Semantic characterization of rational closure: From propositional logic to description logics

Giordano, Laura; Gliozzi, Valentina; Olivetti, Nicola; Pozzato, Gian Luca

doi:10.1016/j.artint.2015.05.001

Cited by 98 publications

(242 citation statements)

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“…The semantics of the T operator can be equivalently formulated in terms of rational models [Giordano et al (2015)]: a model M is any structure ∆ I , <, . I where ∆ I is the domain, < is an irreflexive, transitive, well-founded and modular (for all x, y, z in ∆ I , if x < y then either x < z or z < y) relation over ∆ I .…”

Section: 2mentioning

confidence: 99%

“…A model M in the logic T CL extends standard ALC models by a preference relation among domain elements as in the logic of typicality [Giordano et al (2015)]. In this respect, x < y means that x is "more normal" than y, and that the typical members of a concept C are the minimal elements of C with respect to this relation 8 .…”

Section: T CL : a Logic For Commonsense Concept Combinationmentioning

confidence: 99%

“…A model M can be equivalently defined by postulating the existence of a function k M : ∆ I −→ N, where k M assigns a finite rank to each domain element [Giordano et al (2015)]: the rank of x is the length of the longest chain x 0 < · · · < x from x to a minimal x 0 , i.e. such that there is no x such that x < x 0 .…”

Section: T CL : a Logic For Commonsense Concept Combinationmentioning

confidence: 99%

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

Lieto

Pozzato

2019

Journal of Experimental & Theoretical Artificial Intelligen

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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 ALC + T R , 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 ALC + T R by typicality inclusions of the form p :: T(C) D, whose intuitive meaning is that "we believe with degree p 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 C 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 ALC.

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Section: 2mentioning

confidence: 99%

Section: T CL : a Logic For Commonsense Concept Combinationmentioning

confidence: 99%

Section: T CL : a Logic For Commonsense Concept Combinationmentioning

confidence: 99%

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

Lieto

Pozzato

2019

Journal of Experimental & Theoretical Artificial Intelligen

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“…In this section we consider another option for entailment based on a version of minimality, and derived from the characterisation of rational closure by Giordano et al [17,19]. The general idea is to respect the principle of presumption of typicality (see Section 3), We shall refer to this form of entailment as Presumption of Typicality entailment, shortened to PT-entailment.…”

Section: Pt-entailmentmentioning

confidence: 99%

On rational entailment for Propositional Typicality Logic

Booth

Casini

Meyer

et al. 2019

Artificial Intelligence

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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. * This technical report presents an extended and elaborated version of a paper presented at the 24th International Joint Conference on Artificial Intelligence (IJCAI 2015). Si is a ranked model of KB.Proof:So, at each stage of the algorithm, the current ranked interpretation, when those valuations not satisfying KB are excluded, forms a ranked model of KB. Since the output R * KB takes precisely this form we have the following result. Proposition 5.1 R * KB KB. Proof: Follows from Lemma 5.2 and the construction of R * KB . Next we want to show that for any other ranked model R of KB, we have R * KB LM R. Lemma 5.3 Let R * KB := (L 0 , . . . , L n−1 , L ∞ ) and let R := (M 0 , . . . , M n−1 , M ∞ ) be any other ranked model of KB.From this lemma we can state:Proposition 5.2 Consider any KB and let R be a ranked model of KB. Then R * KB LM R.We are now in a position to define our first form of entailment for PTL. Definition 5.2 (LM-entailment) Let KB ⊆ L • and α ∈ L • . We say KB LM-entails α, denoted KB |≈ LM α, if R * KB α. Its corresponding consequence operator is defined as Cn LM (KB) := {α ∈ L • | R * KB α}.The next result outlines which properties from the previous section are satisfied by Cn LM (·).Theorem 5.2 Cn LM (·) satisfies P1-P7, P9, and P10, but not P8. Proof:For P1, Proposition 5.1 guarantees that R * KB is a model of KB. About P2, by Proposition 5.2, R * KB is the LM-minimum model of KB. If R * KB α, R * KB must also be the LM-minimum model of KB ∪ {α}. For P3, note that R * KB is a ranked model of KB (Lemma 5.1, Item 3, plus Proposition 5.1), and so if α ∈ Cn 0 (KB), then α ∈ R * KB . P4 is an immediate consequence of the satisfaction of P7. 3 P5 is an immediate consequence of the satisfaction o...

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“…if it can be moved, then it is not possible without violating at least one axiom in the KB). We adapt the definition for minimal ranked models here from the work of Giordano et al [71].…”

Section: Definition 18 (Exceptional Subset Sequence)mentioning

confidence: 99%

Practical Reasoning for Defeasible Description Logics

Moodley¹

2017

Preprint

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I, , declare that:1. The research reported in this dissertation, except where otherwise indicated, is my original research.2. This dissertation has not been submitted for any degree or examination at any other university. 5. This dissertation does not contain text, graphics or tables copied and pasted from the Internet, unless specifically acknowledged, and the source being detailed in the dissertation and in the References section.

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Semantic characterization of rational closure: From propositional logic to description logics

Cited by 98 publications

References 28 publications

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

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

On rational entailment for Propositional Typicality Logic

Practical Reasoning for Defeasible Description Logics

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