In this paper, we present variable minimal unsatisfiability (VMU), which is a generalization of minimal unsatisfiability (MU). A characterization of a VMU formula F is that every variable of F is… (More)

We investigate the initial segment complexity of random reals. Let K( ) denote pre4x-free Kolmogorov complexity. A natural measure of the relative randomness of two reals and is to compare complexity… (More)

A degreed is d.r.e. if and only if it contains some difference of r.e. sets A− B. It is an important class of degrees other than that of r.e. degrees. G. E. Sacks [4] showed that the r.e. degrees are… (More)

A set A is computably Lipschitz or cl-reducible, for short, to a set B if A is Turing reducible to B by an oracle Turing machine with use function ϕ such that ϕ is bounded by the identity function up… (More)