turing_machine/bb6_heuristic_2193_notes.txt at master · jhuang97/turing_machine · GitHub

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notes on 1RB1LE_1LB1LC_1RD0LE_---0RB_1RF1LA_0RA0RD

Brick's heuristic: 0.2193 which is kind of low
but mxdys thinks this looks like a cryptid

more readable rules:
1^2 < -> < 1^2
> 1 -> 1 >

0 < 1^2 -> 1 0 >
0 < 0 1^2 -> 1^2 0 >
0 1 < -> < 0 1

L < 1^2 -> L 1 0 >
L 1 < 1 -> L 1^2 0 >

1^2 > 0 -> < 0 1^3
1^2 > R -> < 0 1^3 R

L 1 0 > 0 -> L 1^4 >
L 1^2 0 > 0 -> L 1^2 0 1^2 >

0 1 > 0 -> 1^3 >
1^2 0 > 0 -> < 0 1^4

Let "a ... b [m]> c ... d" = 0^inf 1^a 0 ... 0 1^b 0 1^m > 0 1^c 0 ... 0 1^d 0^inf
Let "a ... b [m]< c ... d" = 0^inf 1^a 0 ... 0 1^b 0 1^m < 0 1^c 0 ... 0 1^d 0^inf
Let L and R continue to denote the 0^inf at the ends.

Rules in this notation:
1. ... a [2k+4]> b ... -> ... a+1 [2k]> b+3 ... (a,b >= 0)
proto-2. L 2 [2k+3]> b ... -> L 2 [2k+1]> b+6 ... (b >= 0)
3. ... a [1]> c ... -> ... [a+c+3]> ... (a,c >= 0)
4. ... a 2k+6 [0]> c ... -> ... a+1 [2k+2]> c+4 ... (a,c >= 0)
5. L 2k+5 [0]> c ... --> L 2 [2k+1]> c+4 (c >= 0)
6. ... a [2]> c ... -> ... a+2 [c+1]> ...
7. L 2 [0]> c ... -> L 2 [c+2]> ...
8. L 4 [0]> c ... -> L [c+8]> ...
9. L 2k+6 [0]> c ... -> L 1 [2k+2]> c+4 ...
10. L 2k_0+3 {2k_i+1} [2k_n+3]> c ... -> L 2 [2k_0+1]> {2k_i+1} 2k_n+1 c+3 ...
  where braces {} contain 0 or more odd numbers 2k_i+1
11. L [2k+5]> c ... -> L 2 [2k+1]> c+3 ...
12. L 2k_0+4 {2k_i+1} [2k_n+3]> c ... -> L 1 [2k_0+2]> {2k_i+1} 2k_n+1 c+3 ...

rule 1 ->
1a. ... a [4k]> b ... -> ... a+k [0]> b+3k ...
1b. ... a [4k+2]> b ... -> ... a+k+2 [b+3k+1]> ...
proto-rule 2 ->
2. L 2 [2k+3]> b ... -> L [b+6k+11]> ...

(  ... [a+2]> R -> ... [a]< 3 R (a >= 0)  )

Proofs or sketches:

Rule 1. ... a [2k+4]> b ... -> ... a+1 [2k]> b+3 ... (a,b >= 0)
cases for small m:
1^a 0 1^4 > 0 1^b
1^a 0 1^2 < 0 1^b+3
1^a 0 < 1^2 0 1^b+3
1^a+1 0 > 0 1^b+3

it doesn't work for odd exponent:
1^a 0 1^5 > 0 1^b
1^a 0 1^3 < 0 1^b+3
1^a 0 1 < 1^2 0 1^b+3
1^a < 0 1^3 0 1^b+3

Rule 2. L 2 [2k+3]> b ... -> L 2 [2k+1]> b+6 ...
L 1^2 0 1^2k+3 > 0 b
L 1^2 0 1^2k+1 < 0 b+3
L 1^2 < 0 1^2k+1 0 b+3
L < 1^2 0 1^2k+1 0 b+3
L 1 0 > 0 1^2k+1 0 b+3
L 1^4 > 1^2k+1 0 b+3
L 1^2k+5 > 0 b+3
L 1^2k+3 < 0 b+6
L 1 < 1^2k+2 0 b+6
L 1^2 0 > 1^2k+1 0 b+6
L 1^2 0 1^2k+1 > 0 b+6

Rule 3. ... a [1]> c ... -> ... [a+c+3]> ...
... 1^a 0 1 > 0 1^c ...
... 1^a+3 > 1^c ...
... 1^a+c+3 > ...

Rule 4. ... a 2k+6 [0]> c ... -> ... a+1 [2k+2]> c+4 ...
... 1^a 0 1^2k+6 0 > 0 1^c ...
... 1^a 0 1^2k+4 < 0 1^c+4 ...
... 1^a 0 < 1^2k+4 0 1^c+4 ...
... 1^a+1 0 > 1^2k+2 0 1^c+4 ...
... 1^a+1 0 1^2k+2 > 0 1^c+4 ...

Rule 5. L 2k+5 [0]> c ... --> L 2 [2k+1]> c+4
L 1^2k+5 0 > 0 1^c ...
L 1^2k+3 < 0 1^c+4 ...
L 1 < 1^2k+2 0 1^c+4 ...
L 1^2 0 > 1^2k+1 0 1^c+4 ...
L 1^2 0 1^2k+1 > 0 1^c+4 ...

Rule 6. ... a [2]> c ... -> ... a+2 [c+1]> ...
... 1^a 0 1^2 > 0 1^c ...
... 1^a 0 < 0 1^c+3 ...
... 1^a+2 0 > 1^c+1 ...
... 1^a+2 0 1^c+1 > ...

Rule 7. L 2 [0]> c ... -> L 2 [c+2]> ...
L 1^2 0 > 0 1^c ...
L 1^2 0 1^2 > 1^c ...
L 1^2 0 1^c+2 > ...

Rule 8. L 4 [0]> c ... -> L [c+8]> ...
L 1^4 0 > 0 1^c ...
L 1^2 < 0 1^c+4 ...
L < 1^2 0 1^c+4 ...
L 1 0 > 0 1^c+4 ...
L 1^4 > 1^c+4 ...
L 1^c+8 > ...

Rule 9. L 2k+6 [0]> c ... -> L 1 [2k+2]> c+4 ...
L 1^2k+6 0 > 0 1^c
L 1^2k+4 < 0 1^c+4
L < 1^2k+4 0 1^c+4
L 1 0 > 1^2k+2 0 1^c+4
L 1 0 1^2k+2 > 0 1^c+4

Rule 10. L 2k_0+3 {2k_i+1} [2k_n+3]> c ... -> L 2 [2k_0+1]> {2k_i+1} 2k_n+1 c+3 ...
L 1^2k_0+3 {0 1^2k_i+1} 0 1^2k_n+3 > 0 1^c
L 1^2k_0+3 {0 1^2k_i+1} 0 1^2k_n+1 < 0 1^c+3
L 1^2k_0+3 {0 1^2k_i+1} < 0 1^2k_n+1 0 1^c+3
L 1^2k_0+3 < {0 1^2k_i+1} 0 1^2k_n+1 0 1^c+3
L 1 < 1^2k_0+2 {0 1^2k_i+1} 0 1^2k_n+1 0 1^c+3
L 1^2 0 > 1^2k_0+1 {0 1^2k_i+1} 0 1^2k_n+1 0 1^c+3
L 1^2 0 1^2k_0+1 > {0 1^2k_i+1} 0 1^2k_n+1 0 1^c+3

Rule 11. L [2k+5]> c ... -> L 2 [2k+1]> c+3 ...
L 1^2k+5 > 0 1^c
L 1^2k+3 < 0 1^c+3
L 1 < 1^2k+2 0 1^c+3
L 1^2 0 > 1^2k+1 0 1^c+3
L 1^2 0 1^2k+1 > 0 1^c+3

Rule 12. L 2k_0+4 {2k_i+1} [2k_n+3]> c ... -> L 1 [2k_0+2]> {2k_i+1} 2k_n+1 c+3 ...
L 1^2k_0+4 {0 1^2k_i+1} 0 1^2k_n+3 > 0 1^c
L 1^2k_0+4 {0 1^2k_i+1} 0 1^2k_n+1 < 0 1^c+3
L 1^2k_0+4 {0 1^2k_i+1} < 0 1^2k_n+1 0 1^c+3
L 1^2k_0+4 < {0 1^2k_i+1} 0 1^2k_n+1 0 1^c+3
L < 1^2k_0+4 {0 1^2k_i+1} 0 1^2k_n+1 0 1^c+3
L 1 0 > 1^2k_0+2 {0 1^2k_i+1} 0 1^2k_n+1 0 1^c+3
L 1 0 1^2k_0+2 > {0 1^2k_i+1} 0 1^2k_n+1 0 1^c+3

Try follow forward simulation:
[8]>
2 [0]> 6
2 [8]>
4 [0]> 6
[14]>
3 [2]> 9
5 [10]>
7 [2]> 6
9 [7]>
2 [7]> 5 3
[28]> 3
7 [0]> 24
2 [3]> 28
[39]>
2 [35] 3
[110]>
29 [82]>
51 [61]>
2 [49]> 59 3
[208]> 3
52 [0]> 159
1 [48]> 163
13 [0]> 199
2 [9]> 203
[232]>
58 [0]> 174
1 [54]> 178
16 [218]>
72 [163]>
1 [70]> 161 3
20 [213]> 3
1 [18]> 211 6
7 [224]> 6
63 [0]> 174
2 [59]> 178
[357]>
2 [353]> 3
[1064]>
266 [0]> 798
1 [262]> 802
68 [998]>
319 [748]>
506 [0]> 561
1 [502]> 565
128 [941]>
1 [126]> 939 3
34 [1033]> 3
1 [32]> 1031 6
9 [0]> 1055 6
2 [5]> 1059 6
[1076]> 6
269 [0]> 813
2 [265]> 817
[1614]>
405 [1210]>
709 [907]>
2 [707]> 905 3
[3028]> 3
757 [0]> 2274
2 [753]> 2278
[4539]>