Locality and Applications to Subsumption Testing in EL and Some of its Extensions

Abstract

In this paper we show that subsumption problems in lightweight description logics (such as E L and E L + ) can be expressed as uniform word problems in classes of semilattices with monotone operators. We use possibilities of efficient local reasoning in such classes of algebras, to obtain uniform PTIME decision procedures for CBox subsumption in E L , E L + and extensions thereof. These locality considerations allow us to present a new family of (possibly many-sorted) logics which extend E L and E L + with n-ary roles and/or numerical domains. As a by-product, this allows us to show that the algebraic models of E L and E L + have ground interpolation and thus that E L , E L + , and their extensions studied in this paper have interpolation. We also show how these ideas can be used for the description logic E L ++ .

Tractable Extensions of the Description Logic ${\mathcal{EL}}$ with Numerical Datatypes

“…Similarly, each set S + and S − contains at most one restriction of the form (≤, x) and at most one restriction of the form (≥, x) as a consequence of the transformation rules (17)-(20). Therefore (10) and (11) hold. Table 5 is safe.…”

“…Another related polynomial extension of EL has been considered by Viorica Sofronie-Stokkermans [10], where in addition to the standard concept constructors in EL, one can use concept of the form ↓m, ↑n, and [m; n], interpreted as (semi-)intervals over a partially ordered set (P, <). The difference with the results discussed before, is that the domain P and therefore the values of the end-point parameters m and n are not fixed but can be chosen arbitrarily for every interpretation.…”

Deciding Unifiability and Computing Local Unifiers in the Description Logic EL without Top Constructor

“…Unification was originally not introduced for Description Logics, but for equational theories [7]. In [26,4] it was shown that equivalence and unification in EL are the same as the word problem and unification, respectively, in the equational theory bSLmO of bounded (meet-)semilattices with monotone operators. The signature Σ bSLmO of this equational theory consists of a binary function symbol ∧, a constant symbol 1, and finitely many unary function symbols f 1 , .…”

О позитивном фрагменте полимодальной логики доказуемости $\mathbf{GLP}$

“…Вопросы сложности модальных логик в ограниченных языках рассматривались ранее преимущественно в контексте дескрипционных логик. В работах [8]- [11] в дескрипционной постановке исследуется сложность задачи, представимой в модальных терминах следующим образом. Пусть формулы построены из переменных, связок ⊤, ⊥, ∧ и не более чем счетного множества связок ♦ .…”