On the Nonexistence of a Strong Minimal Pair
Mingzhong Cai, Yiqun Liu, Yong Liu, Cheng Peng, Yue Yang

TL;DR
This paper proves that strong minimal pairs do not exist among recursively enumerable degrees, using advanced priority arguments that likely require fourth Turing jump complexity.
Contribution
It establishes the nonexistence of strong minimal pairs in r.e. degrees, employing a novel construction that surpasses traditional priority methods.
Findings
No strong minimal pairs in r.e. degrees.
Construction requires $ extbf{0}^{(4)}$-priority arguments.
Advances understanding of the structure of r.e. degrees.
Abstract
Two nonzero recursively enumerable (r.e.) degrees and form a strong minimal pair if and for any nonzero r.e. degree . We prove that there is no strong minimal pair in the r.e. degrees. Our construction goes beyond the usual -priority arguments and we give some evidence to show that it needs -priority arguments.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Advanced Topology and Set Theory · Complexity and Algorithms in Graphs
