In this thesis, we study problems related to the reconstruction (up to bi-interpretability) of first-order theories from various functorial invariants: automorphism groups, endomorphism monoids, (categories of) countable models, and (ultra)categories of models. / Thesis / Master of Science (MSc)

A primeira noção de topos, a de topos de Grothendieck, surgiu há cerca de 50 anos a partir de uma generalização do conceito de feixe na geometria algébrica. Poucos anos mais tarde, uma axiomatização categorial de algumas das propriedades de um topos de Grothendieck deu origem a uma segunda noção de topos, a de topos elementar; e essa descrição permitiu estabelecer ligações entre essas categorias e teoria dos conjuntos e lógica. Neste trabalho, estudamos a teoria de topos com um foco especial na construção da lógica interna dos topoi, e exploramos sua relação com modelos Heyting-valorados. / The first definition of a topos, that of a Grothendieck topos, emerged roughly 50 years ago from a generalization of the notion of sheaves in algebraic geometry. Few years later, a categorical axiomatization of some properties of Grothendieck topoi gave rise to a second notion of topoi, that of an elementary topos; and this description made it possible to establish connections between these categories and set theory and logic. In this work, we study topos theory with a particular focus on the construction of the internal logic of topoi, and explore its relation to Heyting-valued models.

This thesis studies some constructions for building new models of Martin-Löf type theory out of old. We refer to the main techniques as gluing and idempotent splitting. For each we give general conditions under which type constructors exist in the resulting model. These techniques are used to construct some examples of Dialectica models of type theory. The name is chosen by analogy with de Paiva's Dialectica categories, which semantically embody Gödel's Dialectica functional interpretation and its variants. This continues a programme initiated by von Glehn with the construction of the polynomial model of type theory. We complete the analogy between this model and Gödel's original Dialectica by using our techniques to construct a two-level version of this model, equipping the original objects with an extra layer of predicates. In order to do this we have to carefully build up the theory of finite sum types in a display map category. We construct two other notable models. The first is a model analogous to the Diller-Nahm variant, which requires a detailed study of biproducts in categories of algebras. To make clear the generalization from the categories studied by de Paiva, we illustrate the construction of the Diller-Nahm category in terms of gluing an indexed system of types together with a system of predicates. Following this we develop the general techniques needed for the type-theoretic case. The second notable model is analogous to the Dialectica category associated to the error monad as studied by Biering. This model has only weak dependent products. In order to get a model with full dependent products we use the idempotent splitting construction, which generalizes the Karoubi envelope of a category. Making sense of the Karoubi envelope in the type-theoretic case requires us to face up to issues of coherence in our models. We choose the route of making sure all of the constructions we use preserve strict coherence, rather than applying a general coherence theorem to produce a strict model afterwards. Our chosen method preserves more detailed information in the final model.

We investigate categorical models of the unit-free multiplicative and multiplicative-additive fragments of linear logic by representing derivations as particular structures known as dinatural transformations. Suitable categories are considered to satisfy a property known as full completeness if all such entities are the interpretation of a correct derivation. It is demonstrated that certain Hyland-Schalk double glueings [HS03] are capable of transforming large numbers of degenerate models into more accurate ones. Compact closed categories with finite biproducts possess enough structure that their morphisms can be described as forms of linear arrays. We introduce the notion of an extended tensor (or ‘extensor’) over arbitrary semirings, and show that they uniquely describe arrows between objects generated freely from the tensor unit in such categories. It is made evident that the concept may be extended yet further to provide meaningful decompositions of more general arrows. We demonstrate how the calculus of extensors makes it possible to examine the combinatorics of certain double glueing constructions. From this we show that the Hyland-Tan version [Tan97], when applied to compact closed categories satisfying a far weaker version of full completeness, produces genuine fully complete models of unit-free multiplicative linear logic. Research towards the development of a full completeness result for the multiplicative-additive fragment is detailed. The proofs work for categories of finite arrays over certain semirings under both the Hyland-Tan and Schalk [Sch04] constructions. We offer a possible route to finishing this proof. An interpretation of these results with respect to linear logic proof theory is provided, and possible further research paths and generalisations are discussed.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23402号 / 理博第4737号 / 新制||理||1679(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 照井 一成, 教授 牧野 和久, 教授 長谷川 真人 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

Introduced by C.C. Chang in the 1950s, MV algebras are to many-valued (Łukasiewicz) logics what boolean algebras are to two-valued logic. More recently, effect algebras were introduced by physicists to describe quantum logic. In this thesis, we begin by investigating how these two structures, introduced decades apart for wildly different reasons, are intimately related in a mathematically precise way. We survey some connections between MV/effect algebras and more traditional algebraic structures. Then, we look at the categorical structure of effect algebras in depth, and in particular see how the partiality of their operations cause things to be vastly more complicated than their totally defined classical analogues. In the final chapter, we discuss coordinatization of MV algebras and prove some new theorems and construct some new concrete examples, connecting these structures up (requiring a detour through effect algebras!) to boolean inverse semigroups.

Le thème général de cette thèse est l’exploitation de l’interaction entre la sémantique dénotationnelle et la syntaxe. Des sémantiques satisfaisantes ont été découvertes pour les preuves en logique intuitionniste et linéaire, mais dans le cas de la logique classique, la solution du problème est connue pour être particulièrement difficile. Ce travail commence par l’étude d’une interprétation concrète des preuves classiques dans la catégorie des ensembles ordonnés et bimodules, qui mène à l’extraction d’invariants significatifs. Suit une généralisation de cette sémantique concrète, soit l’interprétation des preuves classiques dans une catégorie compacte fermée où chaque objet est doté d’une structure d’algèbre de Frobenius. Ceci nous mène à une définition de réseaux de démonstrations pour la logique classique. Le concept de correction, l’élimination des coupures et le problème de la “full completeness” sont abordés au moyen d’un enrichissement naturel dans les ordres sur la catégorie de Frobenius, produisant une catégorie pour l'élimination des coupures et un concept de ressources pour la logique classique. Revenant sur notre première sémantique concrète, nous montrons que nous avons une représentation fidèle de la catégorie de Frobenius dans la catégorie des ensembles ordonnés et bimodules. / The general theme of this thesis is the exploitation of the fruitful interaction between denotational semantics and syntax. Satisfying semantics have been discovered for proofs in intuitionistic and certain linear logics, but for the classical case, solving the problem is notoriously difficult.This work begins with investigations of concrete interpretations of classical proofs in the category of posets and bimodules, resulting in the definition of meaningful invariants of proofs. Then, generalizing this concrete semantics, classical proofs are interpreted in a free symmetric compact closed category where each object is endowed with the structure of a Frobenius algebra. The generalization paves a way for a theory of proof nets for classical proofs. Correctness, cut elimination and the issue of full completeness are addressed through natural order enrichments defined on the Frobenius category, yielding a category with cut elimination and a concept of resources in classical logic. Revisiting our initial concrete semantics, we show we have a faithful representation of the Frobenius category in the category of posets and bimodules.

Smooth manifolds have been always understood intuitively as spaces that are infinitesimally linear at each point, and thus infinitesimally affine when forgetting about the base point. The aim of this thesis is to develop a general theory of infinitesimal models of algebraic theories that provides us with a formalisation of these notions, and which is in accordance with the intuition when applied in the context of Synthetic Differential Geometry. This allows us to study well-known geometric structures and concepts from the viewpoint of infinitesimal geometric algebra. Infinitesimal models of algebraic theories generalise the notion of a model by allowing the operations of the theory to be interpreted as partial operations rather than total operations. The structures specifying the domains of definition are the infinitesimal structures. We study and compare two definitions of infinitesimal models: actions of a clone on infinitesimal structures and models of the infinitesimalisation of an algebraic theory in cartesian logic. The last construction can be extended to first-order theories, which allows us to define infinitesimally euclidean and projective spaces, in principle. As regards the category of infinitesimal models of an algebraic theory in a Grothendieck topos we prove that it is regular and locally presentable. Taking a Grothendieck topos as a base we study lifts of colimits along the forgetful functor with a focus on the properties of the category of infinitesimally affine spaces. We conclude with applications to Synthetic Differential Geometry. Firstly, with the help of syntactic categories we show that the formal dual of every smooth ring is an infinitesimally affine space with respect to an infinitesimal structure based on nil-square infinitesimals. This gives us a good supply of infinitesimally affine spaces in every well-adapted model of Synthetic Differential Geometry. In particular, it shows that every smooth manifold is infinitesimally affine and that every smooth map preserves this structure. In the second application we develop some basic theory of smooth loci and formal manifolds in naive Synthetic Differential Geometry using infinitesimal geometric algebra.

Cette recherche vise à établir les propriétés fondamentales d'un système formel aux preuves circulaires introduit par Santocanale, auquel on a rajouté la règle de coupure. On démontre, dans un premier temps, qu'il y a une pleine correspondance entre les preuves circulaires et les flèches issues des catégories dites µ-bicomplètes. Ces flèches sont celles que l'on peut définir purement à partir des outils suivants: les produits et coproduits finis, les algèbres initiales et les coalgèbres finales. Dans la catégorie des ensembles, les preuves circulaires dénotent donc les fonctions qu'on peut définir en utilisant les produits cartésiens finis, les unions disjointes finies, l'induction et la coinduction. On décrit également une procédure d'élimination des coupures qui produit, à partir d'une preuve circulaire finie, une preuve sans cycles et sans coupures, mais possiblement infinie. On démontre que l'élimination des coupures fournit une sémantique opérationnelle aux preuves circulaires, c'est-à-dire qu'elle permet de calculer les fonctions dénotées par celles-ci, par le moyen d'une sorte d'automate avec mémoire. Enfin, on s'intéresse au problème de la puissance expressive de cet éliminateur de coupures, c'est-à-dire à la question de caractériser la classe des expressions qu'il peut calculer. On démontre, par une simulation, que l'éliminateur des coupures est strictement plus expressif que les automates à pile d'ordre supérieur. / This research aims at establishing the fundamental properties of a formal system with circular proofs introduced by Santocanale, to which we added the cut rule. We first show that there is a full correspondence between circular proofs and arrows from the so-called µ-bicomplete categories. These arrows are those that can be defined purely from the following tools: finite products and coproducts, initial algebras and final coalgebras. In the category of sets, circular proofs denote functions that one can define by using finite cartesian products, finite disjoint unions, induction and coinduction. We also describe a cut-elimination procedure that produces, from a given finite circular proof, a proof without cycles and cuts, but which may be infinite. We prove that cut-elimination gives an operational semantics to circular proofs, which is to say that they allow to compute the functions denoted by them, by using a sort of automaton with memory. Finally, we are interested in finding the expressive power of that cut-eliminating automaton. In other words, we want to characterize the class of functions that it can compute. We show, through a simulation, that the cut-eliminating automaton is strictly more expressive than higher-order pushdown automata.

Thèse par articles. / La présente thèse fait état des avancées en logique déontique et propose des outils formels pertinents à l'analyse de la validité des inférences légales. D'emblée, la logique vise l'abstraction de différentes structures. Lorsqu'appliquée en argumentation, la logique permet de déterminer les conditions de validité des inférences, fournissant ainsi un critère afin de distinguer entre les bons et les mauvais raisonnements. Comme le montre la multitude de paradoxes en logique déontique, la modélisation des inférences normatives fait cependant face à divers problèmes. D'un point de vue historique, ces difficultés ont donné lieu à différents courants au sein de la littérature, dont les plus importants à ce jour sont ceux qui traitent de l'action et ceux qui visent la modélisation des obligations conditionnelles. La présente thèse de doctorat, qui a été rédigée par articles, vise le développement d'outils formels pertinents à l'analyse du discours juridique. En première partie, nous proposons une revue de la littérature complémentaire à ce qui a été entamé dans Peterson (2011). La seconde partie comprend la contribution théorique proposée. Dans un premier temps, il s'agit d'introduire une logique déontique alternative au système standard. Sans prétendre aller au-delà de ses limites, le système standard de logique déontique possède plusieurs lacunes. La première contribution de cette thèse est d'offrir un système comparable répondant au différentes objections pouvant être formulées contre ce dernier. Cela fait l'objet de deux articles, dont le premier introduit le formalisme nécessaire et le second vulgarise les résultats et les adapte aux fins de l'étude des raisonnements normatifs. En second lieu, les différents problèmes auxquels la logique déontique fait face sont abordés selon la perspective de la théorie des catégories. En analysant la syntaxe des différents systèmes à l'aide des catégories monoïdales, il est possible de lier certains de ces problèmes avec des propriétés structurelles spécifiques des logiques utilisées. Ainsi, une lecture catégorique de la logique déontique permet de motiver l'introduction d'une nouvelle approche syntaxique, définie dans le cadre des catégories monoïdales, de façon à pallier les problèmes relatifs à la modélisation des inférences normatives. En plus de proposer une analyse des différentes logiques de l'action selon la théorie des catégories, la présente thèse étudie les problèmes relatifs aux inférences normatives conditionnelles et propose un système déductif typé. / The present thesis develops formal tools relevant to the analysis of legal discourse. When applied to legal reasoning, logic can be used to model the structure of legal inferences and, as such, it provides a criterion to discriminate between good and bad reasonings. But using logic to model normative reasoning comes with some problems, as shown by the various paradoxes one finds within the literature. From a historical point of view, these paradoxes lead to the introduction of different approaches, such as the ones that emphasize the notion of action and those that try to model conditional normative reasoning. In the first part of this thesis, we provide a review of the literature, which is complementary to the one we did in Peterson (2011). The second part of the thesis concerns our theoretical contribution. First, we propose a monadic deontic logic as an alternative to the standard system, answering many objections that can be made against it. This system is then adapted to model unconditional normative inferences and test their validity. Second, we propose to look at deontic logic from the proof-theoretical perspective of category theory. We begin by proposing a categorical analysis of action logics and then we show that many problems that arise when trying to model conditional normative reasoning come from the structural properties of the logic we use. As such, we show that modeling normative reasoning within the framework of monoidal categories enables us to answer many objections in favour of dyadic and non-monotonic foundations for deontic logic. Finally, we propose a proper typed deontic system to model legal inferences.