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Martin's Conjecture in the Enumeration Degrees

Antonio Nakid Cordero

TL;DR

The paper investigates Martin's Conjecture in the enumeration-degree framework $\mathcal{D}_e$, focusing on $e$-invariant and uniformly $e$-invariant functions. It shows that global behavior in $\mathcal{D}_e$ is far richer than in the Turing setting: there is no Cone Theorem, and one can construct continuum families of cofinal pieces and infinite antichains of uniformly $e$-invariant functions, revealing a complex global landscape. Nevertheless, a sharp local result mirrors the spirit of Martin's Conjecture: uniformly $e$-invariant functions are locally constrained to be constant, increasing, or above the skip, yielding a local analogue of the conjecture. The paper also exhibits a definable non-uniform invariant function using Kalimullin pairs, proving that invariance does not always imply uniform invariance on cones, and thereby clarifying the limitations of a uniformity-based classification in $\mathcal{D}_e$ and outlining key open directions for future work.

Abstract

Martin's Conjecture states that every definable function on the Turing degrees is either constant or increasing, and that every increasing function is an iterate of the Turing jump. This classification has already been corroborated for the class of uniformly invariant functions and a long-standing conjecture by Steel is that every definable function on the Turing degrees is equivalent to a uniformly invariant one. We explore whether a similar classification is possible in the enumeration degrees, an extension of the Turing degrees. We show that the spectrum of behavior is much wider in the enumeration degrees, even for uniformly invariant functions. However, our main result is that uniformly invariant functions behave locally as nicely as possible: they are constant, increasing, or above the skip operator. As a consequence, we show that there is a definable function in the enumeration degrees that is not equivalent to a uniformly invariant one on any cone.

Martin's Conjecture in the Enumeration Degrees

TL;DR

The paper investigates Martin's Conjecture in the enumeration-degree framework , focusing on -invariant and uniformly -invariant functions. It shows that global behavior in is far richer than in the Turing setting: there is no Cone Theorem, and one can construct continuum families of cofinal pieces and infinite antichains of uniformly -invariant functions, revealing a complex global landscape. Nevertheless, a sharp local result mirrors the spirit of Martin's Conjecture: uniformly -invariant functions are locally constrained to be constant, increasing, or above the skip, yielding a local analogue of the conjecture. The paper also exhibits a definable non-uniform invariant function using Kalimullin pairs, proving that invariance does not always imply uniform invariance on cones, and thereby clarifying the limitations of a uniformity-based classification in and outlining key open directions for future work.

Abstract

Martin's Conjecture states that every definable function on the Turing degrees is either constant or increasing, and that every increasing function is an iterate of the Turing jump. This classification has already been corroborated for the class of uniformly invariant functions and a long-standing conjecture by Steel is that every definable function on the Turing degrees is equivalent to a uniformly invariant one. We explore whether a similar classification is possible in the enumeration degrees, an extension of the Turing degrees. We show that the spectrum of behavior is much wider in the enumeration degrees, even for uniformly invariant functions. However, our main result is that uniformly invariant functions behave locally as nicely as possible: they are constant, increasing, or above the skip operator. As a consequence, we show that there is a definable function in the enumeration degrees that is not equivalent to a uniformly invariant one on any cone.
Paper Structure (9 sections, 24 theorems, 32 equations, 1 figure)

This paper contains 9 sections, 24 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Enumeration Degrees and the Skip
  3. Global Failure
  4. Uniform Invariance
  5. A Positive Local Result
  6. A Non-Uniformly Invariant Function
  7. Semicomputable sets and $\mathcal{K}$-pairs
  8. A Borel Invariant Function that is not Uniformly Invariant
  9. Further Questions

Key Result

Theorem 1.1

Assume $AD$. Let $\mathcal{A}\subseteq \mathcal{P}(\mathbb{N})$ be closed under Turing equivalence. Then, either $\mathcal{A}$ or its complement $\overline{\mathcal{A}}$ contains a cone of Turing degrees.

Figures (1)

  • Figure 1: Examples of connected components of three different graphs $\mathcal{G}_{i,j}$ on the degree of computable sets.

Theorems & Definitions (45)

  • Theorem 1.1: Cone Theorem (Martin martinTD1968)
  • Conjecture 1.2: Martin
  • Theorem 1.3: Steel steelClassificationJumpOperators1982, Slaman & Steel slamanDefinableFunctionsDegrees1988
  • Conjecture 1.4: Steel steelClassificationJumpOperators1982
  • Definition 2.1: Friedberg and Rogers, FriedbergRogers1959
  • Proposition 2.2: essentially Medvedev Medvedev
  • Proposition 2.3: Cooper Cooper_1984
  • Proposition 2.4: Andrews et al. Andrews2019OnDegrees
  • Example 3.1
  • Theorem 3.2: joint with Jacobsen-Grocott
  • ...and 35 more