Hardness of busy beaver value BB(15) (2107.12475v2)

Abstract: The busy beaver value BB(n) is the maximum number of steps made by any n-state, 2-symbol deterministic halting Turing machine starting on blank tape. The busy beaver function $n \mapsto \text{BB}(n)$ is uncomputable and, from below, only 4 of its values, BB(1) ... BB(4), are known to date. This leads one to ask: from above, what is the smallest BB value that encodes a major mathematical challenge? Knowing BB(4,888) has been shown by Yedidia and Aaronson [28] to be at least as hard as solving Goldbach's conjecture, with a subsequent improvement, as yet unpublished, to BB(27) [4,1]. We prove that knowing BB(15) is at least as hard as solving the following Collatz-related conjecture by Erd\H{o}s, open since 1979 [9]: for all n > 8 there is at least one digit 2 in the base 3 representation of $2^n$. We do so by constructing an explicit 15-state, 2-symbol Turing machine that halts if and only if the conjecture is false. This 2-symbol Turing machine simulates a conceptually simpler 5-state, 4-symbol machine which we construct first. This makes, to date, BB(15) the smallest busy beaver value that is related to a natural open problem in mathematics, bringing to light one of the many challenges underlying the quest of knowing busy beaver values.

• The paper demonstrates that computing BB(15) is as challenging as resolving Erdős' conjecture on the digit '2' in the base-3 representation of powers of two.
• The authors construct a novel 15-state, 2-symbol Turing machine that simulates a 5-state, 4-symbol machine to perform complex digit-checking routines.
• The findings emphasize the inherent limits of computability and suggest potential applications in optimizing algorithms and automated program analysis.

On the Hardness of Computing Busy Beaver Values: A Case Study of $BB(15)$

This essay examines the implications of a paper that investigates the computational complexity associated with computing busy beaver values, specifically focusing on $BB(15)$. The busy beaver function, introduced by Tibor Radó in 1962, is a well-known function in theoretical computer science that epitomizes the challenges of non-computability. For an $n$-state Turing machine operating with two symbols, $BB(n)$ denotes the maximum number of steps such a machine can take before halting when started on an initially blank tape. Despite the uncomputability of the busy beaver function, its lower values up to $BB(4)$ have been determined. The paper scrutinized here advances our understanding by demonstrating that $BB(15)$ is at least as hard to compute as solving Erdős' conjecture regarding the presence of the digit '2' in the ternary representation of powers of two.

Overview of the Theoretical Context

The quest to determine busy beaver values connects deeply with prominent mathematical conjectures. For instance, $BB(4,888)$ has been linked with Goldbach's conjecture, and subsequently, a later 27-state construction was associated with the same conjecture. The paper contributes to this line of inquiry by associating $BB(15)$ with an Erdős conjecture—a simpler derivation from the Collatz conjecture pattern—stating that for all $n>8$, the number $2^n$ must contain the digit '2' in its base-3 representation.

Main Contributions and Technical Insights

The authors construct an explicit 15-state, 2-symbol Turing machine that simulates a 5-state, 4-symbol machine, both tailored to halt if only Erdős' conjecture is false. This simulation strategy exemplifies the reduction in alphabet size to save on state usage, a nuanced facet of Turing machine construction. The paper meticulously delineates how this machine effectively checks powers of two, transcribed in reverse base-3 on its tape, satisfying complex digit-checking routines for Erdős' conjecture.

Implications and Potential Impact

The implications of this work reside in both theoretical and practical domains of computation theory:

• Theoretical Implications: The association of $BB(15)$ with an unproven mathematical conjecture highlights not only the estimable growth rate of the busy beaver function but also raises questions about the limits of computability and foundational logic constructs like ZFC and Peano arithmetic. It suggests that resolving or disproving Erdős' conjecture would directly influence our understanding of such busy beaver values.
• Practical Implications: On a more practical note, gaining insights into the busy beaver problem aids in comprehending the confines of feasible computation, offering potential applications in optimizing algorithms for determining halting conditions and automated program analysis tools.
• Speculation for the Future: Further developments may involve automating methods to handle similar hard-to-decide mathematical challenges via Turing machine frameworks or leveraging existing Turing machine simulators to verify contested conjectures using similar constructs.

Conclusion

This inquiry into $BB(15)$ demonstrates how busy beaver functions constitute a fertile ground for exploring the frontiers of non-computability and mathematical problem-solving. The results illustrate that even seemingly straightforward numeric conjectures impose immense computational barriers when encoded within Turing machine frameworks, thereby enhancing our comprehension of algorithmic complexity and limits of computation. Future work may involve verifying existing results against new mathematical claims or improving machine efficiency to narrow the computational frontier even further.