Weihrauch Degrees, Omniscience Principles and Weak Computability

Abstract. In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension of this reducibility for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice with the disjoint union of multi-valued functions as greatest lower bound operation. We show that parallelization is a closure operator for this semilattice and the parallelized Weihrauch degrees even form a lattice with the product of multi-valued functions as greatest lower bound operation. We show that the Medvedev lattice and hence the Turing upper semi-lattice can both be embedded into the parallelized Weihrauch lattice in a natural way. The importance of Weihrauch degrees is based on the fact that multi-valued functions on represented spaces can be considered as realizers of mathematical theorems in a very natural way and studying the Weihra...

• Type: Conference
• Year: 2009
• Where: CCA
• Authors: Vasco Brattka, Guido Gherardi
• Keywords: CCA 2009, Lower Bound Operation, Multi-valued Function, Theoretical Computer Science, Weihrauch Degree