Anti-Complex Sets and Reducibilities with Tiny Use
Johanna N. Y. Franklin, Noam Greenberg, Frank Stephan, Guohua Wu · Journal of Symbolic Logic · 2013
Abstract In contrast with the notion of complexity, a setAis called anti-complex if the Kolmogorov complexity of the initial segments ofAchosen 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 setAis anti-complex if and only if it is reducible to another setBwithtiny use, whereby we mean that the use function for reducingAtoBcan 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.