On the Structure of Polynomial Time Reducibility
Journal of the ACM (JACM)
Riemann's Hypothesis and tests for primality
STOC '75 Proceedings of seventh annual ACM symposium on Theory of computing
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
An NP-Complete Number-Theoretic Problem
Journal of the ACM (JACM)
On the Simplification and Equivalence Problems for Straight-Line Programs
Journal of the ACM (JACM)
Reducibility, randomness, and intractibility (Abstract)
STOC '77 Proceedings of the ninth annual ACM symposium on Theory of computing
An NP-complete number-theoretic problem
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
On the distribution of independent formulae of number theory
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
The impact of complexity theory on algorithms for sparse polynomials
ACM SIGSAM Bulletin
Complexity of Ring Morphism Problems
Computational Complexity
Generating graphs that approach a prescribed modularity
Computer Communications
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In this article we show the NP-completeness of some simple number-theoretic problems. Natural simplifications of these problems invariably are known to be in P. Our research was motivated by the question whether one could study non-deterministic computation without loss of generality on a restricted, number theoretically significant class of nondeterministic Turing machines, the nondeterministic diophantine machines defined below [1,2]. The results suggest this is true. Because of the relative difficulty of the reduction of the satisfiability problem used in the proof, and the distinctly number-theoretic character of the problems shown to be NP-complete, we hope that the NP-completeness of these problems will play a role in showing the NP-completeness of further problems of a numerical nature, much as the satisfiability problem has in showing the NP-completeness of combinatorial problems ([3],[5]). The results illustrate an intimate connection between problems in computational theory, such as “P&equil;NP?”, and problems in number theory, e.g. about quadratic congruences in one unknown, which are just beyond the range in which efficient algorithms exist. Thus our work exposes an interface between the state-of-the-art in number theory and in the theory of computation. Also, our results can be seen as the solution to a natural and important revision of Hilbert's 10thproblem: Give a feasible algorithmic procedure to decide whether an arbitrary diophantine equation has solutions. Unless P&equil;NP this is impossible, even for a class of quadratic diophantine equations in two unknowns for which a decision procedure in the original sense is in fact available. Certainly, these results present a striking rigorous demonstration that number theorists' intuitions about where problems about diophantine equations and quadratic congruences start to be truly difficult are justified.