Noncomputable functions in the Blum-Shub-Smale model

TL;DR

This paper investigates the limits of computability within the Blum-Shub-Smale model over real numbers, demonstrating the noncomputability of certain algebraic sets and the halting problem under various oracle conditions.

Contribution

It provides new results on the relative computability of algebraic real number sets and the undecidability of the halting problem in the BSS model, extending previous work by Meer and Ziegler.

Findings

Oracle sets of bounded degree algebraic numbers are insufficient for higher degrees.

The halting problem in the BSS model is undecidable below any countable oracle set.

Algebraic independence techniques are key in proving noncomputability results.

Abstract

Working in the Blum-Shub-Smale model of computation on the real numbers, we answer several questions of Meer and Ziegler. First, we show that, for each natural number d, an oracle for the set of algebraic real numbers of degree at most d is insufficient to allow an oracle BSS-machine to decide membership in the set of algebraic numbers of degree d + 1. We add a number of further results on relative computability of these sets and their unions. Then we show that the halting problem for BSS-computation is not decidable below any countable oracle set, and give a more specific condition, related to the cardinalities of the sets, necessary for relative BSS-computability. Most of our results involve the technique of using as input a tuple of real numbers which is algebraically independent over both the parameters and the oracle of the machine.