# Belief Revision

### Parametrised Difference Revision

Despite the great theoretical advancements in the area of Belief Revision, there has been limited success in terms of implementations. One of the hurdles in implementing revision operators is that their specification (let alone their computation), requires substantial resources. On the other hand, implementing a specific revision operator, like Dalal's operator, would be of limited use. In a recent paper we generalised Dalal's construction defining a whole family of concrete revision operators, called Parametrised Difference revision operators or PD operators for short. This family is wide enough to cover a whole range of different applications, and at the same time it is easy to represent. In this paper we characterise axiomatically the family of PD operators, study its computational complexity, and discuss its benefits for belief revision implementations.

### On Belief Promotion

We introduce a new class of belief change operators, named promotion operators. The aim of these operators is to enhance the acceptation of a formula representing a new piece of information. We give postulates for these operators and provide a representation theorem in terms of minimal change. We also show that this class of operators is a very general one, since it captures as particular cases belief revision, commutative revision, and (essentially) belief contraction.

### Towards Belief Contraction without Compactness

In the AGM paradigm of belief change the background logic is taken to be a supra-classical logic satisfying compactness among other properties. Compactness requires that any conclusion drawn from a set of propositions X is implied by a finite subset of X. There are a number of interesting logics such as Computational Tree Logic (CTL, a temporal logic) which do not possess the compactness property, but are important from the belief change point of view. In this paper we explore AGM style belief contraction in non-compact logics as a starting point, with the expectation that the resulting account will facilitate development of corresponding accounts of belief revision. We show that, when the background logic does not satisfy compactness, as long as the language in question is closed under classical negation and disjunction, AGM style belief contraction functions (with appropriate adjustments) can be constructed. We provide such a constructive account of belief contraction that is characterised exactly by the eight AGM postulates of belief contraction. The primary difference between the classical AGM construction of belief contraction functions and the one presented here is that while the former employs remainders of the belief being removed, we use its complements.

### Propositional Belief Merging with OWA Operators

An Ordered Weighted Averaging (OWA) operator provides a parameterized family of aggregation operators which include many of the well-known operators such as the maximum, the minimum and the mean. We introduce OWA operators as propositional belief merging operators and investigate their logical properties, as well as their relation with IC and pre-IC merging operators.

### A Semantic Perspective on Belief Change in a Preferential Non-Monotonic Framework

Belief change and non-monotonic reasoning are usually viewed as two sides of the same coin, with results showing that one can formally be defined in terms of the other. In this paper we investigate the integration of the two formalisms by studying belief change for a (preferential) non-monotonic framework. We show that the standard AGM approach to belief change can be transferred to a preferential non-monotonic framework in the sense that change operations can be defined on conditional knowledge bases. We take as a point of departure the results presented by Casini and Meyer (2017), and we develop and extend such results with characterisations based on semantics and entrenchment relations, showing how some of the constructions defined for propositional logic can be lifted to our preferential non-monotonic framework.

### 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. Most suggestions have resulted in a form of reductionism' that identifies belief states with orderings of worlds. However, this position has recently been criticised as being unacceptably strong. Other proposals, such as the popular principle (P), aka Independence', characteristic of admissible' operators, remain commendably more modest. In this paper, we supplement the DP postulates and (P) with a number of novel conditions. While the DP postulates constrain the relation between a prior and a posterior conditional belief set, our new principles notably govern the relation between two posterior conditional belief sets obtained from a common prior by different revisions. We show that operators from the resulting family, which subsumes both lexicographic and restrained revision, can be represented as relating belief states associated with a proper ordinal interval' (POI) 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 many AGM era postulates, including Superexpansion, that are not sound for admissible operators in general.

### Incorporating Relevance in Epistemic States in Belief Revision

We present an account of relevance in belief revision where, intuitively, one wants to only consider the relevant part of an agent's epistemic state in a revision. We assume that relevance is a domain-specific notion, and that (ir)relevance assertions are given as part of the agent's epistemic state. Such assertions apply in a given context, and are of the form in the case that formula \sigma holds, the Y part of the agent's epistemic state is independent of the rest of the epistemic state'', where Y is part of the signature of the language. Two approaches are given, one in which (in semantic terms) conditions are placed on a faithful ranking on possible worlds to enforce the (ir)relevance assertions, and a second in which the possible worlds characterising the agent's beliefs may be modified in a revision. These approaches are shown to yield the same resulting belief set. Corresponding postulates and a representation result are given. The overall approach is compared to that of Parikh's for language splitting as well as with multivalued dependencies in relational databases.

### Studies in Credibility-Limited Base Revision

In this paper we present axiomatic characterizations for several classes of credibility-limited base revision functions and establish the interrelation among those classes. We also propose and axiomatically characterize two new base revision functions.

### Axiomatizing a Qualitative Principle of Conditional Preservation for Iterated Belief Change

In their seminal paper, Darwiche and Pearl proposed four axioms for preserving conditional beliefs under iterated belief revision which were recently adapted to iterated belief contraction by Konieczny and Pino Perez. For (semi-)quantitative frameworks like probabilities, Kern-Isberner presented a fully axiomatized principle of conditional preservation for iterated belief change that was shown to cover the axioms both of Darwiche and Pearl, and Konieczny and Pino Perez in the semantic framework of Spohn's ranking function. This paper closes the gap between these works by presenting a purely qualitative principle of conditional preservation for iterated belief change that can be derived from Kern-Isberner's semi-quantitative principle and implies all axioms of the mentioned works, showing in particular that iterated belief revision and belief contraction share common methodological grounds which can be adapted by the respective success condition. Moreover, the approach presented in this paper significantly extends the scope of previous works in that it applies to much more general change problems when epistemic states are changed by sets of conditional beliefs.

### Specifying Plausibility Levels for Iterated Belief Change in the Situation Calculus

We investigate augmenting a theory of belief and actions with qualitative plausibility levels. Shapiro et al. created a framework for modeling iterated belief revision and update which integrated those features with the well-developed theory of action in the situation calculus. However, applying their technique requires associating plausibility levels with initial situations, for which no very convenient mechanism had been proposed. Schwering and Lakemeyer proposed deriving these initial plausibility levels from a set of conditionals, similarly to how models are ranked in Pearl's System Z. However, their approach inherits some limitations of System Z. We consider alternatives, and argue that a perspicuous approach is to measure plausibility by counting the abnormalities in a situation (similarly to cardinality-based circumscription). By allowing abnormalities to change over time, we can also model changing plausibility levels in a natural and simple way, which gives us a flexible approach for handling belief change about predicted and unpredicted exogenous actions.