Closed Left-R.E. Sets

A set is called r-closed left-r.e. iff every set r-reducible to it is also a left-r.e. set. It is shown that some but not all left-r.e. cohesive sets are many-one closed left-r.e. sets. Ascending reductions are many-one reductions via an ascending function; left-r.e. cohesive sets are also ascending closed left-r.e. sets. Furthermore, it is shown that there is a weakly 1-generic many-one closed left-r.e. set. We also consider initial segment complexity of closed left-r.e. sets. We show that initial segment complexity of ascending closed left-r.e. sets is of sublinear order. Furthermore, this is near optimal as for any non-decreasing unbounded recursive function g, there are ascending closed left-r.e. sets A with initial segment complexity C(A(0)A(1) . . . A(n)) ≥ cn/g(n) for some constant c and all n. The initial segment complexity of a conjunctively (or disjunctively) closed left-r.e. set satisfies, for all ε, for almost all n, C(A(0)A(1) . . . A(n)) ≤ (2 + ε) log(n).