[Paper Review] Infinite computations with random oracles

Merlin Carl, Philipp Schlicht|arXiv (Cornell University)|Jun 29, 2013

Computability, Logic, AI Algorithms17 references4 citations

TL;DR

This paper investigates whether a real number computable relative to a large set of oracles—such as those of full measure, positive measure, comeager, or nonmeager Borel sets—is necessarily computable without oracles. It shows that for most infinite time machine models like ITTMs, ITRMs, and α-Turing machines, such uniform computability from large oracle sets implies standard computability, but for ordinal Turing machines (OTMs), the answer is independent of ZFC, highlighting a fundamental divergence in the behavior of these models under randomness assumptions.

ABSTRACT

We consider the following problem for various infinite time machines. If a real is computable relative to large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? We show that the answer is independent from ZFC for ordinal time machines (OTMs) with and without ordinal parameters and give a positive answer for most other machines. For instance, we consider, infinite time Turing machines (ITTMs), unresetting and resetting infinite time register machines (wITRMs, ITRMs), and α-Turing machines for countable admissible ordinals α.

Motivation & Objective

• To determine whether a real number computable relative to a large set of oracles (e.g., full measure, positive measure, comeager, nonmeager Borel) is necessarily computable without oracles.
• To investigate the robustness of the Church-Turing thesis in the context of infinite time computation models.
• To examine the role of randomness and genericity in oracle computation, particularly whether random or generic oracles can yield non-computable reals.
• To compare the behavior of different infinite time machine models—OTMs, ITTMs, ITRMs, α-Turing machines—under various notions of largeness for oracle sets.
• To clarify the boundary between relativized computability and absolute computability in the presence of measure-theoretic or topological largeness conditions.

Proposed method

• Analyzes oracle computation across multiple infinite time machine models: OTMs, ITTMs, ITRMs (unresetting and resetting), and α-Turing machines for countable admissible ordinals α.
• Applies measure-theoretic and topological notions of largeness (e.g., full measure, positive measure, comeager, nonmeager Borel sets) to sets of oracles.
• Uses definability and constructibility techniques, particularly within $L_{eta}$-hierarchies and $L_{ ext{ord}}$-structures, to analyze halting times and computability levels.
• Employs genericity arguments, including Cohen generic reals over $L_{eta}$, to show that certain oracles preserve the admissibility and constructibility levels of the computation.
• Applies forcing and random real theory to show that if a real is computable from all oracles in a set of positive measure or comeager set, then it lies in a low-level constructible hierarchy.
• Uses the fact that mutual generics over $L_{eta}$ yield intersection $L_{eta}$, enabling the conclusion that the computed real is in $L_{eta}$ and thus computable.

Experimental results

Research questions

• RQ1 Does computability from all oracles in a set of positive measure imply standard computability for infinite time Turing machines (ITTM)?
• RQ2 Is the answer to the oracle largeness problem independent of ZFC for ordinal Turing machines (OTMs) with and without ordinal parameters?
• RQ3 For unresetting and resetting infinite time register machines (wITRM, ITRM), does computability from a nonmeager Borel set of oracles imply standard computability?
• RQ4 For α-Turing machines at countable admissible ordinals α, does computability from a set of positive measure or a nonmeager Borel set of oracles imply α-computability?
• RQ5 Under what conditions does mutual genericity of oracles over $L_{eta}$ imply that a real computed from them is in $L_{eta}$ and thus computable?

Key findings

• For infinite time Turing machines (ITTM), writability (eventual writability, accidental writability) in a nonmeager Borel set of oracles implies standard writability.
• For both unresetting and resetting infinite time register machines (wITRM, ITRM), computability from a set of positive measure or a nonmeager Borel set of oracles implies standard computability.
• For all (and unboundedly many) countable admissible ordinals α, computability from a nonmeager Borel set or a set of positive measure of oracles implies α-computability for α-Turing machines.
• For ordinal Turing machines (OTM), the answer to whether large oracle sets imply standard computability is independent of ZFC, both with and without ordinal parameters.
• The supremum of halting times of OTMs, denoted η, satisfies $\bar{\alpha} < \eta$, where $\bar{\alpha}$ is the least ordinal such that $L_{\bar{\alpha}}$ is elementarily equivalent to $L_{\beta}$ for some $\beta > \bar{\alpha}$.
• If a real x is computable from all oracles in a comeager set C of Cohen generics over $L_{\alpha+1}$, then $x \in L_{\alpha}$, and hence x is α-computable.