1RB0RB 1LC1RE 1LF0LD 1RA1LD 1RC1RB ---1LC: Difference between revisions
(describe halting condition) |
(added figure, discussion of why this TM is probably not halting) |
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Q(a,b) := 0^inf 1^(2a+1) <D 0 1^b 0^inf | Q(a,b) := 0^inf 1^(2a+1) <D 0 1^b 0^inf | ||
</pre> | </pre> | ||
[[File:Points x 1 t1k distanceFromHaltingCondition.png|thumb|A log-scale plot of all points Q(x,1) in the first 1000 iterations of the map R. Blue dots are the x values, yellow dots are the difference between x and the nearest halting value beneath x, and green dots are the difference between the nearest halting value above x and x.]] | |||
The TM will halt after N iterations of the map Q if it ever takes the form <math display="inline">Q(2^{N+1} - 2, y)</math>. | The TM will halt after N iterations of the map Q if it ever takes the form <math display="inline">Q(2^{N+1} - 2, y)</math>. This can only occur via P(2a+1) -> Q(a+2,1) or Q(1,2b) -> Q(b+2,1). The figure plots x for all cases (x,1) in the first 1000 iterations of R with a logarithmic scale along the vertical axis. The distance from any halting condition 2^M - 2 is growing exponentially with time, so it is only ever getting harder for this TM to halt. | ||
On the trajectory starting from P(2), the Q(1,y) rules, at least through 10^5 steps, are never triggered. Thus, the implementation of the forward simulation can be made slightly faster (roughly 10% in practice) by, instead of nesting the execution of Q(x,y) within two if statements predicated on x=0 and x=1, only checking if x > 2, applying Q(x,y) if true, and throwing an exception if false. | On the trajectory starting from P(2), the Q(1,y) rules, at least through 10^5 steps, are never triggered. Thus, the implementation of the forward simulation can be made slightly faster (roughly 10% in practice) by, instead of nesting the execution of Q(x,y) within two if statements predicated on x=0 and x=1, only checking if x > 2, applying Q(x,y) if true, and throwing an exception if false. | ||
CoSearch: [https://cosearch.bbchallenge.org/contribution/tpxh8d8d https://cosearch.bbchallenge.org/contricution/tpxh8d8d] | CoSearch: [https://cosearch.bbchallenge.org/contribution/tpxh8d8d https://cosearch.bbchallenge.org/contricution/tpxh8d8d] | ||
Revision as of 23:50, 9 August 2025
1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC (bbch)
Potential BB(6) Cryptid found by @mxdys on 18 Aug 2024. Andrew Ducharme forward simulated the combined map (P(x), Q(x,y)) for 10^8 iterations. The TM did not yet halt. After 10^7 iterations, the TM reached a value (x',1) where x' ~ 10^604100, and after 10^8 iterations, it reached a value x ~ 10^(6.04305 x 10^6).
start: P(2) P(2a) -> P(3a+4) P(2a+1) -> Q(a+2,1) Q(2a+3,b) -> P(b+5a+6) Q(2a+2,b) -> Q(a,b+2a+5) Q(1,2b+1) -> P(3b+8) Q(1,2b) -> Q(b+2,1) Q(0,b) -> halt P(a) := 0^inf 1^a 011 <D 0^inf Q(a,b) := 0^inf 1^(2a+1) <D 0 1^b 0^inf
The TM will halt after N iterations of the map Q if it ever takes the form . This can only occur via P(2a+1) -> Q(a+2,1) or Q(1,2b) -> Q(b+2,1). The figure plots x for all cases (x,1) in the first 1000 iterations of R with a logarithmic scale along the vertical axis. The distance from any halting condition 2^M - 2 is growing exponentially with time, so it is only ever getting harder for this TM to halt.
On the trajectory starting from P(2), the Q(1,y) rules, at least through 10^5 steps, are never triggered. Thus, the implementation of the forward simulation can be made slightly faster (roughly 10% in practice) by, instead of nesting the execution of Q(x,y) within two if statements predicated on x=0 and x=1, only checking if x > 2, applying Q(x,y) if true, and throwing an exception if false.
CoSearch: https://cosearch.bbchallenge.org/contricution/tpxh8d8d