Musings around the geometry of interaction, and coherence

Theoretical Computer Science, 2006

We consider the multiplicative and exponential fragment of linear logic (MELL) and give a Geometry of Interaction (GoI) semantics for it based on unique decomposition categories. We prove a Soundness and Finiteness Theorem for this interpretation. We show that Girard's original approach to GoI 1 via operator algebras is exactly captured in this categorical framework.

2002

We present an axiomatic framework for Girard's Geometry of Interaction based on the notion of linear combinatory algebra. We give a general construction on traced monoidal categories, with certain additional structure, that is sufficient to capture the exponentials of Linear Logic, which produces such algebras (and hence also ordinary combinatory algebras). We illustrate the construction on six standard examples, representing both the 'particle-style'as well as the 'wave-style'Geometry of Interaction.

Mathematical Structures in Computer Science, 2000

Presented to Jim Lambek to mark the occasion of his 75 th birthday Linear bicategories are a generalization of bicategories, in which the one horizontal composition is replaced by two (linked) horizontal compositions. These compositions provide a semantic model for the tensor and par of linear logic: in particular, as composition is fundamentally noncommutative, they provide a suggestive source of models for noncommutative linear logic.

Mathematical Structures in Computer Science, 2002

Category Theory and Computer Science, 1989

1 The pseudo-monoidality quartet Consider the following different (but equivalent!) presentations of the notion monoidal category 1 :

Computer Science Logic, 1993

Electronic Notes in Theoretical Computer Science, 2005

We analyze the categorical foundations of Girard's Geometry of Interaction Program for Linear Logic. The motivation for the work comes from the importance of viewing GoI as a new kind of semantics and thus trying to relate it to extant semantics. In an earlier paper we showed that a special case of Abramsky's GoI situations-ones based on Unique Decomposition Categories (UDC's)-exactly captures Girard's functional analytic models in his first GoI paper, including Girard's original Execution formula in Hilbert spaces, his notions of orthogonality, types, datum, algorithm, etc. Here we associate to a UDC-based GoI Situation a denotational model (a * -autonomous category (without units) with additional exponential structure). We then relate this model to some of the standard GoI models via a fully-faithful embedding into a double-gluing category, thus connecting up GoI with earlier Full Completeness Theorems.

Electronic Notes in Theoretical Computer Science, 1996

The proof-theoretic origins and specialized models of linear logic make it primarily operational in orientation. In contrast first-order logic treats the operational and denotational aspects of general mathematics quite evenhandedly. Here we show that linear logic has models of even broader denotational scope than those of first order logic, namely Chu spaces, the category of which Barr has observed to form a model of linear logic. We have previously argued that every category of n-ary relational structures embeds fully and concretely in the category of Chu spaces over 2 n . The main contributions of this paper are improvements to that argument, and an embedding of every small category in the category of Chu spaces via a symmetric variant of the Yoneda embedding.

In this paper we develops a categorical theory of relations and use this formulation to define the notion of quantization for relations. Categories of relations are defined in the context of symmetric monoidal categories. They are shown to be symmetric monoidal categories in their own right and are found to be isomorphic to certain categories of A−A bicomodules. Properties of relations are defined in terms of the symmetric monoidal structure. Equivalence relations are shown to be commutative monoids in the category of relations. Quantization in our view is a property of functors between monoidal categories. This notion of quantization induce a deformation of all algebraic structures in the category, in particular the ones defining properties of relations like transitivity and symmetry.