Random Continuous Functions

Journal of Logic and Computation

Remmel

et al. 2009

We investigate the notion of K-triviality for closed sets and continuous functions in 2 N. For every K-trivial degreee d, there exists a closed set of degree d and a continuous function of degree d. Every K-trivial closed set contains a K-trivial real. There exists a K-trivial 0 1 class with no computable elements. A closed set is K-trivial if and only if it is the set of zeroes of some K-trivial continuous function. We give a density result for the Medvedev degrees of K-trivial 0 1 sets. If W ≤ T A , then W can compute a path through every A-decidable random closed set if and only if W ≡ T A .

Section: S(abmentioning

confidence: 99%

“…In [6], the notion of randomness was extended to continuous functions on 2 N . Thus, it will be natural to consider K-trivial continuous functions.…”

Section: K-trivial Continuous Functionsmentioning

confidence: 99%

See 1 more Smart Citation

K-Triviality of Closed Sets and Continuous Functions

Barmpalias

Journal of Logic and Computation

Remmel

et al. 2009

“…It was shown in particular that every K-trivial class contains a K-trivial member, but there exist K-trivial Å 0 1 classes with no computable members. The related notion of a random continuous function was introduced in [3]. It was shown that a random continuous function F on 2 N cannot be computable, so that the graph of F cannot be Å 0 1 class.…”

Section: Conclusion and Future Researchmentioning

confidence: 99%

Algorithmic Randomness of Closed Sets

Barmpalias

Brodhead

Journal of Logic and Computation

et al. 2007

We investigate notions of randomness in the space C½2 N of non-empty closed subsets of f0, 1g N . A probability measure is given and a version of the Martin-Lo¨f test for randomness is defined. Å 0 2 random closed sets exist but there are no random Å 0 1 closed sets. It is shown that any random closed set is perfect, has measure 0, and has box dimension log 2 ð4=3Þ. A random closed set has no n-c.e. elements. A closed subset of 2 N may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. If T n ¼ T \ f0, 1g n , then for any random closed set ½T where T has no dead ends, KðT n Þ ! n À Oð1Þ but for any k, KðT n Þ 2 nÀk þ Oð1Þ, where K() is the prefix-free complexity of 2 f0, 1g Ã .

“…In a series of recent papers [2,3,4,5], G. Barmpalias, S. Dashti, R. Weber and the authors have defined a notion of (algorithmic) randomness for closed sets and continuous functions on 2 N . Some definitions are needed.…”

Section: Introductionmentioning

confidence: 99%

Effective Capacity and Randomness of Closed Sets

Electron. Proc. Theor. Comput. Sci.

Brodhead

2010

Self Cite

We investigate the connection between measure and capacity for the space of nonempty closed subsets of {0,1}*. For any computable measure, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets which have nonempty intersection with Q. We prove an effective version of Choquet's capacity theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions that characterize when the capacity of a random closed set equals zero or is >0. We construct for certain measures an effectively closed set with positive capacity and with Lebesgue measure zero