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Explicit Substitution Internal Languages for Autonomous and *Autonomous Categories
 In Proc. Category Theory and Computer Science (CTCS'99), Electron
, 1999
"... We introduce a family of explicit substitution type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the simplytyped calculus with surjective pairing is the internal language for cartesian closed categories. We show tha ..."
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Cited by 7 (2 self)
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We introduce a family of explicit substitution type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the simplytyped calculus with surjective pairing is the internal language for cartesian closed categories. We show that the eight equality and three commutation congruence axioms of the autonomous type theory characterise autonomous categories exactly. The associated rewrite systems are all strongly normalising; modulo a simple notion of congruence, they are also confluent. As a corollary, we solve a Coherence Problem a la Lambek [12]: the equality of maps in any autonomous category freely generated from a discrete graph is decidable. 1 Introduction In this paper we introduce a family of type theories which can be regarded as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the standard simplytyped calculus with surjective pairing is...
Category theory for linear logicians
 Linear Logic in Computer Science
, 2004
"... This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categori ..."
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This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categories and their relation to intuitionistic logic is followed by a consideration of symmetric monoidal closed, linearly distributive and ∗autonomous categories and their relation to multiplicative linear logic. We examine nonsymmetric monoidal categories, and consider them as models of noncommutative linear logic. We introduce traced monoidal categories, and discuss their relation to the geometry of interaction. The necessary aspects of the theory of monads is introduced in order to describe the categorical modelling of the exponentials. We conclude by briefly describing the notion of full completeness, a strong form of categorical completeness, which originated in the categorical model theory of linear logic. No knowledge of category theory is assumed, but we do assume knowledge of linear logic sequent calculus and the standard models of linear logic, and modest familiarity with typed lambda calculus. 0
On the Algebra of Structural Contexts
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2007
"... We discuss a general way of defining contexts in linear logic, based on the observation that linear universal algebra can be symmetrized by assigning an additional variable to represent the output of a term. We give two approaches to this, a syntactical one based on a new, reversible notion of term, ..."
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Cited by 6 (2 self)
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We discuss a general way of defining contexts in linear logic, based on the observation that linear universal algebra can be symmetrized by assigning an additional variable to represent the output of a term. We give two approaches to this, a syntactical one based on a new, reversible notion of term, and an algebraic one based on a simple generalization of typed operads. We relate these to each other and to known examples of logical systems, and show new examples, in particular discussing the relationship between intuitionistic and classical systems. We then present a general framework for extracting deductive systems from a given theory of contexts, and give a generic proof that all these systems have cutelimination.
Type Theories for Autonomous and *Autonomous Categories: I. Type Theories and Rewrite Systems  II. Internal Languages and Coherence Theorems
, 1998
"... We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that simplytyped calculus (augmented by appropriate constructs for products and the terminal object) is the internal language for cartesian clos ..."
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Cited by 5 (4 self)
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We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that simplytyped calculus (augmented by appropriate constructs for products and the terminal object) is the internal language for cartesian closed categories. The rules are presented in the style of Gentzen's Sequent Calculus. A key feature is the systematic treatment of naturality conditions by explicitly representing the categorical composition, or cut in the type theory, by explicit substitution, and the introduction of new letconstructs, one for each of the three type constructors ?;\Omega and (, and a Parigotstyle ¯abstraction to give expression to the involutive negation. The commutation congruences of these theories are precisely those imposed by the naturality conditions. In particular the type theory for autonomous categories may be regarded as a term assignment system for the multiplicative (\Omega ; (;?;?)fragmen...
A Sequent Calculus for Compact Closed Categories
, 1996
"... In this paper, we introduce the system CMLL of sequent calculus and establish its correspondence with compact closed categories. CMLL is equivalent in provability to the system MLL of classical linear logic with the tensor and par connectives identified. We show that the system allows a fairly simpl ..."
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Cited by 5 (0 self)
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In this paper, we introduce the system CMLL of sequent calculus and establish its correspondence with compact closed categories. CMLL is equivalent in provability to the system MLL of classical linear logic with the tensor and par connectives identified. We show that the system allows a fairly simple cutelimination, and the proofs in the system have a natural interpretation in compact closed categories. However, the soundness of the cutelimination procedure in terms of the categorical interpretation is by no means evident. We answer to this question affirmatively and establish the soundness by using the coherence result on compact closed categories by Kelly and Laplaza. 1 Introduction In this paper, we introduce the system CMLL of sequent calculus and establish its correspondence with compact closed categories. CMLL is equivalent in provability to the system MLL of classical linear logic with the tensor ffl and par O connectives identified. Compact closed categories are abundant in ...
Fock Space: A Model of Linear Exponential Types
, 1994
"... It has been observed by several people that, in certain contexts, the free symmetric algebra construction can provide a model of the linear modality ! . This construction arose independently in quantum physics, where it is considered as a canonical model of quantum field theory. In this context, the ..."
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It has been observed by several people that, in certain contexts, the free symmetric algebra construction can provide a model of the linear modality ! . This construction arose independently in quantum physics, where it is considered as a canonical model of quantum field theory. In this context, the construction is known as (bosonic) Fock space. Fock space is used to analyze such quantum phenomena as the annihilation and creation of particles. There is a strong intuitive connection to the principle of renewable resource, which is the philosophical interpretation of the linear modalities. In this paper, we examine Fock space in several categories of vector spaces. We first consider vector spaces, where the Fock construction induces a model of the\Omega ; &; ! fragment in the category of symmetric algebras. When considering Banach spaces, the Fock construction provides a model of a weakening cotriple in the sense of Jacobs. While the models so obtained model a smaller fragment, it is cl...
A specification structure for deadlockfreedom of synchronous processes
 TCS
, 1999
"... Many different notions of "program property", and many different methods of verifying such properties, arise naturally in programming. We present a general framework of Specification Structures for combining different notions and methods in a coherent fashion. We then apply the idea of spe ..."
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Many different notions of "program property", and many different methods of verifying such properties, arise naturally in programming. We present a general framework of Specification Structures for combining different notions and methods in a coherent fashion. We then apply the idea of specification structures to concurrency in the setting of Interaction Categories. As a specific example, a certain specification
Internal Languages for Autonomous and *Autonomous Categories
"... We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the simplytyped calculus with surjective pairing is the internal language for cartesian closed categories. The rules for the typing judgeme ..."
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Cited by 1 (0 self)
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We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the simplytyped calculus with surjective pairing is the internal language for cartesian closed categories. The rules for the typing judgements are presented in the style of Gentzen's Sequent Calculus. A notable feature is the systematic treatment of naturality conditions by expressing the categorical composition, or cut in the type theory, by explicit substitution. We use letconstructs, one for each of the three type constructors ?;\Omega and (, to witness the leftintroduction rules, and a Parigotstyle ¯abstraction to express the involutive negation ?. We show that the eight equality and three commutation congruence axioms of the autonomous type theory characterise autonomous categories exactly. More precisely we prove that there is a canonical interpretation of the (autonomous) type theories in autonomous categorie...
On Double Categories and Multiplicative Linear Logic
, 1999
"... this article, we attack the converse problem of explaining semantics as an artifact of syntax, in other words, of extracting the meaning of a program from syntactical considerations on its dynamics, or the way it interacts with the environment. We start the analysis with a very simple slogan, where ..."
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this article, we attack the converse problem of explaining semantics as an artifact of syntax, in other words, of extracting the meaning of a program from syntactical considerations on its dynamics, or the way it interacts with the environment. We start the analysis with a very simple slogan, where we use module to mean procedure, in the fashion of (Girard 1987b):
A canonical graphical syntax for nonempty finite products and sums
, 2002
"... system for finite products and sums, and proved decidability of equality of morphisms. The question remained ∗ as to whether one can present free categories with finite products and sums in a canonical way, i.e., as a category with morphisms and composition defined directly, rather than modulo equiv ..."
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system for finite products and sums, and proved decidability of equality of morphisms. The question remained ∗ as to whether one can present free categories with finite products and sums in a canonical way, i.e., as a category with morphisms and composition defined directly, rather than modulo equivalence relations. This paper shows that the nonempty case (i.e., omitting initial and final objects) can be treated in a surprisingly simple way: morphisms of the free category can be viewed as certain binary relations, with composition the usual composition of binary relations. In particular, there is a forgetful functor into the category Rel of sets and binary relations. The paper ends by relating these binary relations to proof nets. 1