Computing Bits of Algebraic Numbers

TL;DR

This paper studies the computational complexity of extracting individual bits of real algebraic numbers, showing that the problem is in NC1 with unary input and in the counting hierarchy with binary input, using elementary analysis tools.

Contribution

It establishes the complexity classes for computing bits of algebraic numbers based on input representation, extending previous work on transcendental numbers.

Findings

Bit extraction with unary input is in NC1.

Bit extraction with binary input is in the counting hierarchy.

Proves a lower bound for rational numbers as a limited case.

Abstract

We initiate the complexity theoretic study of the problem of computing the bits of (real) algebraic numbers. This extends the work of Yap on computing the bits of transcendental numbers like \pi, in Logspace. Our main result is that computing a bit of a fixed real algebraic number is in C=NC1\subseteq Logspace when the bit position has a verbose (unary) representation and in the counting hierarchy when it has a succinct (binary) representation. Our tools are drawn from elementary analysis and numerical analysis, and include the Newton-Raphson method. The proof of our main result is entirely elementary, preferring to use the elementary Liouville's theorem over the much deeper Roth's theorem for algebraic numbers. We leave the possibility of proving non-trivial lower bounds for the problem of computing the bits of an algebraic number given the bit position in binary, as our main open question. In this direction we show very limited progress by proving a lower bound for rationals.