Collatz Tag System Emulator

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twinb7
Posts: 190
Joined: February 11th, 2014, 8:08 pm
Location: Ames, Iowa

Collatz Tag System Emulator

Post by twinb7 » January 23rd, 2017, 10:23 pm

This rule emulates a 2-tag system, which in turn emulates the math function 'If the number is even, halve it - otherwise multiply by 3 and add 1'. The Collatz conjecture states that if you apply this function repeatedly to a number, you will eventually reach one.

An equivalent formulation is of a tag system with the transition rules a -> bc, b -> a, c -> aaa.
In other words, begin with a list of 'a's and repeat the following:

If the very first letter is an 'a', append 'bc' to the back. If it is a 'b', append 'a' to the back. If it is a 'c', append 'aaa' to the back. Then cross out the first two letters and repeat. This emulates the Collatz function and repeats it over and over again for a unary number of a's to start with. There's more on it here:https://en.wikipedia.org/wiki/Tag_syste ... _sequences

This rule emulates exactly that. State 1 is 'a', 2 is 'b', and 3 is 'c'. To start a string going, make a pattern of this format:

Code: Select all

x = 20, y = 1, rule = CollatzEmulator
I19A!
Here's the rule table. It's been a long time since I made it so I can't remember exactly how it works.

Code: Select all

@RULE CollatzEmulator
@TABLE
n_states:31
neighborhood:vonNeumann
symmetries:none

#0 none
#1 A
#2 B
#3 C
#4 left eater 0 (eat and sense)
#5 left eater 1.A (eat)
#6 left eater 1.B
#7 left eater 1.C
#8 left eater 2 (resting)
#9 left eater ready (to be duplicated)
#10 left eater emitting A production rule 0 (‘B’)
#11 left eater emitting A production rule 1 (‘C’)
#12 left eater emitting B production rule 0 (‘A’)
#13 left eater emitting C production rule 0 (‘A’)
#14 left eater emitting C production rule 1 (‘A’)
#15 left eater emitting C production rule 2 (‘A’)
#16 ready left moving through A
#17 ready left moving through B
#18 ready left moving through C
#19 ready right moving through A
#20 ready right moving through B
#21 ready right moving through C
#22 A moving through A
#23 B moving through A
#24 C moving through A
#25 A moving through B
#26 B moving through B
#27 C moving through B
#28 A moving through C
#29 B moving through C
#30 C moving through C

var a = {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30}
var b = {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30}
var c = {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30}
var d = {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30}
var e = {16,17,18}
var f = {19,20,21}
var g = {1,2,3}
var h = {4,5,6,7}

#Carrying A
var i = {12,13,14,15,22,25,28}
#Carrying B
var j = {10,23,26,29}
#Carrying C
var k = {11,24,27,30}

#Things that are, at base, A
var l = {1,22,23,24}
#Things that are, at base, B
var m = {2,25,26,27}
#Things that are, at base, C
var n = {3,28,29,30}

var o = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30}
var p = {22,23,24}
var q = {25,26,27}
var r = {28,29,30}

#Carrying A and can create
var s = {22,25,28}
#Carrying B and can create
var t = {23,26,29}
#Carrying C and can create
var u = {24,27,30}

#Info being produced at end, and creating ready left
0,a,b,c,s,1
0,a,b,c,t,2
0,a,b,c,u,3
p,a,0,b,g,16
q,a,0,b,g,17
r,a,0,b,g,18

#Ready moving left and right, duplicating strand

1,a,e,b,c,16
2,a,e,b,c,17
3,a,e,b,c,18
2,a,e,b,c,9
16,a,b,c,d,1
17,a,b,c,d,2
18,a,b,c,d,3
8,a,e,c,d,9
0,9,a,b,c,4
1,a,b,c,9,19
2,a,b,c,9,20
3,a,b,c,9,21
19,a,b,c,d,1
20,a,b,c,d,2
21,a,b,c,d,3
0,19,a,b,c,1
0,20,a,b,c,2
0,21,a,b,c,3
9,a,b,c,d,8
1,a,b,c,f,19
2,a,b,c,f,20
3,a,b,c,f,21

#Cells moving left, and eater responding appropriately
0,a,g,b,c,g
g,a,b,c,0,0
4,a,1,b,c,5
4,a,2,b,c,6
4,a,3,b,c,7
g,a,b,c,h,0
5,a,g,b,c,10
6,a,g,b,c,12
7,a,g,b,c,13

#Eater transitions
10,a,o,c,d,11
11,a,o,c,d,8
12,a,o,c,d,8
13,a,o,c,d,14
14,a,o,c,d,15
15,a,o,c,d,8

#Info transitions through tags
l,a,b,c,i,22
l,a,b,c,j,23
l,a,b,c,k,24
m,a,b,c,i,25
m,a,b,c,j,26
m,a,b,c,k,27
n,a,b,c,i,28
n,a,b,c,j,29
n,a,b,c,k,30
l,a,b,c,d,1
m,a,b,c,d,2
n,a,b,c,d,3



@COLORS
0 0 0 0
1 255 0 0
2 255 128 0
3 255 255 0
4 255 255 255
5 128 0 255
6 0 128 255
7 128 128 255
8 128 128 128
9 128 255 128
16 0 128 128
17 128 0 128
18 128 128 0
19 0 255 255
20 255 0 255
21 255 255 0
22 191 0 64
23 128 0 128
24 64 0 191
25 191 96 64
26 128 64 128
27 64 32 191
28 191 191 64
29 128 128 128
30 64 64 191
tl;dr if you prove all patterns in this CA are bounded then the Collatz Conjecture is true.

EDIT: Wrong rule was added

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BlinkerSpawn
Posts: 1992
Joined: November 8th, 2014, 8:48 pm
Location: Getting a snacker from R-Bee's

Re: Collatz Tag System Emulator

Post by BlinkerSpawn » January 24th, 2017, 12:06 am

Here's a 1DCA I created that (I think) does a pretty good job of visualizing the Collatz tag system in operation:

Code: Select all

@RULE Collatz
0: vacuum
1: a
2: b
3: c
4: a carrying a
5: b carrying a
6: c carrying a
7: a carrying b
8: b carrying b
9: c carrying b
10: a carrying c
11: b carrying c
12: c carrying c
13: a carrying return
14: b carrying return
15: c carrying return
16: crossout
17: append a 1
18: append a 2
19: append bc
20: new signal
@TABLE
n_states:21
neighborhood:oneDimensional
symmetries:none
var as = {4,5,6}
var bs = {7,8,9}
var cs = {10,11,12}
var rs = {13,14,15}
var at = {4,7,10,13}
var bt = {5,8,11,14}
var ct = {6,9,12,15}
var t = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}
var t0 = {0,t}
#a
at,t0,t,1
at,t,t0,1
1,as,t0,4
1,bs,t0,7
1,cs,t0,10
1,t,rs,13
#b
bt,t0,t,2
bt,t,t0,2
2,as,t0,5
2,bs,t0,8
2,cs,t0,11
2,t,rs,14
#c
ct,t0,t,3
ct,t,t0,3
3,as,t0,6
3,bs,t0,9
3,cs,t0,12
3,t,rs,15
#append
0,as,0,19
0,bs,0,13
0,cs,0,17
0,17,0,18
0,18,0,13
0,19,0,15
17,t,0,1
18,t,0,1
19,t,0,2
#crossout
t,0,rs,16
16,0,t,0
t0,16,t,20
20,0,t,0
1,20,t,4
2,20,t,8
3,20,t,12
@COLORS
0   32  32  32
1  192   0   0
2    0 192   0
3    0   0 192
4  255   0   0
5   64 192   0
6   64   0 192
7  192  64   0
8    0 255   0
9    0  64 192
10 192   0  64
11   0 192  64
12   0   0 255
13  96   0   0
14   0  96   0
15   0   0  96
16   0   0   0
17   0 128 128
18 128   0 128
19 128 128   0
20 255 255 255
@ICONS
XPM
/* width height num_colors chars_per_pixel */
"7 140 5 1"
/* colors */
". c #000000"
"A c #FF0000"
"B c #00FF00"
"C c #0000FF"
"N c #FFFFFF"
/* A tag */
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
/* B tag */
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
/* C tag */
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
/* A tag  carrying A */
"AA..AAA"
"AA.A.AA"
"AA.AA.A"
"AA.AAA."
"AA.AA.A"
"AA.A.AA"
"AA..AAA"
/* B tag  carrying A */
"BB..BBB"
"BB.A.BB"
"BB.AA.B"
"BB.AAA."
"BB.AA.B"
"BB.A.BB"
"BB..BBB"
/* C tag  carrying A */
"CC..CCC"
"CC.A.CC"
"CC.AA.C"
"CC.AAA."
"CC.AA.C"
"CC.A.CC"
"CC..CCC"
/* A tag  carrying B */
"AA..AAA"
"AA.B.AA"
"AA.BB.A"
"AA.BBB."
"AA.BB.A"
"AA.B.AA"
"AA..AAA"
/* B tag  carrying B */
"BB..BBB"
"BB.B.BB"
"BB.BB.B"
"BB.BBB."
"BB.BB.B"
"BB.B.BB"
"BB..BBB"
/* C tag  carrying B */
"CC..CCC"
"CC.B.CC"
"CC.BB.C"
"CC.BBB."
"CC.BB.C"
"CC.B.CC"
"CC..CCC"
/* A tag  carrying C */
"AA..AAA"
"AA.C.AA"
"AA.CC.A"
"AA.CCC."
"AA.CC.A"
"AA.C.AA"
"AA..AAA"
/* B tag  carrying C */
"BB..BBB"
"BB.C.BB"
"BB.CC.B"
"BB.CCC."
"BB.CC.B"
"BB.C.BB"
"BB..BBB"
/* C tag  carrying C */
"CC..CCC"
"CC.C.CC"
"CC.CC.C"
"CC.CCC."
"CC.CC.C"
"CC.C.CC"
"CC..CCC"
/* A tag  carrying return */
"AAA..AA"
"AA...AA"
"A....AA"
".....AA"
"A....AA"
"AA...AA"
"AAA..AA"
/* B tag  carrying return */
"BBB..BB"
"BB...BB"
"B....BB"
".....BB"
"B....BB"
"BB...BB"
"BBB..BB"
/* C tag  carrying return */
"CCC..CC"
"CC...CC"
"C....CC"
".....CC"
"C....CC"
"CC...CC"
"CCC..CC"
/* crossout */
"AA...AA"
"AAA.AAA"
".AAAAA."
"..AAA.."
".AAAAA."
"AAA.AAA"
"AA...AA"
/* append a 1 */
"..AAA.."
"..AAA.."
"..AAA.."
"..AAA.."
"..AAA.."
"..AAA.."
"..AAA.."
/* append a 2 */
"AA...AA"
"AA...AA"
"AA...AA"
"AA...AA"
"AA...AA"
"AA...AA"
"AA...AA"
/* append bc */
"......."
"......."
"..BBC.."
"..BCC.."
"..CCB.."
"......."
"......."
/* begin signal */
"N..N..N"
".N.N.N."
"......."
"NN...NN"
"......."
".N.N.N."
"N..N..N"
Enable Cell Icons to really see the inner workings.
Start with a state-20 cell and some number of state-1 cells to the right of it.

Code: Select all

x = 19, y = 1, rule = Collatz
T18A!
Whenever the pattern reduces back to a state-20 cell and a string of state-1 cells, that's the next term in that Collatz sequence.
When the population hits 1 then the sequence has reached 1. (no 1-2-4-1-2-4-... loops)
Last edited by BlinkerSpawn on January 24th, 2017, 12:12 am, edited 2 times in total.
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twinb7
Posts: 190
Joined: February 11th, 2014, 8:08 pm
Location: Ames, Iowa

Re: Collatz Tag System Emulator

Post by twinb7 » January 24th, 2017, 12:10 am

Make sure you're using the right rule - when I first posted it, I accidentally posted the incorrect rule table. Yours is really cool too, but the reason I made mine was to make it visual once completed. It looks like it works in basically the same way, which is pretty cool.

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BlinkerSpawn
Posts: 1992
Joined: November 8th, 2014, 8:48 pm
Location: Getting a snacker from R-Bee's

Re: Collatz Tag System Emulator

Post by BlinkerSpawn » January 24th, 2017, 12:11 am

twinb7 wrote:Make sure you're using the right rule - when I first posted it, I accidentally posted the incorrect rule table.
Ok, grabbed the correct one.
That actually looks pretty cool.
EDIT: I notice my rule has about ten fewer states than yours.
I have an idea for emulating a Collatz tag system in possibly only ten states by using a 2xn work area, which I will attempt to make tomorrow perhaps.
EDIT 2: Don't wait that long.

Code: Select all

@RULE CollatzTape
0: vacuum
1: a
2: b
3: c
4: a head
5: b head
6: c head
7: blank head
8: append 1
9: append 2
@TABLE
n_states:10
neighborhood:Moore
symmetries:none
var t1 = {1,2,3}
var t2 = t1
var t3 = t2
var t = {0,1,2,3}
#a head
4,t1,t2,0,0,0,0,0,t,0
4,t1,0,0,0,0,0,0,t2,0
0,t1,t,0,0,0,0,4,t3,4
0,1,t1,0,0,0,0,7,0,4
#b head
5,t1,t2,0,0,0,0,0,t,0
5,t1,0,0,0,0,0,0,t2,0
0,t1,t,0,0,0,0,5,t3,5
0,2,t1,0,0,0,0,7,0,5
#c head
6,t1,t2,0,0,0,0,0,t,0
6,t1,0,0,0,0,0,0,t2,0
0,t1,t,0,0,0,0,6,t3,6
0,3,t1,0,0,0,0,7,0,6
#append bc
0,0,0,0,0,0,4,t1,0,2
0,0,0,0,0,0,0,4,t1,9
0,0,0,0,0,0,9,2,0,3
0,0,0,0,0,0,0,9,2,7
#append a
0,0,0,0,0,0,5,t1,0,1
0,0,0,0,0,0,0,5,t1,7
#append aaa
0,0,0,0,0,0,6,t1,0,1
0,0,0,0,0,0,0,6,t1,8
0,0,0,0,0,0,8,1,0,1
0,0,0,0,0,0,0,8,1,9
0,0,0,0,0,0,9,1,0,1
0,0,0,0,0,0,0,9,1,7
#append
8,t1,0,0,0,0,0,0,t2,0
9,t1,0,0,0,0,0,0,t2,0
#blank head
7,t1,t2,0,0,0,0,0,t,0
7,t1,t,0,0,0,0,0,t2,0
0,t1,t2,7,0,0,0,0,t,7
t1,0,0,t2,0,7,0,0,0,7
7,0,0,t1,0,0,0,0,0,0
7,0,t1,0,0,0,0,0,0,0
t1,0,0,t2,0,0,0,7,0,0
0,t1,t2,0,0,0,0,0,7,7


@COLORS
0   32  32  32
1  192   0   0
2    0 192   0
3    0   0 192
4  255   0   0
5    0 255   0
6    0   0 255
7  128 128 128
8   96 192   0
9    0  96 192
@ICONS
XPM
/* width height num_colors chars_per_pixel */
"7 63 5 1"
/* colors */
". c #000000"
"A c #FF0000"
"B c #00FF00"
"C c #0000FF"
"N c #FFFFFF"
/* A tag */
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
/* B tag */
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
/* C tag */
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
/* A head */
"...A..."
"..AAA.."
".AAAAA."
"AAAAAAA"
".AAAAA."
"..AAA.."
"...A..."
/* B head */
"...B..."
"..BBB.."
".BBBBB."
"BBBBBBB"
".BBBBB."
"..BBB.."
"...B..."
/* C head */
"...C..."
"..CCC.."
".CCCCC."
"CCCCCCC"
".CCCCC."
"..CCC.."
"...C..."
/* blank head */
"...N..."
"..NNN.."
".NNNNN."
"NNNNNNN"
".NNNNN."
"..NNN.."
"...N..."
/* append 1 */
"..AAA.."
"..AAA.."
"..AAA.."
"..AAA.."
"..AAA.."
"..AAA.."
"..AAA.."
/* append 2 */
"AA...AA"
"AA...AA"
"AA...AA"
"AA...AA"
"AA...AA"
"AA...AA"
"AA...AA"

Code: Select all

x = 13, y = 2, rule = CollatzTape
13A$D!
If every ON cell is red then the current term is one less than the population.
EDIT 3: More complex dynamics, but only nine states and much faster computation:

Code: Select all

@RULE CollatzFuse
0: vacuum
1: a
2: b
3: c
4: a head
5: b head
6: c head
7: append 1
8: append 2
@TABLE
n_states:9
neighborhood:Moore
symmetries:none
var t1 = {1,2,3}
var t2 = t1
var t3 = t2
var t = {0,1,2,3}
var h1 = {4,5,6,7,8}
var h2 = h1
var h = {0,h1}
#pulse
h1,t1,t2,h,0,0,0,0,t,0
h1,0,0,t1,h,0,0,0,0,0
h1,t1,t,0,0,0,0,h,t3,0
h1,1,t1,0,0,0,0,0,h2,4
h1,2,t1,0,0,0,0,0,h2,5
h1,3,t1,0,0,0,0,0,h2,6
t1,0,0,t2,h,h1,0,0,0,h1
0,t1,t,h,0,0,0,h1,t3,h1
0,t1,t2,h,0,0,0,0,h1,h1
#append bc
0,0,0,0,0,0,4,t1,0,2
0,0,0,0,0,0,0,4,t1,8
0,0,0,0,0,0,8,2,0,3
#append a
0,0,0,0,0,0,5,t1,0,1
#append aaa
0,0,0,0,0,0,6,t1,0,1
0,0,0,0,0,0,0,6,t1,7
0,0,0,0,0,0,7,1,0,1
0,0,0,0,0,0,0,7,1,8
0,0,0,0,0,0,8,1,0,1
#append
7,t1,0,0,0,0,0,0,t2,0
8,t1,0,0,0,0,0,0,t2,0
#end
h1,t1,0,0,0,0,0,0,h2,0
@COLORS
0   32  32  32
1  192   0   0
2    0 192   0
3    0   0 192
4  255   0   0
5    0 255   0
6    0   0 255
7   96 192   0
8    0  96 192
@ICONS
XPM
/* width height num_colors chars_per_pixel */
"7 56 4 1"
/* colors */
". c #000000"
"A c #FF0000"
"B c #00FF00"
"C c #0000FF"
/* A tag */
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
"AAAAAAA"
/* B tag */
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
"BBBBBBB"
/* C tag */
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
"CCCCCCC"
/* A head */
"...A..."
"..AAA.."
".AAAAA."
"AAAAAAA"
".AAAAA."
"..AAA.."
"...A..."
/* B head */
"...B..."
"..BBB.."
".BBBBB."
"BBBBBBB"
".BBBBB."
"..BBB.."
"...B..."
/* C head */
"...C..."
"..CCC.."
".CCCCC."
"CCCCCCC"
".CCCCC."
"..CCC.."
"...C..."
/* append 1 */
"..AAA.."
"..AAA.."
"..AAA.."
"..AAA.."
"..AAA.."
"..AAA.."
"..AAA.."
/* append 2 */
"AA...AA"
"AA...AA"
"AA...AA"
"AA...AA"
"AA...AA"
"AA...AA"
"AA...AA"
A starting pattern looks like this, where the total population is the first term:

Code: Select all

x = 7, y = 1, rule = CollatzFuse
D6A!
The rear "head" will continue to be state-4 and then be either state-5 or state-6 for a period of time. The first time the tail is state-4 and in the upper row, that term is the total population plus two for each state-6 cell and one for each state-7 cell.
Example of computational speed-up:
CollatzTape takes 150,255,824 generations to generate the Collatz sequence for n = 41 before terminating.
CollatzFuse takes 162,503 generations to generate the same sequence.
Last edited by BlinkerSpawn on January 24th, 2017, 9:35 pm, edited 1 time in total.
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Rhombic
Posts: 1072
Joined: June 1st, 2013, 5:41 pm

Re: Collatz Tag System Emulator

Post by Rhombic » January 24th, 2017, 5:48 pm

Code: Select all

x = 14, y = 4, rule = CollatzFuse
.D6A3$3ABG9A!
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BlinkerSpawn
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Re: Collatz Tag System Emulator

Post by BlinkerSpawn » January 24th, 2017, 6:54 pm

Rhombic wrote:

Code: Select all

x = 14, y = 4, rule = CollatzFuse
.D6A3$3ABG9A!
I am aware that the minimal transitions means that deliberately false patterns like the bottom one result in erroneous performance.
Was there some other thing you intended to point out?
(Yes, I'm aware of CollatzFuse's particular method of terminating, I didn't feel like trying to fix it.)
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Re: Collatz Tag System Emulator

Post by fluffykitty » January 24th, 2017, 8:03 pm

I think 1000505000 and 501000000 would solve termination.

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Re: Collatz Tag System Emulator

Post by BlinkerSpawn » January 24th, 2017, 9:37 pm

fluffykitty wrote:I think 1000505000 and 501000000 would solve termination.
I added "h1,t1,0,0,0,0,0,0,h2,0" and now the sequence terminates at either 2 or 3.
You shouldn't encounter the second case unless you start with n = 3 * 2^k, however.
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Re: Collatz Tag System Emulator

Post by BlinkerSpawn » January 27th, 2017, 11:04 pm

Collatz fuses for n = 5 to 1000; run with step size ~2^8 for a nice race:

Code: Select all

x = 1000, y = 2986, rule = CollatzFuse
D4A3$D5A3$D6A3$D7A3$D8A3$D9A3$D10A3$D11A3$D12A3$D13A3$D14A3$D15A3$D
16A3$D17A3$D18A3$D19A3$D20A3$D21A3$D22A3$D23A3$D24A3$D25A3$D26A3$D27A
3$D28A3$D29A3$D30A3$D31A3$D32A3$D33A3$D34A3$D35A3$D36A3$D37A3$D38A3$D
39A3$D40A3$D41A3$D42A3$D43A3$D44A3$D45A3$D46A3$D47A3$D48A3$D49A3$D50A
3$D51A3$D52A3$D53A3$D54A3$D55A3$D56A3$D57A3$D58A3$D59A3$D60A3$D61A3$D
62A3$D63A3$D64A3$D65A3$D66A3$D67A3$D68A3$D69A3$D70A3$D71A3$D72A3$D73A
3$D74A3$D75A3$D76A3$D77A3$D78A3$D79A3$D80A3$D81A3$D82A3$D83A3$D84A3$D
85A3$D86A3$D87A3$D88A3$D89A3$D90A3$D91A3$D92A3$D93A3$D94A3$D95A3$D96A
3$D97A3$D98A3$D99A3$D100A3$D101A3$D102A3$D103A3$D104A3$D105A3$D106A3$
D107A3$D108A3$D109A3$D110A3$D111A3$D112A3$D113A3$D114A3$D115A3$D116A
3$D117A3$D118A3$D119A3$D120A3$D121A3$D122A3$D123A3$D124A3$D125A3$D
126A3$D127A3$D128A3$D129A3$D130A3$D131A3$D132A3$D133A3$D134A3$D135A3$
D136A3$D137A3$D138A3$D139A3$D140A3$D141A3$D142A3$D143A3$D144A3$D145A
3$D146A3$D147A3$D148A3$D149A3$D150A3$D151A3$D152A3$D153A3$D154A3$D
155A3$D156A3$D157A3$D158A3$D159A3$D160A3$D161A3$D162A3$D163A3$D164A3$
D165A3$D166A3$D167A3$D168A3$D169A3$D170A3$D171A3$D172A3$D173A3$D174A
3$D175A3$D176A3$D177A3$D178A3$D179A3$D180A3$D181A3$D182A3$D183A3$D
184A3$D185A3$D186A3$D187A3$D188A3$D189A3$D190A3$D191A3$D192A3$D193A3$
D194A3$D195A3$D196A3$D197A3$D198A3$D199A3$D200A3$D201A3$D202A3$D203A
3$D204A3$D205A3$D206A3$D207A3$D208A3$D209A3$D210A3$D211A3$D212A3$D
213A3$D214A3$D215A3$D216A3$D217A3$D218A3$D219A3$D220A3$D221A3$D222A3$
D223A3$D224A3$D225A3$D226A3$D227A3$D228A3$D229A3$D230A3$D231A3$D232A
3$D233A3$D234A3$D235A3$D236A3$D237A3$D238A3$D239A3$D240A3$D241A3$D
242A3$D243A3$D244A3$D245A3$D246A3$D247A3$D248A3$D249A3$D250A3$D251A3$
D252A3$D253A3$D254A3$D255A3$D256A3$D257A3$D258A3$D259A3$D260A3$D261A
3$D262A3$D263A3$D264A3$D265A3$D266A3$D267A3$D268A3$D269A3$D270A3$D
271A3$D272A3$D273A3$D274A3$D275A3$D276A3$D277A3$D278A3$D279A3$D280A3$
D281A3$D282A3$D283A3$D284A3$D285A3$D286A3$D287A3$D288A3$D289A3$D290A
3$D291A3$D292A3$D293A3$D294A3$D295A3$D296A3$D297A3$D298A3$D299A3$D
300A3$D301A3$D302A3$D303A3$D304A3$D305A3$D306A3$D307A3$D308A3$D309A3$
D310A3$D311A3$D312A3$D313A3$D314A3$D315A3$D316A3$D317A3$D318A3$D319A
3$D320A3$D321A3$D322A3$D323A3$D324A3$D325A3$D326A3$D327A3$D328A3$D
329A3$D330A3$D331A3$D332A3$D333A3$D334A3$D335A3$D336A3$D337A3$D338A3$
D339A3$D340A3$D341A3$D342A3$D343A3$D344A3$D345A3$D346A3$D347A3$D348A
3$D349A3$D350A3$D351A3$D352A3$D353A3$D354A3$D355A3$D356A3$D357A3$D
358A3$D359A3$D360A3$D361A3$D362A3$D363A3$D364A3$D365A3$D366A3$D367A3$
D368A3$D369A3$D370A3$D371A3$D372A3$D373A3$D374A3$D375A3$D376A3$D377A
3$D378A3$D379A3$D380A3$D381A3$D382A3$D383A3$D384A3$D385A3$D386A3$D
387A3$D388A3$D389A3$D390A3$D391A3$D392A3$D393A3$D394A3$D395A3$D396A3$
D397A3$D398A3$D399A3$D400A3$D401A3$D402A3$D403A3$D404A3$D405A3$D406A
3$D407A3$D408A3$D409A3$D410A3$D411A3$D412A3$D413A3$D414A3$D415A3$D
416A3$D417A3$D418A3$D419A3$D420A3$D421A3$D422A3$D423A3$D424A3$D425A3$
D426A3$D427A3$D428A3$D429A3$D430A3$D431A3$D432A3$D433A3$D434A3$D435A
3$D436A3$D437A3$D438A3$D439A3$D440A3$D441A3$D442A3$D443A3$D444A3$D
445A3$D446A3$D447A3$D448A3$D449A3$D450A3$D451A3$D452A3$D453A3$D454A3$
D455A3$D456A3$D457A3$D458A3$D459A3$D460A3$D461A3$D462A3$D463A3$D464A
3$D465A3$D466A3$D467A3$D468A3$D469A3$D470A3$D471A3$D472A3$D473A3$D
474A3$D475A3$D476A3$D477A3$D478A3$D479A3$D480A3$D481A3$D482A3$D483A3$
D484A3$D485A3$D486A3$D487A3$D488A3$D489A3$D490A3$D491A3$D492A3$D493A
3$D494A3$D495A3$D496A3$D497A3$D498A3$D499A3$D500A3$D501A3$D502A3$D
503A3$D504A3$D505A3$D506A3$D507A3$D508A3$D509A3$D510A3$D511A3$D512A3$
D513A3$D514A3$D515A3$D516A3$D517A3$D518A3$D519A3$D520A3$D521A3$D522A
3$D523A3$D524A3$D525A3$D526A3$D527A3$D528A3$D529A3$D530A3$D531A3$D
532A3$D533A3$D534A3$D535A3$D536A3$D537A3$D538A3$D539A3$D540A3$D541A3$
D542A3$D543A3$D544A3$D545A3$D546A3$D547A3$D548A3$D549A3$D550A3$D551A
3$D552A3$D553A3$D554A3$D555A3$D556A3$D557A3$D558A3$D559A3$D560A3$D
561A3$D562A3$D563A3$D564A3$D565A3$D566A3$D567A3$D568A3$D569A3$D570A3$
D571A3$D572A3$D573A3$D574A3$D575A3$D576A3$D577A3$D578A3$D579A3$D580A
3$D581A3$D582A3$D583A3$D584A3$D585A3$D586A3$D587A3$D588A3$D589A3$D
590A3$D591A3$D592A3$D593A3$D594A3$D595A3$D596A3$D597A3$D598A3$D599A3$
D600A3$D601A3$D602A3$D603A3$D604A3$D605A3$D606A3$D607A3$D608A3$D609A
3$D610A3$D611A3$D612A3$D613A3$D614A3$D615A3$D616A3$D617A3$D618A3$D
619A3$D620A3$D621A3$D622A3$D623A3$D624A3$D625A3$D626A3$D627A3$D628A3$
D629A3$D630A3$D631A3$D632A3$D633A3$D634A3$D635A3$D636A3$D637A3$D638A
3$D639A3$D640A3$D641A3$D642A3$D643A3$D644A3$D645A3$D646A3$D647A3$D
648A3$D649A3$D650A3$D651A3$D652A3$D653A3$D654A3$D655A3$D656A3$D657A3$
D658A3$D659A3$D660A3$D661A3$D662A3$D663A3$D664A3$D665A3$D666A3$D667A
3$D668A3$D669A3$D670A3$D671A3$D672A3$D673A3$D674A3$D675A3$D676A3$D
677A3$D678A3$D679A3$D680A3$D681A3$D682A3$D683A3$D684A3$D685A3$D686A3$
D687A3$D688A3$D689A3$D690A3$D691A3$D692A3$D693A3$D694A3$D695A3$D696A
3$D697A3$D698A3$D699A3$D700A3$D701A3$D702A3$D703A3$D704A3$D705A3$D
706A3$D707A3$D708A3$D709A3$D710A3$D711A3$D712A3$D713A3$D714A3$D715A3$
D716A3$D717A3$D718A3$D719A3$D720A3$D721A3$D722A3$D723A3$D724A3$D725A
3$D726A3$D727A3$D728A3$D729A3$D730A3$D731A3$D732A3$D733A3$D734A3$D
735A3$D736A3$D737A3$D738A3$D739A3$D740A3$D741A3$D742A3$D743A3$D744A3$
D745A3$D746A3$D747A3$D748A3$D749A3$D750A3$D751A3$D752A3$D753A3$D754A
3$D755A3$D756A3$D757A3$D758A3$D759A3$D760A3$D761A3$D762A3$D763A3$D
764A3$D765A3$D766A3$D767A3$D768A3$D769A3$D770A3$D771A3$D772A3$D773A3$
D774A3$D775A3$D776A3$D777A3$D778A3$D779A3$D780A3$D781A3$D782A3$D783A
3$D784A3$D785A3$D786A3$D787A3$D788A3$D789A3$D790A3$D791A3$D792A3$D
793A3$D794A3$D795A3$D796A3$D797A3$D798A3$D799A3$D800A3$D801A3$D802A3$
D803A3$D804A3$D805A3$D806A3$D807A3$D808A3$D809A3$D810A3$D811A3$D812A
3$D813A3$D814A3$D815A3$D816A3$D817A3$D818A3$D819A3$D820A3$D821A3$D
822A3$D823A3$D824A3$D825A3$D826A3$D827A3$D828A3$D829A3$D830A3$D831A3$
D832A3$D833A3$D834A3$D835A3$D836A3$D837A3$D838A3$D839A3$D840A3$D841A
3$D842A3$D843A3$D844A3$D845A3$D846A3$D847A3$D848A3$D849A3$D850A3$D
851A3$D852A3$D853A3$D854A3$D855A3$D856A3$D857A3$D858A3$D859A3$D860A3$
D861A3$D862A3$D863A3$D864A3$D865A3$D866A3$D867A3$D868A3$D869A3$D870A
3$D871A3$D872A3$D873A3$D874A3$D875A3$D876A3$D877A3$D878A3$D879A3$D
880A3$D881A3$D882A3$D883A3$D884A3$D885A3$D886A3$D887A3$D888A3$D889A3$
D890A3$D891A3$D892A3$D893A3$D894A3$D895A3$D896A3$D897A3$D898A3$D899A
3$D900A3$D901A3$D902A3$D903A3$D904A3$D905A3$D906A3$D907A3$D908A3$D
909A3$D910A3$D911A3$D912A3$D913A3$D914A3$D915A3$D916A3$D917A3$D918A3$
D919A3$D920A3$D921A3$D922A3$D923A3$D924A3$D925A3$D926A3$D927A3$D928A
3$D929A3$D930A3$D931A3$D932A3$D933A3$D934A3$D935A3$D936A3$D937A3$D
938A3$D939A3$D940A3$D941A3$D942A3$D943A3$D944A3$D945A3$D946A3$D947A3$
D948A3$D949A3$D950A3$D951A3$D952A3$D953A3$D954A3$D955A3$D956A3$D957A
3$D958A3$D959A3$D960A3$D961A3$D962A3$D963A3$D964A3$D965A3$D966A3$D
967A3$D968A3$D969A3$D970A3$D971A3$D972A3$D973A3$D974A3$D975A3$D976A3$
D977A3$D978A3$D979A3$D980A3$D981A3$D982A3$D983A3$D984A3$D985A3$D986A
3$D987A3$D988A3$D989A3$D990A3$D991A3$D992A3$D993A3$D994A3$D995A3$D
996A3$D997A3$D998A3$D999A!
The one for n = 937 takes the longest to terminate (7,116,343 steps) although it doesn't have the longest sequence, because runtime for CollatzFuse is proportional to the sum of the terms in the Collatz sequence instead of the sequence's length.
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AbhpzTa
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Re: Collatz Tag System Emulator

Post by AbhpzTa » January 29th, 2017, 3:09 pm

Collatz variant (n>0): if n=1, stop
for n>1:

Code: Select all

if n==0(mod 3), n/3
if n==1(mod 3), 5n-2
if n==2(mod 3), 5n-1
3-tag system:
a -> ccd , b -> efg , c -> aaaaa , d -> a , e -> baaa , f -> bba , g -> bbb

Code: Select all

if n==0(mod 3), n/3
if n==1(mod 3), (5n-2)/3
if n==2(mod 3), (5n-1)/3
starting at n : b followed by n a's

rule table: [n/3 , (5n-2)/3 , (5n-1)/3][1 state_25, 1 state_2, and n state_1's]

Code: Select all

@RULE CollatzVariantModulo3_5n-2_5n-1
@TABLE
n_states:26
neighborhood:Moore
symmetries:none
var a={1,2,3,4,5,6,7}
var b={a}
var c={0,1,2,3,4,5,6,7}
var d={8,9,10,11,12,13,14}
var e={8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}
var f={0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22}
var g={f}
var h={f}
var i={f}
var j={f}
var k={f}
0,0,0,0,0,0,8,a,0,3
0,0,0,0,0,0,9,a,0,5
0,0,0,0,0,0,10,a,0,1
0,0,0,0,0,0,11,a,0,1
0,0,0,0,0,0,13,a,0,2
0,0,0,0,0,0,14,a,0,2
0,0,0,0,0,0,15,1,0,1
0,0,0,0,0,0,15,3,0,3
0,0,0,0,0,0,15,5,0,6
0,0,0,0,0,0,15,6,0,7
0,0,0,0,0,0,16,1,0,1
0,0,0,0,0,0,16,2,0,1
0,0,0,0,0,0,16,3,0,4
0,0,0,0,0,0,17,1,0,1
0,0,0,0,0,0,17,2,0,2
0,0,0,0,0,0,19,a,0,1
0,0,0,0,0,0,0,8,a,15
0,0,0,0,0,0,0,9,a,15
0,0,0,0,0,0,0,10,a,15
0,0,0,0,0,0,0,11,a,22
0,0,0,0,0,0,0,12,a,20
0,0,0,0,0,0,0,13,a,15
0,0,0,0,0,0,0,14,a,17
0,0,0,0,0,0,0,15,1,19
0,0,0,0,0,0,0,15,2,21
0,0,0,0,0,0,0,15,3,16
0,0,0,0,0,0,0,15,5,15
0,0,0,0,0,0,0,15,6,22
0,0,0,0,0,0,0,16,1,17
0,0,0,0,0,0,0,16,2,22
0,0,0,0,0,0,0,16,3,22
0,0,0,0,0,0,0,17,1,22
0,0,0,0,0,0,0,17,2,18
0,0,0,0,0,0,0,18,2,2
0,0,0,0,0,0,0,19,a,16
0,20,0,0,0,0,0,0,0,19
0,21,0,0,0,0,0,0,0,16
0,1,0,0,0,0,0,9,2,0
0,a,c,0,0,0,0,d,b,d
0,a,0,2,0,0,0,d,b,d
0,a,c,0,0,0,0,0,d,d
0,a,0,2,0,0,0,0,d,d
0,a,b,22,0,0,0,0,c,22
0,a,0,2,22,0,0,0,c,22
0,1,c,f,0,0,0,0,25,8
0,2,c,f,0,0,0,0,25,9
0,3,c,f,0,0,0,0,25,10
0,4,c,f,0,0,0,0,25,11
0,5,c,f,0,0,0,0,25,12
0,6,c,f,0,0,0,0,25,13
0,7,c,f,0,0,0,0,25,14
0,25,0,2,0,0,0,0,0,25
18,2,0,0,0,0,0,0,2,22
20,0,0,0,0,0,0,0,a,2
21,0,0,0,0,0,0,0,2,2
a,0,0,c,f,22,0,0,0,23
a,0,0,c,f,0,0,23,0,24
a,0,0,c,f,0,0,0,23,24
a,0,0,c,f,0,0,24,0,25
a,0,0,c,f,0,0,0,24,25
e,f,g,h,i,0,0,j,k,0
Starting at 10:

Code: Select all

x = 12, y = 1, rule = CollatzVariantModulo3_5n-2_5n-1
pAB10A!
Starting at 619 is an EXTREMELY LONG sequence and the maximum value > 10^35.

EDIT:
Exactly 7285 steps.
(1 step = 3k -> k , 3k+1 -> 5k+1 , or 3k+2 -> 5k+3)
The maximum value is 190029051685970780068377034879475196 (1.90029 x 10^35) (index 4466).

rule table for exact numbers

Code: Select all

@RULE CollatzVariantModulo3_5n-2_5n-1_base10
#1 digit 1
#2 digit 2
#3 digit 3
#4 digit 4
#5 digit 5
#6 digit 6
#7 digit 7
#8 digit 8
#9 digit 9
#10 digit 0
#11 stepper
@TABLE
n_states:18
neighborhood:Moore
symmetries:none
var a={3,6,9,10}
var b={1,4,7}
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XPM
"15 255 2 1"
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XPM
"7 119 2 1"
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