A Set-Theoretic Approach to ABox Reasoning Services

Cantone, Domenico; Asmundo, Marianna Nicolosi; Santamaria, Daniele Francesco

doi:10.1007/978-3-319-61252-2_7

Cited by 8 publications

(28 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A set-theoretic approach in DLs has been adopted by Cantone et al in [7] for determining the decidability of higher order conjunctive query answering in the description logic DL 4,× D (where concept and role variables may occur in queries), as well as for developing a tableau based procedure for dealing with several well-known ABox reasoning tasks.…”

Section: Related Workmentioning

confidence: 99%

Adding the power-set to description logics

Giordano

Policriti

2020

Theoretical Computer Science

View full text Add to dashboard Cite

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 ...

show abstract

Section: Related Workmentioning

confidence: 99%

Adding the power-set to description logics

Giordano

Policriti

2020

Theoretical Computer Science

View full text Add to dashboard Cite

show abstract

“…1 In [2] we proved the decidability of the consistency problem of a DL 4,× D -KB and of a relevant ABox reasoning task, namely the Conjunctive Query Answering (CQA) problem for DL 4,× D consisting in computing the answer set of a DL 4,× D -conjunctive query with respect to a DL 4,× D -KB. In [3] we generalized the problem introducing the Higher Order Conjuctive Query Answering (HOCQA) problem for DL 4,× D . Such problem is characterized by Higher Order (HO) DL 4,× D -conjunctive queries admitting variables of three sorts: individual and data type variables, concept variables, and role variables.…”

Section: The Logic DL 4lqs R× (D)mentioning

confidence: 99%

“…Substitutions can be extended to HO DL 4,× D -conjunctive queries in the usual way. Let Q : As illustrated in [3], the HOCQA problem can be instantiated to significant ABox reasoning problems such as (A) role filler retrieval, the problem of retrieving all the fillers x such that the pair (a, x) is an instance of a role R; (B) concept retrieval, the problem of retrieving all concepts which an individual is an instance of; (C) role instance retrieval, the problem of retrieving all roles which a pair of individuals (a, b) is an instance of; and (D) conjunctive query answering, the problem of finding the answer set of a conjunctive query.…”

Section: The Logic DL 4lqs R× (D)mentioning

confidence: 99%

“…In [3] we solved the HOCQA problem just stated by reducing it to the analogous problem formulated in the context of the fragment 4LQS R DL 4,× D (and in turn to the decision procedure for 4LQS R presented in [1]).…”

Section: The Logic DL 4lqs R× (D)mentioning

confidence: 99%

“…The results presented in [2] have been extended in [3] to the main ABox reasoning tasks for DL 4,× D , such as instance checking and concept retrieval. by defining the Higher-Order Conjunctive Query Answering (HOCQA) problem for DL 4,× D .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An Optimized KE-Tableau-Based System for Reasoning in the Description Logic $${\mathcal {DL}}_{{\mathbf {D}}}^{4,\!\times }$$

Cantone

Asmundo

Santamaria

2018

Rules and Reasoning

Self Cite

View full text Add to dashboard Cite

We present a KE-tableau-based procedure for the main TBox and ABox reasoning tasks for the description logic DL 4LQS R,× (D), in short DL 4,× D . The logic DL 4,× D , representable in the decidable multi-sorted quantified set-theoretic fragment 4LQS R , combines the high scalability and efficiency of rule languages such as the Semantic Web Rule Language (SWRL) with the expressivity of description logics. Our algorithm is based on a variant of the KE-tableau system for sets of universally quantified clauses, where the KE-elimination rule is generalized in such a way as to incorporate the γ-rule. The novel system, called KE γ -tableau, turns out to be an improvement of the system introduced in [3] and of standard first-order KE-tableaux [11]. Suitable benchmark test sets executed on C++ implementations of the three mentioned systems show that the performances of the KE γ -tableau-based reasoner are often up to about 400% better than the ones of the other two systems. This a first step towards the construction of efficient reasoners for expressive OWL ontologies based on fragments of computable set-theory.In this paper, the KE-tableau-based procedure defined in [3] is modified, by eliminating the preprocessing phase for universally quantified formulae and replacing the standard KE-elimination rule with a novel elimination rule, called E γ -rule, incorporating the standard rule for treating universally quantified formulae (γ-rule). The resulting system turns out to be more efficient than the and the First-Order (FO) as shown by suitable benchmarking tests executed on C++ implementations of the three systems. The main reason for such a speed-up relies on the fact that the novel E γ -rule does not need to store the instances of universally quantified formulae on the KE-tableau. DL 4,× D -formula ϕ. A 4LQS R DL 4,× D -interpretation is a pair M = (D, M ), where D is a nonempty collection of objects (called domain or universe of M) and M is an assignment

show abstract

Onset and Today’s Perspectives of Multilevel Syllogistic

Cantone,

Omodeo

2024

Lecture Notes in Computer Science

View full text Add to dashboard Cite

A Set-Theoretic Approach to ABox Reasoning Services

Cited by 8 publications

References 17 publications

Adding the power-set to description logics

Adding the power-set to description logics

An Optimized KE-Tableau-Based System for Reasoning in the Description Logic $${\mathcal {DL}}_{{\mathbf {D}}}^{4,\!\times }$$

Onset and Today’s Perspectives of Multilevel Syllogistic

Contact Info

Product

Resources

About