Effectively closed sets and enumerations

Mathematical Theory and Computational Practice

“…Then the sequence of sets W e;n for e, n ∈ ω × ω is an effective enumeration of the Σ 0 1 classes of measure less than 1. This contrasts with the result of [1] that there is no effective numbering of the Π 0 1 classes of measure zero. We next show that any effectively enumerable family of Π 0 1 classes containing all the clopen classes has a Friedberg numbering.…”

“…If m e,t1 [n] at some stage t 1 ≥ t 0 satifies randomness constant c, then we regret having assigned m e a place in our enumeration {α e } e∈ω . To compensate for this regret, we choose a large number p = p c,n and for all stages s ≥ t 1 No, by Theorem 9 constant c. 1 If m e actually does fail randomness constant c, but at a larger length n ′ > n, then because there are infinitely many e ′ with m e ′ = m e we will eventually assign some α d ′ to some such m e ′ at a stage t 2 that is so large that m e ′ ,t2 [n ′ ] = m e ′ [n ′ ]. Thus, each real in R [c+1,∞) will eventually be assigned a permanent α d ′ .…”

Numberings and Randomness

“…If m e,t1 [n] at some stage t 1 ≥ t 0 satifies randomness constant c, then we regret having assigned m e a place in our enumeration {α e } e∈ω . To compensate for this regret, we choose a large number p = p c,n and for all stages s ≥ t 1 No, by Theorem 9 constant c. 1 If m e actually does fail randomness constant c, but at a larger length n ′ > n, then because there are infinitely many e ′ with m e ′ = m e we will eventually assign some α d ′ to some such m e ′ at a stage t 2 that is so large that…”

“…Brodhead and Cenzer [1] showed that there is an effective Friedberg numbering of the Π 0 1 classes in Cantor space 2 ω . They showed that effective numberings exist of the Π 0 1 classes that are homogeneous, and decidable, but not of the families consisting of Π 0 1 classes that are of measure zero, thin, perfect thin, small, very small, or nondecidable, respectively.…”

Complexity for partial computable functions over computable Polish spaces

“…In numerical computations, many important problems such as real number equality, set equality and etc., are undecidable (non-computable). Therefore, studying degrees of noncomputability became recently an active research area in computable analysis (see among others Brattka and Gherardi (2009), Cenzer and Remmel (1998), Brodhead and Cenzer (2008) and Selivanov and Schroeder (2014)).…”