A Uniform Mathematical Representation of Logic and Computation.

The current models of computation share varying levels of correspondence with actual implementation schemes. They can be arranged in a hierarchical structure depending upon their level of abstraction. In classical computing, the circuit model shares closest correspondence with physical implementation, followed by finite automata techniques. The highest level in the abstraction hierarchy is that of the theory of computation.Likewise, there exist computing paradigms based upon a different set of defining principles. The classical paradigm involves computing as has been applied traditionally, and is characterized by Boolean circuits that are irreversible in nature. The reversible paradigm requires invertible primitives in order to perform computation. The paradigm of quantum computing applies the theory of quantum mechanics to perform computational tasks.Our analysis concludes that descriptions at lowest level in the abstraction hierarchy should be uniform across the three paradigms, but the same is not true in case of current descriptions. We propose a mathematical representation of logic and computation that successfully explains computing models in all three paradigms, while making a seamless transition to higher levels of the abstraction hierarchy. This representation is based upon the theory of linear spaces and, hence, is referred to as the linear representation. The representation is first developed in the classical context, followed by an extension to the reversible paradigm by exploiting the well-developed theory on invertible mappings. The quantum paradigm is reconciled with this representation through correspondence that unitary operators share with the proposed linear representation. In this manner, the representation is shown to account for all three paradigms. The correspondence shared with finite automata models is also shown to hold implicitly during the development of this representation. Most importantly, the linear representation accounts for the Hamiltonians that define the dynamics of a computational process, thereby resolving the correspondence shared with underlying physical principles.The consistency of the linear representation is checked against a current existing application in VLSI CAD that exploits the linearity of logic functions for symbolic representation of circuits. Some possible applications and applicability of the linear representation to some open problems are also discussed

Nature as a Network of Morphological Infocomputational Processes for Cognitive Agents

This paper presents a view of nature as a network of infocomputational agents organized in a dynamical hierarchy of levels. It provides a framework for unification of currently disparate understandings of natural, formal, technical, behavioral and social phenomena based on information as a structure, differences in one system that cause the differences in another system, and computation as its dynamics, i.e. physical process of morphological change in the informational structure. We address some of the frequent misunderstandings regarding the natural/morphological computational models and their relationships to physical systems, especially cognitive systems such as living beings. Natural morphological infocomputation as a conceptual framework necessitates generalization of models of computation beyond the traditional Turing machine model presenting symbol manipulation, and requires agent-based concurrent resource-sensitive models of computation in order to be able to cover the whole range of phenomena from physics to cognition. The central role of agency, particularly material vs. cognitive agency is highlighted

Quantum Computers and Decoherence: Exorcising the Demon from the Machine

Decoherence is the main obstacle to the realization of quantum computers. Until recently it was thought that quantum error correcting codes are the only complete solution to the decoherence problem. Here we present an alternative that is based on a combination of a decoherence-free subspace encoding and the application of strong and fast pulses: ``encoded recoupling and decoupling'' (ERD). This alternative has the advantage of lower encoding overhead (as few as two physical qubits per logical qubit suffice), and direct application to a number of promising proposals for the experimental realization of quantum computers.Comment: 15 pages, no figures. Invited contribution to the proceedings of the SPIE Conference on Fluctuations and Noise. Section 8 contains a new result: how to eliminate off-resonant transitions induced by generic "bang-bang" pulses, by using a special type of "bang-bang" pulse