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The Shuffle Hopf Algebra and Noncommutative Full Completeness (1999)

by R. F. Blute, P. J. Scott
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Nuclear and Trace Ideals in Tensored *-Categories

by Samson Abramsky, Richard Blute, Prakash Panangaden , 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored -categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The comp ..."
Abstract - Cited by 39 (12 self) - Add to MetaCart
We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored -categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored -categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored -categories, all morphisms are nuclear, and in the tensored -category of Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps. We also introduce two new examples of tensored -categories, in which integration plays the role of composition. In the first, mor...

Coherent Banach spaces: a continuous denotational semantics

by Jean-yves Girard - Theoretical Computer Science , 1999
"... We present a denotational semantics based on Banach spaces; it is inspired from the familiar coherent semantics of linear logic, the role of coherence being played by the norm: coherence is rendered by a supremum, whereas incoherence is rendered by a sum, and cliques are rendered by vectors of norm ..."
Abstract - Cited by 19 (3 self) - Add to MetaCart
We present a denotational semantics based on Banach spaces; it is inspired from the familiar coherent semantics of linear logic, the role of coherence being played by the norm: coherence is rendered by a supremum, whereas incoherence is rendered by a sum, and cliques are rendered by vectors of norm at most 1. The basic constructs of linear (and therefore intuitionistic) logic are implemented in this framework: positive connectives yield ℓ 1-like norms and negative connectives yield ℓ ∞-like norms. The problem of non-reflexivity of Banach spaces is handled by specifying the dual in ¡ advance, whereas the exponential connectives (i.e. intuitionistic implication) are handled by means of analytical functions on the open unit ball. The fact that this ball is open (and not closed) explains the absence of a simple solution to the question of a topological cartesian closed

Chu Spaces as a Semantic Bridge Between Linear Logic and Mathematics

by Vaughan Pratt - Theoretical Computer Science , 1998
"... The motivating role of linear logic is as a "logic behind logic." We propose a sibling role for it as a logic of transformational mathematics via the self-dual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soun ..."
Abstract - Cited by 16 (2 self) - Add to MetaCart
The motivating role of linear logic is as a "logic behind logic." We propose a sibling role for it as a logic of transformational mathematics via the self-dual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soundly interpreting linear logic while fully and concretely embedding a comprehensive range of concrete categories of mathematics. Our main goal is to treat each end of this bridge in expository detail. In addition we introduce the dialectic lambda-calculus, and show that dinaturality semantics is not fully complete for the Chu interpretation of linear logic. 1 Introduction Linear logic was introduced by J.-Y. Girard as a "logic behind logic." It separates logical reasoning into a core linear part in which formulas are merely moved around, and an auxiliary nonlinear part in which formulas may be deleted and copied. The core, multiplicative linear logic (MLL), is a substructural logic whose basic connect...

Category theory for linear logicians

by Richard Blute, Philip Scott - 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 ..."
Abstract - Cited by 13 (2 self) - Add to MetaCart
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
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...n by concatenation of strings. We denote the length of word w by |w|. Let k[X ∗ ] be the free k-vector space generated by X. We consider k[X ∗ ] endowed with the following Hopf algebra structure (cf. =-=[BS2]-=-): (i) A = k[X ∗ ] is an algebra, i.e. comes equipped with an associative k-linear multiplication (with unit) m : A ⊗ A → A: w ⊗ w ′ ↦→ w .w ′ = � 40 u∈Sh(w,w ′ ) u (7)swhere Sh(w, w ′ ) denotes the s...

Entropic hopf algebras and models of non-commutative linear logic

by Richard F. Blute, FRANCOIS LAMARCHE, PAUL RUET - THEORY AND APPLICATIONS OF CATEGORIES 10 , 2002
"... We give a definition of categorical model for the multiplicative fragment of non-commutative logic. We call such structures entropic categories. We demonstrate the soundness and completeness of our axiomatization with respect to cut-elimination. We then focus on several methods of building entropic ..."
Abstract - Cited by 6 (3 self) - Add to MetaCart
We give a definition of categorical model for the multiplicative fragment of non-commutative logic. We call such structures entropic categories. We demonstrate the soundness and completeness of our axiomatization with respect to cut-elimination. We then focus on several methods of building entropic categories. Our first models are constructed via the notion of a partial bimonoid acting on a cocomplete category. We also explore an entropic version of the Chu construction, and apply it in this setting. It has recently been demonstrated that Hopf algebras provide an excellent framework for modeling a number of variants of multiplicative linear logic, such as commutative, braided and cyclic. We extend these ideas to the entropic setting by developing a new type of Hopf algebra, which we call entropic Hopf algebras. We show that the category of modules over an entropic Hopf algebra is an entropic category, (possibly after application of the Chu construction). Several examples are discussed, based first on the notion of a bigroup. Finally the Tannaka-Krein reconstruction theorem is extended to the entropic setting.

Types as Processes, via Chu spaces

by Vaughan Pratt - EXRESS'97 Proceedings , 1997
"... We match up types and processes by putting values in correspondence with events, coproduct with (noninteracting) parallel composition, and tensor product with orthocurrence. We then bring types and processes into closer correspondence by broadening and unifying the semantics of both using Chu spaces ..."
Abstract - Cited by 3 (0 self) - Add to MetaCart
We match up types and processes by putting values in correspondence with events, coproduct with (noninteracting) parallel composition, and tensor product with orthocurrence. We then bring types and processes into closer correspondence by broadening and unifying the semantics of both using Chu spaces and their transformational logic. Beyond this point the connection appears to break down; we pose the question of whether the failures of the corrrespondence are intrinsic or cultural. 1 Introduction Types-as-processes modernizes data-as-programs. It is the Curry-Howard propositions-as-types correspondence with propositions replaced by processes. To the extent that types and processes are both part of the working programmer 's toolkit, even more than propositions, the types-as-processes correspondence is more central to the practice of programming than propositions-astypes. Moreover the connection works out very well mathematically, at least up to a point. The similarities and differences ...
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...full completeness result in the sense of Abramsky and Jagadeesan [1] and Hyland and Pratt Ong [11], but with their game semantics replaced by dinaturality semantics along the lines of Blute and Scott =-=[4,5]-=-. This summer with Gordon Plotkin we have been able to remove the restriction on two variables. This entailed strengthening dinaturality to logicality, necessitated by the presence of at least four di...

Edinburgh, Scotland

by unknown authors , 1998
"... Most tensored \Lambda-categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact ..."
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Most tensored \Lambda-categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored \Lambda-categories, all morphisms are nuclear, and in the tensored \Lambda-category of Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps. We also introduce two new examples of tensored \Lambda-categories, in which integration plays the role of composition. In the first, morphisms are a special class of distributions, which we call tame distributions. We also introduce a category of probabilistic relations.

Categorical Semantics of Linear Logic Paul-André

by Equipe Preuves, Programmes Et Systèmes, Université Paris , 2007
"... Proof Theory is the result of a tumultuous history, developed on the periphery of mainstream mathematics. Hence, its language is often idiosyncratic: sequent calculus, cutelimination, subformula property, etc. This survey is designed to guide the novice reader and the itinerant mathematician on a sm ..."
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Proof Theory is the result of a tumultuous history, developed on the periphery of mainstream mathematics. Hence, its language is often idiosyncratic: sequent calculus, cutelimination, subformula property, etc. This survey is designed to guide the novice reader and the itinerant mathematician on a smooth and engaging path through the subject – focused on the symbolic mechanisms of cut-elimination, and their transcription as coherence diagrams in categories with structure. This spiritual journey at the meeting point of linguistic and algebra is demanding at times, but unusually rewarding: to date, no language (formal or informal) has been studied as thoroughly in mathematics as the language of proofs. We start the survey by a short introduction to Proof Theory (Chapter 1) followed by an informal explanation of the principles of Denotational Semantics (Chapter 2) analogous to a Representation Theory for proofs, generating invariants modulo cutelimination. After describing in full detail the cut-elimination procedure of linear logic (Chapter 3), we explain how to transcribe it into the language of categories with structure. We review two alternative constructions of a ∗-autonomous category, or monoidal category with duality (Chapter 4). After giving a 2-categorical account of lax and colax monoidal adjunctions (Chapter 5) and recalling the notions of monoids and monads (Chapter 6) we relate four different categorical axiomatizations of propositional linear logic appearing in the litterature (Chapter 7).

Ludics Characterization of Multiplicative-Additive Linear

by Christophe Fouqueré, Myriam Quatrini
"... ar ..."
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...hat is a shuffle of paths. To our knowledge, the shuffle operation, largely used in combinatorics and to study parallelism (see for example [6, 11]), appears in Logics only to study non-commutativity =-=[2, 3]-=-. We give a few properties concerning orthogonality in terms of path travelling, introducing visitable paths, i.e., paths that are visited by orthogonality. In section 3 additive and multiplicative op...

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