202 research outputs found
Undecidability of the unification and admissibility problems for modal and description logics
We show that the unification problem `is there a substitution instance of a
given formula that is provable in a given logic?' is undecidable for basic
modal logics K and K4 extended with the universal modality. It follows that the
admissibility problem for inference rules is undecidable for these logics as
well. These are the first examples of standard decidable modal logics for which
the unification and admissibility problems are undecidable. We also prove
undecidability of the unification and admissibility problems for K and K4 with
at least two modal operators and nominals (instead of the universal modality),
thereby showing that these problems are undecidable for basic hybrid logics.
Recently, unification has been introduced as an important reasoning service for
description logics. The undecidability proof for K with nominals can be used to
show the undecidability of unification for boolean description logics with
nominals (such as ALCO and SHIQO). The undecidability proof for K with the
universal modality can be used to show that the unification problem relative to
role boxes is undecidable for Boolean description logic with transitive roles,
inverse roles, and role hierarchies (such as SHI and SHIQ)
Admissibility of Î <sub>2</sub>-inference rules: Interpolation, model completion, and contact algebras
We devise three strategies for recognizing admissibility of non-standard inference rules via interpolation, uniform interpolation, and model completions. We apply our machinery to the case of symmetric implication calculus S2IC, where we also supply a finite axiomatization of the model completion of its algebraic counterpart, via the equivalent theory of contact algebras. Using this result we obtain a finite basis for admissible Î 2-rules
Computing Minimal EL-Unifiers is Hard
Unification has been investigated both in modal logics and in description logics, albeit with different motivations. In description logics, unification can be used to detect redundancies in ontologies. In this context, it is not sufficient to decide unifiability, one must also compute appropriate unifiers and present them to the user. For the description logic EL, which is used to define several large biomedical ontologies, deciding unifiability is an NP-complete problem. It is known that every solvable EL-unification problem has a minimal unifier, and that every minimal unifier is a local unifier. Existing unification algorithms for EL compute all minimal unifiers, but additionally (all or some) non-minimal local unifiers. Computing only the minimal unifiers would be better since there are considerably less minimal unifiers than local ones, and their size is usually also quite small. In this paper we investigate the question whether the known algorithms for EL-unification can be modified such that they compute exactly the minimal unifiers without changing the complexity and the basic nature of the algorithms. Basically, the answer we give to this question is negative
Dismatching and Local Disunification in EL
Unification in Description Logics has been introduced as a means to detect redundancies in ontologies. We try to extend the known decidability results for unification in the Description Logic EL to disunification since negative constraints on unifiers can be used to avoid unwanted unifiers. While decidability of the solvability of general EL-disunification problems remains an open problem, we obtain NP-completeness results for two interesting special cases: dismatching problems, where one side of each negative constraint
must be ground, and local solvability of disunification problems, where we restrict the attention to solutions that are built from so-called atoms occurring in the input problem. More precisely, we first show that dismatching can be reduced to local disunification, and then provide two complementary NP-algorithms for finding local solutions of (general) disunification problems
About the type of modal logics for the unification problem
Dans cette thÚse, nous étudierons le problÚme de l'unification dans les logiques modales ordinaires, les fusions de deux logiques modales et les logiques épistémiques multi-modales. Relativement à une logique propositionnelle L, étant donnée une formule A, nous devons trouver des substitutions s telle que s(A) est dans L. Lorsqu'elles existent, ces substitutions sont appelées unifieurs de A dans L. Nous étudions différentes méthodes pour construire des ensembles minimaux complets d'unifieurs d'une formule donnée A et, en fonction de la cardinalité des ces ensembles minimaux complets, nous discutons du type de l'unification de A. Enfin, nous déterminons les types de l'unification de plusieurs logiques propositionnelles.In this thesis, we shall investigate on the unification problem in ordinary modal logics, fusions of two modal logics and multi-modal epistemic logics. With respect to a propositional logic L, given a formula A, we have to find substitutions s such that s(A) is in L. When they exist, these substitutions are called unifiers of A in L. We study different methods for the construction of minimal complete sets of unifiers of a given formula A and according to the cardinality of these minimal complete sets, we shall discuss on the unification type of A. Then, we determine the unification types of several propositional logics
Dismatching and Local Disunification in EL
Unification in Description Logics has been introduced as a means to detect redundancies in ontologies. We try to extend the known decidability results for unification in the Description Logic EL to disunification since negative constraints on unifiers can be used to avoid unwanted unifiers. While decidability of the solvability of general EL-disunification problems remains an open problem, we obtain NP-completeness results for two interesting special cases: dismatching problems, where one side of each negative constraint must be ground, and local solvability of disunification problems, where we restrict the attention to solutions that are built from so-called atoms occurring in the input problem. More precisely, we first show that dismatching can be reduced to local disunification, and then provide two complementary NP-algorithms for finding local solutions of (general) disunification problems
Mechanised Uniform Interpolation for Modal Logics K, GL, and iSL
The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) Gödel-Löb logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong Löb logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic
Mechanised Uniform Interpolation for Modal Logics K, GL, and iSL
The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) Gödel-Löb logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong Löb logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic
Automated proof search in non-classical logics : efficient matrix proof methods for modal and intuitionistic logics
In this thesis we develop efficient methods for automated proof search within
an important class of mathematical logics. The logics considered are the varying,
cumulative and constant domain versions of the first-order modal logics
K, K4, D, D4, T, S4 and S5, and first-order intuitionistic logic. The use of
these non-classical logics is commonplace within Computing Science and Artificial
Intelligence in applications in which efficient machine assisted proof search
is essential.
Traditional techniques for the design of efficient proof methods for classical
logic prove to be of limited use in this context due to their dependence on
properties of classical logic not shared by most of the logics under consideration.
One major contribution of this thesis is to reformulate and abstract some of these
classical techniques to facilitate their application to a wider class of mathematical
logics.
We begin with Bibel's Connection Calculus: a matrix proof method for classical
logic comparable in efficiency with most machine orientated proof methods
for that logic. We reformulate this method to support its decomposition into
a collection of individual techniques for improving the efficiency of proof search
within a standard cut-free sequent calculus for classical logic. Each technique
is presented as a means of alleviating a particular form of redundancy manifest
within sequent-based proof search. One important result that arises from this
anaylsis is an appreciation of the role of unification as a tool for removing certain
proof-theoretic complexities of specific sequent rules; in the case of classical
logic: the interaction of the quantifier rules.
All of the non-classical logics under consideration admit complete sequent
calculi. We anaylse the search spaces induced by these sequent proof systems
and apply the techniques identified previously to remove specific redundancies
found therein. Significantly, our proof-theoretic analysis of the role of unification
renders it useful even within the propositional fragments of modal and
intuitionistic logic
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