Qualitative Reasoning about Incomplete Categorization Rules Based on Interpolation and Extrapolation in Conceptual Spaces

Schockaert, Steven; Prade, Henri

doi:10.1007/978-3-642-23963-2_24

Cited by 5 publications

(7 citation statements)

References 15 publications

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“…Another qualitative approach to the same problem has recently been proposed by the authors [25], using the notion of betweenness as primitive in the setting of Gärdenfors conceptual spaces. In this latter approach, the basic inference pattern involves two rules, while in the approach presented here, it relies on three rules (which is also to be compared with case-based reasoning where a current situation is matched with only one rule at a time).…”

Section: Discussionmentioning

confidence: 99%

Completing rule bases in symbolic domains by analogy making

Prade¹,

Schockaert²

2011

Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011)

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The paper considers the problem of completing a set of parallel if-then rules that provides a partial description of how a conclusion variable depends on the values of condition variables, where each variable takes its value among a finite ordered set of labels. The proposed approach does not require the use of fuzzy sets for the interpretation of these labels or for defining similarity measures, but rather relies on the extrapolation of missing rules on the basis of analogical proportions that hold for each variable between the labels of several parallel rules. The analogical proportions are evaluated for binary and multiple-valued variables on the basis of a logical expression involving Łukasiewicz implication. The underlying assumption is that the mapping partially specified by the given rules is as regular as suggested by these rules. A comparative discussion with other approaches is presented.

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Section: Discussionmentioning

confidence: 99%

Completing rule bases in symbolic domains by analogy making

Prade¹,

Schockaert²

2011

Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011)

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“…[ Derrac and Schockaert, 2014, Derrac and Schockaert, 2015, Schockaert and Prade, 2011 have thoroughly studied betweenness in conceptual spaces as a basis for common-sense reasoning. They argue that betweenness is invariant under changes in context, which are typically reflected by changes of the salience weights in a conceptual space.…”

Section: Betweennessmentioning

confidence: 99%

“…They argue that betweenness is invariant under changes in context, which are typically reflected by changes of the salience weights in a conceptual space. [Schockaert and Prade, 2011] generalize the crisp betweenness relation from points to regions. [Derrac and Schockaert, 2014] propose different soft notions of betweenness for points and subsequently generalize them to regions as well.…”

Section: Betweennessmentioning

confidence: 99%

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Formal ways for measuring relations between concepts in conceptual spaces^†

Bechberger

Kühnberger

2018

Expert Systems

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The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points in a high-dimensional space and concepts are represented by regions in this space. In this article, we extend our recent mathematical formalization of this framework by providing quantitative mathematical definitions for measuring relations between concepts: We develop formal ways for computing concept size, subsethood, implication, similarity, and betweenness. This considerably increases the representational capabilities of our formalization and makes it the most thorough and comprehensive formalization of conceptual spaces developed so far.The parameter W = W ∆ , {W δ } δ∈∆ contains two parts: W ∆ is the set of positive domain weights w δ with δ∈∆ w δ = |∆|. Moreover, W contains for each domain δ ∈ ∆ a set W δ of dimension weights w d with d∈δ w d = 1.The similarity of two points in a conceptual space is inversely related to their distance. This can be written as follows :Sim(x, y) = e −c·d(x,y) with a constant c > 0 and a given metric dBetweenness is a logical predicate B(x, y, z) that is true if and only if y is considered to be between x and z. It can be defined based on a given metric d:The betweenness relation based on d E results in the line segment connecting the points x and z, whereas the betweenness relation based on d M results in an axis-parallel cuboid between the points x and z. One can define convexity and star-shapedness based on the notion of betweenness:Definition 2. (Star-shapedness) A set S ⊆ CS is star-shaped under a metric d with respect to a set P ⊆ S : ⇐⇒ ∀p ∈ P, z ∈ S, y ∈ CS : (B d (p, y, z) → y ∈ S)

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“…Another qualitative approach, without any grade, relies at the semantic level on the conceptual spaces framework [63] where the possibility of expressing spatial-like localization such as "being in between" (by means of a ternary relation), or "parallelism" (by means of a quaternary relation) provides a basis for capturing interpolative and extrapolative reasoning respectively [103]. This proposal comes close to logical reasoning with analogical proportions [98], which are quaternary statements of the form "a is to b as c is to d" (involving Boolean, or graded, properties), but remains more cautious.…”

Section: Similaritymentioning

confidence: 99%