Bandi, Mahesh M
(2002)
A Uniform Mathematical Representation of Logic and Computation.
Master's Thesis, University of Pittsburgh.
(Unpublished)
Abstract
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 welldeveloped 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.
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Details
Item Type: 
University of Pittsburgh ETD

Status: 
Unpublished 
Creators/Authors: 

ETD Committee: 

Date: 
4 September 2002 
Date Type: 
Completion 
Defense Date: 
3 July 2002 
Approval Date: 
4 September 2002 
Submission Date: 
7 July 2002 
Access Restriction: 
No restriction; Release the ETD for access worldwide immediately. 
Institution: 
University of Pittsburgh 
Schools and Programs: 
Swanson School of Engineering > Electrical Engineering 
Degree: 
MSEE  Master of Science in Electrical Engineering 
Thesis Type: 
Master's Thesis 
Refereed: 
Yes 
Uncontrolled Keywords: 
classical computation; computation; logic; mathematical representation; quantum computation; reversible computation 
Other ID: 
http://etd.library.pitt.edu:80/ETD/available/etd07072002002050/, etd07072002002050 
Date Deposited: 
10 Nov 2011 19:50 
Last Modified: 
15 Nov 2016 13:45 
URI: 
http://dscholarship.pitt.edu/id/eprint/8287 
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