Modal logic is the foundation for a versatile and well-established class of knowledge representation for-malisms in artificial intelligence. Enriching modal logics with non-monotonic reasoning capabilities such as preferential reasoning as developed by Lehmann and colleagues would therefore constitute a natural extension of such KR formalisms. Nevertheless, there is at present no generally accepted semantics, with corresponding syntactic characterization, for preferential consequence in modal logics. In this paper we fill this gap by providing a natural and intuitive semantics for preferential and rational modal consequence. We do so by placing a preference order on possible worlds indexed by Kripke models they belong to. We also prove representation results for both preferential and rational consequence, which paves the way for effective decision procedures for modal preferential reasoning. We then illustrate applications of our constructions to modal logics widely used in AI, notably in the contexts of reasoning about actions, knowledge and beliefs. We argue that our semantics constitutes the foundation on which to explore preferential reasoning in modal logics in general.

Abstract Standard belief change assumes an underlying logic containing full classical propositional logic. However, there are good reasons for considering belief change in less expressive logics as well. In this paper we build on recent investigations by Delgrande on contraction for Horn logic. We show that the standard basic form of contraction, partial meet, is too strong in the Horn case. This result stands in contrast to Delgrande's conjecture that orderly maxichoice is the appropriate form of contraction for Horn logic. We then define a more appropriate notion of basic contraction for the Horn case, influenced by the convexity property holding for full propositional logic and which we refer to as infra contraction. The main contribution of this work is a result which shows that the construction method for Horn contraction for belief sets based on our infra remainder sets corresponds exactly to Hansson's classical kernel contraction for belief sets, when restricted to Horn logic. This result is obtained via a detour through contraction for belief bases. We prove that kernel contraction for belief bases produces precisely the same results as the belief base version of infra contraction. The use of belief bases to obtain this result provides evidence for the conjecture that Horn belief change is best viewed as a 'hybrid' version of belief set change and belief base change. One of the consequences of the link with base contraction is the provision of a representation result for Horn contraction for belief sets in which a version of the Core-retainment postulate features.

Historically, approaches to defeasible reasoning have been concerned mostly with one aspect of defeasibility, viz. that of arguments, in which the focus is on normal-ity of the premise. In this paper we are interested in an-other aspect of defeasibility, namely that of defeasible modes of reasoning. We do this by adopting a preferen-tial modal semantics that we defined in previous work and which allows us to refer to the relative normality of accessible worlds. This leads us to define preferen-tial versions of the traditional notions of knowledge, be-liefs, obligations and actions, to name a few, as studied in modal logics. The resulting preferential modal logics make it possible to capture, and reason with, aspects of defeasibility heretofore beyond the reach of modal for-malisms.

Nonmonotonic logics are usually characterized by the pres-ence of some notion of ‘conditional ’ that fails monotonicity. Research on nonmonotonic logics is therefore largely con-cerned with the defeasibility of argument forms and the as-sociated normality (or abnormality) of its constituents. In contrast, defeasible modes of inference aim to formalize the defeasible aspects of modal notions such as actions, obli-gations and knowledge. In this work we enrich the stan-dard possible worlds semantics with a preference ordering on worlds in Kripke models. The resulting family of modal logics allow for the elegant expression of defeasible modali-ties. We also propose a tableau calculus which is sound and complete with respect to our preferential semantics. Keywords Knowledge representation and reasoning; modal logic; pref-erential semantics; defeasible modes of inference

abstract. We introduce Propositional Typicality Logic (PTL), a logic for reasoning about typicality. We do so by enriching classical proposi-tional logic with a typicality operator of which the intuition is to capture the most typical (or normal) situations in which a given formula holds. The semantics is in terms of ranked models as studied in KLM-style preferential reasoning. This allows us to show that KLM-style rational consequence relations can be embedded in our logic. Moreover we show that we can define consequence relations on the language of PTL itself, thereby moving beyond the propositional setting. Building on the exist-ing link between propositional rational consequence and belief revision, we show that the same correspondence holds in the case of rational con-sequence and belief revision defined on the language of PTL. Finally we also investigate different notions of entailment for PTL and propose two appropriate candidates.

Abstract. In this work we present External Transaction Logic, a logic that extends Transaction logic with the ability to model and execute transactions requiring interactions with external entities, as e.g. external web-source, web-services or agents. Transactions are defined in a logic programming style by the composition of internal and external primitives. These primitives are incorporated in a quite general manner, as a parameter of the External Transaction Logic theory, allowing the specification of transactions integrating knowledge and actions from multiple sources and semantics. Since one has different control over internal and external domains, different transaction properties are ensured depending on where actions are executed. Namely, internal actions executed in a knowledge base that we fully control, follow the standard model of transactions, where the failure of the transaction leaves the knowledge unaffected. On the other hand, transactional properties over actions executed externally need to be relaxed, as it is impossible to roll back actions executed in a domain that is external. To deal with this, external actions can be defined along with compensating operations. If a transaction fails after executing some external action, then these compensations are executed in a backward order to achieve a relaxed model of atomicity. We provide a model theory for External Transaction Logic, that can be used to reason about the conditions of execution of transactions that require the issuing of both internal and external actions on abstract knowledge bases with potentially different state semantics. We also present here a corresponding proof theory (sound and complete w.r.t. the model theory) that provides means to execute such transactions in a top-down manner.