Efficient minimal preference change

In this article, we study a minimal change approach to preference dynamics. We treat a set of preferences as a special kind of theory, and define minimal change preference contraction and revision operations in the spirit of the Alchourrón, Gärdenfors, and Makinson theory of belief revision. We characterise minimal contraction of preference sets by a set of postulates and prove a representation theorem. We also give a linear time algorithm which implements minimal contraction by a single preference. We then define minimal contraction by a set of preferences, and show that the problem of a minimal contraction by a set of preferences is NP-hard

Extending the Harper identity to iterated belief change

The field of iterated belief change has focused mainly on revision, with the other main operator of AGM belief change theory, i.e., contraction receiving relatively little attention. In this paper we extend the Harper Identity from single-step change to define iterated contraction in terms of iterated revision. Specifically, just as the Harper Identity provides a recipe for defining the belief set resulting from contracting A in terms of (i) the initial belief set and (ii) the belief set resulting from revision by ¬A, we look at ways to define the plausibility ordering over worlds resulting from contracting A in terms of (iii) the initial plausibility ordering, and (iv) the plausibility ordering resulting from revision by ¬A. After noting that the most straightforward such extension leads to a trivialisation of the space of permissible orderings, we provide a family of operators for combining plausibility orderings that avoid such a result. These operators are characterised in our domain of interest by a pair of intuitively compelling properties, which turn out to enable the derivation of a number of iterated contraction postulates from postulates for iterated revision. We finish by observing that a salient member of this family allows for the derivation of counterparts for contraction of some well known iterated revision operators, as well as for defining new iterated contraction operators

Admissible and Restrained Revision

As partial justification of their framework for iterated belief revision Darwiche and Pearl convincingly argued against Boutiliers natural revision and provided a prototypical revision operator that fits into their scheme. We show that the Darwiche-Pearl arguments lead naturally to the acceptance of a smaller class of operators which we refer to as admissible. Admissible revision ensures that the penultimate input is not ignored completely, thereby eliminating natural revision, but includes the Darwiche-Pearl operator, Nayaks lexicographic revision operator, and a newly introduced operator called restrained revision. We demonstrate that restrained revision is the most conservative of admissible revision operators, effecting as few changes as possible, while lexicographic revision is the least conservative, and point out that restrained revision can also be viewed as a composite operator, consisting of natural revision preceded by an application of a "backwards revision" operator previously studied by Papini. Finally, we propose the establishment of a principled approach for choosing an appropriate revision operator in different contexts and discuss future work

Elementary iterated revision and the Levi Identity

Recent work has considered the problem of extending to the case of iterated belief change the so-called ‘Harper Identity’ (HI), which defines single-shot contraction in terms of single-shot revision. The present paper considers the prospects of providing a similar extension of the Levi Identity (LI), in which the direction of definition runs the other way. We restrict our attention here to the three classic iterated revision operators–natural, restrained and lexicographic, for which we provide here the first collective characterisation in the literature, under the appellation of ‘elementary’ operators. We consider two prima facie plausible ways of extending (LI). The first proposal involves the use of the rational closure operator to offer a ‘reductive’ account of iterated revision in terms of iterated contraction. The second, which doesn’t commit to reductionism, was put forward some years ago by Nayak et al. We establish that, for elementary revision operators and under mild assumptions regarding contraction, Nayak’s proposal is equivalent to a new set of postulates formalising the claim that contraction by not-A should be considered to be a kind of ‘mild’ revision by A. We then show that these, in turn, under slightly weaker assumptions, jointly amount to the conjunction of a pair of constraints on the extension of (HI) that were recently proposed in the literature. Finally, we consider the consequences of endorsing both suggestions and show that this would yield an identification of rational revision with natural revision. We close the paper by discussing the general prospects for defi ing iterated revision in terms of iterated contraction

AGM 25 years: twenty-five years of research in belief change

The 1985 paper by Carlos Alchourrón (1931–1996), Peter Gärdenfors, and David Makinson (AGM), “On the Logic of Theory Change: Partial Meet Contraction and Revision Functions” was the starting-point of a large and rapidly growing literature that employs formal models in the investigation of changes in belief states and databases. In this review, the first twenty five years of this development are summarized. The topics covered include equivalent characterizations of AGM operations, extended representations of the belief states, change operators not included in the original framework, iterated change, applications of the model, its connections with other formal frameworks, computatibility of AGM operations, and criticism of the model.info:eu-repo/semantics/publishedVersio

From iterated revision to iterated contraction: extending the Harper Identity

The study of iterated belief change has principally focused on revision, with the other main operator of AGM belief change theory, namely contraction, receiving comparatively little attention. In this paper we show how principles of iterated revision can be carried over to iterated contraction by generalising a principle known as the ‘Harper Identity’. The Harper Identity provides a recipe for defining the belief set resulting from contraction by a sentence A in terms of (i) the initial belief set and (ii) the belief set resulting from revision by ¬A. Here, we look at ways to similarly define the conditional belief set resulting from contraction by A. After noting that the most straightforward proposal of this kind leads to triviality, we characterise a promising family of alternative suggestions that avoid such a result. One member of that family, which involves the operation of rational closure, is noted to be particularly theoretically fruitful and normatively appealing

On strengthening the logic of iterated belief revision: proper ordinal interval operators

Darwiche and Pearl’s seminal 1997 article outlined a number of baseline principles for a logic of iterated belief revision. These principles, the DP postulates, have been supplemented in a number of alternative ways. However, most of the suggestions for doing so have been radical enough to result in a dubious ‘reductionist’ principle that identifies belief states with orderings of worlds. The present paper offers a more modest strengthening of Darwiche and Pearl’s proposal. While the DP postulates constrain the relation between a prior and a posterior conditional belief set, our new principles govern the relation between two posterior conditional belief sets obtained from a common prior by different revisions. We show that operators from the family that these principles characterise, which subsumes both lexicographic and restrained revision, can be represented as relating belief states that are associated with a ‘proper ordinal interval’ assignment, a structure more fine-grained than a simple ordering of worlds. We close the paper by noting that these operators satisfy iterated versions of a large number of AGM era postulates

The Boltzmann Machine: a Connectionist Model for Supra-Classical Logic

This thesis moves towards reconciliation of two of the major paradigms of artificial intelligence: by exploring the representation of symbolic logic in an artificial neural network. Previous attempts at the machine representation of classical logic are reviewed. We however, consider the requirements of inference in the broader realm of supra-classical, non-monotonic logic. This logic is concerned with the tolerance of exceptions, thought to be associated with common-sense reasoning. Biological plausibility extends these requirements in the context of human cognition. The thesis identifies the requirements of supra-classical, non-monotonic logic in relation to the properties of candidate neural networks. Previous research has theoretically identified the Boltzmann machine as a potential candidate. We provide experimental evidence supporting a version of the Boltzmann machine as a practical representation of this logic. The theme is pursued by looking at the benefits of utilising the relationship between the logic and the Boltzmann machine in two areas. We report adaptations to the machine architecture which select for different information distributions. These distributions correspond to state preference in traditional logic versus the concept of atomic typicality in contemporary approaches to logic. We also show that the learning algorithm of the Boltzmann machine can be adapted to implement pseudo-rehearsal during retraining. The results of machine retraining are then utilised to consider the plausibility of some current theories of belief revision in logic. Furthermore, we propose an alternative approach to belief revision based on the experimental results of retraining the Boltzmann machine