What happens at the Boundary of Computation?
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In this video, we look inside the bizarre busy beaver function.
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REFERENCE NOTES
As mentioned, [1] was the primary source. The first video and this one were originally inspired by reading [1], so many of the ideas can be traced to that point. Anyone still interested should continue their search there. Sources [3]-[4] were essential for understanding the exact nature of the busy beaver and champion machines. In fact, I should have also thanks Pascal Michel in the video. He has maintained a detailed recorded of which machines were champions at which points and I found myself referencing it regularly. Sources [5]-[7] plus some conversations with the author were essential for my understanding of the Collatz-like behavior.
REFERENCES
[1] S. Aaronson, "The Busy Beaver Frontier", https://www.scottaaronson.com/papers/...
[2] T. Radó. "On non-computable functions". Bell System Technical Journal. 41 (3): 877–884, 1962
[3] P. Michel. "The Busy Beaver competition: a historical survey". 2019. https://arxiv.org/abs/0906.3749
[4] P. Michel. "Problems in Number Theory from the Busy Beaver Competition". Logical Methods in Computer Science
Vol. 11(4:10), pp. 1–35, 2015
[5] S. Ligocki, "BB(6, 2) is greater than 10^10...^10", 2022, https://www.sligocki.com/2022/06/21/b...
[6] S. Ligocki, "Collatz-like behavior of Busy Beavers", 2021, https://www.sligocki.com/2021/07/17/b...
[7] S. Ligocki, "BBB Complex Collatz-like Behavior", 2021, https://www.sligocki.com/2022/02/22/c...
TIME CODES
0:00 Introduction
0:42 Reviewing the Basics
2:11 How does it grow faster than anything computable?
2:58 Using Collatz for Absurd Growth
3:59 Collatz in the 5-state machine
7:31 Exponential Collatz in the 6-state machine
9:05 The Busy Beavers answer famous open problems
11:23 The Busy Beavers are unknowable by any mathematical system
14:10 The Conjectures
14:26 Thank You's
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