Mass Problems

Informally, mass problems are similar to decision problems. The difference is that, while a decision problem has only one solution, a mass problem is allowed to have more than one solution. Many concepts which apply to decision problems apply equally well to mass problems. For instance, a mass problem is said to be solvable if it has at least one computable solution. Also, one mass problem is said to be reducible to another mass problem if, given a solution of the second problem, we can use it to find a solution of the first problem. Many unsolvable mathematical problems are most naturally viewed as mass problems rather than decision problems. For example, let CPA be the problem of finding a completion of Peano Arithmetic. A well-known theorem going back to Gödel and Tarski says that CPA is unsolvable, in the sense that there are no computable completions of Peano Arithmetic. Note that, in describing CPA as a “problem” whose “solutions” are the completions of Peano Arithmetic, we are implicitly viewing CPA as a mass problem rather than a decision problem. This is because Peano Arithmetic has many different completions, with many different Turing degrees. There is no single Turing degree which can be said to measure the amount of unsolvability which is inherent in CPA. Formally, a mass problem is defined to be an arbitrary set of Turing oracles, i.e., an arbitrary subset of the Baire space, N. Let P and Q be mass problems. We say that P is solvable if P has at least one recursive element. We say that P is Muchnik reducible to Q if for all g ∈ Q there exists f ∈ P such that f is Turing reducible to g. A Muchnik degree is an equivalence class of mass problems under mutual Muchnik reducibility. The Muchnik degrees are partially ordered in the obvious way, by Muchnik reducibility. Under this partial ordering, it is straightforward to show that the Muchnik degrees form a complete distributive lattice. There is an obvious, natural embedding of the upper semilattice of Turing degrees into the lattice of Muchnik degrees, obtained by mapping the Turing degree of f to the Muchnik degree of the singleton set {f}. The lattice of Muchnik degrees was first introduced by Albert Muchnik (of Friedberg/Muchnik fame) in 1963. Subsequently Muchnik degrees have been