Graphtheoretic arguments in lowlevel complexity
 Leslie G. Valiant
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Abstract
We have surveyed one approach to understanding complexity issues for certain easily computable natural functions. Shifting graphs have been seen to account accurately and in a unified way for the superlinear complexity of several problems for various restricted models of computation. To attack "unrestricted" models (in the present context combinational circuits or straightline arithmetic programs,) a first attempt, through superconcentrators, fails to provide any lower bounds although it does give counterexamples to alternative approaches. The notion of rigidity, however, does offer for the first time a reduction of relevant computational questions to noncomputional properties. The "reduction" consists of the conjunction of Corollary 6.3 and Theorem 6.4 which show that "for most sets of linear forms over the reals the stated algebraic and combinatorial reasons account for the fact that they cannot be computed in linear time and depth O(log n) simultaneously." We have outlined some problem areas which our preliminary results raise, and feel that further progress on most of these is humanly feasible. We would be interested in alternative approaches also.
Problem 6 Propose reductions of relevant complexity issues to noncomputational properties, that are more promising or tractable than the ones above.
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 Title
 Graphtheoretic arguments in lowlevel complexity
 Book Title
 Mathematical Foundations of Computer Science 1977
 Book Subtitle
 Proceedings, 6th Symposium, Tatranská Lomnica September 5–9, 1977
 Pages
 pp 162176
 Copyright
 1977
 DOI
 10.1007/3540083537_135
 Print ISBN
 9783540083535
 Online ISBN
 9783540372851
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 53
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
 Additional Links
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 Editors
 Authors

 Leslie G. Valiant ^{(1)}
 Author Affiliations

 1. Computer Science Department, University of Edinburgh, Edinburgh, Scotland
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