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

Implementation of Two Layers Type Theory in Dedukti and Application to Cubical Type Theory

International audienceIn this paper, we make a substantial step towards an encoding of Cubical Type Theory (CTT) in the Dedukti logical framework. Type-checking CTT expressions features a decision procedure in a de Morgan algebra that so far could not be expressed by the rewrite rules of Dedukti. As an alternative, 2 Layer Type Theories are variants of Martin-Lf Type Theory where all or part of the definitionalequality can be represented in terms of a so-called external equality. We propose to split the encodingby giving an encoding of 2 Layer Type Theories (2LTT) in Dedukti, and a partial encoding of CTTin 2LTT

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

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

Functorial Models of Differential Linear Logic

There are two types of duality in Linear Logic. The first one is negation, balancing positive and negative formulas. The second one is the duality between linear and non-linear proofs, made symmetrical by Differential Linear Logic. The first duality has been here since the beginning of Linear Logic and has a significance in terms of proof-search and programming operation. However, the computational content of the second is still to understand. In this paper, we reexpress models of Differential Linear Logic in terms of models of Polarized Linear Logic, hoping to get closer to the understanding of Differential Linear Logic. It formalizes Differentiation in terms of a functor on a coslice category, within the linear-non-linear adjunction models of Linear Logic. It is also a first step in studying models mixing polarization and differentiation, concrete instances of which arise in functional analysis

Functorial Models of Differential Linear Logic

There are two types of duality in Linear Logic. The first one is negation, balancing positive and negative formulas. The second one is the duality between linear and non-linear proofs, made symmetrical by Differential Linear Logic. The first duality has been here since the beginning of Linear Logic and has a significance in terms of proof-search and programming operation. However, the computational content of the second is still to understand. In this paper, we reexpress models of Differential Linear Logic in terms of models of Polarized Linear Logic, hoping to get closer to the understanding of Differential Linear Logic. It formalizes Differentiation in terms of a functor on a coslice category, within the linear-non-linear adjunction models of Linear Logic. It is also a first step in studying models mixing polarization and differentiation, concrete instances of which arise in functional analysis