Two Sides of the Same Coin? Abstract. Peter G~rdenfors. Cognitive Science. Department of Philosophy. University of Lund. S-223 50 Lund, Sweden. 1 Program.
Belief Revision and Nonmonotonic Logic: Two Sides of the Same Coin? Abstract Peter G~rdenfors Cognitive Science Department of Philosophy University of Lund S-223 50 Lund, Sweden
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Program
Belief revision and nonmonotonic logic are motivated by quite different ideas. The theory of belief revision deals with the dynamics of belief states, that is, it aims at modelling how an agent or a computer system updates its state of belief as a result of receiving new information. Of particular interest is the case where the new information is incompatible with the old state of belief. Nonmonotonic logic, on the other hand, is concerned with a systematic study of how we jump to conclusions from what we believe. By using default assumptions, generalizations etc. we tend to believe in things that do not follow from our knowledge by the classical rules of logic. A thorough understanding of this process is desirable since we want AI-systems to be able to perform the same kind of reasoning. Despite the differences in motivation for the theories of belief revision and nonmonotonic logic, I argue in this paper that the formal structures of the two theory areas, as they have developed, are surprisingly similar. My aim is to show that it is possible to translate concepts, models, and results from one area to the other. Establishing such a translation will hopefully lead to a cross-fertilization of the two research areas. The translation between belief revision and nonmonotonic logic was first given in Makinson and G~rdenfors (1990).
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Belief revision
A presentation of the theory of belief revision as can be found in, for example, Alchourron, G~denfors, and Makinson (1985) and G~denfors (1988). In this theory, states of belief axe modelled by belief sets which axe sets of sentences from a given language.Belief sets model the statics of epistemic states. For their dynamics we need methods for updating
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belief sets. Three kinds of updates are central: (i) Expansion: A new sentence together with its logical consequences is added to a belief set K. The belief set that results from expanding K by a sentence A will be denoted K -k A. (ii) Revision: A new sentence that is inconsistent with a belief set K is added, but in order that the resulting belief set be consistent some of the old sentences of K are deleted. The result of revising K by a sentence A will be denoted K ] . (iii) Contraction: Some sentence in K is retracted without adding any new beliefs. In order that the resulting belief set be closed under logical consequences some other sentences from K must be given up. The result of contracting K with respect to A will be denoted K - A. There ave two methods of attacking the problem of specifying revision and contraction operations. One is to present rationality postulates for the processes. The second method is to adopt a more constructive approach and build computationally oriented models of belief revision which can take a belief set together with a sentences to be added as input and which gives a revised belief set as output. In the theory of belief revision the rationality postulates and the model approach are connected via representation theorems which say that all models in a certain class satisfy a certain set of postulates (for example, the postulates K*I-K*8 for revision as presented in G£rdenfors (1988)), and vice versa, any revision method satisfying these postulates can be identified with one of the models in the given class.
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Nonmonotonic logic
When discussing the merits and drawbacks of various models of nonmonotonic logic one needs to adopt an abstract perspective. In the same way as for the theory of belief revision one can discuss postulates for nonmonotonic reasoning. It was Gabbay (1985) who initiated this kind of investigation and it has been elaborated by Makinson (1990) and Kraus, Lehmann, and Magidor (1989) who investigate a large class of such postulates and their consequences.
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Translating postulates
When comparing the theory of belief revision to nonmontonic logic I focus on the level of postulates. G~rdenfors and Makinson (1990) suggest a method of translating postulates for belief revision into postulates for non-monotonic logic, and vice versa. The key idea for the translation from belief revision to nonmonotonic logic is t h a t a statement of the form B E K~ is seen as a nonmonotonic inference from A to B given the set K of sentences as background (default) information. So the statement B E K~ for belief revision is translated into the statement A ~ B for nonmonotonic logic (or into A ~,,KB, if one wants to emphasize the role of the background beliefs). Conversely, a statement of the form A ~ B for nonmonotonic logic is translated in to a statement of the form B E K~ for belief revision, where K is introduced as a fixed belief set. Using this recipe it is possible to translate all the postulates K*I-K*8 for belief revision into conditions for nonmonotonic logic. It turns out that every postulate translates into a condition on t h a t is valid in some kinds of nonmonotonic inferences in the literature. Conversely, every postulate on t h a t is known in the literature translates into a condition on belief revision that is a consequence of K*I-K*8.
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Connections between models
Turning finally to the connection between models for belief revision and models for nonmonotonic inferences, there are several constructions in the two areas that seem to be closely related. More det~led investigations into the connections between the two kinds of models have just begun. I believe that the connection that has been established between postulates for belief revision and nonmonotonic logic will be useful when looking for connections between the two areas on the model level. By utilizing the translation method outlined here, the two research areas will be able to cross-fertilize each other. An extended abstract of this paper was published on pp. 768-773 in Proceedings of ECAI 90, ed. by Luigia Carlucci Aiello, Pitman Publishing Co., London 1990.
Note
References Alchourron, C.E., P. G~rdenfors, and D. Makinson (1985), 'On the logic of theory change: Partial meet contraction and revision functions', The Journal of Symbolic Logic, 50, 510-530. Gabbay, D. (1985), 'Theoretical foundations for nonmonotonic reasoning in expert systems', in Logic and Models of Concurrent Systems, K. Apt ed. (Berlin: SpringerVerlag). G£rdenfors P. (1988), Knowledge in Flux: Modeling the Dynamics of Epistemic States, (Cambridge, MA: The MIT Press, Bradford Books). G£rdenfors, P. and D. Makinson. (1988), 'Revisions of knowledge systems using epistemic entrenchment', in Proceedings of the Second Conference on Theoretical Aspects of Reasoning about Knowledge, M. Vardi ed. (Los Altos, CA: Morgan Kanfmann). Kraus, S., D. Lehmann, and M. Magidor, (1990), 'Nonmonotonic reasoning, preferential models and cumulative logics', Artificial Intelligence, 44, 167-207. Makinson, D. (1990), 'General patterns in nonmonotonic reasoning', to appear as chapter 2 of Handbook of Logic in Artificial Intelligence and Logic Programming, Volume II: Non-Monotonic and Uncertain Reasoning, (Oxford: Oxford University Press). Makinson, D. and P. G£rdenfors (1990): 'Relations between the logic of theory change and nonmonotonic logic', to appear in the proceedings of the Konstanz workshop on belief revision, ed. by A. Fuhrmann and M. Morrean (Berlin: Springer Verlag).
