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

Synthesis and Optimization of Reversible Circuits - A Survey

Reversible logic circuits have been historically motivated by theoretical research in low-power electronics as well as practical improvement of bit-manipulation transforms in cryptography and computer graphics. Recently, reversible circuits have attracted interest as components of quantum algorithms, as well as in photonic and nano-computing technologies where some switching devices offer no signal gain. Research in generating reversible logic distinguishes between circuit synthesis, post-synthesis optimization, and technology mapping. In this survey, we review algorithmic paradigms --- search-based, cycle-based, transformation-based, and BDD-based --- as well as specific algorithms for reversible synthesis, both exact and heuristic. We conclude the survey by outlining key open challenges in synthesis of reversible and quantum logic, as well as most common misconceptions.Comment: 34 pages, 15 figures, 2 table

A Framework for Verification of Signal Propagation Through Sequential Nanomagnet Logic Devices

Nanomagnet Logic is an emerging technology for low-power, highly-scalable implementation of quantum-dot cellular automata. Feedback permits reuse of logical subroutines, which is a desired functionality of any computational device. Determining whether feedback is feasible is essential to assessing the robustness of nanomagnet logic in any pipelined computing design. Therefore, development of a quantitative approach for verification of feedback paths is critical for development of design and synthesis tools for nanomagnet logic structures. In this paper, a framework for verification of sequential nanomagnet logic devices is presented. A set of definitions for canonical alignment and state definitions for NML paths are presented, as well as mathematical operations for determining the resulting states. The simulation results are presented for quantification of the NML magnetization angles for horizontal, vertical, negative-diagonal, and positive diagonal geometric alignments. The presented framework may be used as the basis for defining a representation of signal propagation for design and verification for robust NML devices and preventing deadlock resulting from improper implementation