The Cardinality of an Oracle in Blum-Shub-Smale Computation
Logic in Computer Science
2010-06-03 v1
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.
Keywords
Cite
@article{arxiv.1006.0396,
title = {The Cardinality of an Oracle in Blum-Shub-Smale Computation},
author = {Wesley Calvert and Ken Kramer and Russell Miller},
journal= {arXiv preprint arXiv:1006.0396},
year = {2010}
}