Description Logics in the Calculus of Structures

We introduce a new proof system for the description logic ALC in the framework of the calculus of structures, a structural proof theory that employs deep inference. This new formal presentation introduces positive proofs for description logics. Moreover, this result makes possible the study of sub-structural refinements of description logics, for which a semantics can now be defined. 1 A calculus of structures for description logics Proof systems in the calculus of structures are defined by a set of deep inference rules operating on structures[1]. The rules are said to be deep because unlike the sequent calculus for which rules must be applied at the root of sequents, the rules of the calculus of structures can be applied at any depth inside a structure. As noted by Schild[2], ALC is a syntactic variant of propositional multimodal logic K(m). Therefore, since this logic involves no interaction between its modalities, its proof system in the calculus of structures can be straightforwardly extended from a proof system of unimodal K in the calculus of structures, such as the cut-free proof system SKSgK described in [3]. Let A be a countable set equipped with a bijective function ·̄ : A → A, such that Ā = A, and Ā 6= A for every A ∈ A. The elements of A are called primitive concepts, and two of them are denoted by > and ⊥ such that >̄ := ⊥ and ⊥̄ := >. The set R of prestructures of ALC concepts is defined by the following grammar, where A is a primitive concept and R is a role name: C,D ::= > | ⊥ | A | C̄ | (C,D) | [C,D] | ∃R.C | ∀R.C . On the setR, the relation = is defined to be the smallest congruence relation induced by the following equations. Associativity (C, (D,E)) = ((C,D), E) [C, [D,E]] = [[C,D], E] Commutativity