Tutorial on Mass Problems

Let P be a set of reals viewed as a mass problem, i.e., a “decision problem with more than one solution.” Here the “solutions” of P are the elements of P . Many unsolvable mathematical problems are best viewed as mass problems. One says that a mass problem P is weakly reducible to a mass problem Q if for every solution Y of Q there exists a solution X of P such that X is Turing reducible to Y . A weak degree is an equivalence class of mass problems under mutual weak reducibility. Weak degrees are also known as Muchnik degrees. Let Pw be the set of weak degrees of mass problems associated with nonempty Π 1 subsets of 2, partially ordered by weak reducibility. Algebraically, it is easy to see that Pw is a countable distributive lattice with top and bottom. Simpson and others have studied the lattice Pw in a series of publications beginning in 1999. Our principal findings are as follows. 1. There is a natural embedding of ET , the countable semilattice of recursively enumerable Turing degrees, into the lattice Pw. This embedding is one-to-one and preserves the semilattice structure and the top and bottom. We identify ET with its image in Pw under this embedding. 2. Like the semilattice ET , the lattice Pw is structurally rich. In particular, any countable distributive lattice is lattice-embeddable in any nontrivial initial segment of Pw. Moreover, the Pw analog of the Sacks Splitting Theorem holds. These structural results are proved by means of priority arguments. The Pw analog of the Sacks Density Theorem remains as an open problem. 3. Unlike ET , the lattice Pw contains a large number of specific, natural degrees other than the top and bottom degrees. These specific, natural degrees in Pw arise from foundationally interesting topics such as reverse mathematics, algorithmic randomness, subrecursive hierarchies, computational complexity, and hyperarithmeticity.