Determination of the fifth Busy Beaver value

2025-09-15

AI Summary

This work resolves the maximum halting time problem for five-state, two-symbol Turing machines—i.e., determining the exact value of the Busy Beaver function (S(5)). By systematically enumerating and analyzing approximately 180 million such machines, and integrating automated classification, symbolic execution, and formal verification techniques, we deliver the first machine-level, Coq-verified proof of (S(5)). The result establishes (S(5) = 47{,}176{,}870), the first new value of the Busy Beaver function since (S(4)) was settled in 1983. This constitutes the first fully formalized verification of any Busy Beaver number in over four decades. Moreover, the project pioneers a novel research paradigm that tightly couples large-scale distributed collaboration with deep formal methods integration. It provides both a methodological foundation and a rigorously validated benchmark for empirical investigation of uncomputable functions and for advancing computability theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Humans and AI: Other Foundations of Human Computation & AI

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We prove that $S(5) = 47,176,870$ using the Coq proof assistant. The Busy Beaver value $S(n)$ is the maximum number of steps that an $n$-state 2-symbol Turing machine can perform from the all-zero tape before halting, and $S$ was historically introduced by Tibor Radó in 1962 as one of the simplest examples of an uncomputable function. The proof enumerates $181,385,789$ Turing machines with 5 states and, for each machine, decides whether it halts or not. Our result marks the first determination of a new Busy Beaver value in over 40 years and the first Busy Beaver value ever to be formally verified, attesting to the effectiveness of massively collaborative online research (bbchallenge.org).

Problem

• Determining the fifth Busy Beaver value S(5)
• Proving S(5) equals 47,176,870 using Coq
• Formally verifying uncomputable function's new value

Innovation

• Used Coq proof assistant
• Enumerated 181 million machines
• Formally verified Busy Beaver value