127 research outputs found
Living Without Beth and Craig: Definitions and Interpolants in Description Logics with Nominals and Role Inclusions
The Craig interpolation property (CIP) states that an interpolant for an
implication exists iff it is valid. The projective Beth definability property
(PBDP) states that an explicit definition exists iff a formula stating implicit
definability is valid. Thus, the CIP and PBDP transform potentially hard
existence problems into deduction problems in the underlying logic. Description
Logics with nominals and/or role inclusions do not enjoy the CIP nor PBDP, but
interpolants and explicit definitions have many potential applications in
ontology engineering and ontology-based data management. In this article we
show the following: even without Craig and Beth, the existence of interpolants
and explicit definitions is decidable in description logics with nominals
and/or role inclusions such as ALCO, ALCH and ALCHIO. However, living without
Craig and Beth makes this problem harder than deduction: we prove that the
existence problems become 2ExpTime-complete, thus one exponential harder than
validity. The existence of explicit definitions is 2ExpTime-hard even if one
asks for a definition of a nominal using any symbol distinct from that nominal,
but it becomes ExpTime-complete if one asks for a definition of a concept name
using any symbol distinct from that concept name.Comment: We have added results on description logics with role inclusions and
an ExpTime-completeness result for the explicit definability of concept
names. The title has been modified by adding role inclusions. This paper has
been accepted for AAAA 202
Craig Interpolation for Decidable First-Order Fragments
We show that the guarded-negation fragment (GNFO) is, in a precise sense, the
smallest extension of the guarded fragment (GFO) with Craig interpolation. In
contrast, we show that the smallest extension of the two-variable fragment
(FO2), and of the forward fragment (FF) with Craig interpolation, is full
first-order logic. Similarly, we also show that all extensions of FO2 and of
the fluted fragment (FL) with Craig interpolation are undecidable.Comment: Submitted for FoSSaCS 2024. arXiv admin note: substantial text
overlap with arXiv:2304.0808
Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via Nested Sequents
We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants
On -Interpolation in Local Theory Extensions and Applications to the Study of Interpolation in the Description Logics
We study the problem of -interpolation, where is a set of binary
predicate symbols, for certain classes of local extensions of a base theory.
For computing the -interpolating terms, we use a hierarchic approach: This
allows us to compute the interpolating terms using a method for computing
interpolating terms in the base theory. We use these results for proving
-interpolation in classes of semilattices with monotone operators; we
show, by giving a counterexample, that -interpolation does not hold if by
"shared" symbols we mean just the common symbols. We use these results for the
study of -interpolation in the description logics and
.Comment: 33 page
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