We show how the AGM framework for belief change (expansion, revision, contraction) can be extended to deal with conditioning in the so-called Desirability-Indifference framework, based on abstract notions of accepting and rejecting options, as well as on abstract notions of events. This level of abstraction allows us to deal simultaneously with classical and quantum probability theory.

Belief contraction is the operation of removing from the set K of initial beliefs a particular belief φ . One reason for doing so is, for example, the discovery that some previously trusted evidence supporting φ was faulty. For instance, a prosecutor might form the belief that the defendant is guilty on the basis of his confession; if the prosecutor later discovers that the confession was extorted, she might abandon the belief of guilt, that is, become open minded about whether the defendant is guilty or not. In their seminal contribution to belief change, Alchourrón, Gärdenfors and Makinson ([1]) defined the notion of "rational and minimal" contraction by means of a set of eight properties, known as the AGM axioms or postulates. They did so within a syntactic approach where the initial belief set K is a consistent and deductively closed set of propositional formulas and the result of removing φ from K is a new set of propositional formulas, denoted by K φ . We provide a new characterization of AGM belief contraction based on a so-far-unnoticed connection between the notion of belief contraction and the Stalnaker-Lewis theory of conditionals ([34, 21]).

Belief change studies how to update knowledge bases used for reasoning. Traditionally belief revision has been based on full propositional logic. However, reasoning with full propositional knowledge bases is computationally hard, whereas reasoning with Horn knowledge bases is fast. In the past several years, there has been considerable work in belief revision theory on developing a theory of belief contraction for knowledge represented in Horn form. Our main focus here is the computational complexity of belief contraction, and, in particular, of various methods and approaches suggested in the literature.

When a belief state is represented as a probability function P, the resulting belief state of the contraction of a sentence (belief) from the original belief state P can be given by the probabilistic version of the Harper Identity. Specifically, the result of contracting P by a sentence h is taken to be the mixture of two states: the original state P, and the resultant state P* ~h of revising P by the negation of h. What proportion of P and P* ~h should be used in this mixture remains an open issue and is largely ignored in literature. In this paper, we first classify different belief states by their stability, and then exploit the quantitative nature of probabilities and combine it with the basic ideas of argumentation theory to determine  the mixture proportions. We, therefore, propose a novel approach to probabilistic belief contraction using argumentation.

Belief change studies how to update knowledge bases used for reasoning. Traditionally belief revision has been based on full propositional logic. However, reasoning with full propositional knowledge bases is computationally hard, whereas reasoning with Horn knowledge bases is fast. In the past several years, there has been considerable work in belief revision theory on developing a theory of belief contraction for knowledge represented in Horn form. Our main focus here is the computational complexity of belief contraction, and, in particular, of various methods and approaches suggested in the literature. This is a natural and important question, especially in connection with one of the primary motivations for considering Horn representation: efficiency. The problems considered lead to questions about Horn envelopes (or Horn LUBs), introduced earlier in the context of knowledge compilation. This work gives a syntactic characterization of the remainders of a Horn belief set with respect to a consequence to be contracted, as the Horn envelopes of the belief set and an elementary conjunction corresponding to a truth assignment satisfying a certain explicitly given formula. This gives an efficient algorithm to generate all remainders, each represented by a truth assignment. On the negative side, examples are given of Horn belief sets and consequences where Horn formulas representing the result of contraction, based either on remainders or on weak remainders, must have exponential size for almost all possible choice functions (i.e., different possible choices of partial meet contraction). Therefore using the Horn framework for belief contraction does not by itself give us computational efficiency. Further work is required to explore the possibilities for efficient belief change methods.