Atomic flows are a geometric invariant of classical propositional proofs in deep inference. In this thesis we use atomic flows to describe new normal forms of proofs, of which the traditional normal forms are special cases, we also give several normalisation procedures for obtaining the normal forms. We define, and use to present our results, a new deep-inference formalism called the functorial calculus, which is more flexible than the traditional calculus of structures. To our surprise we are able to 1) normalise proofs without looking at their logical connectives or logical rules; and 2) normalise proofs in less than exponential time.

\prefacesection{Abstract} Multi-object tracking (MOT) is a key building block in video analytics and finds extensive use in surveillance, search and rescue, and autonomous driving applications. Object detection, a crucial stage in MOT, dominates in the overall tracking inference time due to its reliance on Deep Neural Networks (DNNs). Despite the superior performance of cutting-edge object detectors, their extensive computational demands limit their real-time application on embedded devices that possess constrained processing capabilities. Hence, we aim to reduce the computational burdens of object detection while maintaining tracking performance. As the first approach, we adapt frame resolutions to reduce computational complexity. During inference, frame resolutions can be tuned according to the complexity of visual scenes. We present DeepScale, a model-agnostic frame resolution selection approach that operates on top of existing fully convolutional network-based trackers. By analyzing the effect of frame resolution on detection performance, DeepScale strikes good trade-offs between detection accuracy and processing speed by adapting frame resolutions on-the-fly. Our second approach focuses on enhancing the efficiency of a tracker by model adaptation. We introduce AttTrack to expedite tracking by interleaving the execution of object detectors of different model sizes in inference. A sophisticated network (teacher) runs for keyframes only while, for non-keyframe, knowledge is transferred from the teacher to a smaller network (student) to improve the latter’s performance. Our third contribution involves exploiting temporal-spatial redundancies to enable real-time multi-camera tracking. We propose the MVSparse pipeline which consists of a central processing unit that aggregates information from multiple cameras (on an edge server or in the cloud) and distributed lightweight Reinforcement Learning (RL) agents running on individual cameras that predict the informative blocks in the current frame based on past frames on the same camera and detection results from other cameras. / Thesis / Doctor of Science (PhD)

In this thesis we see deductive systems for classical propositional and predicate logic which use deep inference, i.e. inference rules apply arbitrarily deep inside formulas, and a certain symmetry, which provides an involution on derivations. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they enjoy various new interesting properties. Not only the identity axiom, but also cut, weakening and even contraction are reducible to atomic form. This leads to inference rules that are local, meaning that the effort of applying them is bounded, and finitary, meaning that, given a conclusion, there is only a finite number of premises to choose from. The systems also enjoy new normal forms for derivations and, in the propositional case, a cut elimination procedure that is drastically simpler than the ones for sequent systems.

In this thesis we see deductive systems for classical propositional and predicate logic which use deep inference, i.e. inference rules apply arbitrarily deep inside formulas, and a certain symmetry, which provides an involution on derivations. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they enjoy various new interesting properties. Not only the identity axiom, but also cut, weakening and even contraction are reducible to atomic form. This leads to inference rules that are local, meaning that the effort of applying them is bounded, and finitary, meaning that, given a conclusion, there is only a finite number of premises to choose from. The systems also enjoy new normal forms for derivations and, in the propositional case, a cut elimination procedure that is drastically simpler than the ones for sequent systems.

Les questions <EM>"Qu'est-ce qu'une preuve?"</EM> et <EM>"Quand deux preuves sont-elles identiques?"</EM> sont fondamentales pour la théorie de la preuve. Mais pour la logique classique propositionnelle --- la logique la plus répandue --- nous n'avons pas encore de réponse satisfaisante. C'est embarrassant non seulement pour la théorie de la preuve, mais aussi pour l'informatique, où la logique classique joue un rôle majeur dans le raisonnement automatique et dans la programmation logique. De même, l'architecture des processeurs est fondée sur la logique classique. Tous les domaines dans lesquels la recherche de preuve est employée peuvent bénéficier d'une meilleure compréhension de la notion de preuve en logique classique, et le célèbre problème NP-vs-coNP peut être réduit à la question de savoir s'il existe une preuve courte (c'est-à-dire, de taille polynomiale) pour chaque tautologie booléenne. Normalement, les preuves sont étudiées comme des objets syntaxiques au sein de systèmes déductifs (par exemple, les tableaux, le calcul des séquents, la résolution, ...). Ici, nous prenons le point de vue que ces objets syntaxiques (également connus sous le nom d'arbres de preuve) doivent être considérés comme des représentations concrètes des objets abstraits que sont les preuves, et qu'un tel objet abstrait peut être représenté par un arbre en résolution ou dans le calcul des séquents. Le thème principal de ce travail est d'améliorer notre compréhension des objets abstraits que sont les preuves, et cela se fera sous trois angles différents, étudiés dans les trois parties de ce mémoire: l'algèbre abstraite (chapitre 2), la combinatoire (chapitres 3 et 4), et la complexité (chapitre 5).

In this thesis I study several deductive systems for linear logic, its fragments, and some noncommutative extensions. All systems will be designed within the calculus of structures, which is a proof theoretical formalism for specifying logical systems, in the tradition of Hilbert's formalism, natural deduction, and the sequent calculus. Systems in the calculus of structures are based on two simple principles: deep inference and top-down symmetry. Together they have remarkable consequences for the properties of the logical systems. For example, for linear logic it is possible to design a deductive system, in which all rules are local. In particular, the contraction rule is reduced to an atomic version, and there is no global promotion rule. I will also show an extension of multiplicative exponential linear logic by a noncommutative, self-dual connective which is not representable in the sequent calculus. All systems enjoy the cut elimination property. Moreover, this can be proved independently from the sequent calculus via techniques that are based on the new top-down symmetry. Furthermore, for all systems, I will present several decomposition theorems which constitute a new type of normal form for derivations.

In this thesis I study several deductive systems for linear logic, its fragments, and some noncommutative extensions. All systems will be designed within the calculus of structures, which is a proof theoretical formalism for specifying logical systems, in the tradition of Hilbert's formalism, natural deduction, and the sequent calculus. Systems in the calculus of structures are based on two simple principles: deep inference and top-down symmetry. Together they have remarkable consequences for the properties of the logical systems. For example, for linear logic it is possible to design a deductive system, in which all rules are local. In particular, the contraction rule is reduced to an atomic version, and there is no global promotion rule. I will also show an extension of multiplicative exponential linear logic by a noncommutative, self-dual connective which is not representable in the sequent calculus. All systems enjoy the cut elimination property. Moreover, this can be proved independently from the sequent calculus via techniques that are based on the new top-down symmetry. Furthermore, for all systems, I will present several decomposition theorems which constitute a new type of normal form for derivations.