Unification of Concept Terms in Description Logics

Baader, Franz; Narendran, Paliath

doi:10.1006/jsco.2000.0426

Cited by 90 publications

(122 citation statements)

References 31 publications

(42 reference statements)

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“…5 yield a new ExpTime-algorithm for deciding solvability of linear language equations. While the original decision procedure [1] constructs a tree automaton of exponential size and uses a linear-time emptiness test, our algorithm constructs a polynomial-size propagation net and uses an algorithm that is worst-case exponential, but exhibits a better behavior if the constructed set of propagation rules contains few nondeterministic clauses.…”

Section: Discussionmentioning

confidence: 99%

“…, t n over F and x. 1 We assume that the reader is familiar with Herbrand interpretations (see, e.g., [10]). We call a Herbrand interpretation H over the above signature finite if it interprets every predicate P ∈ P by a finite set P H .…”

Section: Propagation Rulesmentioning

confidence: 99%

“…A solution assigns finite sets of words to the variables X i such that the equation holds. Deciding whether such an equation has a solution is ExpTime-complete [1].…”

Section: Hardnessmentioning

confidence: 99%

“…The unification problem in this logic was shown to be ExpTime-complete in [1]. There, solving unification in FL 0 is shown to be equivalent to solving linear language equations.…”

Section: Introductionmentioning

confidence: 98%

“…It is worst-case exponential, but there are cases in which our algorithm runs in polynomial time. Thus, it has advantages over the previous algorithm [1], which is always exponential.…”

Section: Introductionmentioning

confidence: 99%

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Finding Finite Herbrand Models

Borgwardt

Morawska

2012

Logic for Programming, Artificial Intelligence, and Reasoning

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Abstract. We show that finding finite Herbrand models for a restricted class of first-order clauses is ExpTime-complete. A Herbrand model is called finite if it interprets all predicates by finite subsets of the Herbrand universe. The restricted class of clauses consists of anti-Horn clauses with monadic predicates and terms constructed over unary function symbols and constants. The decision procedure can be used as a new goal-oriented algorithm to solve linear language equations and unification problems in the description logic FL0. The new algorithm has only worst-case exponential runtime, in contrast to the previous one which was even best-case exponential.

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Section: Discussionmentioning

confidence: 99%

Section: Propagation Rulesmentioning

confidence: 99%

“…A solution assigns finite sets of words to the variables X i such that the equation holds. Deciding whether such an equation has a solution is ExpTime-complete [1].…”

Section: Hardnessmentioning

confidence: 99%

“…The unification problem in this logic was shown to be ExpTime-complete in [1]. There, solving unification in FL 0 is shown to be equivalent to solving linear language equations.…”

Section: Introductionmentioning

confidence: 98%

“…It is worst-case exponential, but there are cases in which our algorithm runs in polynomial time. Thus, it has advantages over the previous algorithm [1], which is always exponential.…”

Section: Introductionmentioning

confidence: 99%

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Finding Finite Herbrand Models

Borgwardt

Morawska

2012

Logic for Programming, Artificial Intelligence, and Reasoning

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Nonstandard Inferences in Description Logics: The Story So Far

Baader

Küsters

2006

International Mathematical Series

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Description logics (DLs) are a successful family of logic-based knowledge representation formalisms that can be used to represent the terminological knowledge of an application domain in a structured and formally well-founded way. DL systems provide their users with inference procedures that allow to reason about the represented knowledge. Standard inference problems (such as the subsumption and the instance problem) are now well-understood. Their computational properties (such as decidability and complexity) have been investigated in detail, and modern DL systems are equipped with highly optimized implementations of these inference procedures, which-in spite of their high worst-case complexity-perform quite well in practice.In applications of DL systems it has turned out that building and maintaining large DL knowledge bases can be further facilitated by procedures for other, nonstandard inference problem, such as computing the least common subsumer and the most specific concept, and rewriting and matching of concepts. While the research concerning these non-standard inferences is not as mature as the one for the standard inferences, it has now reached a point where it makes sense to motivate these inferences within a uniform application framework, give an overview of the results obtained so far, describe the remaining open problems, and give perspectives for future research in this direction.

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Strict Language Inequalities and Their Decision Problems

Okhotin

2005

Mathematical Foundations of Computer Science 2005

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Abstract. Systems of language equations of the form {ϕ(X1, . . . , Xn) = ∅, ψ(X1, . . . , Xn) = ∅} are studied, where ϕ, ψ may contain set-theoretic operations and concatenation; they can be equivalently represented as strict inequalities ξ(X1, . . . , Xn) ⊂ L0. It is proved that the problem whether such an inequality has a solution is Σ2-complete, the problem whether it has a unique solution is in (Σ3 ∩ Π3) \ (Σ2 ∪ Π2), the existence of a regular solution is a Σ1-complete problem, while testing whether there are finitely many solutions is Σ3-complete. The class of languages representable by their unique solutions is exactly the class of recursive sets, though a decision procedure cannot be algorithmically constructed out of an inequality, even if a proof of solution uniqueness is attached.

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Unification of Concept Terms in Description Logics

Cited by 90 publications

References 31 publications

Finding Finite Herbrand Models

Finding Finite Herbrand Models

Nonstandard Inferences in Description Logics: The Story So Far

Strict Language Inequalities and Their Decision Problems

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