A Uniform Tableaux-Based Method for Concept Abduction and Contraction in Description Logics

We present algorithms for Concept Abduction and Concept Contraction, two reasoning services in Description Logics (DL) recently proposed to model how several supplies fit a demand (all described by concepts), and vice versa, in e-commerce. An extended version of the paper is in [2]. Recent papers on tableaux for description logics use a labeling function L to map an individualx to a set of concepts L(x) such that for every concept C, C ∈ L(x) stands for the formula C(x), and similarly for roles R ∈ L(x, y). Here we distinguish between formulas labeled “true” and formulas labeled “false” in the tableaux, hence we use two labeling functions T() and F(), both going from individuals to sets ofconcepts, and from pairs of individuals to sets of roles. A (usual) tableau branch is now represented by two functions T() and F(). Moreover, we write in the name of an individual x its history, i.e., the string identifying x is made up of integers and role symbols, such as x = 1R3Q7, which means that individual x is used for concepts in a quantification involving role R, and inside, a quantification involving role Q. Integers in between roles make sure that such strings are unique, i.e., there can be two individuals with the same role sequence, but not with the same integer sequence [3]. Given an individual x in a tableau, an interpretation (∆I , ·I) satisfies two tableau labels T(x) and F(x) if, for every concept C ∈ T(x) and every concept D ∈ F(x), it is xI ∈ CI and xI ∈ DI respectively. Similarly, (∆I , ·I) satisfies two tableau labels T(x, y) and F(x, y) if for every role R ∈ T(x, y) and for every role Q ∈ F(x, y) it holds (xI , yI) ∈ RI and (xI, yI) ∈ QI . We note however that for the DL we adopt, every role Q that appears in a label F(x, y) is of the form ¬R, hence Q ∈ F(x, y) means, in fact, (xI , yI) ∈ RI too. An interpretation satisfies a tableau branch if it satisfies T(x), F(x), T(x, y) and F(x, y) for every individual x, and for every pair of individuals x, y in the branch. Each rule has a precondition, and an action modifying the tableau. When the precondition is met, the action can be performed. In order to simplify the preconditions, we assume that, for each different instance of the preconditions, each rule is applied at most once. We also assume that concepts are always simplified in Negation Normal Form (NNF, see [1, ch.2]), so that negations come only in front of concept names. Without NNF, the number of rules should be doubled. In what follows, given a concept C, we denote with C the NNF of ¬C. Rules come in pairs, first the (usual) version with a construct in the T-constraints, then the dual construct in the F-constraints. However, in what follows we omit rules marked with an asterisk (*), because the correspondent formulae do not appear in our tableaux for ALN .