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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| Main Authors: | , , |
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| Format: | eBook Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
2020
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| Edition: | 1 |
| Series: | Memoirs of the American Mathematical Society |
| Subjects: | |
| ISBN: | 9781470441623, 1470441624 |
| ISSN: | 0065-9266, 1947-6221 |
| Online Access: | Get full text |
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| Summary: | 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. |
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| Bibliography: | 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 |