Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, \emph{the Muchnik reals}, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists w\,\forall x\,A(x,wx)$ and a \emph{bounding principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists z\,\forall x\,\exists y\,(y\le_{\mathrm{T}}(x,z)\land A(x,y))$ where $x,y,z$ range over Muchnik reals, $w$ ranges over functions from Muchnik reals to Muchnik reals, and $A(x,y)$ is a formula not containing $w$ or $z$. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an English translation of Muchnik's 1963 paper on Muchnik degrees.Comment: 44 page

Realization of CoFeB|MgO|CoFeB magnetic tunnel junction devices through materials analysis, process integration and circuit simulation

Spin based magnetic tunnel junctions (MTJs) consist of two ferromagnetic thin films separated by a nonmagnetic insulating barrier. The MTJ exhibits two switchable resistive states, making them ideal candidates for non-volatile memory. The discovery of high Tunneling Magnetoresistance (TMR) in MgO based MTJs has brought spintronics into the forefronts of modern technology. A device structure CoFeB|MgO|CoFeB achieved by physical vapor deposition (PVD) has revolutionized the hard-drive industry to go beyond densities of gigabyte per square inch. There is increasing interest in the application of these devices toward other technical areas, such as sensors, logic and reconfigurable computing. In these structures, the thicknesses of the layers are in the order of a few nanometers. For integration of these devices in other platforms, particularly on silicon, to augment the well-developed CMOS technology, it is imperative to (1) investigate processing constraints, (2) develop appropriate physical models, and (3) build circuit models for effective circuit implementation. The work presented in this dissertation focuses on these three important aspects for the realization of CoFeB|MgO|CoFeB MTJs on silicon. A systematic annealing study has been carried out to investigate the role of boron in the device structure. It has been shown using electron energy loss spectroscopy (EELS), and 2D x-ray diffraction (2D XRD) that boron diffuses into MgO with an activation energy of 1.30.4 eV and facilitates the crystallization of CoFe with (200) out-of-plane oriented crystals, with MgO as a template. The grain size of CoFe has been definitively shown to be smaller than the grain size of MgO, which were otherwise believed to be the same. A process temperature of 385°C has been determined to be the optimum limit of processing. A low temperature (\u3c385°C) process employing standard integrated circuit fabrication techniques has been developed. The partial crystallization of CoFe necessitates the modification of the tunneling model. A new model that combines the Julliëre\u27s, free electron and tight-binding model with the probabilistic distribution of grains on either side of the tunneling barrier has been proposed. This model explains the variation of TMR as a function of temperature in devices made by PVD. A generalized circuit macromodel has been developed representing field-switchable magnetic tunnel junctions (MTJs) characterized by two distinct voltage-dependent resistance values in parallel and antiparallel states. General-purpose subcircuit implementations are designed for a switchable voltage-dependent resistor capable of implementation using any version of SPICE. Transient simulation of a flash-comparator circuit using multiple MTJs in series is successfully demonstrated showing the robustness of the model

Generalized explosion principles

Paraconsistency is commonly defined and/or characterized as the failure of a principle of explosion. The various standard forms of explosion involve one or more logical operators or connectives, among which the negation operator is the most frequent and primary. In this article, we start by asking whether a negation operator is essential for describing explosion and paraconsistency. In other words, is it possible to describe a principle of explosion and hence a notion of paraconsistency that is independent of connectives? A negation-free paraconsistency resulting from the failure of a generalized principle of explosion is presented first. We also derive a notion of quasi-negation from this and investigate its properties. Next, more general principles of explosion are considered. These are also negation-free; moreover, these principles gradually move away from the idea that an explosion requires a statement and its opposite. Thus, these principles can capture the explosion observed in logics where a statement and its negation explode only in the presence of additional information, such as in the logics of formal inconsistency.Comment: 28 pages, 1 figure. The final version of the article has been submitted for publication in the Special Issue of Studia Logica on Paraconsistenc

Duration of classicality of an inhomogeneous quantum field with repulsive contact self-interactions

Quantum fields with large degeneracy are often approximated as classical fields. Here, we show how the quantum and classical evolution of a highly degenerate quantum field with repulsive contact self-interactions differ fromeach other. Initially, the field is taken to be homogeneous except for a small plane-wave perturbation in only one mode. In quantum field theory, modes satisfying both momentum and energy conservation of the quasiparticles, grow exponentially with time. However, in the classical field approximation, the system is stable. We calculate the time scale after which the classical field description becomes invalid