Abstract
Cupping nonzero computably enumerable (c.e. for short) degrees to 0′ in various structures has been one of the most important topics in the development of classical computability theory. An incomplete c.e. degree a is cuppable if there is an incomplete c.e. degree b such that a∪b=0′, and noncuppable if there is no such degree b. Sacks splitting theorem shows the existence of cuppable degrees. However, Yates(unpublished) and Cooper [3] proved that there are noncomputable noncuppable degrees. After that, Harrington and Shelah were able to employ the cupping/noncupping properties to show that the theory of the c.e. degrees under relation ≤ is undecidable. Cuppable and noncuppable degrees were further studied later. See Harrington [7], Miller [10], Fejer and Soare [6], Ambos-Spies, Lachlan and Soare [1], etc..
A. Li is partially supported by NSF Grant No. 60325206 (China). G. Wu is partially supported by a Start-Up Grant M48110008 from Nanyang Technological University. All three authors are partially supported by the International Joint Project No. 60310213 of NSFC of China. This work was done partially while the authors were visiting the Institute for Mathematical Sciences, National University of Singapore in 2005. The visit was supported by the Institute.
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Li, A., Song, Y., Wu, G. (2006). Universal Cupping Degrees. In: Cai, JY., Cooper, S.B., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2006. Lecture Notes in Computer Science, vol 3959. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11750321_68
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