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<p>The thesis consists of three research projects concerning mathematical models for analog computers, originally developed by John Tucker and Jeff Zucker. The models are capable of representing systems that essentially “diverge,” exhibiting no valid behaviour---much the way that digital computers are capable of running programs that never halt. While there is no solution to the general Halting Problem, there are certainly theorems that identify large collections of instances that are guaranteed to halt. For example, if we use a simplified language featuring only assignment, branching, algebraic operations, and loops whose bounds must be fixed in advance (i.e. at “compile time”), we know that all instances expressible in this language will halt.</p> <p>In this spirit, one of the major objectives of all three thesis projects is identify a large class of instances of analog computation (analog computer + input) that are guaranteed to “converge.” In our semantic models, this convergence is assured if a certain operator (representing the computer and its input) has a unique fixed point. The first project is based on an original fixed point construction, while the second and third projects are based on Tucker and Zucker's construction. The second project narrows the scope of the model to a special case in order to concretely identify a class of operators with well-behaved fixed points, and considers some applications. The third project goes the opposite way: widening the scope of the model in order to generalize it.</p> / Doctor of Philosophy (PhD)

This work is concerned with the implementation and implication of non-volatile charge storage on VLSI system design. To that end, the floating-gate pFET (fg-pFET) is considered in the context of large-scale arrays. The programming of the element in an efficient and predictable way is essential to the implementation of these systems, and is thus explored. The overhead of the control circuitry for the fg-pFET, a key scalability issue, is examined. A light-weight, trend-accurate model is absolutely necessary for VLSI system design and simulation, and is also provided. Finally, several reconfigurable and reprogrammable systems that were built are discussed.

The memristor is a two-terminal semiconductor device that is able to mimic the conductance response of synapses and can be utilized in next-generation computing platforms that will compute similarly to the mammalian brain. The initial memristor implementation is operated by the digital formation and dissolution of a highly conductive filament. However, an analog memristor is necessary to mimic analog synapses in the mammalian brain. To understand the mechanisms of operation and impact of different device designs, analog memristors were fabricated, modeled, and characterized. To realize analog memristors, lithiated transition metal oxides were grown by molecular beam epitaxy, RF sputtering, and liquid phase electro-epitaxy. Analog memristors were modeled using a finite element model simulation and characterized with X-ray absorption spectroscopy, impedance spectroscopy, and other electrical methods. It was shown that lithium movement facilitates analog memristance and nanoscopic ionic-electronic memristors with ion-soluble electrodes can be key enabling devices for brain-inspired computing.

Il existe de nombreux modèles de calcul sur les réels. Ces différents modèles calculent diverses fonctions, certains sont plus puissants que d'autres, certains sont deux à deux incomparables. Le calcul sur les réels est donc de ce point de vue bien différent du calcul sur les entiers qui est unifié par la thèse de Church-Turing affirmant que tous les modèles raisonnables calculent les mêmes fonctions. Nous montrons des équivalences entre les fonctions récursivement calculables et une certaine classe de fonctions R-récursives et entre les fonctions GPAC-calculables et les fonctions récursivement calculables. Nous montrons également une hiérarchie de classes de fonctions R-récursives qui caractérisent les fonctions élémentairement calculables, les fonctions de la hiérarchie de Grzegorczyk et les fonctions récursivement calculables à l'aide d'un opérateur de limite. Ces résultats constituent donc une avancée vers une éventuelle unification des modèles de calcul sur les réels / Computation on the real numbers can be modelised in several different ways. There indeed exist a lot of different computation models on the reals. However, there are few results for comparing those models, and most of these results are incomparability results. The case of computation over the reals hence is quite different from the classical case where Church thesis claims that all reasonable models compute exactly the same functions. We show that recursively computable functions (in the sense of computable analysis) can be shown equivalent to some adequately defined class of R-recursive functions, and also to GPAC-computable functions. More than an analog characterization of recursively enumerable functions, we show that the limit operator we defined can be used to provide an analog characterization of elementarily computable functions and functions from Grzegorczyk's hierarchy. Those results can be seen as a first step toward a unification of computable functions over the reals