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Characterising SJT reducibility

Noam Greenberg, Andre Nies, Dan Turetsky

Abstract

SJT reducibility between sets $A,B \subseteq \mathbb N$ is defined by $A \le_{SJT} B$ if for each computable function $h$ that is unbounded and nondecreasing, there is an $h$-bounded uniformly $B$-c.e.\ trace $(T_n)_{n \in \mathbb N} $ such that for each $n$, the value $J^A(n)$ of the jump is in $T_n$, if defined. This reducibility is slightly weaker than Turing reducibility. We study SJT reducibility, and as a main result give several characterisations of it on the $K$-trivial sets. This is the first case of extending the three lowness paradigms, weak as an oracle, computed by many, and inert, to the setting of weak reducibilities.

Characterising SJT reducibility

Abstract

SJT reducibility between sets is defined by if for each computable function that is unbounded and nondecreasing, there is an -bounded uniformly -c.e.\ trace such that for each , the value of the jump is in , if defined. This reducibility is slightly weaker than Turing reducibility. We study SJT reducibility, and as a main result give several characterisations of it on the -trivial sets. This is the first case of extending the three lowness paradigms, weak as an oracle, computed by many, and inert, to the setting of weak reducibilities.
Paper Structure (17 sections, 10 theorems, 15 equations)

This paper contains 17 sections, 10 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Weak reducibilities
  3. Three lowness paradigms
  4. Characterising SJT-reducibility according to the three paradigms.
  5. The difficulties with partial relativisation
  6. Benign cost functions
  7. A proof of \ref{['thm:A_to_A_join_B']}
  8. Proof of (b)$\Rightarrow$(a) of \ref{['thm:cost_function_characterisation']}
  9. Proof of (a)$\Rightarrow$(b) of \ref{['thm:cost_function_characterisation']}
  10. Computing with random sets
  11. A proof of (a)$\Rightarrow$(b) of \ref{['thm:random_computing_characterisation']}
  12. A proof of (b)$\Rightarrow$(a) of \ref{['thm:random_computing_characterisation']}, cases (1)--(4).
  13. Proof of (b)$\Rightarrow$(a) of \ref{['thm:random_computing_characterisation']}, case (5)
  14. Concluding remarks and open questions
  15. The structure of the SJT degrees
  16. ...and 2 more sections

Key Result

Theorem 1.4

Suppose that both $A$ and $B$ are jump-traceable. The following are equivalent:

Theorems & Definitions (30)

  • Definition 1.1
  • Definition 1.2: SJT reducibility, Nies:book, 8.4.37
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2: Greenberg.Nies:11
  • Lemma 2.3
  • Claim 1
  • ...and 20 more