M i n i m a l Change and M a x i m a l Coherence: A Basis for Belief Revision and Reasoning about Actions A n a n d S. Rao Australian AI Institute Carlton, Vic-3053 Australia Email:
[email protected] Abstract The study of belief revision and reasoning about actions have been two of the most active areas of research in A I . Both these areas involve reasoning about change. However very little work has been done in analyzing the principles common to both these areas. This paper presents a formal characterization of belief revision, based on the principles of minimal change and maximal coherence. This formal theory is then used to reason about actions. The resulting theory provides an elegant solution to the conceptual frame and ramification problems. It also facilitates reasoning in dynamic situations where the world changes during the execution of an action. The principles of minimal change and maximal coherence seem to unify belief revision and reasoning about actions and may form a fundamental core for reasoning about other dynamic processes that involve change.
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Introduction
Belief revision is the process by which an agent revises her set of beliefs at the current instant of time, based on some input from the external world, to move into the next instant of time, possibly with a different set of beliefs. Thus the essential problem in belief revision is determining what beliefs the agent must acquire and what beliefs the agent must give up. Reasoning about actions usually involves determining the state of the world after an action has been performed. Again this involves determining what beliefs should change and what beliefs should not change when moving from one state to another. The latter problem has long been known as the frame problem [McCarthy and Hayes, 1969]. Associated with this is the problem of specifying all the sideeffects or ramifications of an action and making sure that the agent acquires (or gives up) beliefs about those side-effects as well. Both belief revision and reasoning about actions,
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N o r m a n Y. Foo University of Sydney Sydney, NSW-2006 Australia Email:
[email protected] involve reasoning about change. However very little work has been done in analyzing the principles common to both these areas. Work on belief revision has for the most part concentrated on building efficient systems that perform belief revision [Doyle, 1979, de Kleer, 1986]. Work on reasoning about actions has addressed both the computational and foundational aspects [Brown, 1987, Georgeff and Lansky, 1987], but has been studied independently of belief revision. In this paper, we present a unified picture of both areas by formalizing the underlying principles of belief revision and reasoning about actions. Minimal change and maximal coherence are two of the most important principles involved in reasoning about change. Minimization has been used in AI and philosophical logic in a variety of ways. In this paper, by minimizing change we mean minimizing the acquisition of beliefs or minimizing the loss of beliefs when an agent moves from one state to another. By maximizing coherence we mean retaining as many coherent beliefs as possible during a state change. Theories of belief revision can be divided into two broad categories - the coherence and foundational theories of belief revision. A detailed analysis of both these theories has been carried out elsewhere [Rao and Foo, 1989]. In this paper we give a brief overview of the coherence theory of belief revision, which is based on the principles of minimal change and maximal coherence. The axiomatization of the theory is based on the work by Alchourron, Gardenfors and Makinson [1985] (henceforth called the AGM-theory) and the semantics is based on the possible-worlds formulation. This theory of belief revision is then used to describe a theory of actions which has a clear semantics and also solves the conceptual frame and ramification problems. 1 One of the advantages of having a uniform framework for belief revision and reasoning about actions, is that it facilitates reasoning in dynamic situations, where the world changes during the execution of an action. As the paper is part of the first author's thesis, inNo claims are being made regarding the computational efficiency of our solution. We address only the conceptual problem in this paper.
than s i t u a t i o n calculus [ M c C a r t h y and Hayes, 1969] and is equivalent in expressive power to the extended situation calculus discussed by GeorgefF [1987]. T h e semantics of d y n a m i c operators is based on selection functions, which select some possible worlds as being closer to the current w o r l d t h a n the others. W h e n the agent performs expansion or cont r a c t i o n , he is said to move i n t o one of these closer worlds and designate these worlds as the worlds of the next t i m e instant(s). T h e interpretation for this language and the semantics of the d y n a m i c operations are f o r m a l l y defined as follows: D e f i n i t i o n : T h e dynamic coherence interpretation is a tuple, CI = < C, A K I ) , where and are expansion and contraction functions t h a t map W (worlds), A C (agent constants), T D ( t i m e points), and (set of worlds) to a 2 T £ ) (set of t i m e points) and A K I is an autoepistemic K r i p k e i n t e r p r e t a t i o n . D e f i n i t i o n : A dynamic coherence i n t e r p r e t a t i o n C I , satisfies a well-formed f o r m u l a at w o r l d w and t i m e point 3 t ( w r i t t e n a s C I , w , t g i v e n the following conditions,
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T h e o r e m : T h e coherence m o d a l system or the C S - m o d a l system is sound and complete w i t h respect to the class of coherence models.
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Reasoning about Actions
In this section we shall treat actions as a special type of d y n a m i c operation and define it in terms of the other d y n a m i c operators, EXP, CON and REV. T h u s reasoning about actions is also based on the i n t u i t i v e principles of m i n i m a l change and m a x i m a l coherence. As expansions and contractions can be viewed as the semantic counterparts of the addlist and deletelist of S T R I P S [Fikes and Nilsson, 1971], the logic described here can be viewed as a semantic theory for S T R I P S and the principles of m i n i m a l change and m a x i m a l coherence as a generalization of the S T R I P S assumption. In a d d i t i o n to predicate letters the logic of actions consists of action letters, which are used to f o r m action formulas. T h e symbols ' ; ' and denote sequential and parallel actions. T h e m o d a l operator A C T is used to denote the performance of an action. If the m o d a l A C T f o r m u l a is satisfiable/provable then the action is said to be successful; otherwise the action is said to have failed. T h i s allows reasoning about b o t h successful and failed actions which is crucial for reasoning about real world domains [Georgeff, 1987]. A well-formed action formula can either be a simple action f o r m u l a or a sequential or parallel action f o r m u l a . A simple action f o r m u l a is denoted by where is an action letter, are first-order terms and a is the agent who is perf o r m i n g the action. T h e m o d a l f o r m u l a A C T u) is read as 'at t i m e t, action f o r m u l a is carried out to reach t i m e p o i n t . T h e action f o r m u l a can be a c o m b i n a t i o n of sequential or parallel action formulas. Actions have usually been defined in terms ot preconditions, which must hold before the action is performed, and the consequents, which must hold after the action has been performed. T h i s approach has been used in s i t u a t i o n calculus [ M c C a r t h y and Hayes, 1969] and S T R I P S [Fikes and Nilsson, 1971]. T h u s , associated w i t h every simple action f o r m u l a ifr, are two formulas p and c, which are respectively the precondition and consequent of . Given the current w o r l d , a t i m e point and a simple action f o r m u l a , the action function A selects the closest possible world(s) at the next t i m e p o i n t ( s ) , t h a t satisfy the f o l l o w i n g conditions: • the precondition f o r m u l a of the action is believed in the current world • the consequent of the action must be believed at the next t i m e instant, with minimal change and maximal coherence with respect to the original
world. T h e interesting part of the condition is the italicized p a r t which helps in solving the frame and ramificat i o n problems. In the case of sequential action for-
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the agent NATURE at time t1 with respect to the formula-on(B, table). It is impossible to have a single time point where the move action by the agent and the above revision be satisfiable. As the agent has no control over the changes caused by NATURE, we can impose a partial order on the dynamic operations and conclude that the revision performed by the agent NATURE succeeds and the move action by the agent fails. Thus the unified theory of belief revision and reasoning about actions provides a framework for reasoning about dynamic, real-world situations, where the world changes during the performance of an action.
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ory of actions provides an inferential solution and a model-theoretic solution to the frame and r a m i f i cation problems based on the principles of m i n i m a l change and m a x i m a l coherence. T h e example presented above is a very simple one and has been used m a i n l y to illustrate the theory. More complicated, multiple-agent, dynamic-world problems are discussed in the complete report [Rao, 1989]. For example consider the situation where block B is an ice-block and it melts before the block A can be placed on top of i t . As this is a change caused by nature, we essentially have a revision by
Comparison
A G M Theory There are significant differences between the coherence theory of belief revision as outlined by Gardenfors et. al. [1985, 1988] and that of section 2. Firstly, AGM-theory carries out the dynamic analysis at a meta-level and hence lacks a model-theory and the soundness and completeness result as outlined in this paper. Also unlike the theory outlined in section 2, AGM-theory does not handle nested beliefs and does not exhibit static nonmonotonic properties [Rao, 1989]. Finally, AGM-theory d oes not consider reasoning about actions. Lifschitz's A p p r o a c h Lifschitz [1987] introduces two predicates, success and affects, to define the notion of a successful action and the notion of what fluents are affected by an action. Using these, he defines the law of change(LC) and law of inertia(LI) By circumscribing the predicate causes, Lifschitz is able to solve the frame problem. This is hardly surprising because causes describes the dynamics of the belief system by specifying what changes when the agent moves from one state to another as a result of an action. The presence of (LC) and minimization of causes is essentially equivalent to minimizing the change caused by an action. Similarly, the presence of (LI) and minimization of causes is equivalent to maximizing the coherence between state changes. Also the definitions of causes and precond are analogous to the axiom of simple actions. Hence Lifschitz's theory essentially embodies the principles of minimal change and maximal coherence. However there are significant differences between our approach and that of Lifschitz. Firstly, Lifschitz's approach does not solve the ramification problem. Also the minimization process of Lifschitz is local [Hanks and McDermott, 1987] (i.e., with respect to a particular predicate) and relies on the user to specify the world in certain ways, whereas in the theory proposed here the minimization process is global (i.e., with respect to all the beliefs of the agent) and is independent of the syntax. Finally, the approach is incapable of handling dynamic situations. This is because the ontology is not powerful enough to represent or reason about changes caused
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by the external world. Shoham's A p p r o a c h Based on the notion of preferential logics, Shoham [1987] introduces the logic of chronological ignorance. The underlying logic is a modal logic of temporal knowledge. A model M2 of the above logic is preferred over another model M\} if Mi is chronologically more ignorant than M\. The preferred models in the logic of chronological ignorance are the chronologically maximally ignorant (c.m.i.) models. By considering only the c.m.i. models of the given problem Shoham is able to solve the qualification problem (see the complete report [Rao, 1989] for more details) and extended prediction problem (a generalized form of the frame problem). The notion of c.m.i. models can be simplified as follows - (1) prefer maximally ignorant models at each state (or time point) and (2) prefer maximally ignorant models when moving from one time point to another or during a state change. The first solves the qualification problem and the second solves the frame problem. Notice that in both cases one need not worry about a sequence of time points chrono5 logically backward from the current time point. The net effect of revision is to believe as much of the original beliefs as possible and acquire as few beliefs as required. Acquiring as few beliefs as possible is the same as minimization of beliefs or maximization of non-beliefs (or ignorance in the case of knowledge). Thus (2) above, embodies the principle of minimal change. The main differences between Shoham's approach and the one outlined in this paper are as follows. Firstly, his theory does not provide an axiomatic approach to the frame and ramification problems. Secondly, the principles of minimal change and maximal coherence are more general than the notion of chronologically maximizing ignorance. As a result our theory can be applied to database updates, belief revision, and reasoning about actions. However, it should be noted that Shoham defines a special class of theories called causal theories, which have an unique c.m.i model, and gives an algorithm for computing it. As mentioned before, this paper does not address the issue of efficient algorithms for computing contractions and revisions. Reiter's Approach In default logic [Reiter, 1980] one assumes by default that all actions and facts are normal. Actions which change the truth-value of certain facts are considered to be abnormal with respect to those facts. Thus given the default rules and the abnormality conditions the theory is supposed to predict the facts that will hold after the action has been performed. Unfortunately, this approach and other similar approaches using circumscription lead to counter-intuitive results as shown by Hanks and McDermott [1987]. 5
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This fact was also noted by Lifschitz [1987].
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In our opinion, the classical nonmonotonic theories like default logic and circumscription are not suitable for reasoning about change because they do not take into account the interaction with the environment. They are suitable only for static reasoning where the agent "jumps to conclusions" based on incomplete information ]Rao, 1989]. Attempts to rectify the problem and make them suitable for dynamic reasoning, require extensive reformulation as done by Lifschitz and in any case has to incorporate some principles of change as discussed in this paper. Possible-worlds A p p r o a c h Ginsberg and Smith [1987] first proposed the use of the possible-worlds model for solving the qualification, frame, and ramification problems. They provide a constructive procedure for computing the closest possible worlds. Their approach has certain problems [Winslett, 1988] as they minimize the formulas in the world and not the beliefs of the agent at a particular world as done in this paper. The significant difference between the possible-models approach (PMA) [Winslett, 1988] and our approach is the inability of the PMA to represent uncommitted states of the agent. This makes the notion of minimal change different in the two theories. None of the above approaches are capable of han6 dling dynamic situations (the ice block example), where the state of the world changes during the execution of an action. This requires an explicit coupling of belief revision and reasoning about actions as carried out in this paper. Such situations will become more common when one starts addressing real-world problems.
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Conclusion
This paper unifies the work in belief revision and reasoning about actions by analyzing the principles of minimal change and maximal coherence which forms a core part of both the areas. A sound and complete theory of belief revision is provided which is then used to represent and reason about actions. This theory of action solves the conceptual frame and ramification problems by utilizing the intuitive principles of minimal change and maximal coherence. It also provides a framework to reason about dynamic situations. A c k n o w l e d g e m e n t : We would like to thank Mike Georgeff, Dave Wilkins and the anonymous referees' for their helpful comments. The first author was supported by the Sydney University Postgraduate Scholarship and the Research Foundation for Information Technology. Thanks are also due to IBM Corp. for their generous support during our stay at IBM T.J. Watson Research Center and IBM Thornwood respectively. 6
More recently, Lifschitz and Rabinov [1989] have proposed a circumscriptive formalism to deal with dynamic situations. But their solution still suffers from some of the other drawbacks mentioned earlier and is incapable of representing multiple-agent domains.
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Axioms for Contraction CON BEL
BEL CON From A x i o m of Revision
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