On the computational complexity of algebraic numbers: the Hartmanis--Stearns problem revisited

TL;DR

This paper investigates the computational complexity of algebraic irrational numbers' base expansions, showing they cannot be generated by certain automata, thus advancing understanding of their inherent computational difficulty.

Contribution

It proves that algebraic irrational numbers' base-$b$ expansions cannot be generated by deterministic pushdown automata or tag machines with dilation factor greater than one.

Findings

Algebraic irrationals cannot be generated by deterministic pushdown automata.

Such numbers cannot be generated by tag machines with dilation factor > 1.

The Hartmanis--Stearns problem is addressed for multistack machines.

Abstract

We consider the complexity of integer base expansions of algebraic irrational numbers from a computational point of view. We show that the Hartmanis--Stearns problem can be solved in a satisfactory way for the class of multistack machines. In this direction, our main result is that the base-$b$ expansion of an algebraic irrational real number cannot be generated by a deterministic pushdown automaton. We also confirm an old claim of Cobham proving that such numbers cannot be generated by a tag machine with dilation factor larger than one.