Analysis and synthesis of abstract data types through generalization from examples

The discovery of general patterns of behavior from a set of input/output examples can be a useful technique in the automated analysis and synthesis of software systems. These generalized descriptions of the behavior form a set of assertions which can be used for validation, program synthesis, program testing and run-time monitoring. Describing the behavior is characterized as a learning process in which general patterns can be easily characterized. The learning algorithm must choose a transform function and define a subset of the transform space which is related to equivalence classes of behavior in the original domain. An algorithm for analyzing the behavior of abstract data types is presented and several examples are given. The use of the analysis for purposes of program synthesis is also discussed

Analysis and synthesis of abstract data types through generalization from examples

The discovery of general patterns of behavior from a set of input/output examples can be a useful technique in the automated analysis and synthesis of software systems. These generalized descriptions of the behavior form a set of assertions which can be used for validation, program synthesis, program testing, and run-time monitoring. Describing the behavior is characterized as a learning process in which the set of inputs is mapped into an appropriate transform space such that general patterns can be easily characterized. The learning algorithm must chose a transform function and define a subset of the transform space which is related to equivalence classes of behavior in the original domain. An algorithm for analyzing the behavior of abstract data types is presented and several examples are given. The use of the analysis for purposes of program synthesis is also discussed

Hamiltonian properties of graphs with large neighborhood unions

AbstractLet G be a graph of order n, σk = min{ϵi=1kd(νi): {ν1,…, νk} is an independent set of vertices in G}, NC = min{|N(u)∪ N(ν)|: uν∉E(G)} and NC2 = min{|N(u)∪N(ν)|: d(u,ν)=2}. Ore proved that G is hamiltonian if σ2⩾n⩾3, while Faudree et al. proved that G is hamiltonian if G is 2-connected and NC⩾13(2n−1). It is shown that both results are generalized by a recent result of Bauer et al. Various other existing results in hamiltonian graph theory involving degree-sums or cardinalities of neighborhood unions are also compared in terms of generality. Furthermore, some new results are proved. In particular, it is shown that the bound 13(2n−1) on NC in the result of Faudree et al. can be lowered to 13(2n−1), which is best possible. Also, G is shown to have a cycle of length at least min{n, 2(NC2)} if G is 2-connected and σ3⩾n+2. A Dλ-cycle (Dλ-path) of G is a cycle (path) C such that every component of G−V(C) has order smaller than λ. Sufficient conditions of Lindquester for the existence of Hamilton cycles and paths involving NC2 are extended to Dλ-cycles and Dλ-paths

Dynamics over Signed Networks

A signed network is a network with each link associated with a positive or negative sign. Models for nodes interacting over such signed networks, where two different types of interactions take place along the positive and negative links, respectively, arise from various biological, social, political, and economic systems. As modifications to the conventional DeGroot dynamics for positive links, two basic types of negative interactions along negative links, namely the opposing rule and the repelling rule, have been proposed and studied in the literature. This paper reviews a few fundamental convergence results for such dynamics over deterministic or random signed networks under a unified algebraic-graphical method. We show that a systematic tool of studying node state evolution over signed networks can be obtained utilizing generalized Perron-Frobenius theory, graph theory, and elementary algebraic recursions.Comment: In press, SIAM Revie

Fault-tolerant gates on hypergraph product codes

L’un des défis les plus passionnants auquel nous sommes confrontés aujourd’hui est la perspective de la construction d’un ordinateur quantique de grande échelle. L’information quantique est fragile et les implémentations de circuits quantiques sont imparfaites et su- jettes aux erreurs. Pour réaliser un tel ordinateur, nous devons construire des circuits quan- tiques tolérants aux fautes capables d’opérer dans le monde réel. Comme il sera expliqué plus loin, les circuits quantiques tolérant aux fautes nécessitent plus de ressources que leurs équivalents idéaux, sans bruit. De manière générale, le but de mes recherches est de minimiser les ressources nécessaires à la construction d’un circuit quantique fiable. Les codes de correction d’erreur quantiques protègent l’information des erreurs en l’encodant de manière redondante dans plusieurs qubits. Bien que la redondance requière un plus grand nombre de qubits, ces qubits supplé- mentaires jouent un rôle de protection: cette redondance sert de garantie. Si certains qubits sont endommagés en raison d’un circuit défectueux, nous pourrons toujours récupérer l’informations. Préparer et maintenir des qubits pendant des durées suffisamment longues pour effectuer un calcul s’est révélé être une tâche expérimentale difficile. Il existe un écart important entre le nombre de qubits que nous pouvons contrôler en laboratoire et le nombre requis pour implementer des algorithmes dans lesquels les ordinateurs quantiques ont le dessus sur ceux classiques. Par conséquent, si nous voulons contourner ce problème et réaliser des circuits quantiques à tolérance aux fautes, nous devons rendre nos constructions aussi efficaces que possible. Nous devons minimiser le surcoût, défini comme le nombre de qubits physiques nécessaires pour construire un qubit logique. Dans un article important, Gottesman a montré que, si certains types de codes de correction d’erreur quantique existaient, cela pourrait alors conduire à la construction de circuits quantiques tolérants aux fautes avec un surcoût favorable. Ces codes sont appelés codes éparses. La proposition de Gottesman décrivait des techniques pour exécuter des opérations logiques sur des codes éparses quantiques arbitraires. Cette proposition était limitée à certains égards, car elle ne permettait d’exécuter qu’un nombre constant de portes logiques par unité de temps. Dans cette thèse, nous travaillons avec une classe spécifique de codes éparses quantiques appelés codes de produits d’hypergraphes. Nous montrons comment effectuer des opérations sur ces codes en utilisant une technique appelée déformation du code. Notre technique généralise les codages basés sur les défauts topologiques dans les codes de surface aux codes de produits d’hypergraphes. Nous généralisons la notion de perforation et montrons qu’elle peut être exprimée naturellement dans les codes de produits d’hypergraphes. Comme cela sera expliqué en détail, les défauts de perforation ont eux-mêmes une portée limitée. Pour réaliser une classe de portes plus large, nous intro- duisons un nouveau défaut appelé trou de ver basé sur les perforations. À titre d’exemple, nous illustrons le fonctionnement de ce défaut dans le contexte du code de surface. Ce défaut a quelques caractéristiques clés. Premièrement, il préserve la propriété éparses du code au cours de la déformation, contrairement à une approche naïve qui ne garantie pas cette propriété. Deuxièmement, il généralise de manière simple les codes de produits d’hypergraphes. Il s’agit du premier cadre suffisamment riche pour décrire les portes tolérantes aux fautes de cette classe de codes. Enfin, nous contournons une limitation de l’approche de Gottesman qui ne permettait d’effectuer qu’un certain nombre de portes logiques à un moment donné. Notre proposition permet d’opérer sur tous les qubits encodés à tout moment.One of the most exciting challenges that faces us today is the prospect of building a scalable quantum computer. Implementations of quantum circuits are imperfect and prone to error. In order to realize a scalable quantum computer, we need to construct fault-tolerant quantum circuits capable of working in the real world. As will be explained further below, fault-tolerant quantum circuits require more resources than their ideal, noise-free counterparts. Broadly, the aim of my research is to minimize the resources required to construct a reliable quantum circuit. Quantum error correcting codes protect information from errors by encoding our information redundantly into qubits. Although the number of qubits that we require increases, this redundancy serves as a buffer – in the event that some qubits are damaged because of a faulty circuit, we will still be able to recover our information. Preparing and maintaining qubits for durations long enough to perform a computation has proved to be a challenging experimental task. There is a large gap between the number of qubits we can control in the lab and the number required to implement algorithms where quantum computers have the upper hand over classical ones. Therefore, if we want to circumvent this bottleneck, we need to make fault-tolerant quantum circuits as efficient as possible. To be precise, we need to minimize the overhead, defined as the number of physical qubits required to construct a logical qubit. In an important paper, Gottesman showed that if certain kinds of quantum error correcting codes were to exist, then this could lead to constructions of fault-tolerant quantum circuits with favorable overhead. These codes are called quantum Low-Density Parity-Check (LDPC) codes. Gottesman’s proposal described techniques to perform gates on generic quantum LDPC codes. This proposal limited the number of logical gates that could be performed at any given time. In this thesis, we work with a specific class of quantum LDPC codes called hypergraph product codes. We demonstrate how to perform gates on these codes using a technique called code deformation. Our technique generalizes defect-based encodings in the surface code to hypergraph product codes. We generalize puncture defects and show that they can be expressed naturally in hypergraph product codes. As will be explained in detail, puncture defects are themselves limited in scope; they only permit a limited set of gates. To perform a larger class of gates, we introduce a novel defect called a wormhole that is based on punctures. As an example, we illustrate how this defect works in the context of the surface code. This defect has a few key features. First, it preserves the LDPC property of the code over the course of code deformation. At the outset, this property was not guaranteed. Second, it generalizes in a straightforward way to hypergraph product codes. This is the first framework that is rich enough to describe fault-tolerant gates on this class of codes. Finally, we circumvent a limitation in Gottesman’s approach which only allowed a limited number of logical gates at any given time. Our proposal allows to access the entire code block at any given time