Computability Limits of Sequential Hypothesis Testing

2026-05-04

AI Summary

This study investigates the existence of computable decision rules in sequential hypothesis testing under the “finite-errors” criterion—requiring that, almost surely, only finitely many errors are made and the procedure eventually stabilizes on the correct conclusion. By integrating Cover’s theorem, notions from limit computability (Δ⁰₂ sets), computable analysis, and probabilistic methods, the paper provides the first complete characterization of necessary and sufficient conditions for the existence of computable finite-error sequential tests, both for subsets of the rationals and for any effectively presented countable family of real-valued means. This work precisely delineates the computable boundary within which empirical methods can converge to the truth, thereby establishing fundamental limits on the computability of sequential learning.

Abstract

Sequential hypothesis testing asks for decision rules that update as data arrive. A natural goal is \emph{eventual correctness}: the rule may change its mind early on, but it should make only finitely many wrong decisions almost surely. Starting from Cover's theorem, which guarantees such behavior for membership in a countable set of candidate means, we ask a sharper question: \emph{which sets actually admit computable sequential decision procedures with finitely many errors?} We answer this optimally by giving a complete characterization both necessary and sufficient of the subsets of $\Q$ that admit a computable finite-error sequential membership test. We further extend the characterization to any \emph{effectively presented} countable family of real means, exactly the setting in which Cover's identification rule can be implemented computably. Beyond the technical boundary, the results clarify within a precise probabilistic setting what it can mean for inquiry to ``converge to the truth,'' and they formalize a limit to which empirical methods can be expected to succeed when only eventual stabilization (rather than fixed-time guarantees) is demanded. keywords: Cover's theorem, sequential decision procedures, finite error learning, limit computability, $Δ^0_2$ sets.

Problem

• sequential hypothesis testing
• computable decision procedures
• finite error learning
• limit computability
• Δ⁰₂ sets

Innovation

• sequential decision procedures
• finite error learning
• limit computability
• Δ⁰₂ sets
• Cover's theorem