24,061 research outputs found
Globular: an online proof assistant for higher-dimensional rewriting
This article introduces Globular, an online proof assistant for the
formalization and verification of proofs in higher-dimensional category theory.
The tool produces graphical visualizations of higher-dimensional proofs,
assists in their construction with a point-and- click interface, and performs
type checking to prevent incorrect rewrites. Hosted on the web, it has a low
barrier to use, and allows hyperlinking of formalized proofs directly from
research papers. It allows the formalization of proofs from logic, topology and
algebra which are not formalizable by other methods, and we give several
examples
Chasing diagrams in cryptography
Cryptography is a theory of secret functions. Category theory is a general
theory of functions. Cryptography has reached a stage where its structures
often take several pages to define, and its formulas sometimes run from page to
page. Category theory has some complicated definitions as well, but one of its
specialties is taming the flood of structure. Cryptography seems to be in need
of high level methods, whereas category theory always needs concrete
applications. So why is there no categorical cryptography? One reason may be
that the foundations of modern cryptography are built from probabilistic
polynomial-time Turing machines, and category theory does not have a good
handle on such things. On the other hand, such foundational problems might be
the very reason why cryptographic constructions often resemble low level
machine programming. I present some preliminary explorations towards
categorical cryptography. It turns out that some of the main security concepts
are easily characterized through the categorical technique of *diagram
chasing*, which was first used Lambek's seminal `Lecture Notes on Rings and
Modules'.Comment: 17 pages, 4 figures; to appear in: 'Categories in Logic, Language and
Physics. Festschrift on the occasion of Jim Lambek's 90th birthday', Claudia
Casadio, Bob Coecke, Michael Moortgat, and Philip Scott (editors); this
version: fixed typos found by kind reader
Sequential products in effect categories
A new categorical framework is provided for dealing with multiple arguments
in a programming language with effects, for example in a language with
imperative features. Like related frameworks (Monads, Arrows, Freyd
categories), we distinguish two kinds of functions. In addition, we also
distinguish two kinds of equations. Then, we are able to define a kind of
product, that generalizes the usual categorical product. This yields a powerful
tool for deriving many results about languages with effects
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
Linear logic with idempotent exponential modalities: a note
In this note we discuss a variant of linear logic with idempotent exponential
modalities. We propose a sequent calculus system and discuss its semantics. We
also give a concrete relational model for this calculus
Adjunctions for exceptions
An algebraic method is used to study the semantics of exceptions in computer
languages. The exceptions form a computational effect, in the sense that there
is an apparent mismatch between the syntax of exceptions and their intended
semantics. We solve this apparent contradiction by efining a logic for
exceptions with a proof system which is close to their syntax and where their
intended semantics can be seen as a model. This requires a robust framework for
logics and their morphisms, which is provided by categorical tools relying on
adjunctions, fractions and limit sketches.Comment: In this Version 2, minor improvements are made to Version
- …