Finiteness spaces

2005 ◽  
Vol 15 (4) ◽  
pp. 615-646 ◽  
Author(s):  
THOMAS EHRHARD
Keyword(s):  
Linear Logic ◽  
Discrete Analogue ◽  
Relational Model ◽  
Formal Similarity ◽  
Closure Condition ◽  
Linear Topology ◽  
Game Semantics ◽  
Primitive Recursion ◽  
Continuous Maps ◽  
Köthe Space

We investigate a new denotational model of linear logic based on the purely relational model. In this semantics, webs are equipped with a notion of ‘finitary’ subsets satisfying a closure condition and proofs are interpreted as finitary sets. In spite of a formal similarity, this model is quite different from the usual models of linear logic (coherence semantics, hypercoherence semantics, the various existing game semantics…). In particular, the standard fix-point operators used for defining the general recursive functions are not finitary, although the primitive recursion operators are. This model can be considered as a discrete analogue of the Köthe space semantics introduced in a previous paper: we show how, given a field, each finiteness space gives rise to a vector space endowed with a linear topology, a notion introduced by Lefschetz in 1942, and we study the corresponding model where morphisms are linear continuous maps (a version of Girard's quantitative semantics with coefficients in the field). In this way we obtain a new model of the recently introduced differential lambda-calculus.

scholarly journals Games and full completeness for multiplicative linear logic

1994 ◽  
Vol 59 (2) ◽  
pp. 543-574 ◽  
Author(s):  
Samson Abramsky ◽  
Radha Jagadeesan
Keyword(s):  
Linear Logic ◽  
Winning Strategy ◽  
Game Semantics ◽  
Categorical Model ◽  
Natural Notion ◽  
Winning Strategies

AbstractWe present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cut-free proof net. A key role is played by the notion of history-free strategy; strong connections are made between history-free strategies and the Geometry of Interaction. Our semantics incorporates a natural notion of polarity, leading to a refined treatment of the additives. We make comparisons with related work by Joyal, Blass, et al.

A parallel game semantics for Linear Logic

1997 ◽  
Vol 36 (3) ◽  
pp. 189-217
Author(s):  
Stefano Baratella ◽  
Stefano Berardi
Keyword(s):  
Linear Logic ◽  
Game Semantics

Hopf algebras and linear logic

1996 ◽  
Vol 6 (2) ◽  
pp. 189-212 ◽  
Author(s):  
Richard F. Blute
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Linear Logic ◽  
Vector Spaces ◽  
Monoidal Categories ◽  
Simple Manner ◽  
Linear Topology ◽  
Topological Vector Spaces ◽  
Expository Paper ◽  
Involutive Negation

It has recently become evident that categories of representations of Hopf algebras provide fundamental examples of monoidal categories. In this expository paper, we examine such categories as models of (multiplicative) linear logic. By varying the Hopf algebra, it is possible to model several variants of linear logic. We present models of the original commutative logic, the noncommutative logic of Lambek and Abrusci, the braided variant due to the author, and the cyclic logic of Yetter. Hopf algebras provide a unifying framework for the analysis of these variants. While these categories are monoidal closed, they lack sufficient structure to model the involutive negation of classical linear logic. We recall work of Lefschetz and Barr in which vector spaces are endowed with an additional topological structure, called linear topology. The resulting category has a large class of reflexive objects, which form a *-autonomous category, and so model the involutive negation. We show that the monoidal closed structure of the category of representations of a Hopf algebra can be extended to this topological category in a natural and simple manner. The models we obtain have the advantage of being nondegenerate in the sense that the two multiplicative connectives, tensor and par, are not equated. It has been recently shown by Barr that this category of topological vector spaces can be viewed as a subcategory of a certain Chu category. In an Appendix, Barr uses this equivalence to analyze the structure of its tensor product.

scholarly journals A game semantics for linear logic

1992 ◽  
Vol 56 (1-3) ◽  
pp. 183-220 ◽  
Author(s):  
Andreas Blass
Keyword(s):  
Linear Logic ◽  
Game Semantics

Convenient antiderivatives for differential linear categories

2020 ◽  
Vol 30 (5) ◽  
pp. 545-569
Author(s):  
Jean-Simon Pacaud Lemay
Keyword(s):  
Natural Transformation ◽  
Linear Logic ◽  
Natural Isomorphism ◽  
Relational Model ◽  
Vector Spaces ◽  
Integration Operator ◽  
Categorical Models ◽  
Differential Categories ◽  
Linear Category ◽  
Fundamental Theorems

AbstractDifferential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation , which all differential categories have, is a natural isomorphism. Differential categories with antiderivatives come equipped with a canonical integration operator such that generalizations of the Fundamental Theorems of Calculus hold. In this paper, we show that Blute, Ehrhard, and Tasson's differential category of convenient vector spaces has antiderivatives. To help prove this result, we show that a differential linear category – which is a differential category with a monoidal coalgebra modality – has antiderivatives if and only if one can integrate over the monoidal unit and such that the Fundamental Theorems of Calculus hold. We also show that generalizations of the relational model (which are biproduct completions of complete semirings) are also differential linear categories with antiderivatives.

Execution time of λ-terms via denotational semantics and intersection types

2017 ◽  
Vol 28 (7) ◽  
pp. 1169-1203 ◽  
Author(s):  
DANIEL DE CARVALHO
Keyword(s):  
Execution Time ◽  
Linear Logic ◽  
Type System ◽  
Denotational Semantics ◽  
Relational Model ◽  
Intersection Types

The multiset-based relational model of linear logic induces a semantics of the untyped λ-calculus, which corresponds with a non-idempotent intersection type system, System R. We prove that, in System R, the size of type derivations and the size of types are closely related to the execution time of λ-terms in a particular environment machine, Krivine's machine.

scholarly journals Jump from parallel to sequential proofs: exponentials

2016 ◽  
Vol 28 (7) ◽  
pp. 1204-1252
Author(s):  
PAOLO DI GIAMBERARDINO
Keyword(s):  
Distinctive Feature ◽  
Linear Logic ◽  
The Other ◽  
Cut Elimination ◽  
Strong Normalization ◽  
Structural Rules ◽  
New Class ◽  
Game Semantics ◽  
Proof Nets

In previous works, by importing ideas from game semantics (notably Faggian–Maurel–Curien'sludics nets), we defined a new class of multiplicative/additive polarized proof nets, calledJ-proof nets. The distinctive feature of J-proof nets with respect to other proof net syntaxes, is the possibility of representing proof nets which are partially sequentialized, by usingjumps(that is, untyped extra edges) as sequentiality constraints. Starting from this result, in the present work, we extend J-proof nets to the multiplicative/exponential fragment, in order to take into account structural rules: More precisely, we replace the familiar linear logic notion of exponential box with a less restricting one (calledcone) defined by means of jumps. As a consequence, we get a syntax for polarized nets where, instead of a structure of boxes nested one into the other, we have one of cones which can bepartially overlapping. Moreover, we define cut-elimination for exponential J-proof nets, proving, by a variant of Gandy's method, that even in case of ‘superposed’ cones, reduction enjoys confluence and strong normalization.

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