270 research outputs found
Labelled Natural Deduction for Substructural Logics
In this paper a uniform methodology to perform Natural Deduction over the family of linear, relevance and intuitionistic logics is proposed. The methodology follows the Labelled Deductive Systems (LDS) discipline, where the deductive process manipulates declarative units { formulas labelled according to a labelling algebra. In the system de-scribed here, labels are either ground terms or variables of a given labelling language and inference rules manipulate formulas and labels simultaneously, generating (whenever necessary) constraints on the labels used in the rules. A set of natural deduction style inference rules is given, and the notion of a derivation is dened which associates a la-belled natural deduction style \structural derivation " with a set of generated constraints. Algorithmic procedures, based on a technique called resource abduction, are dened to solve the constraints generated within a derivation, and their termination conditions dis-cussed. A natural deduction derivation is correct with respect to a given substructural logic, if, under the condition that the algorithmic procedures terminate, the associated set of constraints is satised with respect to the underlying labelling algebra. This is shown by proving that the natural deduction system is sound and complete with respect to the LKE tableaux system [DG94].
Labelled natural deduction for substructural logics
In this paper a uniform methodology to perform Natural Deduction over the family of linear, relevance and intuitionistic logics is proposed. The methodology follows the Labelled Deductive Systems (LDS) discipline, where the deductive process manipulates declarative units - formulas labelled according to a labelling algebra. In the system described here, labels are either ground terms or variables of a given labelling language and inference rules manipulate formulas and labels simultaneously, generating (whenever necessary) constraints on the labels used in the rules. A set of natural deduction style inference rules is given, and the notion of a derivation is defined which associates a labelled natural deduction style "structural derivation" with a set of generated constraints. Algorithmic procedures, based on a technique called resource abduction, are defined to solve the constraints generated within a derivation, and their termination conditions discussed. A natural deduction derivation is correct with respect to a given substructural logic, if, under the condition that the algorithmic procedures terminate, the associated set of constraints is satisfied with respect to the underlying labelling algebra. This is shown by proving that the natural deduction system is sound and complete with respect to the LKE tableaux system
A Labelled Analytic Theorem Proving Environment for Categorial Grammar
We present a system for the investigation of computational properties of
categorial grammar parsing based on a labelled analytic tableaux theorem
prover. This proof method allows us to take a modular approach, in which the
basic grammar can be kept constant, while a range of categorial calculi can be
captured by assigning different properties to the labelling algebra. The
theorem proving strategy is particularly well suited to the treatment of
categorial grammar, because it allows us to distribute the computational cost
between the algorithm which deals with the grammatical types and the algebraic
checker which constrains the derivation.Comment: 11 pages, LaTeX2e, uses examples.sty and a4wide.st
An Alternative Natural Deduction for the Intuitionistic Propositional Logic
A natural deduction system NI, for the full propositional intuitionistic logic, is proposed. The operational rules of NI are obtained by the translation from Gentzen’s calculus LJ and the normalization is proved, via translations from sequent calculus derivations to natural deduction derivations and back.This work is supported by the Ministary of Science and Technology of Serbia, grant number ON174026
An Abstract Approach to Consequence Relations
We generalise the Blok-J\'onsson account of structural consequence relations,
later developed by Galatos, Tsinakis and other authors, in such a way as to
naturally accommodate multiset consequence. While Blok and J\'onsson admit, in
place of sheer formulas, a wider range of syntactic units to be manipulated in
deductions (including sequents or equations), these objects are invariably
aggregated via set-theoretical union. Our approach is more general in that
non-idempotent forms of premiss and conclusion aggregation, including multiset
sum and fuzzy set union, are considered. In their abstract form, thus,
deductive relations are defined as additional compatible preorderings over
certain partially ordered monoids. We investigate these relations using
categorical methods, and provide analogues of the main results obtained in the
general theory of consequence relations. Then we focus on the driving example
of multiset deductive relations, providing variations of the methods of matrix
semantics and Hilbert systems in Abstract Algebraic Logic
Non-normal modalities in variants of Linear Logic
This article presents modal versions of resource-conscious logics. We
concentrate on extensions of variants of Linear Logic with one minimal
non-normal modality. In earlier work, where we investigated agency in
multi-agent systems, we have shown that the results scale up to logics with
multiple non-minimal modalities. Here, we start with the language of
propositional intuitionistic Linear Logic without the additive disjunction, to
which we add a modality. We provide an interpretation of this language on a
class of Kripke resource models extended with a neighbourhood function: modal
Kripke resource models. We propose a Hilbert-style axiomatization and a
Gentzen-style sequent calculus. We show that the proof theories are sound and
complete with respect to the class of modal Kripke resource models. We show
that the sequent calculus admits cut elimination and that proof-search is in
PSPACE. We then show how to extend the results when non-commutative connectives
are added to the language. Finally, we put the logical framework to use by
instantiating it as logics of agency. In particular, we propose a logic to
reason about the resource-sensitive use of artefacts and illustrate it with a
variety of examples
Algebraic proof theory for LE-logics
In this paper we extend the research programme in algebraic proof theory from
axiomatic extensions of the full Lambek calculus to logics algebraically
captured by certain varieties of normal lattice expansions (normal LE-logics).
Specifically, we generalise the residuated frames in [16] to arbitrary
signatures of normal lattice expansions (LE). Such a generalization provides a
valuable tool for proving important properties of LE-logics in full uniformity.
We prove semantic cut elimination for the display calculi D.LE associated with
the basic normal LE-logics and their axiomatic extensions with analytic
inductive axioms. We also prove the finite model property (FMP) for each such
calculus D.LE, as well as for its extensions with analytic structural rules
satisfying certain additional properties
Natural deduction systems for some non-commutative logics
Varieties of natural deduction systems are introduced for Wansing’s paraconsistent non-commutative substructural logic, called a constructive sequential propositional logic (COSPL), and its fragments. Normalization, strong normalization and Church-Rosser theorems are proved for these systems. These results include some new results on full Lambek logic (FL) and its fragments, because FL is a fragment of COSPL
Goal-directed proof theory
This report is the draft of a book about goal directed proof theoretical formulations of non-classical logics. It evolved from a response to the existence of two camps in the applied logic (computer science/artificial intelligence) community. There are those members who believe that the new non-classical logics are the most important ones for applications and that classical logic itself is now no longer the main workhorse of applied logic, and there are those who maintain that classical logic is the only logic worth considering and that within classical logic the Horn clause fragment is the most important one. The book presents a uniform Prolog-like formulation of the landscape of classical and non-classical logics, done in such away that the distinctions and movements from one logic to another seem simple and natural; and within it classical logic becomes just one among many. This should please the non-classical logic camp. It will also please the classical logic camp since the goal directed formulation makes it all look like an algorithmic extension of Logic Programming. The approach also seems to provide very good compuational complexity bounds across its landscape
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