On the disjunctive rational closure of a conditional knowledge base

Let D be an integral domain with quotient field K and E a subset of K. The set Int R ( E , D ) : = { ψ ∈ K ( X ) ; ψ ( E ) ⊆ D } is called the ring of integer-valued rational functions on E over D. We first introduce the notion of the rational closure of E, defined as c l D R ( E ) : = { x ∈ K ; ψ ( x ) ∈ D for all ψ ∈ Int R ( E , D ) } . We then compare the rational closure with the divisorial closure when D is a Krull domain. Furthermore, we investigate the polynomial closure in Mori domains.

Approximations of system W for inference from strongly and weakly consistent belief bases

The only recently introduced System W is a nonmonotonicinductive inference operator exhibiting some notable proper-ties like extending rational closure and satisfying syntax split-ting postulates for inference from conditional belief bases.A semantic model of system W is given by its underlyingpreferred structure of worlds, a strict partial order on the setof propositional interpretations, also called possible worlds,over the signature of the belief base. Existing implementa-tions of system W are severely limited by the number ofpropositional variables that occur in a belief base becauseof the exponentially growing number of possible worlds. Inthis paper, we present an approach to realizing nonmono-tonic reasoning with system W by using partial maximumsatisfiability (PMaxSAT) problems and exploiting the powerof current PMaxSAT solvers. An evaluation of our approachdemonstrates that it outperforms previous implementations ofsystem W and scales reasoning with system W up to a newdimension.

The recently introduced notion of an inductive inference operator captures the process of completing a given conditional belief base to an inference relation. System W is such an inductive inference operator exhibiting some notable properties like extending rational closure and satisfying syntax splitting for inference from conditional belief bases. However, the definition of system W and the shown results regarding its properties only take belief bases into account that satisfy a strong notion of consistency where no worlds may be completely infeasible. In this paper, we lift this limitation and extend the definition of system W to also cover belief bases that force some worlds to be infeasible. We establish the position of the extended system W within a map of other inductive inference operators being able to deal with the presence of infeasible worlds, including system Z and multipreference closure. For placing lexicographic inference in this map, we show that the definition of lexicographic inference must be slightly modified so that it is an inductive inference operator satisfying direct inference even when there are worlds that are infeasible. Furthermore, we show that, like its unextended version, the extended system W enjoys other desirable properties such as still fully complying with syntax splitting.

Situated conditional reasoning

Defeasible RDFS via rational closure

Upon Reflection

Abstract Multifluid model (MFM) simulations have been carried out on liquid–solid fluidized beds (LSFB) consisting of binary and higher‐order polydisperse particle mixtures. The role of particle–particle interactions was found to be as crucial as the drag force under laminar and homogenous LSFB flow regimes. The commonly used particle–particle closure models are designed for turbulent and heterogeneous gas–solid flow regimes and thus exhibit limited to no success when implemented for LSFB operating under laminar and homogenous conditions. A need is perceived to carry out direct numerical simulations of liquid–solid flows and extract data from them to develop rational closure terms to account for the physics of LSFB. Finally, a recommendation flow regime map signifying the performance of the MFM has been proposed. This map will act as a potential guideline to identify whether or not the bed expansion characteristics of a given polydisperse LSFB can be correctly simulated using MFM closures tested.

The question of conditional inference, i.e., of which conditional sentences of the form ``if A then, normally, B'' should follow from a set KB of such sentences, has been one of the classic questions of AI, with several well-known solutions proposed. Perhaps the most notable is the rational closure construction of Lehmann and Magidor, under which the set of inferred conditionals forms a rational consequence relation, i.e., satisfies all the rules of preferential reasoning, *plus* Rational Monotonicity. However, this last named rule is not universally accepted, and other researchers have advocated working within the larger class of *disjunctive* consequence relations, which satisfy the weaker requirement of Disjunctive Rationality. While there are convincing arguments that the rational closure forms the ``simplest'' rational consequence relation extending a given set of conditionals, the question of what is the simplest *disjunctive* consequence relation has not been explored. In this paper, we propose a solution to this question and explore some of its properties.

We extend the expressivity of classical conditional reasoning by introducing context as a new parameter. The enriched conditional logic generalises the defeasible setting in the style of Kraus, Lehmann and Magidor, and allows for a more refined representation of an agent’s epistemic state, distinguishing, for example, between expectations and counterfactuals. In this paper we introduce the language for the enriched logic, and define an appropriate semantic framework for it. We analyse which properties generally associated with conditional reasoning are still satisfied by the new semantic framework, provide an appropriate representation result, and define an entailment relation based on Lehmann and Magidor’s notion of Rational Closure.

Reasoning about exceptions in ontologies is nowadays one of the challenges the description logics community is facing. The paper describes a preferential approach for dealing with exceptions in Description Logics, based on the rational closure. The rational closure has the merit of providing a simple and efficient approach for reasoning with exceptions, but it does not allow independent handling of the inheritance of different defeasible properties of concepts. In this work we outline a possible solution to this problem by introducing a weaker variant of the lexicographical closure, that we call skeptical closure, which requires to construct a single base. We develop a bi-preference semantics for defining a characterization of the skeptical closure.

Datalog is a declarative logic programming language that uses classical logical reasoning as its basic form of reasoning. Defeasible reasoning is a form of non-classical reasoning that is able to deal with exceptions to general assertions in a formal manner. The KLM approach to defeasible reasoning is an axiomatic approach based on the concept of plausible inference. Since Datalog uses classical reasoning, it is currently not able to handle defeasible implications and exceptions. We aim to extend the expressivity of Datalog by incorporating KLM-style defeasible reasoning into classical Datalog. We present a systematic approach for extending the KLM properties and a well-known form of defeasible entailment: Rational Closure. We conclude by exploring Datalog extensions of less conservative forms of defeasible entailment: Relevant and Lexicographic Closure. We provide algorithmic definitions for these forms of defeasible entailment and prove that the definitions are LM-rational.

A reconstruction of multipreference closure

Under the assumption that the rational closure of a group algebra of a left-ordered group in the ring of operators of the module of formal Malcev series is a division ring, we find a canonical form of nonsingular matrices of this division ring. Bibliography: 10 titles.

In this thesis we study KLM-style rational reasoning in defeasible Description Logics. We illustrate that many recent approaches to derive consequences under Rational Closure (and its stronger variants, lexicographic and relevant closure) suffer the fatal drawback of neglecting defeasible information in quantified concepts. We propose novel model-theoretic semantics that are able to derive the missing entailments in two differently strong flavours. Our solution introduces a preference relation to distinguish sets of models in terms of their typicality (amount of defeasible information derivable for quantified concepts). The semantics defined through the most typical (most preferred) sets of models are proven superior to previous approaches in that their entailments properly extend previously derivable consequences, in particular, allowing to derive defeasible consequences for quantified concepts. The dissertation concludes with an algorithmic characterisation of this uniform maximisation of typicality, which accompanies our investigation of the computational complexity for deriving consequences under these new semantics.

ABSTRACT We propose a nonmonotonic Description Logic of typicality able to account for the phenomenon of the combination of prototypical concepts. The proposed logic relies on the logic of typicality , whose semantics is based on the notion of rational closure, as well as on the distributed semantics of probabilistic Description Logics, and is equipped with a cognitive heuristic used by humans for concept composition. We first extend the logic of typicality by typicality inclusions of the form , whose intuitive meaning is that ‘we believe with degree about the fact that typical Cs are Ds’. As in the distributed semantics, we define different scenarios containing only some typicality inclusions, each one having a suitable probability. We then exploit such scenarios in order to ascribe typical properties to a concept obtained as the combination of two prototypical concepts. We also show that reasoning in the proposed Description Logic is ExpTime-complete as for the underlying standard Description Logic .

On rational entailment for Propositional Typicality Logic

From iterated revision to iterated contraction: Extending the Harper Identity

Let G be a connected simply connected nilpotent Lie group with discrete uniform subgroup \(\Gamma \). A connected closed subgroup H of G is called \(\Gamma \)-rational if \(H\cap \Gamma \) is a discrete uniform subgroup of H. For a closed connected subgroup H of G, let \(\mathcal {I}(H, \Gamma )\) denote the identity component of the closure of the subgroup generated by H and \(\Gamma \). In this paper, we prove that \(\mathcal {I}(H, \Gamma )\) is the smallest normal \(\Gamma \)-rational connected closed subgroup containing H. As an immediate consequence, we obtain that \(\mathcal {I}(H, \Gamma )\) depends only on the commensurability class of \(\Gamma \). As applications, we give two results. In the first, we determine explicitly the smallest \(\Gamma \)-rational connected closed subgroup containing H. The second is a characterization of ergodicity of nilflow \( (G/\Gamma , H)\) in terms of \(\mathcal {I}(H, \Gamma )\). Furthermore, a characterization of the irreducible unitary representations of G for which the restriction to \(\Gamma \) remain irreducible is given.