Weakly Complete Problems are Not Rare

Abstract

Certain natural decision problems are known to be intractable because they are complete for E, the class of all problems decidable in exponential time. Lutz recently conjectured that many other seemingly intractable problems are not complete for E, but are intractable nonetheless because they are weakly complete for E. The main result of this paper shows that Lutz's intuition is at least partially correct; many more problems are weakly complete for E than are complete for E. The main result of this paper states that weakly complete problems are not rare in the sense that they form a non-measure 0 subset of E. This extends a recent result of Lutz that establishes the existence of problems that are weakly complete, but not complete, for E. The proof of Lutz's original result employs a sophisticated martingale diagonalization argument. Here we simplify and extend Lutz's argument to prove the main result. This simplified martingale diagonalization argument may be applicable to other quest..