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Extra resources for Conditionals in Nonmonotonic Reasoning and Belief Revision: Considering Conditionals as Agents

Example text

AGM *2) A ∈ K ∗ A. (AGM *3) K ∗ A ⊆ K + A. (AGM *4) If ¬A ∈ / K then K + A ⊆ K ∗ A. (AGM *5) K ∗ A is inconsistent iff A is contradictory. (AGM *6) If A and B are logically equivalent, then K ∗ A = K ∗ B. (AGM *7) K ∗ A ∧ B ⊆ (K ∗ A) + B. (AGM *8) If ¬B ∈ / K ∗ A then (K ∗ A) + B ⊆ K ∗ A ∧ B. 2 Belief Revision 15 Unlike for expansion, these postulates are not sufficient to describe uniquely one optimal revision operator; they only outline the scope of reasonable belief revision. Katsuno and Mendelzon [KM91a] rephrased the AGM-revision axioms more concisely for propositional logic (where K is supposed to be a single formula, too): (AGM’ *1) K ∗ A implies A.

Let K be a propositional belief set, that is, K ⊆ L is a set of propositions which is closed under classical consequence Cn. Let A be some proposition representing the newly acquired information which K is to be revised by. e. if A does not contradict any of the beliefs in K. This type of revision is called expansion, denoted by +. G¨ ardenfors [G¨ ar88] lists intuitive postulates for an expansion which are apt to characterize expansion uniquely within a classical logical framework: AGM-postulates for expansion: (AGM +1) K + A is a belief set.

It is generalized by the notion of cross-entropy Q(ω) Q(ω) log R(Q, P ) = P (ω) ω∈Ω = ∞ for Q(ω) = 0) between two distri(with 0 log 00 = 0 and Q(ω) log Q(ω) 0 butions Q and P . If P0 denotes the uniform distribution P0 (ω) = 1/m for all worlds ω, then R(Q, P0 ) = −H(Q) + log m relates absolute and relative entropy. Cross-entropy is a well-known information-theoretic measure of dissimilarity between two distributions and has been studied extensively (see, for instance, [Csi75, HHJ92, Jay83a, Kul68]; for a brief, but informative introduction and further references, cf.

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