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Reflexive Spaces of Smooth Functions: a Logical Account of Linear Partial Reflexive spaces of smooth functions : a logical account of linear partial differential equations Marie Kerjean To cite this version: Marie Kerjean. Reflexive spaces of smooth functions : a logical account of linear partial differential equations. Logic in Computer Science [cs.LO]. Université Sorbonne Paris Cité, 2018. English. NNT : 2018USPCC144. tel-02386294 HAL Id: tel-02386294 https://tel.archives-ouvertes.fr/tel-02386294 Submitted on 29 Nov 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. I Titre: Espaces réflexifs de fonctions lisses: un compte rendu logique des équations aux dérivées partielles linéaires. Résumé La théorie de la preuve se développe depuis la correspondance de Curry-Howard suivant deux sources d’inspirations: les langages de programmation, pour lesquels elle agit comme une théorie des types de données, et l’étude séman- tique des preuves. Cette dernière consiste à donner des modèles mathématiques pour les comportements des preuves/programmes. En particulier, la sémantique dénotationnelle s’attache à interpréter ceux-ci comme des fonctions entre les types, et permet en retour d’affiner notre compréhension des preuves/programmes. La logique linéaire (LL) donne une interprétation logique des notions d’algèbre linéaire, quand la logique linéaire différentielle (DiLL) permet une compréhension logique de la notion de différentielle. Cette thèse s’attache à renforcer la correspondance sémantique entre théorie de la preuve et analyse fonction- nelle, en insistant sur le caractère involutif de la négation dans DiLL. La première partie consiste en un rappel des notions de linéarité, polarisation et différentiation en théorie de la preuve, ainsi qu’un exposé rapide de théorie des espaces vectoriels topologiques. La deuxième partie donne deux modèles duaux de la logique linéaire dif- férentielle, interprétant la négation d’une formule respectivement par le dual faible et le dual de Mackey. Quand la topologie faible ne permet qu’une interprétation discrète des preuves sous forme de série formelle, la topologie de Mackey nous permet de donner un modèle polarisé et lisse de DiLL. Enfin, la troisième partie de cette thèse s’attache à interpréter les preuves de DiLL par des distribuitons à support compact. Nous donnons un modèle polarisé de DiLL où les types négatifs sont interprétés par des espaces Fréchet Nucléaires. Nous montrons que enfin la résolution des équations aux dérivées partielles linéaires à coefficients constants obéit à une syntaxe qui généralise celle de DiLL, que nous détaillons. Mots-clefs : théorie de la preuve, logique linéaire, sémantique dénotationnelle, espaces vectoriels topologies, théorie des distributions. Title: Reflexive spaces of smooth functions: a logical account of linear partial differential equations. Abstract Around the Curry-Howard correspondence, proof-theory has grown along two distinct fields: the theory of programming languages, for which formulas acts as data types, and the semantic study of proofs. The latter consists in giving mathematical models of proofs and programs. In particular, denotational semantics distinguishes data types which serves as input or output of programs, and allows in return for a finer understanding of proofs and programs. Linear Logic (LL) gives a logical interpretation of the basic notions of linear algebra, while Differential Linear Logic allows for a logical understanding of differentiation. This manuscript strengthens the link between proof-theory and functional analysis, and highlights the role of linear involutive negation in DiLL. The first part of this thesis consists in a quick overview of prerequisites on the notions of linearity, polarisation and differentiation in proof-theory, and gives the necessary background in the theory of locally convex topological vector spaces. The second part uses two classic topologies on the dual of a topological vector space and gives two models of DiLL: the weak topology allows only for a discrete interpretation of proofs through formal power series, while the Mackey topology on the dual allows for a smooth and polarised model of DiLL. Finally, the third part interprets proofs of DiLL by distributions. We detail a polarized model of DiLL in which negatives are Fréchet Nuclear spaces, and proofs are distributions with compact support. We also show that solving linear partial differential equations with constant coefficients can be typed by a syntax similar to the one of DiLL, which we detail. key-words : proof-theory, linear logic, denotational semantics, locally convex topological vector spaces, distribu- tion theory. II III Contents 1 Introduction 2 I Preliminaries 9 2 Linear Logic, Differential Linear Logic and their models 10 2.1 The Curry-Howard correspondence .................................. 12 2.1.1 Minimal logic and the lambda-calculus ........................... 12 2.1.2 LK and classical Logic .................................... 14 2.2 Linear Logic .............................................. 16 2.2.1 Syntax and cut -elimination ................................. 16 2.2.2 Categorical semantics .................................... 16 2.2.3 An example: Köthe spaces .................................. 22 2.3 Polarized Linear Logic ........................................ 23 2.3.1 LLpol and LLP. ....................................... 24 2.3.2 Categorical semantics .................................... 26 2.3.2.1 Mixed chiralities .................................. 27 2.3.2.2 Negative chiralities. ................................ 28 2.3.2.3 Interpreting MALLpol ............................... 29 2.3.2.4 Interpreting LLpol ................................. 31 2.4 Differential Linear Logic ....................................... 34 2.4.1 Syntax and cut-elimination .................................. 34 2.4.1.1 The syntax ..................................... 34 2.4.1.2 Typing: intuitions behind the exponential rules. 36 2.4.1.3 Cut-elimination and sums ............................. 37 2.4.2 Categorical semantics .................................... 40 2.4.2.1 ˚-autonomous Seely categories with biproduct and co-dereliction . 40 2.4.2.2 Invariance of the semantics over cut-elimination . 43 2.4.2.3 Exponential structures ............................... 45 2.4.3 A Smooth intuitionistic model: Convenient spaces ..................... 45 2.5 Polarized Differential Linear Logic .................................. 49 2.5.1 A sequent calculus ...................................... 49 2.5.2 Categorical semantics .................................... 50 2.5.2.1 Dereliction and co-dereliction as functors ..................... 50 2.5.2.2 A polarized biproduct ............................... 51 2.5.2.3 Categorical models of DiLLpol .......................... 52 3 Topological vector spaces 53 Closure and orthogonalities ...................................... 55 Filters and topologies ......................................... 55 3.1 First definitions ............................................ 56 3.1.1 Topologies on vector spaces ................................. 56 3.1.2 Metrics and semi-norms ................................... 57 3.1.3 Compact and precompact sets ................................ 58 IV 3.1.4 Projective and inductive topolgies .............................. 58 3.1.5 Completeness ......................................... 59 3.2 Examples: sequences and measures ................................. 60 3.2.1 Spaces of continuous functions ................................ 60 3.2.2 Spaces of smooth functions ................................. 60 3.2.3 Sequence spaces ....................................... 61 3.3 Linear functions and their topologies ................................. 62 3.3.1 Linear continuous maps ................................... 62 3.3.2 Weak properties and dual pairs ................................ 63 3.3.3 Dual pairs ........................................... 65 3.4 Bornologies and uniform convergence ................................ 66 3.4.1 Polars and equicontinuous sets ................................ 66 3.4.2 Boundedness for sets and functions ............................. 67 3.4.3 Topologies on spaces of linear functions ........................... 68 3.4.4 Barrels ............................................ 70 3.5 Reflexivity ............................................... 70 3.5.1 Weak, Mackey and Arens reflexivities ............................ 71 3.5.2 Strong reflexivity ....................................... 72 3.5.3 The duality of linear continuous functions .......................... 73 3.6 Topological tensor products and bilinear maps ............................ 74 3.6.1 The projective, inductive and B tensor products ....................... 74 3.6.2 The injective tensor product ................................. 76 3.6.3 The ε tensor product ..................................... 78 II Classical models of DiLL 79 4 Mackey and Weak topologies as left and right adjoint to pairing 80 Quantitative semantics ........................................ 80 CHU and its adjunctions to TOPVEC ................................
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