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1、The Logic of Conditional Doxastic Actions: A theory of dynamic multi-agent belief revision Alexandru Baltag Computing Laboratory, Oxford University. Alexandru.Baltagcomlab.ox.ac.uk Sonja Smets Center for Logic and Philosophy of Science, Vrije Universiteit Brussel. sonsmetsvub.ac.be Abstract.We prese
2、nt a logic of conditional doxastic actions, obtained by incorporating ideas from belief revision theory into the usual dynamic logic of epistemic actions. We do this by extending to actions the setting of epistemic plausibility models, developed in Baltag and Smets (2006) for representing (static) c
3、onditional beliefs. We introduce a natural extension of the notion of update product from Baltag and Moss (2004) to plausibility models. 1Introduction In this paper, we extend the semantic setting proposed in Baltag and Smets (2006) for “static” multi-agent belief revision to defi ne a notion of (mu
4、lti-agent) belief update with an action, i.e. a dynamic notion of belief revision. This improves on the work in Baltag et al. (1998) and Baltag and Moss (2004), by incorporating ideas from belief revision into dynamic-epistemic logic. In Baltag and Smets (2006), we proposed two equivalent semantic s
5、ettings for “static” belief revision, and proved them to be equivalent with each other and with a multi-agent epistemic version of the AGM belief revision theory: conditional doxastic models and epistemic plausibility models. We argued extensively that these settings provided the “right” qualitative
6、 semantics for multi-agent belief revision, forming the basis of a conditional doxastic logic (CDL, for short), that captured the main “laws” of hypothetical beliefs. We went beyond static revision, using CDL to explore a restricted notion of “dynamic” belief revision, by modeling and axiomatizing m
7、ulti-agent belief updates induced by public and private announcements. In this paper, we go further and describe a full dynamic logic of conditional doxastic actions, which subsumes all the other approaches known to us that combine dynamic-epistemic logic with belief revision. We do this by putting
8、together the ideas from Baltag et al. (1998), Baltag and Moss (2004) on the logic of epistemic programs with the ideas from Baltag and Smets (2006). In particular, we adopt the view of “actions” as having a similar underlying doxas- tic/epistemic structure as the “states”: thus, we use the same kind
9、 of structures (plausibility frames) to model both actions and states. We also adopt the fundamental idea of “update product” from Baltag and Moss (2004), extending it naturally from epistemic Kripke models to plausibility models. Proceedings of the Workshop on Rationality and Knowledge, ESSLLI 2006
10、 Sergei Artemov hence, the quantitative fl avor. 2 i.e. a refl exive and transitive relation. 2 elements.3Using the notation MinT := t T : t t0for all t0 T for the set of minimal elements of T, the last condition says that: for every T S, if T 6= then MinT 6= . Plausibility frames for only one agent
11、 and without the epistemic relations have been used as models for conditionals and belief revision in Grove (1988), Gardenfors (1986), Gardenfors (1988), Segerberg (1998) etc. Observe that the conditions on the preorder aare (equivalent to) Groves conditions for the (relational version of) his model
12、s in Grove (1988). The standard formulation of Grove models (in terms of a “system of spheres”, weakening the similar notion in Lewis (1973) was proved in Grove (1988) to be equivalent to the above relational formulation.4 Given a plausibility frame S, an S-proposition is any subset P S. We say that
13、 the state s satisfi es the proposition P if s P. Observe that a plausibility frame is just a special case of a Kripke frame. So, as is standard for Kripke frames, we can defi ne an epistemic plausibility model to be an epistemic plausibility frame S together with a valuation map kk : P(S), mapping
14、every element of a given set of “atomic sentences” into S-propositions. Notation: strict plausibility, doxastic indistinguishability. As with any preorder, the (“non-strict”) plausibility relation aabove has a “strict” (i.e. asymmetric) version , taking program terms and sentences into other sentenc
15、es. As in Baltag and Moss (2004), the conditional doxastic maps on the signature induce in a natural way conditional doxastic maps on basic programs in CDL(): we put ( ) a := 0 : 0 a. The given listing can be used to assign syntactic preconditions for basic programs, by putting: pre(i ) := i, and pr
16、e( ) := (the trivially true sentence) if is not in the listing. Thus, the basic programs of the form form a (fi nite) syntactic CDAM13 . Every given interpretation | | : CDL() Prop of sentences as doxastic propositions will convert this syntactic model into a “real” (semantic) CDAM, called |. To giv
17、e the semantics, we defi ne by induction two interpretation maps, one taking any sentence to a doxastic proposition | Prop, the second taking any program term to a (possibly non-deterministic) doxastic “program”, i.e. a set of basic actions in some CDAM. The defi nition is completely similar to the
18、one in Baltag and Moss (2004), so we skip the details here. Suffi ce to say that the semantics of basic dynamic modalities is given by the inverse map: | |S= ? |S ?1 | S | Notation.To state our proof system, we encode the notion of post-conditional contextual appearance of an action in our syntax. F
19、or sentences , and basic program = , we put: := _ Ka 06 Ka ! This notation can be justifi ed by observing that it semantically matches the modality corre- sponding to post-conditional contextual appearance: | |S= s S : s ? ( |S)s,|S a ?1 |S | Theorem. A complete proof system for the logic CDL() is o
20、btained by adding to the above axioms and rules of CDL the following Reduction Axioms: ppre() p pre() ( ) B a pre() B a a( ) where p is any atomic sentence, ,0are programs and is a basic program in L(). Acknowledgments. During the writing of this paper, the second authors research was spon- sored by
21、 the Flemish Fund for Scientifi c Research (FWO, Brussel). 13A syntactic CDAM is just a conditional doxastic frame endowed with a syntactic precondition map, asso- ciating sentences to basic action. For justifi cation and examples, in the context of epistemic action models, see Baltag and Moss (2004
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