81 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
Living without Beth and Craig: Definitions and Interpolants in the Guarded and Two-Variable Fragments
In logics with the Craig interpolation property (CIP) the existence of an interpolant for an implication follows from the validity of the implication. In logics with the projective Beth definability property (PBDP), the existence of an explicit definition of a relation follows from the validity of a formula expressing its implicit definability. The two-variable fragment, FO2, and the guarded fragment, GF, of first-order logic both fail to have the CIP and the PBDP. We show that nevertheless in both fragments the existence of interpolants and explicit definitions is decidable. In GF, both problems are 3ExpTime-complete in general, and 2ExpTime-complete if the arity of relation symbols is bounded by a constant c not smaller than 3. In FO2, we prove a coN2ExpTime upper bound and a 2ExpTime lower bound for both problems. Thus, both for GF and FO2 existence of interpolants and explicit definitions are decidable but harder than validity (in case of FO2 under standard complexity assumptions)
Exact query reformulation over databases with first-order and description logics ontologies
We study a general framework for query rewriting in the presence of an arbitrary first-order logic ontology over a database signature. The framework supports deciding the existence of a safe-range first-order equivalent reformulation of a query in terms of the database signature, and if so, it provides an effective approach to construct the reformulation based on interpolation using standard theorem proving techniques (e.g., tableau). Since the reformulation is a safe-range formula, it is effectively executable as an SQL query. At the end, we present a non-trivial application of the framework with ontologies in the very expressive ALCHOIQ description logic, by providing effective means to compute safe-range first-order exact reformulations of queries
Synthesizing Nested Relational Queries from Implicit Specifications
Derived datasets can be defined implicitly or explicitly. An implicit
definition (of dataset in terms of datasets ) is a logical
specification involving the source data and the interface data .
It is a valid definition of in terms of , if any two models of the
specification agreeing on agree on . In contrast, an explicit
definition is a query that produces from . Variants of Beth's
theorem state that one can convert implicit definitions to explicit ones.
Further, this conversion can be done effectively given a proof witnessing
implicit definability in a suitable proof system. We prove the analogous
effective implicit-to-explicit result for nested relations: implicit
definitions, given in the natural logic for nested relations, can be
effectively converted to explicit definitions in the nested relational calculus
NRC. As a consequence, we can effectively extract rewritings of NRC queries in
terms of NRC views, given a proof witnessing that the query is determined by
the views
Interpolation in Linear Logic and Related Systems
We prove that there are continuum-many axiomatic extensions of the full
Lambek calculus with exchange that have the deductive interpolation property.
Further, we extend this result to both classical and intuitionistic linear
logic as well as their multiplicative-additive fragments. None of the logics we
exhibit have the Craig interpolation property, but we show that they all enjoy
a guarded form of Craig interpolation. We also exhibit continuum-many axiomatic
extensions of each of these logics without the deductive interpolation
property
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