[Paper Review] Harrington's Solution to McLaughlin's Conjecture and Non-uniform Self-moduli

Peter Gerdes|arXiv (Cornell University)|Dec 15, 2010

Computability, Logic, AI Algorithms17 references3 citations

TL;DR

This paper presents Harrington's unpublished method for solving McLaughlin's conjecture on implicit definability in computability theory, establishing the existence of a uniformly effective sequence of $Π^0_1$ singletons that are $Low_\alpha$ but not computable from the effective join of the $\alpha$ jumps of the others for any $\alpha <_{\mathcal{O}} \omega^{ck}_1$. The key contribution is demonstrating that the growth rate required for non-uniform computation of a Turing degree can be separated by an arbitrary number of jumps from the rate needed for uniform computation, using a novel construction of nice ordinal notations and forcing over trees.

ABSTRACT

While much work has been done to characterize the Turing degrees computing members of various collections of fast growing functions, much less has been done to characterize the rate of growth necessary to compute particular degrees. Prior work has shown that every degree computed by all sufficiently fast growing functions is uniformly computed by all sufficiently fast growing functions. We show that the rate of growth sufficient for a function to uniformly compute a given Turing degree can be separated by an arbitrary number of jumps from the rate of growth that suffices for a function to non-uniformly compute the degree. These results use the unpublished method Harrington developed to answer McLaughlin's conjecture so we begin the paper with a rigorous presentation of the approach Harrington sketched in his handwritten notes on the conjecture. We also provide proofs for the important computability theoretic results Harrington noted were corollaries of this approach. In particular we provide the first published proof of Harrington's result that there is an effectively given sequence of Π^0_1 singletons that are Low_α none of which is computable in the effective join of the α jumps of the others for every computable ordinal α.

Motivation & Objective

• To present and formalize Harrington's unpublished method for solving McLaughlin's conjecture on implicit definability in computability theory.
• To establish the existence of a uniformly effective sequence of $\Pi^0_1$ singletons that are $Low_\alpha$ but not computable from the effective join of the $\alpha$ jumps of the others for all $\alpha <_{\mathcal{O}} \omega^{ck}_1$.
• To demonstrate that the growth rate required for non-uniform computation of a Turing degree can be separated by an arbitrary number of jumps from the rate required for uniform computation.
• To provide the first published proof of Harrington's result on non-uniform self-moduli using a novel construction of nice ordinal notations and forcing over trees.

Proposed method

• Constructing a canonical, effective path through Kleene's $\mathcal{O}$ using a $\Pi^1_1$ cofinal sequence of notations to ensure uniform computability of associated functions.
• Defining a recursive, monotonic mapping from strings to notations that preserves the order type and allows for effective path construction in $\mathcal{O}$.
• Introducing the notation $\beta^{\blacktriangleleft}$ to represent the initial segment of $\beta^\diamond$ that defines the path to $\beta$, enabling effective comparison of notations.
• Using local forcing relations $\mathrel{{\Vdash}_{T}}$ and $\mathrel{{\Vdash}^{X}_{T}}$ on trees $T$ to simulate forcing over paths, with relativization to $\mathbf{0}^{(\beta)}$ for higher jump levels.
• Applying the effective sum operation $+$ on notations to build new notations $\kappa_{i+1} = \kappa_i + 1 + \alpha_i'$, ensuring that each $\kappa_i$ is a nice notation.
• Proving that for any $\beta$, the notation $\beta^\prime$ is effectively constructible from $\beta^\blacktriangleleft$, ensuring the entire construction remains computable.

Experimental results

Research questions

• RQ1 Can Harrington's unpublished method for solving McLaughlin's conjecture be formally reconstructed and published?
• RQ2 Is there an effectively given sequence of $\Pi^0_1$ singletons that are $Low_\alpha$ but not computable from the effective join of the $\alpha$ jumps of the others for all $\alpha <_{\mathcal{O}} \omega^{ck}_1$?
• RQ3 Can the growth rate required for non-uniform computation of a Turing degree be separated by an arbitrary number of jumps from the rate required for uniform computation?
• RQ4 What is the role of nice ordinal notations in constructing non-uniform self-moduli and separating growth rates in Turing degree computation?

Key findings

• The paper provides the first published proof of Harrington's result that there exists an effectively given sequence of $\Pi^0_1$ singletons that are $Low_\alpha$ but not computable from the effective join of the $\alpha$ jumps of the others for any $\alpha <_{\mathcal{O}} \omega^{ck}_1$.
• It is shown that for any given Turing degree, the growth rate sufficient for non-uniform computation can be separated by an arbitrary number of jumps from the rate sufficient for uniform computation.
• The construction of a $\Pi^1_1$ path through $\mathcal{O}$ ensures that all required computations for $\beta^\diamond$ and $\ell(\beta)$ are uniformly computable from the path's notations.
• The method establishes that $\beta^{\blacktriangleleft}$ is effectively constructible from $\beta$, and that $\beta^\prime$ is a nice notation, ensuring the entire construction remains effective and computable.
• The paper proves that $\kappa_{i+1}$ is a nice notation whenever $\kappa_i$ and $\alpha_i'$ are nice, ensuring the inductive construction of the path remains valid.
• It is shown that $\beta <_{\mathcal{O}} \gamma$ if and only if $\beta^{\blacktriangleleft} <_L \gamma^{\blacktriangleleft}$, establishing a computable characterization of the order on notations.