Anti-complex sets and reducibilities with tiny use
Franklin, Johanna N. Y.
Date of Issue2013
School of Physical and Mathematical Sciences
In contrast with the notion of complexity, a set A is called anti-complex if the Kolmogorov complexity of the initial segments of A chosen by a recursive function is always bounded by the identity function. We show that, as for complexity, the natural arena for examining anti-complexity is the weak-truth table degrees. In this context, we show the equivalence of anti-complexity and other lowness notions such as r.e. traceability or being weak truth-table reducible to a Schnorr trivial set. A set A is anti-complex if and only if it is reducible to another set B with tiny use, whereby we mean that the use function for reducing A to B can be made to grow arbitrarily slowly, as gauged by unbounded nondecreasing recursive functions. This notion of reducibility is then studied in its own right, and we also investigate its range and the range of its uniform counterpart.
The Journal of Symbolic Logic
© 2013 Association for Symbolic Logic. This paper was published in The Journal of Symbolic Logic and is made available as an electronic reprint (preprint) with permission of Association for Symbolic Logic. The paper can be found at the following official DOI: [http://dx.doi.org/10.2178/jsl/1067620177]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.