The Description Logic SHIQ with a Flexible Meta-Modelling Hierarchy

Motz, Regina; Rohrer, Edelweis; Severi, Paula

doi:10.2139/ssrn.3198920

Cited by 1 publication

(2 citation statements)

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“…Pan et al in [28] and Motz et al in [26] define extensions of OWL DL and of SHIQ (respectively), based on semantics interpreting concepts as well-founded sets. In particular, [26] adds to SHIQ meta-modelling axioms equating individuals to concepts, without requiring that the instances of a concept need to stay in the same layer, and develop a tableau algorithm as an extension of the one for SHIQ.…”

Section: Related Workmentioning

confidence: 99%

“…Many other approaches to metamodeling have been proposed in the literature, including membership among concepts. Most of them [3,11,19,22,16] are based on a Hilog semantics, while [28,26] define extensions of OWL DL and of SHIQ (respectively), based on semantics interpreting concepts as well-founded sets-i.e. sets with no cycles or infinite descending chains of ∈-related sets.…”

Section: Introductionmentioning

confidence: 99%

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Adding the power-set to description logics

Giordano

Policriti

2020

Theoretical Computer Science

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We explore the relationships between Description Logics and Set Theory. The study is carried on using, on the set-theoretic side, a very rudimentary axiomatic set theory Ω, consisting of only four axioms characterizing binary union, set difference, inclusion, and the power-set. An extension of ALC, ALC Ω , is then defined in which concepts are naturally interpreted as sets living in Ω-models. In ALC Ω not only membership between concepts is allowed-even admitting circularity-but also the power-set construct is exploited to add metamodeling capabilities. We investigate translations of ALC Ω into standard description logics as well as a set-theoretic translation. A polynomial encoding of ALC Ω in ALCOI proves the validity of the finite model property as well as an EXP-TIME upper bound on the complexity of concept satisfiability. We develop a settheoretic translation of ALC Ω in the theory Ω, exploiting a technique proposed for translating normal modal and polymodal logics into Ω. Finally, we show that the fragment LC Ω of ALC Ω , which does not admit roles and individual names, is as expressive as ALC ΩAdding the Power-Set to Description Logicscompleted with the standard deduction rules of generalization and modus-ponens.The above theory must be intended as a minimal Hilbert-style axiomatic system for set theory. When thinking of specific models of Ω, however, we can clearly think of structures satisfying extra axioms. In particular, for example, the familiar well-founded models of set theories, are perfectly legitimate models of Ω, in which the extra axiom of well-foundedness-implying that ∈ cannot form cycles or infinite descending chainsholds. For instance let x = {∅, {∅}}. x is a finite well-founded set, and the sets ∅ and {∅} are elements of x. Instead, the set y = {∅, {∅, {∅, {. . .}}} is finite but not wellfounded.Whatever the axioms satisfied by the Ω-model under consideration are, however, everything in the domain of such a model is supposed to be a set. As a consequence, a set will have (only) sets as its elements. Moreover, as observed, circular definitions of sets are not forbidden. That is, for example, there are models of Ω in which there are sets admitting themselves as elements. For instance, the set y above could simply be defined as y = {∅, y} and has elements ∅ and y itself.Finally, not postulating in Ω any explicit "axiomatic link" between membership ∈ and equality-more precisely: having no extensionality axiom-, there exist Ω-models in which there are different sets with equal collection of elements. One (elementary) consequence of the extensionality axiom is the familiar fact that if a ⊆ b and b ⊆ a, then a = b. In non-extensional models, instead, there can be pairwise distinct sets included in each other. The set x ′ = {a, b, {b, c}} with a, b and c pairwise distinct and such that a, b, c ⊆ ∅, does not satisfy extensionality as a, b and c are different sets with the same (empty) extension.Definition 3. Ω-models are first order interpretations M = (U, · M ) satisfying the axioms of the theory ...

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%