## Soup propagation distance

For general discussion about Conway's Game of Life.
Kiran
Posts: 285
Joined: March 4th, 2015, 6:48 pm

### Soup propagation distance

Imagine an infinite plane full of soup, the plane is iterated in some with the {0,0} cell set to on and with that cell set to off, at every time step corresponding cells in the two planes are xored and the results compose a third evolving plane. The average maximum Euclidean (EDIT: changed from Moore) distance a 1 achieves from {0,0} is the soup propagation distance of that rule.

Besides rules like B3/S23, B36/S23 and B3/S1248, it would be interesting to calculate to reasonable accuracy the soup propagation distance of B1/S45678, B2/S45678 and B35678/S34567.
Last edited by Kiran on November 25th, 2015, 7:10 pm, edited 1 time in total.
Kiran Linsuain

Kazyan
Posts: 959
Joined: February 6th, 2014, 11:02 pm

### Re: Soup propagation distance

Interesting idea! I'm pretty sure the soup propagation distance is infinite for a lot of explosive rules, but perhaps not for Life.

We can also take a look at soup propagation speed, which would be the slope of the graph of generation by current soup propagation distance, with whatever caveats necessary to not make the speed zero over a soup that stabilizes. I'd guess that the propagation speed correlates tightly to transitions in the rulestring--that is, when only one of two sequential integers appears. For example, Life would have survival transitions (1,2) and (3,4), and birth transitions (2,3) and (3,4). Although, some transitions would have a greater effect than others for the same reason that B3/S238 could fool you into thinking it's Life for 1000+ generations, but B3/S235 couldn't. Over any transition (X,X+1), there are 8CX ways for a cell to have X live neighbors, and (8-X) ways to add a new live neighbor to each of those 8CX options. So there's a total of (8-X)[sub]8[/sub]C[sub]X[/sub] ways for a variable cell to matter.

Now we can weight every possible transition by plugging it into that expression:

(0,1) = 8
(1,2) = 56
(2,3) = 168
(3,4) = 280
(4,5) = 280
(5,6) = 168
(6,7) = 56
(7,8) = 8

Then, for the rulestring in question, we just add up the weights of every transition to get what I'll call the sensitivity for the string. I suspect that a rulestring's sensitivity correlates tightly with the soup propagation speed.

Hmm...maybe the distribution of propagation distances and speeds for specific-but-random soups--that is, we don't take the average of what happens to every soup; we just present all the data as a graph--could be used to define the classes of CA more formally?
Tanner Jacobi
Coldlander, a novel, available in paperback and as an ebook.

Kiran
Posts: 285
Joined: March 4th, 2015, 6:48 pm

### Re: Soup propagation distance

I suspect that a rulestring's sensitivity correlates tightly with the soup propagation speed.
This may work well for the first generation and, for many rules, reasonably well for the first few, but in many rules soup gradually becomes structured and the ratios of neighborhoods no longer apply. For example in B3/S23 soup quickly fragments into clumps and the density plummets. Also distance is speed times time, which means the rules I mentioned in the first post probably have a finite soup propagation distance because eventually information stops spreading even if it initially spreads quickly.
could be used to define the classes of CA more formally?
I don't think so because the last three rules mentioned in my first post are clearly exploding, but they almost certainly have a finite soup propagation distance. Explosivity is a behaviour of the edges while the soup propagation distance is defined inside an infinite soup and depends mostly on the long term internal structure of soup.

EDIT:
Also, the rule B468/S1357 has sensitivity 3/4 yet it cannot even expand beyond it's building box while B1/S45678 has a sensitivity of only 43/256 but is very explosive. Explosivity is not ambiguous for non B0 rules that have B1 or B2 and for rules without any of B01234, it can be ambiguous for B3 and B0 rules.
Kiran Linsuain

Kazyan
Posts: 959
Joined: February 6th, 2014, 11:02 pm

### Re: Soup propagation distance

I wrote my first meaningful golly script to estimate soup propagation distances of arbitrary rules.

Code: Select all

``````# soup-propagation-distance.py
# Estimates the soup propagation distance of a rule. Estimates approaching 4 digits are unrealiable.
# Tanner Jacobi, August 2015

import golly as g

# Clear out layer 0 for use
g.setlayer(0)
g.addlayer()
g.setlayer(0)
g.dellayer()

# Pick your desired rule and level of patience
g.setrule(g.getstring("Which rule to use?", "B3/S23"))
tedium = int(g.getstring("How many soups to average over?", "10"))

disttotal = 0
g.show("Performing trial 1 of " + str(tedium) + ". Current Mean: 0.000")

for x in range(1, tedium+1):
# Set up layer 1. Empty layer 0 already here.
g.addlayer()
g.setlayer(1)

# Run two random 1999x1999 soups that differ only by the central cell for 30,000 generations
g.select([-999,-999,1999,1999])
g.randfill(50)
soup = g.getcells(g.getselrect())
g.setcell(0,0,1)
g.setlayer(0)
g.putcells(soup,0,0)
g.setcell(0,0,0)
g.run(30000)
g.setlayer(1)
g.run(30000)

# Xor the results.
WhereToXor = g.getrect()
LayerOneAsh = g.getcells(WhereToXor)
g.setlayer(0)
g.putcells(LayerOneAsh,0,0,1,0,0,1,"xor")

# Find propagation distance from the xor result and accumulate
dist = g.getrect()
bugdist = g.getrect()
if dist: # If dist is empty, propagation was zero, so we'll skip evaluating and accumulating
dist[2] = dist[0] + dist[2] - 1
dist[3] = dist[1] + dist[3] - 1
dist[0] = abs(dist[0])
dist[1] = abs(dist[1])
disttotal = disttotal + max(dist)

# Progress update before next trial
g.show("Performing trial " + str(x+1) + " of " + str(tedium) + ". Current Mean: " + '%.3f'%(disttotal/float(x)))

# Remove layers; it's faster than deleting everything for large patterns. Leave with an empty layer 0.
g.addlayer()
g.movelayer(g.getlayer(),0)
g.setlayer(2)
g.dellayer()
g.dellayer()

# When all the tedium is done, report average
g.show("Done. Estimate for rule " + g.getrule() + ": " + str('%.3f'%(disttotal/float(tedium))))``````
On my machine, it takes ~4.5 seconds per trial in Life. For explosive rules, it takes...well, don't use it for explosive rules.

From running this script, Life seems to have a soup propagation distance of about 444.8, from 1,000 trials. For the same amount of trials on B34/S26 (my pet rule), it's about 43.6.
Tanner Jacobi
Coldlander, a novel, available in paperback and as an ebook.

Bullet51
Posts: 556
Joined: July 21st, 2014, 4:35 am

### Re: Soup propagation distance

Kazyan wrote: From running this script, Life seems to have a soup propagation distance of about 444.8, from 1,000 trials.
I have ran 1000 trials and the soup propagation distance seems to be 444.2 .
What will happen when we use bigger soups? I have tried some 1199x1199 soups, and the soup propagation distance is near 610.
Still drifting.

Kiran
Posts: 285
Joined: March 4th, 2015, 6:48 pm

### Re: Soup propagation distance

Are you using a toroidal universe, as long as the propagation does not come close to meeting itself it should yield very accurate results? Perhaps a script should give a warning if the propagation comes within a certain distance of the edge.
EDIT:
This would also make it somewhat practical to test B1/S45678, B2/S45678 and B35678/S34567, although the toruses would have to be very large.
Kiran Linsuain

Bullet51
Posts: 556
Joined: July 21st, 2014, 4:35 am

### Re: Soup propagation distance

Kiran wrote:The average maximum Moore distance a 1 achieves from {0,0} is the soup propagation distance of that rule.
By superimposing 338 xored results, it is shown that that the probability for a 1 to appear at some place only depends on its Euclidean distance to the center. What about changing the Moore distance into the Euclidean distance?
Attachments
Difference pattern of b3/s23 soups
difference.jpg (254.78 KiB) Viewed 18014 times
Still drifting.

Kazyan
Posts: 959
Joined: February 6th, 2014, 11:02 pm

### Re: Soup propagation distance

To run the script on a torus, append :T1999,1999 to the rulestring you enter. However, Golly does not "support" toruses so much as it "tolerates" them, to the tune of this script taking several orders of magnitude longer to complete, so I didn't make that a default.
Tanner Jacobi
Coldlander, a novel, available in paperback and as an ebook.

calcyman
Posts: 2200
Joined: June 1st, 2009, 4:32 pm

### Re: Soup propagation distance

Kazyan wrote:However, Golly does not "support" toruses so much as it "tolerates" them
Haha, that's so hilariously true. This might actually be my favourite remark ever made on these forums.
By superimposing 338 xored results, it is shown that that the probability for a 1 to appear at some place only depends on its Euclidean distance to the center.
To leading order, yes, that appears to be true. This is unsurprising if you approximate a soup by a colony of radioactive ants performing random walks on Z^2; the envelope is exactly circular in the limit.
What do you do with ill crystallographers? Take them to the mono-clinic!

Kiran
Posts: 285
Joined: March 4th, 2015, 6:48 pm

### Re: Soup propagation distance

I tried the script for a 1999x1999 torus with 10 trials, got 403.000.
Also, to better visualize the effect a four state rule can be used with black (0,0) cells, red (0,1) cells, green (1,0) cells and white (1,1) cells that together simulate two independent universes at once.
A script to generate the four state rule table for any life-like rule could be more efficient than writing them by hand.
Kiran Linsuain

Bullet51
Posts: 556
Joined: July 21st, 2014, 4:35 am

### Re: Soup propagation distance

Kiran wrote: Also, to better visualize the effect a four state rule can be used with black (0,0) cells, red (0,1) cells, green (1,0) cells and white (1,1) cells that together simulate two independent universes at once.
Something like this?

Code: Select all

``````@RULE doublelife
@TABLE

n_states:4
neighborhood:Moore
symmetries:permute

var a={3,1}
var a3=a
var a2=a
var a1=a
var a4=a
var a5=a
var a6=a
var a7=a

var b={0,2}
var b3=b
var b2=b
var b1=b
var b4=b
var b5=b
var b6=b
var b7=b

var c={2,1}
var c3=c
var c2=c
var c1=c
var c4=c
var c5=c
var c6=c
var c7=c

var d={0,3}
var d3=d
var d2=d
var d1=d
var d4=d
var d5=d
var d6=d
var d7=d

var x={0,3,2,1}
var x3=x
var x2=x
var x1=x
var x4=x
var x5=x
var x6=x
var x7=x
var x8=x

x,3,3,3,2,2,2,0,0,1
x,3,3,1,2,2,0,0,0,1
x,3,1,1,2,0,0,0,0,1
x,1,1,1,0,0,0,0,0,1

3,2,2,2,3,3,0,0,0,1
3,2,2,1,3,0,0,0,0,1
3,2,1,1,0,0,0,0,0,1

2,3,3,3,2,2,0,0,0,1
2,3,3,1,2,0,0,0,0,1
2,3,1,1,0,0,0,0,0,1

1,3,3,2,2,2,0,0,0,1
1,3,1,2,2,0,0,0,0,1
1,1,1,2,0,0,0,0,0,1

1,3,3,3,2,2,0,0,0,1
1,3,3,1,2,0,0,0,0,1
1,3,1,1,0,0,0,0,0,1

1,3,3,2,2,0,0,0,0,1
1,3,1,2,0,0,0,0,0,1
1,1,1,0,0,0,0,0,0,1

x,a,a3,a2,b3,b2,b1,b4,b5,3
x,a,a3,a2,b3,b2,b1,b4,b5,3
3,a,a3,b3,b2,b1,b4,b5,b6,3
1,a,a3,b3,b2,b1,b4,b5,b6,3

x,c,c3,c2,d3,d2,d1,d4,d5,2
x,c,c3,c2,d3,d2,d1,d4,d5,2
2,c,c3,d3,d2,d1,d4,d5,d6,2
1,c,c3,d3,d2,d1,d4,d5,d6,2

x,x3,x2,x1,x4,x5,x6,x7,x8,0

@COLORS

0 0   0   0
1 255 255 255
2 0   255 0
3 255 0   0
``````
Still drifting.

Kiran
Posts: 285
Joined: March 4th, 2015, 6:48 pm

### Re: Soup propagation distance

Something like this?
Exactly!
Kiran Linsuain

dvgrn
Moderator
Posts: 6735
Joined: May 17th, 2009, 11:00 pm
Location: Madison, WI
Contact:

### Re: Soup propagation distance

Very nice! Here's a big sample soup to try out -- nothing special about this example, but it's fun to watch:

Code: Select all

``````x = 250, y = 250, rule = doublelife
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2A3.A.A10.A3.2A4.A4.3A!``````
It turned out to be somewhat amusing to convert LifeHistory patterns into DoubleLife. You end up with a highly ordered pattern superimposed on a highly chaotic one:

Code: Select all

``````x = 130, y = 179, rule = doublelife
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Kiran
Posts: 285
Joined: March 4th, 2015, 6:48 pm

### Re: Soup propagation distance

Can it be done for other rules?
Kiran Linsuain

dvgrn
Moderator
Posts: 6735
Joined: May 17th, 2009, 11:00 pm
Location: Madison, WI
Contact:

### Re: Soup propagation distance

Kiran wrote:Can it be done for other rules?
Certainly. Sometimes a lot more lines of rules will be needed, though. There's a minor combinatorial explosion if you have birth and survival conditions with lots of different neighbor counts, instead of just the three conditions for Conway's Life.

For example, I think HighLife would need 4*3+3 = 15 groups of rules, because it has four birth survival conditions, instead of regular Life's 3 survival conditions and 3*2+3 = 9 groups of rules. And the groups would sometimes be larger when the birth or survival is on a 4 or 5 neighbor count, because there are more different ways to overlap the independent states.

Maybe we could agree on a standardizable name for these double rules -- e.g., "DoubleB36S23" for HighLife.

I found it a little confusing that the combined state was mapped onto state 1. I had initially assumed that the mapping would be something like red = 1 (0x01), green = 2 (0x10), and white = 3 (0x11).

Below is a translated "DoubleB3S23" rule with the binary-based state mapping, and some added comments to make it clear what the various rules are doing. I've also attached a ZIP file that can be opened in Golly to load the rule and display a test pattern, showing (I hope) that I did the conversion correctly and the rule is now symmetric with respect to state 1 and state 2:
DoubleB3S23.zip
DoubleB3S23 rule with test chaotic+ordered pattern
(7.76 KiB) Downloaded 388 times
If someone wants to try tackling the problem, it should be fairly straightforward to write a Golly script that generates Double rules on demand for any Lifelike CA. Might be easier to skip the variables and write out all the specific state transitions. Could be a little trickier (though definitely a nicer result) to use variables to boil down the rule file to a minimum number of lines, as Bullet51 has done for B3/S23.

Code: Select all

``````@RULE DoubleB3S23
@TABLE

n_states:4
neighborhood:Moore
symmetries:permute

# 0x01 bit ON
var a={1,3}
var aa=a
var ab=a
var ac=a
var ad=a
var ae=a
var af=a
var ag=a

# 0x01 bit OFF
var b={0,2}
var ba=b
var bb=b
var bc=b
var bd=b
var be=b
var bf=b
var bg=b

# 0x10 bit ON
var c={2,3}
var ca=c
var cb=c
var cc=c
var cd=c
var ce=c
var cf=c
var cg=c

# 0x10 bit OFF
var d={0,1}
var da=d
var db=d
var dc=d
var dd=d
var de=d
var df=d
var dg=d

# don't-care cells
var x={0,1,2,3}
var xa=x
var xb=x
var xc=x
var xd=x
var xe=x
var xf=x
var xg=x
var xh=x

# birth for both 0x01 and 0x10 bits
x,1,1,1,2,2,2,0,0,3
x,1,1,3,2,2,0,0,0,3
x,1,3,3,2,0,0,0,0,3
x,3,3,3,0,0,0,0,0,3

# survival for 0x01 bit, birth for 0x10 bit
1,2,2,2,1,1,0,0,0,3
1,2,2,3,1,0,0,0,0,3
1,2,3,3,0,0,0,0,0,3

# birth for 0x01 bit, survival for 0x10 bit
2,1,1,1,2,2,0,0,0,3
2,1,1,3,2,0,0,0,0,3
2,1,3,3,0,0,0,0,0,3

# 2-neighbor survival for 0x01 bit, 3-neighbor survival for 0x10 bit
3,1,1,2,2,2,0,0,0,3
3,1,3,2,2,0,0,0,0,3
3,3,3,2,0,0,0,0,0,3

# 3-neighbor survival for 0x01 bit, 2-neighbor survival for 0x10 bit
3,1,1,1,2,2,0,0,0,3
3,1,1,3,2,0,0,0,0,3
3,1,3,3,0,0,0,0,0,3

# 2-neighbor survival for both 0x01 and 0x10 bits
3,1,1,2,2,0,0,0,0,3
3,1,3,2,0,0,0,0,0,3
3,3,3,0,0,0,0,0,0,3

# birth for 0x01 bit, death for 0x10 bit
x,a,aa,ab,ba,bb,bc,bd,be,1
x,a,aa,ab,ba,bb,bc,bd,be,1
1,a,aa,ba,bb,bc,bd,be,bf,1
3,a,aa,ba,bb,bc,bd,be,bf,1

# death for 0x01 bit, birth for 0x10 bit
x,c,ca,cb,da,db,dc,dd,de,2
x,c,ca,cb,da,db,dc,dd,de,2
2,c,ca,da,db,dc,dd,de,df,2
3,c,ca,da,db,dc,dd,de,df,2

# death for both bits
x,xa,xb,xc,xd,xe,xf,xg,xh,0

@COLORS

0 0   0   0
1 255 0   0
2 0   255 0
3 255 255 255``````

Kiran
Posts: 285
Joined: March 4th, 2015, 6:48 pm

### Re: Soup propagation distance

I found it a little confusing that the combined state was mapped onto state 1.
This makes it easier to set up a soup propagation test because the soup can be generated using golly's random fill in a 2-state rule then the rule can be changed and finally the origin is changed to a green or red cell.
Kiran Linsuain

dvgrn
Moderator
Posts: 6735
Joined: May 17th, 2009, 11:00 pm
Location: Madison, WI
Contact:

### Re: Soup propagation distance

Kiran wrote:
I found it a little confusing that the combined state was mapped onto state 1.
This makes it easier to set up a soup propagation test because the soup can be generated using golly's random fill in a 2-state rule then the rule can be changed and finally the origin is changed to a green or red cell.
Ah, got it -- didn't think of that.

That series of steps could be cut down to a single step with a Python script, and then it wouldn't matter which state was 1. Here's a sample script for the DoubleB3S23 rule:

makedoublesoup.py (but don't use this one, use the second version below):

Code: Select all

``````import golly as g
import itertools

answer = g.getstring("Enter x y half-diameters for soup propagation trial:",
"100 100",
"Soup size")
xy = answer.split()
# extract x and y amounts
if len(xy) == 0: xy = [100,100]
if len(xy) == 1: g.exit("Supply x and y amounts separated by a space.")
x = int(xy[0])
y = int(xy[1])
if x==0: x=100
if y==0: y=100

if g.getname()!="DoubleSoupTest":
g.addlayer()
g.setname("DoubleSoupTest")
else:
g.new("DoubleSoupTest")
g.select([-x,-y,x*2+1, y*2+1])
g.setrule("B3S23")
g.randfill(50)
g.setrule("DoubleB3S23")
cells=g.getcells(g.getselrect())

coords = zip(cells[0::3],cells[1::3],[3]*len(cells[2::3]))
flatclist = list(itertools.chain.from_iterable(coords))
if len(coords)%2==0: flatclist += [0]
# if the above line is not included, the list gets interpreted
#   as a two-state list half the time -- very weird when you first see it!

g.putcells(g.join(flatclist, [0,0,1]))``````
EDIT: Okay, that first attempt made for horribly slow state-1 to state-3 conversions for even moderately large soups. Here's the up-to-date fix using a custom rule table to change the state:

Code: Select all

``````import golly as g
import itertools
import os

# Blazingly fast way to convert one state to another
# -- used in this case to move DoubleB3S23 state 1, from g.randfill(), to state 3

def CreateRule():
fname = os.path.join(g.getdir("rules"), "Double123StateFixer.rule")
if not os.path.exists(fname):
with open(fname, 'wb') as f:
f.write('@RULE Double123StateFixer\n\n')
f.write('@TABLE\nn_states:4\nneighborhood:vonNeumann\nsymmetries:none\n\n')
f.write('var a={0,1,2,3}\nvar b={a}\nvar c={a}\nvar d={a}\nvar f={a}\n\n')
f.write('1,a,b,c,d,3\n')

CreateRule()

answer = g.getstring("Enter x y half-diameters for soup propagation trial:",
"500 500",
"Soup size")
xy = answer.split()
# extract x and y amounts
if len(xy) == 0: xy = [500,500]
if len(xy) == 1: g.exit("Supply x and y amounts separated by a space.")
x = int(xy[0])
y = int(xy[1])
if x==0: x=500
if y==0: y=500

if g.getname()!="DoubleSoupTest":
g.addlayer()
g.setname("DoubleSoupTest")
else:
g.new("DoubleSoupTest")
g.select([-x,-y,x*2+1, y*2+1])
g.setrule("B3S23")
g.randfill(50)

oldrule = g.setrule("Double123StateFixer")
g.run(1)
step = g.getstep()
g.setrule("DoubleB3S23")
g.setstep(step)
g.setgen("-1")
g.setcell(0,0,1)``````
The script is still slow for half-diameters above 1000 or so, but Golly's g.randfill() function is what's taking almost all of the time.

I bet there's a way to create a random 50% fill a couple of orders of magnitude faster, by writing a macrocell pattern file directly, and then opening it...! Should be fairly easy at least for power-of-two-sized square soups.

Tropylium
Posts: 406
Joined: May 31st, 2011, 7:12 pm
Location: Finland

### Re: Soup propagation distance

As far as I've looked into this matter before (or some concepts resembling it), soup propagation speed in Life is almost always lightspeed for connected clusters of cells, and dominated by cluster growth rate outside of them. This makes things very sensitive to the "cloud count" and geometry of the evolving pattern…

Kiran
Posts: 285
Joined: March 4th, 2015, 6:48 pm

### Re: Soup propagation distance

A custom-built circuit board with the rule hard-wired into the integrated circuits could possibly test rules like B1/S45678 in a reasonable time, but that sounds a bit impractical.
...Life is almost always lightspeed for connected clusters of cells, and dominated by cluster growth rate outside of them.
In B1/S45678 it would be dominated by the crystallization time, would it be practical to estimate the propagation distance by testing propagation speeds in different stages of crystallization and combining the results into an estimate?

Last bumped by Kiran on December 27th, 2015, 9:02 pm.
Kiran Linsuain