Bouncer: Difference between revisions
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'''Shift rules:''' | '''Shift rules:''' | ||
* <code> | * B0 A1: <code>B 10< →[2] B <01</code>, | ||
* <code> | * A0 B1: <code>A >10 →[2] A 01></code>. | ||
'''Bounce rule:''' | '''Bounce rule:''' | ||
* | * '''B'''<sub>n</sub> := A0 (B0 A1)<sup>2n+1</sup> B0 (A0 B1)<sup>2n+2</sup>: <code>A 00(10<sup>2n</sup>)>00 →[8n+8] A (10<sup>2(n+1)</sup>)></code> | ||
In particular, | In particular, we have | ||
'''B'''<sub>n</sub>: <code>A 0<sup>∞</sup>(10<sup>2n</sup>)>0<sup>∞</sup> →[8n+8] 0<sup>∞</sup>(10<sup>2(n+1)</sup>)>0<sup>∞</sup></code>, | |||
which by induction proves that the transcript of <code>1RB0LB_1LA0RA</code> on the all-zeros tape is ('''B'''<sub>n</sub>)<sup>n≥0</sup>. This shows that <code>1RB0LB_1LA0RA</code> is a bouncer. Moreover, we have shown that there are 8''n'' + 8 steps between the ''n''<sup>th</sup> and the (''n''+1)<sup>st</sup> bounce. | |||
== See also == | == See also == | ||
Revision as of 14:23, 13 November 2024
1RB0LB_1LA0RA (bbch) with 2 states, the smallest number of states for which a bouncer can appear.A bouncer is a Turing machine whose tape head, roughly speaking, alternates back and forth between the two edges of the tape in a linear fashion, growing the tape along one or both edges with each iteration. A bouncer is a possible classification of non-halting Turing machines.
Example
1RB0LB_1LA0RA (bbch) is an example of a bouncer, and its spacetime diagram is shown in the picture on the right.
Analysis
Shift rules:
- B0 A1:
B 10< →[2] B <01, - A0 B1:
A >10 →[2] A 01>.
Bounce rule:
- Bn := A0 (B0 A1)2n+1 B0 (A0 B1)2n+2:
A 00(102n)>00 →[8n+8] A (102(n+1))>
In particular, we have
Bn: A 0∞(102n)>0∞ →[8n+8] 0∞(102(n+1))>0∞,
which by induction proves that the transcript of 1RB0LB_1LA0RA on the all-zeros tape is (Bn)n≥0. This shows that 1RB0LB_1LA0RA is a bouncer. Moreover, we have shown that there are 8n + 8 steps between the nth and the (n+1)st bounce.
See also
- Section 7 of bbchallenge's deciders write-up.