Prime Implicate-based Belief Revision Operators

A belief revision operator can be seen as a function which takes as input a set of beliefsK and an input formula φ and outputs a new set of beliefs K ?φ. Many of the belief revision schemes that have been defined in the literature require additional input. The extra information they need comes in various forms: relations over subsets of the sets of beliefs [2], epistemic entrenchment relations [1], system of spheres [6], faithful orderings [7], etc. In many applications we do not have such background information, which is why there is a need for revision operators which give good results without it. Unfortunately, the above approaches appear ill-suited to cases where we do not have any information regarding the relative importance of different beliefs or models. For example, if we accord equal importance to each of the beliefs (or each model of the beliefs or non-beliefs), which seems the most reasonable thing to do if we have no preference information, then these approaches result in the infamous drastic revision operator which gives up all old beliefs whenever the incoming information contradicts them. All of the above belief revision schemes are insensitive to syntax: logically equivalent sets of beliefs are revised in the same way, and logically equivalent input formulas lead to the same result. The so-called formula-based approaches, like the full meet [5, 4] and cardinality-maximizing base revision operators [5, 9], abandon the postulate of insensitivity to syntax, and allow e.g. the set of beliefs K1 = {a, b} to be revised differently from K2 = {a ∧ b}. Such approaches can do without extra information: they do not collapse into the drastic revision operator. There are only very few belief revision operators that are both insensitive to syntax and independent of extra information. The most prominent one is Dalal’s [3]. It is often called model-based: revision is identified with a move from the models of K to those models of φ that are closest in terms of the Hamming distance. Two other modelbased revision operators exist: Weber’s [12] and Satoh’s [11]. In this paper we propose two revision operators which are formulabased yet syntax-insensitive, and do not rely on background information. Our operators are obtained by first replacing the belief base by its set of prime implicates and then applying either the full meet or the cardinality-maximizing base revision operators. The prime implicates of a belief base, defined as its logically strongest clausal consequences, can be seen as the primitive semantic components of the belief base, from which all other beliefs can be derived. We argue that when no extra information is available, prime implicates provide a natural and interesting way of representing a set of beliefs. Moreover, the fact that equivalent sets of formulae have the same sets of