46 research outputs found
Glueing and Orthogonality for Models of Linear Logic
We present the general theory of the method of glueing and associated technique of orthogonality for constructing categorical models of all the structure of linear logic: in particular we treat the exponentials in detail. We indicate simple applications of the methods and show that they cover familiar examples.
On Linear Information Systems
Scott's information systems provide a categorically equivalent, intensional
description of Scott domains and continuous functions. Following a well
established pattern in denotational semantics, we define a linear version of
information systems, providing a model of intuitionistic linear logic (a
new-Seely category), with a "set-theoretic" interpretation of exponentials that
recovers Scott continuous functions via the co-Kleisli construction. From a
domain theoretic point of view, linear information systems are equivalent to
prime algebraic Scott domains, which in turn generalize prime algebraic
lattices, already known to provide a model of classical linear logic
Linear Realisability and Cobordisms
Cobordism categories are known to be compact closed. They can therefore be
used to define non-degenerate models of multiplicative linear logic by
combining the Int construction with double glueing. In this work we detail such
construction in the case of low-dimensional cobordisms, and exhibit a connexion
between those models and the model of Interaction graphs introduced by Seiller.
In particular, we exhibit how the so-called trefoil property is a consequence
of the associativity of composition of higher structures, providing a first
step toward establishing models as obtained from a double glueing construction.
We discuss possible extensions to higher-dimensional cobordisms categorie
Algebraic totality, towards completeness
Finiteness spaces constitute a categorical model of Linear Logic (LL) whose
objects can be seen as linearly topologised spaces, (a class of topological
vector spaces introduced by Lefschetz in 1942) and morphisms as continuous
linear maps. First, we recall definitions of finiteness spaces and describe
their basic properties deduced from the general theory of linearly topologised
spaces. Then we give an interpretation of LL based on linear algebra. Second,
thanks to separation properties, we can introduce an algebraic notion of
totality candidate in the framework of linearly topologised spaces: a totality
candidate is a closed affine subspace which does not contain 0. We show that
finiteness spaces with totality candidates constitute a model of classical LL.
Finally, we give a barycentric simply typed lambda-calculus, with booleans
and a conditional operator, which can be interpreted in this
model. We prove completeness at type for
every n by an algebraic method