Weihrauch Degrees, Omniscience Principles and Weak Computability

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 prove that parallelization is a closure operator for this semi-lattice 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 can be embedded into the parallelized Weihrauch lattice in a natural way, even into the sublattice of total continuous multi-valued functions on Baire space and such that greatest lower bounds and least upper bounds are preserved. As a consequence we obtain that Turing degrees can be embedded into the single-valued part of this sublattice. 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 Weihrauch reductions between theorems in this sense means to ask which theorems can be transformed continuously or computably into each other. This allows a new purely topological or computational approach to metamathematics that sheds new light on the nature of theorems. As crucial corner points of this classification scheme we study the limited principle of omniscience LPO, the lesser limited principle of omniscience LLPO and their parallelizations. We recall that the parallelized version of LPO is complete for limit computable functions (which are exactly the effectively Σ 2 –measurable functions in the Borel hierarchy). We prove that parallelized LLPO is equivalent to Weak Kőnig’s Lemma and hence to the Hahn-Banach Theorem in this new and very strong sense. We call a multi-valued function weakly computable if it is reducible to the Weihrauch degree of parallelized LLPO and we present a new proof that the class of weakly computable operations is closed under composition. This proof is based on a computational version of Kleene’s ternary logic. Moreover, we characterize weakly computable operations on computable metric spaces as operations that admit upper semi-computable compact-valued selectors and we prove that any single-valued weakly computable operation is already computable in the ordinary sense. 2000 Mathematics Subject Classification. 03F60,03D30,03B30,03E15.