Asymptotic density and the coarse computability bound

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2016-02-11
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Hirschfeldt, Dennis
Jockusch, Carl
Schupp, Paul
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McNicholl, Timothy
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Computer Science

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The Computer Science Department was officially established in 1969, with Robert Stewart serving as the founding Department Chair. Faculty were composed of joint appointments with Mathematics, Statistics, and Electrical Engineering. In 1969, the building which now houses the Computer Science department, then simply called the Computer Science building, was completed. Later it was named Atanasoff Hall. Throughout the 1980s to present, the department expanded and developed its teaching and research agendas to cover many areas of computing.

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Mathematics
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For r is an element of [0, 1] we say that a set A subset of omega is coarsely computable at density r if there is a computable set C such that {n: C(n) = A(n)} has lower density at least r. Let gamma (A) = sup{r : A is coarsely computable at density r}. We study the interactions of these concepts with Turing reducibility. For example, we show that if r is an element of (0, 1] there are sets A(0), A(1) such that gamma(A(0)) = gamma(A(1)) = r where A(0) is coarsely computable at density r while A(1) is not coarsely computable at density r. We show that a real r is an element of [0, 1] is equal to gamma (A) for some c.e. set A if and only if r is left-Sigma(0)(3). A surprising result is that if G is a Delta(0)(2) 1-generic set, and A < = (T) G with gamma(A) = 1, then A is coarsely computable at density 1.

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This is a manuscript of an article published as Hirschfeldt, Denis R., Carl G. Jockusch Jr, Timothy H. McNicholl, and Paul E. Schupp. "Asymptotic density and the coarse computability bound." Computability 5, no. 1 (2016): 13-27, doi:10.3233/COM-150035. Posted with permission.

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Fri Jan 01 00:00:00 UTC 2016
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