6 research outputs found
A Coding Theoretic Study on MLL proof nets
Coding theory is very useful for real world applications. A notable example
is digital television. Basically, coding theory is to study a way of detecting
and/or correcting data that may be true or false. Moreover coding theory is an
area of mathematics, in which there is an interplay between many branches of
mathematics, e.g., abstract algebra, combinatorics, discrete geometry,
information theory, etc. In this paper we propose a novel approach for
analyzing proof nets of Multiplicative Linear Logic (MLL) by coding theory. We
define families of proof structures and introduce a metric space for each
family. In each family, 1. an MLL proof net is a true code element; 2. a proof
structure that is not an MLL proof net is a false (or corrupted) code element.
The definition of our metrics reflects the duality of the multiplicative
connectives elegantly. In this paper we show that in the framework one
error-detecting is possible but one error-correcting not. Our proof of the
impossibility of one error-correcting is interesting in the sense that a proof
theoretical property is proved using a graph theoretical argument. In addition,
we show that affine logic and MLL + MIX are not appropriate for this framework.
That explains why MLL is better than such similar logics.Comment: minor modification
LNL polycategories and doctrines of linear logic
We define and study LNL polycategories, which abstract the judgmental
structure of classical linear logic with exponentials. Many existing structures
can be represented as LNL polycategories, including LNL adjunctions, linear
exponential comonads, LNL multicategories, IL-indexed categories, linearly
distributive categories with storage, commutative and strong monads,
CBPV-structures, models of polarized calculi, Freyd-categories, and skew
multicategories, as well as ordinary cartesian, symmetric, and planar
multicategories and monoidal categories, symmetric polycategories, and linearly
distributive and *-autonomous categories. To study such classes of structures
uniformly, we define a notion of LNL doctrine, such that each of these classes
of structures can be identified with the algebras for some such doctrine. We
show that free algebras for LNL doctrines can be presented by a sequent
calculus, and that every morphism of doctrines induces an adjunction between
their 2-categories of algebras
Classical linear logic of implications
Abstract. We give a simple term calculus for the multiplicative exponential fragment of Classical Linear Logic, by extending Barber and Plotkin’s system for the intuitionistic case. The calculus has the nonlinear andlinear implications as the basic constructs, andthis design choice allows a technically managable axiomatization without commuting conversions. Despite this simplicity, the calculus is shown to be sound andcomplete for category-theoretic models given by ∗-autonomous categories with linear exponential comonads.
Under consideration for publication in Math. Struct. in Comp. Science Classical Linear Logic of Implications
We give a simple term calculus for the multiplicative exponential fragment of Classical Linear Logic, by extending Barber and Plotkin’s dual-context system for the intuitionistic case. The calculus has the non-linear and linear implications as the basic constructs, and this design choice allows a technically manageable axiomatization without commuting conversions. Despite this simplicity, the calculus is shown to be sound and complete for category-theoretic models given by ∗-autonomous categories with linear exponential comonads. 1