This paper establishes and proves representation theorems for cumulative propositional dependence logic and for cumulative propositional logic with team semantics. Cumulative logics are famously given by System~C. For propositional dependence logic, we show that System C entailments are exactly captured by cumulative models from Kraus, Lehmann and Magidor. On the other hand, we show that entailment in cumulative propositional logics with team semantics is exactly captured by cumulative and asymmetric models. For the latter, we also obtain equivalence with cumulative logics based on propositional logic with classical semantics. The proofs will be useful for proving representation theorems for other cumulative logics without negation and material implication.

We propose a framework for reasoning from inconsistent knowledge bases using minimal hitting sets, i. e., sets of interpretations such that each formula of the knowledge base is satisfied by at least one those interpretations. By additionally considering preference orders over minimal hitting sets, we can define a wide variety of non-monotonic inference relations. We consider concrete preference orders based on set inclusion, cardinality, the number of conflicting atoms within the hitting set, and using the Hamming distance between pairs of interpretations. We compare the resulting inference relations, characterize their logical properties, and position them relative to classical inference from maximal consistent subsets. Finally, we show that inference based on minimal conflicting atoms coincides with reasoning in Priest’s 3-valued logic.

This paper considers the complexity and properties of KLM-style preferential reasoning in the setting of propositional logic with team semantics and dependence atoms, also known as propositional dependence logic. Preferential team-based reasoning is shown to be cumulative, yet violates System P. We give intuitive conditions that fully characterise those cases where preferential propositional dependence logic satisfies System P. We show that these characterisations do, surprisingly, not carry over to preferential team-based propositional logic. Furthermore, we show how classical entailment and dependence logic entailment can be expressed in terms of non-trivial preferential models. Finally, we present the complexity of preferential team-based reasoning for two natural representations. This includes novel complexity results for classical (non-team-based) preferential reasoning.

We establish a generic, model-theoretic characterization of rational belief revision operators implementing the paradigm of minimal change according to the seminal work by Alchourrón, Gärdenfors, and Makinson (AGM). Our characterization applies to all Tarskian logics, that is, all logics with a classical model-theoretic semantics, and hence a wide variety of formalisms were used in knowledge representation and beyond, including many for which a model-theoretic characterization has hitherto been lacking. Our starting point is the approach by Katsuno and Mendelzon (K&M), who provided such a characterization for propositional logic over finite signatures. We generalize K&M’s approach to the setting of AGM-style revision over bases in arbitrary Tarskian logics, where base may refer to one of the various ways of representing an agent’s beliefs (such as belief sets, arbitrary or finite sets of sentences, or single sentences). Our first core result is a representation theorem providing a two-way correspondence between AGM-style revision operators and specific assignments : functions associating every base to a “preference” relation over interpretations, which must be total but is—in contrast to prior approaches—not always transitive. As our second core contribution, we provide a characterization of all logics for which our result can be strengthened to assignments producing transitive preference relations (as in K&M’s original work). Alongside these main contributions, we discuss diverse variants of our findings as well as ramifications for other areas of belief revision theory.

This paper studies the realizability of belief revision and belief contraction operators in epistemic spaces. We observe that AGM revision and AGM contraction operators for epistemic spaces are only realizable in precisely determined epistemic spaces. We define the class of linear change operators, which are a special kind of maxichoice operators. When AGM revision, respectively, AGM contraction, is realizable, linear change operators are a canonical realization.

According to Boutillier, Darwiche, Pearl and others, principles for iterated revision can be characterised in terms of changing beliefs about conditionals. For iterated contraction a similar formulation is not known. This is especially because for iterated belief change the connection between revision and contraction via the Levi and Harper identity is not straightforward, and therefore, characterisation results do not transfer easily between iterated revision and contraction. In this article, we develop an axiomatisation of iterated contraction in terms of changing conditional beliefs. We prove that the new set of postulates conforms semantically to the class of operators like the ones given by Konieczny and Pino P\'erez for iterated contraction.