A Note On the Turing Degrees of Divergence Bounded Computable Reals
نویسندگان
چکیده
The Turing degree of a real number is defined as the Turing degree of its binary expansion. In this note we apply the double witnesses technique recently developed by Downey, Wu and Zheng [2] and show that there exists a ∆2-Turing degree which contains no divergence bounded computable real numbers. This extends the result of [2] that not every ∆2-Turing degree contains a d-c.e. real.
منابع مشابه
A Hierarchy of Turing Degrees for Divergence Bounded Computable Real Numbers
A real number x is f-bounded computable (f-bc, for short) for a function f if there is a computable sequence (xs) of rational numbers which converges to x f-bounded effectively in the sense that, for any natural number n, the sequence (xs) has at most f (n) non-overlapping jumps of size larger than 2−n. f-bc reals are called divergence bounded computable if f is computable. In this paper we giv...
متن کاملOn the Turing Degrees of Divergence Bounded Computable Reals
The d-c.e. (difference of c.e.) and dbc (divergence bounded computable) reals are two important subclasses of ∆2-reals which have very interesting computability-theoretical as well as very nice analytical properties. Recently, Downey, Wu and Zheng [2] have shown by a double witness technique that not every ∆2-Turing degree contains a d-c.e. real. In this paper we show that the classes of Turing...
متن کاملThe computable Lipschitz degrees of computably enumerable sets are not dense
The computable Lipschitz reducibility was introduced by Downey, Hirschfeldt and LaForte under the name of strong weak truthtable reducibility [6]. This reducibility measures both the relative randomness and the relative computational power of real numbers. This paper proves that the computable Lipschitz degrees of computably enumerable sets are not dense. An immediate corollary is that the Solo...
متن کاملDegree-Theoretic Aspects of Computably Enumerable Reals
A real is computable if its left cut, L( ); is computable. If (qi)i is a computable sequence of rationals computably converging to ; then fqig; the corresponding set, is always computable. A computably enumerable (c.e.) real is a real which is the limit of an increasing computable sequence of rationals, and has a left cut which is c.e. We study the Turing degrees of representations of c.e. real...
متن کاملLimitwise monotonic sets of reals
1. f (x, s) f (x, s + 1) for all x and s; 2. sups f (x, s) < ∞ for every x ; 3. F(x) = sups f (x, s). A set A ⊆ N is limitwise monotonic if A equals to the range of some limitwise monotonic function. If we replace here the computable functions f by X -computable functions for some Turing oracle X then we get the notions of X -limitwise monotonic functions and sets, respectively. Note that a sim...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Electr. Notes Theor. Comput. Sci.
دوره 120 شماره
صفحات -
تاریخ انتشار 2005