The Cardinality of an Oracle in Blum-Shub-Smale Computation

TL;DR

This paper investigates the limitations of BSS-reducibility in real computation, proving that the halting problem cannot be reduced to countable sets, thus highlighting the importance of oracle set size in computational power.

Contribution

It establishes a new theorem linking the cardinality of oracle sets to the complexity of sets they can compute in BSS-models, confirming that countable sets are insufficient for certain problems.

Findings

Countable sets cannot decide the BSS halting problem.

The cardinality of the oracle set constrains the computational power.

The results connect set size with computational complexity in real computation.

Abstract

We examine the relation of BSS-reducibility on subsets of the real numbers. The question was asked recently (and anonymously) whether it is possible for the halting problem H in BSS-computation to be BSS-reducible to a countable set. Intuitively, it seems that a countable set ought not to contain enough information to decide membership in a reasonably complex (uncountable) set such as H. We confirm this intuition, and prove a more general theorem linking the cardinality of the oracle set to the cardinality, in a local sense, of the set which it computes. We also mention other recent results on BSS-computation and algebraic real numbers.