## Works by Paul Brodhead

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1. Algorithmic randomness of continuous functions.George Barmpalias, Paul Brodhead, Douglas Cenzer, Jeffrey B. Remmel & Rebecca Weber - 2008 - Archive for Mathematical Logic 46 (7-8):533-546.
We investigate notions of randomness in the space ${{\mathcal C}(2^{\mathbb N})}$ of continuous functions on ${2^{\mathbb N}}$ . A probability measure is given and a version of the Martin-Löf test for randomness is defined. Random ${\Delta^0_2}$ continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any ${y \in 2^{\mathbb N}}$ , (...)
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2. Continuity of capping in C bT.Paul Brodhead, Angsheng Li & Weilin Li - 2008 - Annals of Pure and Applied Logic 155 (1):1-15.
A set Asubset of or equal toω is called computably enumerable , if there is an algorithm to enumerate the elements of it. For sets A,Bsubset of or equal toω, we say that A is bounded Turing reducible to reducible to) B if there is a Turing functional, Φ say, with a computable bound of oracle query bits such that A is computed by Φ equipped with an oracle B, written image. Let image be the structure of the c.e. bT-degrees, (...)
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3. Effectively closed sets and enumerations.Paul Brodhead & Douglas Cenzer - 2008 - Archive for Mathematical Logic 46 (7-8):565-582.
An effectively closed set, or ${\Pi^{0}_{1}}$ class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of ${\Pi^{0}_{1}}$ classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of ${\Pi^{0}_{1}}$ classes and (...)
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