Cited By
View all- Lipton R(1975)Polynomials with 0-1 coefficients that are hard to evaluateProceedings of the 16th Annual Symposium on Foundations of Computer Science10.1109/SFCS.1975.25(6-10)Online publication date: 13-Oct-1975
Complexity measures and hierarchies for the evaluation of integers, polynomials, and n-linear forms
Pages 1 - 5
Abstract
The difficulty of evaluating integers and polynomials has been studied in various frameworks ranging from the addition-chain approach [5] to integer evaluation to recent efforts aimed at generating polynomials that are hard to evaluate [2,8,10]. Here we consider the classes of integers and polynomials that can be evaluated within given complexity bounds and prove the existence of proper hierarchies of complexity classes. The framework in which our problems are cast is general enough to allow any finite set of binary operations rather than just addition, subtraction, multiplication, and division. The motivation for studying complexity classes rather than specific integers or polynomials is analogous to why complexity classes are studied in automata-based complexity: (i) the immense difficulty associated with computing the complexity of a specific integer or polynomial; (ii) the important insight obtained from discovering the structure of the complexity classes.
References
[1]
A. Brauer. Bulletin of the AMS 45:736-739, 1939.
[2]
A. Borodin and S. Cook. On the number of additions to compute specific polynomials. Conference Record of the Sixth ACM Symposium on the Theory of Computing, Seattle, Washington, May 1974.
[3]
S. Cook. Linear time simulation of deterministic two-way push down automata. Proceedings of IFIP Congress 71, TA-2, North-Holland, Amsterdam, 172-179.
[4]
P. Erdös. Acta Arithmetica 6:77-81, 1960.
[5]
D. Knuth. The Art of Computer Programming, Volume II: Seminumerical Algorithms. Addison-Wesley, Reading, Massachusetts, 1969.
[6]
E. Landau. Über einige neuere Fortschritte der additiven Zahlentheorie. Cambridge, 2nd edition, 1937.
[7]
I. Niven and H. Zuckerman. An Introduction to the Theory of Numbers. John Wiley and Sons, New York, 2nd edition, 1966.
[8]
M. S. Paterson and L. J. Stockmeyer. On the number of nonscalar multiplications necessary to evaluate polynomials. SIAM J. Computing 2(1):60-66, March 1973.
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J. Savage. An Algorithm for the computation of linear forms. SIAM J. Computing 3:150-158, 1974.
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V. Strassen. Polynomials with rational coefficients which are hard to compute. SIAM J. Computing 3:128-149, 1974.
Index Terms
- Complexity measures and hierarchies for the evaluation of integers, polynomials, and n-linear forms
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Published In
May 1975
265 pages
ISBN:9781450374194
DOI:10.1145/800116
Copyright © 1975 ACM.
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Published: 05 May 1975
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STOC '75 Paper Acceptance Rate 31 of 87 submissions, 36%;
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View all- Lipton R(1975)Polynomials with 0-1 coefficients that are hard to evaluateProceedings of the 16th Annual Symposium on Foundations of Computer Science10.1109/SFCS.1975.25(6-10)Online publication date: 13-Oct-1975
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