Joint Revision of Beliefs and Intention

We present a formal semantical model to capture action, belief and intention, based on the “database perspective” (Shoham 2009). We then provide postulates for belief and intention revision, and state a representation theorem relating our postulates to the formal model. Our belief postulates are in the spirit of the AGM theory; the intention postulates stand in rough correspondence with the belief postulates. Introduction and Motivation While there is an extensive literature developing logical models to reason about changing informational attitudes (eg., belief, knowledge, certainty), other mental states have received less attention1. However, this is changing with recent articles introducing dynamic logics of intention (van der Hoek, Jamroga, and Wooldridge 2007; Herzig and Lorini 2008)2. These papers take as a starting point logical frameworks derived from Cohen and Levesque’s seminal paper (Cohen and Levesque 1990) aimed at formalizing Bratman’s planning theory of intention (Bratman 1987). In this paper we take a different angle on intentions, focusing on intention revision as it relates to, and is intertwined with, belief revision. We view the problem of intention revision as a database management problem (see (Shoham 2009) for more on the conceptual underpinnings of this standpoint). At any given moment, an agent must keep track of a number of facts about the current situation. This includes beliefs about the current state, beliefs about possible future states, which actions are available now and in the future, and also what the agent plans to do at future moments. It is important that all of this information be jointly consistent at any given moment and furthermore can be modified as needed while maintaining consistency. In the following we introduce a simple logic that formally models such a “database”. That is, consistency in this logic Copyright c © 2009, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. A notable exception is work on logics of preferences and preference change. See (van Benthem 2009) for a survey of recent work. See also a recent discussion of “goal dynamics” in (Castelfranchi and Paglieri 2007). is meant to represent not only that the agent’s beliefs are consistent and the agent’s future plan is consistent, but also that the agent’s beliefs and intentions together form a coherent picture of what may happen, and of how the agent’s own actions will play a role in what happens. Many of the BDIstyle logics emanating from (Cohen and Levesque 1990) can be viewed as addressing this issue (Rao and Georgeff 1992; Meyer, van der Hoek, and van Linder 1999, are two examples). Our primary contribution in this article (in line with the recent articles on dynamic BDI logics mentioned above) is to focus also on how the database is to be modified, and in the process to provide a clear picture of how intentions and beliefs relate. What can cause an agent’s database to change? In this paper, we focus on two main sources: 1. The agent makes some observation, e.g. from sensory input. If the new observation is inconsistent with the agent’s beliefs, these beliefs will have to be revised to accommodate it. While we are fully aware of their shortcomings, in particular when it comes to iterated revision, our account of belief revision follows the classical AGM postulates (Alchourrón, Gärdenfors, and Makinson 1985) rather closely. The goal is thus to give general conditions on a single revision with new information that the agent has already committed to incorporating. 2. The agent forms a new intention. Here we focus on future directed intentions, understood as time-labelled actions that might make up a plan. Analogously to belief revision, it is assumed the agent has already committed to a new intention, so it must be accommodated by any means short of revising beliefs. The force of the theory is in restricting how this can be accomplished. To be more precise, we purport to model an intelligent database, which receives instructions from some planner (e.g. a STRIPS-like planner) that is itself engaged in some form of practical reasoning. The job of the database is to maintain consistency and coherence between intentions and beliefs. This simple description, however, obscures some important subtleties in the interaction between beliefs and intentions, subtleties we would also like to capture. The following will serve as a running example. Suppose an agent intends to drive to the city at 6:00 this evening. Upon adopting this intention, the agent will come to have new beliefs based on the predicted success of this intention, e.g. that he will be in the city by 7:00. These further beliefs are important in the course of further planning, for instance, what he will do in the city. The intention is also supported by the absence of certain beliefs. It would be irrational to form this intention if the agent believed his car was not working and this was the only means of getting there. Likewise, even if originally the agent thought his car might be working, upon learning that it is not and lacking other ideas of how to get there, the intention to go to the city should be dropped. Yet, by dropping this intention that was based on the now-dropped belief, other beliefs, including the belief that he will be in the city by 7:00, should also be dropped, which may in turn force other intentions and beliefs to be dropped. And so on. To deal with these subtleties, we treat intentioncontingent beliefs, or simply contingent beliefs, and concrete “physical” beliefs, or non-contingent beliefs, separately. Non-contingent beliefs concern the world as it is, independent of the agent’s future plans, but including what (sequences of) actions will be physically possible. Thus, in addition to non-contingent atomic facts, the agent will have beliefs about what the preconditions and postconditions of actions are, and about which sequences of actions might be possible. Our treatment of contingent beliefs is similar to the notion of weak belief in (van der Hoek, Jamroga, and Wooldridge 2007), but differs in an important respect. We assume that the postconditions of intended actions are believed in this stronger sense, but that the preconditions need not be believed. The intuition behind this decision is that, from the perspective of a planner, the postconditions of intended actions are justifiably believed merely by the fact that the agent has committed to bringing them about. In this way, these beliefs are contingent on the success of the agent’s plans. The preconditions, on the other hand, may still present a practical problem yet to be solved by the planner. To say that they are believed underrates the fact that they are not directly justified by any future intended action. Hence, contingent beliefs are simply derived from the agent’s non-contingent beliefs by adding the postconditions (and all consequences) of any intended actions. These kinds of beliefs might also be called “optimistic” beliefs, since the agent assumes the success of the action without ensuring the preconditions hold. In this way, our account avoids the potentially infinite regress alluded to above by allowing belief revision to trigger intention revision, but restricting intention revision to trigger belief revision only in this stronger, derivative sense of contingent belief. In the next section, we describe the belief and intention revision postulates on an informal level before going into formal details and definitions. We then define the logic underlying the database, as a simple temporal logic with transitions labeled by actions. The models of this logic are then used to give a semantic characterization of our revision operations, which are shown in the next section to represent our main postulates. Finally in the last section before the conclusion, we define a notion of contingent beliefs, as described above, and provide postulates for revision of these beliefs, as derived from the separate postulates for beliefs and intentions. A proof sketch of our representation theorem appears in an appendix. A Preview of the Postulates The main aim of our framework is to implement the “database perspective” on intentions in the form of a dual logical theory of belief and intention revision. In this section, before going into the formalities of our framework, we offer a preview of the revision postulates that we will be working toward for the rest of the paper. Relevant definitions of key terms like coherence will come later. If B is a set of non-contingent beliefs (i.e. a set of formulas, which by definition does not depend on intentions) and I is a set of intentions (which shall be action/time pairs (a, t), including an empty pair !), we shall define a class of intention revision operators ◦ that adhere to the following restrictions when 〈B, I〉 ◦ (a, t) = 〈B′, I ′〉 for some proposed new intention (a, t). 1. 〈B′, I ′〉 is coherent; 2. If 〈B, {(a, t)}〉 is coherent, then (a, t) ∈ I ′; 3. If 〈B, I ∪ {(a, t)}〉 is coherent, then I ∪ {(a, t)} ⊆ I ′; 4. I ′ ⊆ I ∪ {(a, t)};