Reversal-Bounded Multicounter Machines and Their Decision Problems
Journal of the ACM (JACM)
NP-complete decision problems for quadratic polynomials
STOC '76 Proceedings of the eighth 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
A New Public-Key Cipher System Based Upon the Diophantine Equations
IEEE Transactions on Computers
The complexity of the equivalence problem for counter machines, semilinear sets, and simple programs
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
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Systems of nonlinear equations of the form D: A&ymarc; &equil; &sgr;marc;.(x), where A is an m×n matrix of rational constants and &ymarc; &equil; (Y1,...,yn), &sgr;(x) &equil; (&sgr;1(x),..., &sgr;m (x)) are column vectors are considered. Each &sgr;i(x) is of the form ri(x) or @@@@ri(x)@@@@, where ri(x) is a rational function of x with rational coefficients. It is shown that the problem of determining for a given system D whether there exists a nonnegative integral solution (y1,...,yn,X) satisfying D is decidable. In fact, the problem is NP-complete when restricted to systems D in which the maximum degree of the polynomials defining the &sgr;i(x)'s is bounded by some fixed polynomial in the length of the representation of D. Some recent results connecting Diophantine equations and counter machines are briefly mentioned.