380 research outputs found
Weak topologies for Linear Logic
We construct a denotational model of linear logic, whose objects are all the
locally convex and separated topological vector spaces endowed with their weak
topology. The negation is interpreted as the dual, linear proofs are
interpreted as continuous linear functions, and non-linear proofs as sequences
of monomials. We do not complete our constructions by a double-orthogonality
operation. This yields an interpretation of the polarity of the connectives in
terms of topology
Lifting Coalgebra Modalities and Model Structure to Eilenberg-Moore Categories
A categorical model of the multiplicative and exponential fragments of
intuitionistic linear logic (), known as a \emph{linear
category}, is a symmetric monoidal closed category with a monoidal coalgebra
modality (also known as a linear exponential comonad). Inspired by Blute and
Scott's work on categories of modules of Hopf algebras as models of linear
logic, we study categories of algebras of monads (also known as Eilenberg-Moore
categories) as models of . We define a lifting
monad on a linear category as a Hopf monad -- in the Brugui{\`e}res, Lack, and
Virelizier sense -- with a special kind of mixed distributive law over the
monoidal coalgebra modality. As our main result, we show that the linear
category structure lifts to the category of algebras of lifting
monads. We explain how groups in the category of coalgebras of the monoidal
coalgebra modality induce lifting monads and provide a source
for such groups from enrichment over abelian groups. Along the way we also
define mixed distributive laws of symmetric comonoidal monads over symmetric
monoidal comonads and lifting differential category structure.Comment: An extend abstract version of this paper appears in the conference
proceedings of the 3rd International Conference on Formal Structures for
Computation and Deduction (FSCD 2018), under the title "Lifting Coalgebra
Modalities and Model Structure to Eilenberg-Moore Categories
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
Applying quantitative semantics to higher-order quantum computing
Finding a denotational semantics for higher order quantum computation is a
long-standing problem in the semantics of quantum programming languages. Most
past approaches to this problem fell short in one way or another, either
limiting the language to an unusably small finitary fragment, or giving up
important features of quantum physics such as entanglement. In this paper, we
propose a denotational semantics for a quantum lambda calculus with recursion
and an infinite data type, using constructions from quantitative semantics of
linear logic
On the linear structure of cones
For encompassing the limitations of probabilistic coherence spaces which do not seem to provide natural interpretations of continuous data types such as the real line, Ehrhard and al. introduced a model of probabilistic higher order computation based on (positive) cones, and a class of totally monotone functions that they called "stable". Then Crubillé proved that this model is a conservative extension of the earlier probabilistic coherence space model. We continue these investigations by showing that the category of cones and linear and Scott-continuous functions is a model of intuitionistic linear logic. To define the tensor product, we use the special adjoint functor theorem, and we prove that this operation is and extension of the standard tensor product of probabilistic coherence spaces. We also show that these latter are dense in cones, thus allowing to lift the main properties of the tensor product of probabilistic coherence spaces to general cones. Last we define in the same way an exponential of cones and extend measurability to these new operations
A Bicategorical Model for Finite Nondeterminism
Finiteness spaces were introduced by Ehrhard as a refinement of the relational model of linear logic. A finiteness space is a set equipped with a class of finitary subsets which can be thought of being subsets that behave like finite sets. A morphism between finiteness spaces is a relation that preserves the finitary structure. This model provided a semantics for finite non-determism and it gave a semantical motivation for differential linear logic and the syntactic notion of Taylor expansion. In this paper, we present a bicategorical extension of this construction where the relational model is replaced with the model of generalized species of structures introduced by Fiore et al. and the finiteness property now relies on finite presentability
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