Rational Closure Research Articles

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

In this article, we investigate approximations of the inductive inference operator system W that has been shown to exhibit desirable inference properties and to extend both system Z, and thus rational closure, and c-inference. For versions of these inference operators that are extended to also cover inference from belief bases that are only weakly consistent, we first show that extended system Z and extended c-inference are captured by extended system W. Then we introduce general functions for generating inductive inference operators: the combination of two inductive inference operators by union, and the completion of an inductive inference operator by an arbitrary set of axioms. We construct the least inductive inference operator extending system Z and c-inference that is closed under system P and show that it is still strictly extended by extended system W. Furthermore, we introduce an inductive inference operator that strictly extends extended system W and that is strictly extended by lexicographic inference. This leads to a comprehensive map of inference relations between rational closure and extended c-inference on the one side and lexicographic inference on the other side with extended system W and its approximations at its centre, where all relationships also hold for the unextended versions.

Partial MaxSAT Approach to Nonmonotonic Reasoning with System W

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.

Conditionals, Infeasible Worlds, and Reasoning with System W

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

Conditionals are useful for modelling many forms of everyday human reasoning but are not always sufficiently expressive to represent the information we want to reason about. In this paper, we make a case for a form of situated conditional. By ‘situated’, we mean that there is a context, based on an agent's beliefs and expectations, that works as background information in evaluating a conditional, and we allow such a context to vary. These conditionals are able to distinguish, for example, between expectations and counterfactuals. Formally, they are shown to generalise the conditional setting in the style of Kraus, Lehmann, and Magidor. We show that situated conditionals can be described in terms of a set of rationality postulates. We then propose an intuitive semantics for these conditionals and present a representation result which shows that our semantic construction corresponds exactly to the description in terms of postulates. With the semantics in place, we define a form of entailment for situated conditional knowledge bases, which we refer to as minimal closure. Finally, we proceed to show that it is possible to reduce the computation of minimal closure to a series of propositional entailment and satisfiability checks. While this is also the case for rational closure, it is somewhat surprising that the result carries over to minimal closure.

Defeasible RDFS via rational closure

In the field of non-monotonic logics, the notion of Rational Closure (RC) is acknowledged as a notable approach. In recent years, RC has gained popularity in the context of Description Logics (DLs), the logic underpinning the standard semantic Web Ontology Language OWL 2, whose main ingredients are classes, the relationship among classes and roles, which are used to describe the properties of classes.In this work, we show instead how to integrate RC within the triple language RDFS (Resource Description Framework Schema), which together with OWL 2 is a major standard semantic web ontology language.To do so, we start from ρdf, a minimal, but significant RDFS fragment that covers the essential features of RDFS, and then extend it to ρdf⊥, allowing to state that two entities are incompatible/disjoint with each other. Eventually, we propose defeasible ρdf⊥ via a typical RC construction allowing to state default class/property inclusions.Furthermore, to overcome the main weaknesses of RC in our context, i.e., the “drowning problem” (viz. the “inheritance blocking problem”), we further extend our construction by leveraging Defeasible Inheritance Networks (DIN) defining a new non-monotonic inference relation that combines the advantages of both (RC and DIN). To the best of our knowledge this is the first time of such an attempt.In summary, the main features of our approach are: (i) the defeasible ρdf⊥ we propose here remains syntactically a triple language by extending it with new predicate symbols with specific semantics; (ii) the logic is defined in such a way that any RDFS reasoner/store may handle the new predicates as ordinary terms if it does not want to take account of the extra non-monotonic capabilities; (iii) the defeasible entailment decision procedure is built on top of the ρdf⊥ entailment decision procedure, which in turn is an extension of the one for ρdf via some additional inference rules favouring a potential implementation; (iv) the computational complexity of deciding entailment in ρdf and ρdf⊥ are the same; and (v) defeasible entailment can be decided via a polynomial number of calls to an oracle deciding ground triple entailment in ρdf⊥ and, in particular, deciding defeasible entailment can be done in polynomial time.

Upon Reflection

Upon Reflection

Segregation and intermixing in polydisperse liquid–solid fluidized beds: A multifluid model validation study

AbstractMultifluid 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.

Conditional Inference under Disjunctive Rationality

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.

Contextual Conditional Reasoning

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: from the Lexicographic Closure to the Skeptical 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.

Algorithmic definitions for KLM-style defeasible disjunctive Datalog

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

The paper describes a preferential approach for dealing with exceptions in KLM preferential logics, based on the rational closure. It is well known that rational closure does not allow an independent handling of inheritance of different defeasible properties of concepts. In this work, we consider an alternative closure construction, called Multi Preference closure (MP-closure), which has been first considered for reasoning with exceptions in DLs. We reconstruct the notion of MP-closure in the propositional case and show that it is a natural (weaker) variant of Lehmann's lexicographic closure, which appears to be too bold in some cases. The MP-closure defines a preferential consequence relation that, although weaker than lexicographic closure, is stronger than Relevant Closure.

Stable decomposability of matrices over the rational closure of a group algebra of an ordered group

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.

A Lightweight Defeasible Description Logic in Depth

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.

A description logic framework for commonsense conceptual combination integrating typicality, probabilities and cognitive heuristics

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

Propositional Typicality Logic (PTL) is a recently proposed logic, obtained by enriching classical propositional logic with a typicality operator capturing the most typical (alias normal or conventional) situations in which a given sentence holds. The semantics of PTL is in terms of ranked models as studied in the well-known KLM approach to preferential reasoning and therefore KLM-style rational consequence relations can be embedded in PTL. In spite of the non-monotonic features introduced by the semantics adopted for the typicality operator, the obvious Tarskian definition of entailment for PTL remains monotonic and is therefore not appropriate in many contexts. Our first important result is an impossibility theorem showing that a set of proposed postulates that at first all seem appropriate for a notion of entailment with regard to typicality cannot be satisfied simultaneously. Closer inspection reveals that this result is best interpreted as an argument for advocating the development of more than one type of PTL entailment. In the spirit of this interpretation, we investigate three different (semantic) versions of entailment for PTL, each one based on the definition of rational closure as introduced by Lehmann and Magidor for KLM-style conditionals, and constructed using different notions of minimality.

From iterated revision to iterated contraction: Extending the Harper Identity

The study of iterated belief change has principally focused on revision, with the other main operator of AGM belief change theory, namely contraction, receiving comparatively little attention. In this paper we show how principles of iterated revision can be carried over to iterated contraction by generalising a principle known as the ‘Harper Identity’. The Harper Identity provides a recipe for defining the belief set resulting from contraction by a sentence A in terms of (i) the initial belief set and (ii) the belief set resulting from revision by ¬A. Here, we look at ways to similarly define the conditional belief set resulting from contraction by A. After noting that the most straightforward proposal of this kind leads to triviality, we characterise a promising family of alternative suggestions that avoid such a result. One member of that family, which involves the operation of rational closure, is noted to be particularly theoretically fruitful and normatively appealing.

On the rational closure of connected closed subgroups of connected simply connected nilpotent Lie groups

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.

Contextual rational closure for defeasible $\mathcal {ALC}$

Description logics have been extended in a number of ways to support defeasible reasoning in the KLM tradition. Such features include preferential or rational defeasible concept inclusion, and defeasible roles in complex concept descriptions. Semantically, defeasible subsumption is obtained by means of a preference order on objects, while defeasible roles are obtained by adding a preference order to role interpretations. In this paper, we address an important limitation in defeasible extensions of description logics, namely the restriction in the semantics of defeasible concept inclusion to a single preference order on objects. We do this by inducing a modular preference order on objects from each modular preference order on roles, and using these to relativise defeasible subsumption. This yields a notion of contextualised rational defeasible subsumption, with contexts described by roles. We also provide a semantic construction for rational closure and a method for its computation, and present a correspondence result between the two.

On Conway mutation and link homology

We give a new, elementary proof that Khovanov homology with Z/2Z–coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that δ–graded knot Floer homology is mutation–invariant. Using the Clifford module structure on HFK˜ induced by basepoint maps, we carry out this strategy for mutations on a large class of tangles. Let L′ be a link obtained from L by mutating the tangle T. Suppose some rational closure of T corresponding to the mutation is the unlink on any number of components. Then L and L′ have isomorphic δ–graded HFKˆ groups over Z/2Z as well as isomorphic Khovanov homology over Q. We apply these results to establish mutation–invariance for the infinite families of Kinoshita-Terasaka and Conway knots. Finally, we give sufficient conditions for a general Khovanov-Floer theory to be mutation–invariant.