Once upon a time, over 40 years ago, a horde of computer scientists descended on the West German city of Dortmund. They were competing to catch an elusive quarry — only four of its kind had ever been captured. Over 100 competitors dragged in the strangest creatures they could find, but they still fell short. The fifth busy beaver had escaped their clutches. Of course, that slippery beast and its relatives aren’t actually rodents. They’re simple-looking computer programs that take a surprisingly long time to run.
The search for these unusually active programs has connections to some of the most famous open questions in mathematics, and roots in an unsolvable problem as old as computer science itself. That’s precisely what makes the hunt so compelling. Three of the Dortmund participants summed up the prevailing attitude in a postmortem report: “Though we know we cannot win the war against the mathematical law, we would like to win a battle.”
The spiritual sequel to the Dortmund hunt began two years ago, when a graduate student named Tristan Stérin launched a website announcing the Busy Beaver Challenge. This time, the participants would cooperate, and everyone was welcome. Over time, the online community grew to include more than 20 contributors from around the world, most of them without traditional academic credentials. Today, the team declared victory. They’ve finally verified the true value of a number called BB(5), which quantifies just how busy that fifth beaver is. They obtained the result — 47,176,870 — using a piece of software called the Coq proof assistant which certifies that mathematical proofs are free of errors.
The Busy Beaver Challenge Unveils BB(5)
“The sociological and mathematical engineering that they’ve done to get this far is really impressive,” said Cristopher Moore, a computer scientist at the Santa Fe Institute. “I’m surprised how fast they did it,” said Damien Woods, a computer scientist at Maynooth University in Ireland and Stérin’s adviser. “That’s really like Usain Bolt territory.”
The search for the busy beaver is ultimately a trophy hunt. The specific value of BB(5) doesn’t have applications in other areas of computer science. But for busy beaver hunters, the hard-fought victory over mathematical impossibility is its own reward. It may be the last battle they’ll ever win.
The Halting Problem and Turing Machines
The programs that interest busy beaver hunters aren’t written in any ordinary programming language — they’re instructions for venerable (and theoretical) computers called Turing machines. The pioneering computer scientist Alan Turing conceived of these hypothetical devices in 1936 as a way to mathematically model the process of computation. Turing machines perform computations by reading and writing 0s and 1s on an infinite tape divided into square cells, using a “head” that operates on one cell at a time.
Image Description: An illustration showcasing a Turing machine in operation, reading and writing binary digits on an infinite tape.
Naively, one might think determining whether a Turing machine halts should be simple, but Turing proved this problem undecidable in general. Some Turing machines halt quickly, others fall into easily detectable infinite loops, but some evade this classification, cycling through complex behavior that makes it challenging to determine if they will ever halt.
The Historical Hunt for Busy Beavers
Beavers enter the story with the mathematician Tibor Radó, who became intrigued by the theory of computation late in life. Radó imagined sorting Turing machines into groups based on the number of rules they had and defined the “Busy Beaver game” in a 1962 paper. In each group, a Turing machine blessed with BB(n) steps before halting clinches the prize for being the busiest beaver.
Stérin’s approach modernized this hunt by leveraging collective effort and sophisticated software tools. His team’s verification of BB(5) at 47,176,870 steps involved eliminating redundancies, utilizing self-contained proofs for non-halting machines, and leveraging a collaborative online platform for visualization and communication.
The Busy Beaver Community and Tools
The community expanded their toolkit with Marxen's "closed tape language method," identifying non-halting Turing machines through a broad class of patterns. Contributors like Justin Blanchard optimized this method, making it run much faster and tackling a significant portion of difficult Turing machines termed "Skelet machines."
Image Description: A space-time diagram visualizing the behavior of a Turing machine during the Busy Beaver Challenge.
Maja Kądziołka used the Coq software to translate proofs into formally verified logic, providing the highest standard of rigor. This effort culminated in confirming that no five-rule Turing machine runs longer than Marxen and Buntrock’s busy beaver, solidifying BB(5) as 47,176,870.
Reflecting on the Journey and the Future
As the Busy Beaver Challenge team drafts a formal academic paper, they face a potential insurmountable barrier for BB(6) — a six-rule machine whose halting problem is analogous to the Collatz conjecture. This barrier could signify the end of solvable busy beaver challenges without significant mathematical breakthroughs.
Contributors like Stérin and mei are realizing new aspirations. Stérin aims to develop collaborative software tools, while mei is exploring a career as a train driver, demonstrating the diverse potential pathways for individuals involved in such niche research.
Conclusion and Takeaways for Startups
Remember these 3 key ideas for your startup:
Collaborative Efforts Yield High Rewards: The Busy Beaver Challenge succeeded through international collaboration and diverse contributions, proving that collective intelligence is a powerful tool in problem-solving. Emulate this in your startup to find innovative solutions.
Embrace Technological Tools: Utilizing tools like Coq for formal proofs or visual interfaces for complex data helped solidify results efficiently. Integrate advanced tools and methodologies in your workflows to optimize productivity and accuracy.
Flexible Contributions Can Lead to Breakthroughs: Contributors with non-traditional backgrounds significantly advanced the Busy Beaver Challenge. Encourage diverse perspectives in your team to drive breakthroughs and creative solutions.
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This summary encapsulates the extraordinary journey of the Busy Beaver Challenge, emphasizing the value of collaboration, technological innovation, and diverse contributions for SMEs and startups.
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