A Suggestion for an n-ary Description Logic

A restriction most Description Logics (DLs) have in common with most Modal Logics is their restriction to unary and binary predicates. To our knowledge, the only DLs that overcome these restrictions and allow for arbitrary n-ary relations are NaryKandor 8] and the very expressive DL DLR 2]. In the eld of Modal Logics, there are two generali-sations that allow for n-ary predicates: Polyadic Modal Logics 9] and the more expressive Guarded Fragment 1], which was shown to be ExpTime-complete in 5] and for which a resolution based decision procedure exists 3]. Unfortunately, when extended by operators that are standard in DLs such as number restrictions, features, or transitive roles, this logic becomes undecidable. In this paper, we present a new DL, GF1 ? , that was designed to meet three goals: It should allow for n-ary relations, \concept" subsumption and satissability should be in PSpace, and it should allow the extension with number restrictions and/or transitive roles (without losing decid-ability). GF1 ? is a fragment of the rst Guarded Fragment, which in turn is a fragment of rst order logic. Quan-tiied variables in the rst Guarded Fragment must be guarded, i.e., formulae are restricted to those of the form 9x:(R(x; y) ^ (x)) and 8x:(R(x; y)) (x)); where x; y are variable vectors, R is an atom (the so-called guard), and is a formulae of the rst guarded fragment with free variables x. GF1 ? is obtained from the rst guarded fragment by restricting the way in which variables may occur in guards: The variables of each atom are divided into two parts, and all quantiied variables must ll exactly one of these parts. It can easily be seen that GF1 ? extends both ALCI and polyadic Modal Logics. For example, we can describe unmarried women whose parents are married and who have only married brothers by the following GF1 ?-formula. 9w:M(w; x 0)](F(w)))) In this preliminary report, we present a PSpace tableaux algorithm for GF1 ? , which we believe can be extended to handle, for example, number restrictions. 2 Preliminaries In this Section, we introduce GF1 ? , explain the syntactic restrictions, deene inference problems, and explain why we believe GF1 ? to be a DL. Deenition 1 Let X be a set of variables. The set free() X denotes the set of variables that occur free in a formula. For a variable vector x 2 X n …