Recently epistemic spaces have been introduced to formalize instantiations of iterated belief change operators and their translations from one concrete representation (epistemic space) to another. In this work, we build on these notions to deepen the understanding of iterated belief change and propose a general method for comparing the expressiveness of existing epistemic spaces. We introduce the notion of canonicity for epistemic spaces as a tool for identifying those that are sufficient, or necessary, to realize certain classes of iterated belief change operators. In particular, we give the canonical epistemic space (up to equivalence) that allows one to instantiate any iterated belief change operator.

We introduce a new epistemic space: the space of rational rankings. This space is very useful for understanding some aspects of belief dynamics. In particular, the issues which concern improving the new information. Thus, we define in a very clear and succinct way a class of operators capturing the fact that the new information is improved. An interesting feature of this space is that the behavior of these operators can be characterized through a few equations and inequalities which are very simple and whose meaning is transparent. We prove that these operators are indeed improvement operators. Moreover, we show that these operators have good behavior when they undergo a sufficient number of iterations. In such a case, they become Darwiche and Pearl revision operators.

We propose new operators for weighted propositional belief merging operators. We introduce distance-based operators that use the product as aggregation function. In social choice theory, the product, called the Nash welfare function, is known to be a more equitable social welfare function than the classical utilitarian welfare function (based on a sum). We study which properties are satisfied by the obtained corresponding weighted merging operators. In particular, we show that, unlike the Nash welfare function, distance-based operators using the product do not satisfy the Pigou-Dalton property. Then, we introduce a new family of weighted merging operators, which we call utility-based weighted merging operators, where the utility is roughly the converse of a distance for distance-based operators. For most well-known distance-based operators, it is easy to find the corresponding utility-based merging operators. But an interesting result is that the utility-based weighted merging operator based on the product does not correspond to any standard distance-based weighted merging operator, and this operator satisfies the Pigou-Dalton property.

In standard propositional belief merging, one implicit assumption is that all sources have exactly the same importance. But there are many situations where the sources have different importance/reliability/expertise that have to be taken into account in the merging process. In this work we study the problem of weighted merging operators, which aimed to take these weights into account in a sensible way. We give a syntactical characterization of these operators, and then we state a representation theorem in terms of plausibility preorders on interpretations. We also propose a general method to build weighted distance-based merging operators, and provide some concrete examples, using two different weight functions.

In this work, we address one important problem of Katsuno and Mendelzon update operators, that is to require that any updated belief base must entail any new input in a consistent way. This assumes that any situation can be updated into one satisfying that input, which is unrealistic. To solve this problem, we must relax either the success or the consistency principle. Each case leads to a distinct family of update operators, that we semantically characterize by plausibility relations over possible worlds, considering a credibility limit that aims to forbid unrealistic changes. We discuss in which cases one family is more adequate than the other one.

The behavior of Iterated Belief Revision operators with respect to iteration has been characterized by a set of four postulates proposed by Darwiche and Pearl. These postulates give constraints on a single iteration step, and this is not enough to forbid some pathological operators. In this paper, we propose a generalization of these postulates to solve this issue and we study its implications. One surprising consequence is that, for TPO-representable operators (i.e., for operators defined as transitions on total pre-orders on interpretations), there are very few operators that satisfy this generalization.

Belief revision aims at incorporating, in a rational way, a new piece of information into the beliefs of an agent. Most works in belief revision suppose a classical logic setting, where the beliefs of the agent are consistent. Moreover, the consistency postulate states that the result of the revision should be consistent if the new piece of information is consistent. But in real applications it may easily happen that (some parts of) the beliefs of the agent are not consistent. In this case then it seems reasonable to use paraconsistent logics to derive sensible conclusions from these inconsistent beliefs. However, in this context, the standard belief revision postulates trivialize the revision process. In this work we discuss how to adapt these postulates when the underlying logic is Priest's LP logic, in order to model a rational change, while being a conservative extension of AGM/KM belief revision. This implies, in particular, to adequately adapt the notion of expansion. We provide a representation theorem and some examples of belief revision operators in this setting.

In this work, we explore the links between the Borda voting rule and belief merging operators. More precisely, we define two families of merging operators inspired by the definition of the Borda voting rule. We also introduce a notion of cancellation in belief merging, inspired by the axiomatization of the Borda voting rule proposed by Young. This allows us to provide a characterization of the drastic merging operator.

In this work we introduce a 3-valued logic with modalities, with the aim of having a clear and precise representation of epistemic states, thus the formulas of this logic will be our epistemic states. Indeed, these formulas are identified with ranking functions of 3 values, a generalization of total preorders of three levels. In this framework we analyze some types of changes of these epistemic structures and give syntactical characterizations of them in the introduced logic. In particular, we introduce and study carefully a new operator called Cautious Improvement operator. We also characterize all operators that are definable in this framework.

With the aim of studying social properties of belief merging and having a better understanding of impossibility, we extend in three ways the framework of logic-based merging introduced by Konieczny and Pino Perez. First, at the level of representation of the information, we pass from belief bases to complex epistemic states. Second, the profiles are represented as functions of finite societies to the set of epistemic states (a sort of vectors) and not as multisets of epistemic states. Third, we extend the set of rational postulates in order to consider the epistemic versions of the classical postulates of social choice theory: standard domain, Pareto property, independence of irrelevant alternatives and absence of dictator. These epistemic versions of social postulates are given, essentially, in terms of the finite propositional logic. We state some representation theorems for these operators. These extensions and representation theorems allow us to establish an epistemic and very general version of Arrow's impossibility theorem. One of the interesting features of our result, is that it holds for different representations of epistemic states; for instance conditionals, ordinal conditional functions and, of course, total preorders.

In this paper we introduce and study credibility-limited improvement operators. The idea is to accept the new piece of information if this information is judged credible by the agent, so in this case a revision is performed. When the new piece of information is not credible then it is not accepted (no revision is performed), but its plausibility is still improved in the epistemic state of the agent, similarly to what is done by improvement operators. We use a generalized definition of Darwiche and Pearl epistemic states, where to each epistemic state can be associated, in addition to the set of accepted formulas (beliefs), a set of credible formulas. We provide a syntactic and semantic characterization of these operators.

Abstract One of the main tools in the study of nonmonotonic consequence relations is the representation of such relations in terms of preferential models. In this paper we give an unified and simpler framework to obtain such representation theorems.

Lehmann, Magidor and others have investigated the effects of adding the non-Horn rule of rational monotony to the rules for preferential inference in nonmonotonic reasoning. In particular, they have shown that every inference relation satisfying those rules is generated by some ranked preferential model. We explore the effects of adding a number of other non-Horn rules that are stronger than or incomparable with rational monotony, but which are still weaker than plain monotony. Distinguished among these is a rule of determinacy preservation, equivalent to one of rational transitivity, for which we establish a representation theorem in terms of quasi-linear preferential models. An important tool in the proof of the representation theorem is the following purely semantic result, implicit in work of Freund, but here established by a more direct argument: every ranked preferential model generates the same inference relation as some ranked preferential model that is collapsed, in the sense of being both injective and such that each of its states is minimal for some formula. We also consider certain other non-Horn rules which are incomparable with monotony but arc implied by conditional excluded middle, and establish a representation result for a central one among them, which we call fragmented disjunction, equivalent to fragmented conjunction, in terms of almost linear preferential models. Finally, we consider briefly some curious Horn rules beyond the preferential ones but weaker than monotony, notably those which we call conjunctive insistence and n-monotony.