Arithmetic complexity

Authors:
Lou Van Den Dries;Yiannis N. Moschovakis
Affiliations:
University of Illinois, Urbana-Champaign, Urbana, IL;University of California, Los Angeles, CA and University of Athens, Greece
Venue:
ACM Transactions on Computational Logic (TOCL)
Year:
2009

Citing 6
Cited 1

A lower bound for integer greatest common divisor computations

Journal of the ACM (JACM)
Lower bounds for arithmetic problems

Information Processing Letters
Lower bounds for computations with the floor operation

SIAM Journal on Computing
Modern computer algebra

Modern computer algebra
The Art of Computer Programming, 2nd Ed. (Addison-Wesley Series in Computer Science and Information

The Art of Computer Programming, 2nd Ed. (Addison-Wesley Series in Computer Science and Information
On primitive recursive algorithms and the greatest common divisor function

Theoretical Computer Science

Lower bounds for decision problems in imaginary, norm-Euclidean quadratic integer rings

Journal of Symbolic Computation

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Abstract

We obtain lower bounds on the cost of computing various arithmetic functions and deciding various arithmetic relations from specified primitives. This includes lower bounds for computing the greatest common divisor and deciding coprimeness of two integers, from primitives like addition, subtraction, division with remainder and multiplication. Some of our results are in terms of recursive programs, but they generalize directly to most (plausibly all) algorithms from the specified primitives. Our methods involve some elementary number theory as well as the development of some basic notions and facts about recursive algorithms.