Prime forms and minimal change in propositional belief bases

Abstract

This paper proposes to use prime implicants and prime implicates normal forms to represent belief sets. This representation is used, on the one hand, to define syntactical versions of belief change operators that also satisfy the rationality postulates but present better complexity properties than those proposed in the literature and, on the other hand, to propose a new minimal distance that adopts as a minimal belief unit a "fact", defined as a prime implicate of the belief set, instead of the usually adopted Hamming distance, i.e., the number of propositional symbols on which the models differ. Some experiments are also presented that show that this new minimal distance allows to define belief change operators that usually preserve more information of the original belief set.

Judgment aggregation in dynamic logic of propositional assignments

“…A term D is an implicant of ϕ if and only if D | ϕ. We present now the formal definition of prime implicants as given by [33]: Every formula ϕ can be rewritten as a conjunction of negations of the prime implicants of ¬ϕ [34]. Since in particular this holds for IC, we assume in the following that integrity constraints are written in this syntactical form.…”

On the Merit of Selecting Different Belief Merging Operators

“…However, ∆ does not produce useful and informative results since they will be a tautology when the profile is inconsistent, so, the conformity, fairness and satisfaction relations are insufficient. For this reason we need some way to classify the degree of "useful and informative" merging results and so we propose to use the notion of strength introduced in [13]. With this notion we can say that ∆ is weaker than the other operators since its merging results are weaker.…”

Prime implicates and relevant belief revision

“…The set K M as defined above is used for a specific kind of model-based revision in [10]the authors link their paper to Parikh's and Makinson's work on the notion of relevant belief change, but do not explicitly discuss the relation between K M and K N . In [12], the same authors propose a solution to the problem of relevance in belief revision in terms of preferences over prime implicants, minimal conjunctions of literals that entail K. The idea of defining revision of K in view of its prime implicate set was put forward in [3], where it is conjectured that this revision obeys relevance.…”

“…It might be possible to take one step further, and define a continuum of finest L-splittings, where L is any subclassical logic. 10 Note that the logic L 0 for which Cn L 0 (K) = K for all K is such a logic. Other, perhaps more interesting subclassical logics could be all sorts of paraconsistent and paracomplete logics.…”