The Collatz Process Embeds a Base Conversion Algorithm

Stérin, Tristan; Woods, Damien

doi:10.1007/978-3-030-61739-4_9

Cited by 4 publications

(2 citation statements)

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“…Beyond making complex, albeit pretty, pictures there is deeper connection here: the small tile set of [4] can be shown to simulate the base 3, mul2, Finite State Transducer that we introduce in Section 3, Figure 2, and our small Turing machines use it to look for counterexamples to Erdős' conjecture. The tile set also simulates the inverse of the mul2 FST (which computes the operation x → x/2 in ternary) which was used to build a 3-state 4-symbol non-halting machine that runs the Collatz map on any ternary input [11] and finally, it also simulates the dual of that FST (which computes the operation x → 3x + 1 in binary) which can be used to simultaneously compute the Collatz map both in binary and ternary [16]. So these three closely-related FSTs are all encoded within that small tile set, that in turn encodes the three conjectures.…”

Section: Erdős' Conjecture and Its Relationship To The Collatz And We...mentioning

confidence: 99%

“…The construction begins with the mul2 finite state transducer (FST) in Figure 2. A similar FST, and its 'dual', can be used to compute iterations of the Collatz map [16] (Appendix B). The fact that there is an FST that multiplies by 2 in base 3 is not surprising, since there is one for any affine transformation in any natural-number base ≥ 1 [2].…”

Section: Five States Four Symbols Turing Machinementioning

confidence: 99%

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On the hardness of knowing busy beaver values BB(15) and BB(5,4)

Stérin¹,

Woods²

2021

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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, and BB(n, k) denotes its k-symbol generalisation to k ≥ 2. The busy beaver function n → BB(n) is uncomputable and its values have been linked to hard open problems in mathematics and notions of unprovability.In this paper, we show that there are two explicit Turing machines, one with 15 states and 2 symbols, the other with 5 states and 4 symbols, that halt if and only if the following Collatz-related conjecture by Erdős [7] does not hold: for all n > 8 there is at least one digit 2 in the base 3 representation of 2 n . This result implies that knowing the values of BB(15) or BB(5,4) is at least as hard as solving Erdős' conjecture and makes, to date, BB(15) the smallest busy beaver value that is related to a natural open problem in mathematics. For comparison, Yedidia and Aaronson [20] show that knowing BB(4,888) and BB(5,372) are as hard as solving Goldbach's conjecture and the Riemann hypothesis, respectively (later informally improved to BB(27) and BB( 744)). Finally, our result puts a finite, albeit large, bound on Erdős' conjecture, by making it equivalent to the following finite statement: for all 8 < n ≤ min(BB(15), BB(5, 4)) there is at least one digit 2 in the base 3 representation of 2 n .

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Section: Erdős' Conjecture and Its Relationship To The Collatz And We...mentioning

confidence: 99%

Section: Five States Four Symbols Turing Machinementioning

confidence: 99%