧¬K2¬h (2 does not know h);K1(¬K2h∧¬K2¬h) (1 knows that 2 does not know h); B2(¬K1h∧¬K1¬h) (2 believes that 1 does not know h); K1(B2(¬K1h∧¬K1¬h)) (1 knows that 2 believes that 1 does not know h); etc. Intuitively, and as well accepted, the distinction between knowledge and beliefs is that knowledge must be true in the actual world while beliefs could be false. But when one looks at the formulations in (Baltag and Moss 2004; Baltag, Moss, and Solecki 1998) the Kripke models have only one accessibility relation and only one modality is defined; in (Baltag and Moss 2004) it is referred to as “knowledge”. As one can notice, replacing B2(¬K1h ∧ ¬K1¬h) by K2(¬K1h ∧ ¬K1¬h) would not be intuitive. Also, the action and update model papers do not analyze the properties of the resulting Kripke models. In this paper, we propose a method for drawing conclusions regarding both knowledge and beliefs of agents (e.g., the above mentioned formulae) after the execution of a sequence of actions, in the spirit of belief updates (and update models) rather than the belief revision based approaches2 used in (Segerberg 1998; Baltag and Smets 2008; van Benthem 2007; van Ditmarsch 2005). We achieve this goal by noting that knowledge is reducible to true belief in certain logics (Halpern, Samet, and Segev 2009). To see how this works, let us revisit the introductory example. The initial state of beliefs of agents 1 and 2 can be represented by the pointed Kripke model on the top-left of Fig. 1 (the actual world has a double circle). The logic of this model, together with the axiom Kiφ ↔ Biφ ∧ φ (indicating knowledge is true belief), has¬Kih∧¬Ki¬h (i = 1, 2) as a consequence. The update model representing “1 peeks into the box while 2 is looking away” is given in the bottom-left of Fig. 1 which encodes the view of the action occurrence of each agent. Intuitively, it says that 1 sees the action occurs (denoted by the event σ) while 2 does not (τ ). The result of the update is the pointed Kripke model on the right. It is easy to check that the logic specified by the result of the update (with the The belief revision based approach needs an additional ordering relation and is orthogonal to our approach. Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence