Minimal Weak Truth Table Degrees and Computably Enumerable Turing Degrees

Two of the central concepts for the study of degree structures in computability theory are computably enumerable degrees and minimal degrees. For strong notions of reducibility, such as m-deducibility or truth table reducibility, it is possible for computably enumerable degrees to be minimal. For we...

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Hauptverfasser: Downey, Rodney G., Ng, Keng Meng, Solomon, Reed
Format: E-Book Buch
Sprache:Englisch
Veröffentlicht: Providence, Rhode Island American Mathematical Society 2020
Ausgabe:1
Schriftenreihe:Memoirs of the American Mathematical Society
Schlagworte:
Computable functions
Recursively enumerable sets
Unsolvability (Mathematical logic)
Computably enumerable set
Turing degree
weak truth table degree
minimal degree
ISBN:9781470441623, 1470441624
ISSN:0065-9266, 1947-6221
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Zusammenfassung:Two of the central concepts for the study of degree structures in computability theory are computably enumerable degrees and minimal degrees. For strong notions of reducibility, such as m-deducibility or truth table reducibility, it is possible for computably enumerable degrees to be minimal. For weaker notions of reducibility, such as weak truth table reducibility or Turing reducibility, it is not possible to combine these properties in a single degree. We consider how minimal weak truth table degrees interact with computably enumerable Turing degrees and obtain three main results. First, there are sets with minimal weak truth table degree which bound noncomputable computably enumerable sets under Turing reducibility. Second, no set with computable enumerable Turing degree can have minimal weak truth table degree. Third, no \Delta^0_2 set which Turing bounds a promptly simple set can have minimal weak truth table degree.
Bibliographie:May 2020, volume 265, number 1284 (first of 7 numbers)
Includes bibliographical references
ISBN:9781470441623
1470441624
ISSN:0065-9266
1947-6221
DOI:10.1090/memo/1284