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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 blanktape, and BB($n,k$) denotes its $k$-symbol generalisation to $kgeq 2$. Thebusy beaver function $n mapsto text{BB}(n)$ is uncomputable and its valueshave been linked to hard open problems in mathematics and notions ofunprovability.
In this paper, we show that there are two explicit Turing machines, one with15 states and 2 symbols, the other with 5 states and 4 symbols, that halt ifand only if the following Collatz-related conjecture by Erdős [1979] doesnot hold does not hold: for all $n>8$ there is at least one digit 2 in the base3 representation of $2^n$. This result implies that knowing the values ofBB(15) or BB(5,4) is at least as hard as solving Erdős' conjecture andmakes, to date, BB(15) the smallest busy beaver value that is related to anatural open problem in mathematics. For comparison, Yedidia and Aaronson[2016] show that knowing BB(4,888) and BB(5,372) are as hard as solvingGoldbach's conjecture and the Riemann hypothesis, respectively (laterinformally improved to BB(27) and BB(744)). Finally, our result puts a finite,albeit large, bound on Erdős' conjecture, by making it equivalent to thefollowing finite statement: for all $8 < n leq min(text{BB}(15),text{BB}(5,4))$ there is at least one digit 2 in the base 3 representation of$2^n$.

The Busy Beaver PDF Free Download
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The Busy Beaver Pdf free. download full

Kids Learn Today the Alphabet & Two Words for Each Letter in this Busy Beavers Collection with a Fun & Catchy Melody. Babies & Toddlers will also Have Fun Si. Analogues of one of Rado’s busy beaver functions from classical computability,´ and show, in analogy with Rado’s results, that they grow faster than a wide class of´ inﬁnite-time computable functions. Chapter 4 is tasked with developing a system of ordinal notations via a natural approach involving inﬁnite-time computation, 1.

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From: Tristan Stérin [view email]
[v1]Mon, 26 Jul 2021 21:02:44 UTC (520 KB)
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