Degrees of unsolvability within a regressive isol
نویسندگان
چکیده
منابع مشابه
Degrees of Unsolvability: A Tutorial
Given a problem P , one associates to P a degree of unsolvability, i.e., a quantity which measures the amount of algorithmic unsolvability which is inherent in P . We focus on two degree structures: the semilattice of Turing degrees, DT, and its completion, Dw = D̂T, the lattice of Muchnik degrees. We emphasize specific, natural degrees and their relationship to reverse mathematics. We show how ...
متن کاملDegrees of Unsolvability
Modern computability theory began with Turing [Turing, 1936], where he introduced the notion of a function computable by a Turing machine. Soon after, it was shown that this definition was equivalent to several others that had been proposed previously and the Church-Turing thesis that Turing computability captured precisely the informal notion of computability was commonly accepted. This isolat...
متن کاملSome fundamental issues concerning degrees of unsolvability
Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT . The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is > 0 and < 0 ′. The second issue is to find a “smallness property” of an infinite, co-recursively enumerable set A ⊆ ω which ensures that the...
متن کاملDegrees of unsolvability of continuous functions
We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substr...
متن کاملThe Degrees of Unsolvability of Continuous Functions
Computable analysis provides a standard notion of computability for continuous functions on the real numbers. This notion was first explicitly formulated and studied by Lacombe and Grzegorczyk in the 1950’s, although it can be traced back to Turing and beyond that to Brouwer. However, a satisfactory notion of the degrees of unsolvability of continuous functions has only recently been introduced...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Fundamenta Mathematicae
سال: 1974
ISSN: 0016-2736,1730-6329
DOI: 10.4064/fm-86-1-29-40