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What algorithms could not be
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What algorithms could not be
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Dean
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Walter H.
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Walter H. Dean
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author
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Matthews
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Robert
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Robert Matthews
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Stanley
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Jason
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Jason Stanley
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Paul
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Paul Benacerraf
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Parikh
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Rohit
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Rohit Parikh
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Rutgers University
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Graduate School - New Brunswick
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theses
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2007
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2007
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English
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electronic
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viii, 369 pages
Abstract
This dissertation addresses a variety of foundational issues pertaining to the notion of algorithm employed in mathematics and computer science. In these settings, an algorithm is taken to be an effective mathematical procedure for solving a previously stated mathematical problem. Procedures of this sort comprise the notional subject matter of the subfield of computer science known as
algorithmic analysis. In this context, algorithms are referred to via proper names (e.g. Mergesort) of which computational properties are directly predicated (e.g. Mergesort has running time
O(n log(n))). Moreover, many formal results are traditionally stated by explicitly quantifying over algorithms (e.g. there is a polynomial time primality algorithm; there is no linear time comparison sorting algorithm).
These observations motivate the view that algorithms are themselves abstract mathematical objects -- a view I refer to as
algorithmic realism. The status of this view is clearly related to that of Church's Thesis -- i.e. the claim that a function is computable by an algorithm just in case it is partial recursive.
But whereas Church's Thesis is widely accepted on the basis of several well-known mathematical analyses of algorithmic computability, there is presently no consensus on how (or if) we can uniformly identify individual algorithms with mathematical objects.
In the first chapter of this work, I attempt to illustrate the theoretical significance of algorithmic realism. I suggest that this view is not only of foundational significance to computer science, but also worth highlighting due to the role algorithms play in the fixation of mathematical knowledge and their relationship to intensional entities such as propositions and properties. Chapter Two presents a formal framework for reducing computational discourse to mathematical discourse modeled on contemporary discussion of mathematical nominalism. Chapter Three is centered on a case study intended to illustrate the technical exigencies involved with defending algorithmic realism. Chapter Four explores various ways in
which algorithms might be identified with models of computation. And finally, Chapter Five argues that no such identification can uniformly satisfy all statements of algorithmic identity and non-identity affirmed by computational practice. I suggest that the technical crux of algorithmic realism lies in the necessity of reducing recursive specifications of algorithms to iterative models of computation in a manner which preserves algorithmic identity.
algorithmic analysis. In this context, algorithms are referred to via proper names (e.g. Mergesort) of which computational properties are directly predicated (e.g. Mergesort has running time
O(n log(n))). Moreover, many formal results are traditionally stated by explicitly quantifying over algorithms (e.g. there is a polynomial time primality algorithm; there is no linear time comparison sorting algorithm).
These observations motivate the view that algorithms are themselves abstract mathematical objects -- a view I refer to as
algorithmic realism. The status of this view is clearly related to that of Church's Thesis -- i.e. the claim that a function is computable by an algorithm just in case it is partial recursive.
But whereas Church's Thesis is widely accepted on the basis of several well-known mathematical analyses of algorithmic computability, there is presently no consensus on how (or if) we can uniformly identify individual algorithms with mathematical objects.
In the first chapter of this work, I attempt to illustrate the theoretical significance of algorithmic realism. I suggest that this view is not only of foundational significance to computer science, but also worth highlighting due to the role algorithms play in the fixation of mathematical knowledge and their relationship to intensional entities such as propositions and properties. Chapter Two presents a formal framework for reducing computational discourse to mathematical discourse modeled on contemporary discussion of mathematical nominalism. Chapter Three is centered on a case study intended to illustrate the technical exigencies involved with defending algorithmic realism. Chapter Four explores various ways in
which algorithms might be identified with models of computation. And finally, Chapter Five argues that no such identification can uniformly satisfy all statements of algorithmic identity and non-identity affirmed by computational practice. I suggest that the technical crux of algorithmic realism lies in the necessity of reducing recursive specifications of algorithms to iterative models of computation in a manner which preserves algorithmic identity.
Note
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Ph.D.
Note
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Includes bibliographical references (p. 360-368).
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Philosophy
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Algorithms
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Graduate School - New Brunswick Electronic Theses and Dissertations
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.15811
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doi:10.7282/T34M9501
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ETD doctoral
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Walter Dean
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Rutgers University. Graduate School - New Brunswick
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Author Agreement License
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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