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Over 30 years ago L. Chua proposed the existence of a new class of passive circuit elements, which he called memristors and memristive devices. The unique electrical characteristics associated with them, along with the advantages of crossbar structures, have the potential to revolutionize computing architectures. A well-defined and effective memristor model for circuit design combined with a design paradigm based on well-understood underlying logic design principles would certainly accelerate research on nanoscale circuits and systems. Toward this goal, we propose a memristor crossbar circuit design paradigm in which memristors are modeled using the quantum mechanical phenomenon of tunneling. We use this circuit model to design and simulate various logic circuit designs capable of universal computation. Finally, we develop and present a new design paradigm for memristor-based crossbar circuits.