rahul khandelwal
580 California St., Suite 400
San Francisco, CA, 94104
THEORY OF COMPUTATION LECTURE NOTES
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Abstract
This document does not claim any originality and cannot be used as a substitute for prescribed textbooks. The information presented here is merely a collection by the committee members for their respective teaching assignments. Various sources as mentioned at the end of the document as well as freely available material from internet were consulted for preparing this document. The ownership of the information lies with the respective authors or institutions.
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The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.
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The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
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We study the Weihrauch degrees of closed choice for finite sets, closed choice for convex sets and sorting infinite sequences over finite alphabets. Our main result is that choice for finite sets of cardinality i + 1 is reducible to choice for convex sets in dimension j, which in turn is reducible to sorting infinite sequences over an alphabet of size k + 1, iff i ≤ j ≤ k. Our proofs invoke Kleene's recursion theorem, and we describe in some detail how Kleene's recursion theorem gives rise to a technique for proving separations of Weihrauch degrees.
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In most accounts of realization of computational processes by physical mechanisms, it is presupposed that there is one-to-one correspondence between the causally active states of the physical process and the states of the computation. Yet such proposals either stipulate that only one model of computation is implemented, or they do not reflect upon the variety of models that could be implemented physically. In this paper, I claim that mechanistic accounts of computation should allow for a broad variation of models of computation. In particular, some non-standard models should not be excluded a priori. The relationship between mathematical models of computation and mechanistically adequate models is studied in more detail.
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