54,114 research outputs found
Analysis of a variational model for nematic shells
We analyze an elastic surface energy which was recently introduced by G.
Napoli and L.Vergori to model thin films of nematic liquid crystals. We show
how a novel approach that takes into account also the extrinsic properties of
the surfaces coated by the liquid crystal leads to considerable differences
with respect to the classical intrinsic energy. Our results concern three
connected aspects: i) using methods of the calculus of variations, we establish
a relation between the existence of minimizers and the topology of the surface;
ii) we prove, by a Ginzburg-Landau approximation, the well-posedness of the
gradient flow of the energy; iii) in the case of a parametrized axisymmetric
torus we obtain a stronger characterization of global and local minimizers,
which we supplement with numerical experiments.Comment: Revised version. Includes referee's comments. Some proofs are
changed. To appear on Mathematical Models and Methods in Applied Sciences
(M3AS
Canonical Proof nets for Classical Logic
Proof nets provide abstract counterparts to sequent proofs modulo rule
permutations; the idea being that if two proofs have the same underlying
proof-net, they are in essence the same proof. Providing a convincing proof-net
counterpart to proofs in the classical sequent calculus is thus an important
step in understanding classical sequent calculus proofs. By convincing, we mean
that (a) there should be a canonical function from sequent proofs to proof
nets, (b) it should be possible to check the correctness of a net in polynomial
time, (c) every correct net should be obtainable from a sequent calculus proof,
and (d) there should be a cut-elimination procedure which preserves
correctness. Previous attempts to give proof-net-like objects for propositional
classical logic have failed at least one of the above conditions. In [23], the
author presented a calculus of proof nets (expansion nets) satisfying (a) and
(b); the paper defined a sequent calculus corresponding to expansion nets but
gave no explicit demonstration of (c). That sequent calculus, called LK\ast in
this paper, is a novel one-sided sequent calculus with both additively and
multiplicatively formulated disjunction rules. In this paper (a self-contained
extended version of [23]), we give a full proof of (c) for expansion nets with
respect to LK\ast, and in addition give a cut-elimination procedure internal to
expansion nets - this makes expansion nets the first notion of proof-net for
classical logic satisfying all four criteria.Comment: Accepted for publication in APAL (Special issue, Classical Logic and
Computation
Expansion Trees with Cut
Herbrand's theorem is one of the most fundamental insights in logic. From the
syntactic point of view it suggests a compact representation of proofs in
classical first- and higher-order logic by recording the information which
instances have been chosen for which quantifiers, known in the literature as
expansion trees.
Such a representation is inherently analytic and hence corresponds to a
cut-free sequent calculus proof. Recently several extensions of such proof
representations to proofs with cut have been proposed. These extensions are
based on graphical formalisms similar to proof nets and are limited to prenex
formulas.
In this paper we present a new approach that directly extends expansion trees
by cuts and covers also non-prenex formulas. We describe a cut-elimination
procedure for our expansion trees with cut that is based on the natural
reduction steps. We prove that it is weakly normalizing using methods from the
epsilon-calculus
A proof-theoretic analysis of the classical propositional matrix method
The matrix method, due to Bibel and Andrews, is a proof procedure designed for automated theorem-proving. We show that underlying this method is a fully structured combinatorial model of conventional classical proof theory. © 2012 The Author, 2012. Published by Oxford University Press
Focusing and Polarization in Intuitionistic Logic
A focused proof system provides a normal form to cut-free proofs that
structures the application of invertible and non-invertible inference rules.
The focused proof system of Andreoli for linear logic has been applied to both
the proof search and the proof normalization approaches to computation. Various
proof systems in literature exhibit characteristics of focusing to one degree
or another. We present a new, focused proof system for intuitionistic logic,
called LJF, and show how other proof systems can be mapped into the new system
by inserting logical connectives that prematurely stop focusing. We also use
LJF to design a focused proof system for classical logic. Our approach to the
design and analysis of these systems is based on the completeness of focusing
in linear logic and on the notion of polarity that appears in Girard's LC and
LU proof systems
Analyzing Individual Proofs as the Basis of Interoperability between Proof Systems
We describe the first results of a project of analyzing in which theories
formal proofs can be ex- pressed. We use this analysis as the basis of
interoperability between proof systems.Comment: In Proceedings PxTP 2017, arXiv:1712.0089
- …