Visualization 2001 Conference (Acm
Visualization 2001 Conference (Acm
Computability and Randomness
| Hi-index | 0.00 |
Polynomial clone reducibilities are generalizations of the truth-table reducibilities. A polynomial clone is a set of functions over a finite set X that is closed under composition and contains all the constant and projection functions. For a fixed polynomial clone $${\fancyscript{C}}$$ C , a sequence $${B\in X^{\omega}}$$ B 驴 X 驴 is $${\fancyscript{C}}$$ C -reducible to $${A \in {X}^{\omega}}$$ A 驴 X 驴 if there is an algorithm that computes B from A using only effectively selected functions from $${\fancyscript{C}}$$ C . We show that if A is Kurtz random and $${\fancyscript{C}_{1} \nsubseteq \fancyscript{C}_{2}}$$ C 1 驴 C 2 are polynomial clones, then there is a B that is $${\fancyscript{C}_{1}}$$ C 1 -reducible to A but not $${\fancyscript{C}_{2}}$$ C 2 -reducible to A. This implies a generalization of a result first proved by Lachlan (Z Math Logik Grundlagen Math 11:17---44, 1965) for the case |X| = 2. We also show that the same result holds if Kurtz random is replaced by Kolmogorov---Loveland stochastic.