Abstract
In Part II, we proved that besides computable problems there are also incomputable ones. So, given a computational problem, it makes sense to talk about its degree of unsolvability. Of course, at this point we only know of two such degrees: one is shared by all computable problems, and the other is shared by all incomputable ones. (This will change, however, in the next chapter.) Nevertheless, the main aim of this chapter is to formalize the intuitive notion of the degree of unsolvability. Building on the concept of the oracle Turing machine, we will first define the concept of the Turing reduction, the most general reduction between computational problems. We will then proceed in a natural way to the definition of Turing degree—the formal counterpart of the intuitive notion of the degree of unsolvability.
Degree indicates the extent to which something happens or the amount something is felt.
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© 2015 Springer-Verlag Berlin Heidelberg
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Robič, B. (2015). Degrees of Unsolvability. In: The Foundations of Computability Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44808-3_11
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DOI: https://doi.org/10.1007/978-3-662-44808-3_11
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Online ISBN: 978-3-662-44808-3
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