Zone of Influence

[Nathaniel]

Can someone explain the definition of zone of influence provided in the Game of Life dictionary? In particular, to respond to the comments here? I too don't understand what the motivation for the definition with clipped corners is.

[ebcube]

I'm not sure if that holds in Life, but if the definition is CA-agnostic (that is, it applies to any CA) a broader definition, accounting for range-2 neighbourhoods, Von Neumann and the like, seems necessary.

[Wojowu]

It would be hard to explain without picture, so I'm posting .png image (what would we do without Paint ).
So, here is range two zone of influence, marked with gray. "Jump" is here a move by one cell orthogonally or diagonally. Any cell is in zone of influence of given cell if we can get here in at least n possible ways making not more than n jumps. On first grid I've shown that two cells are in range two zone of influence, because we can get here using 2 different ways. On second grid I've shown that white cell is outside this zone, because there is only one way with two jumps to get here, using two diagonal jumps. On third grid I've shown that middle cell is in its zone of influence. But in that way we can show that it is outside its range 1 zone! But I think it is only a small mistake in definition I still can't explain what is said on forum, but I hope I helped a bit.

[137ben]

But in that way we can show that it is outside its range 1 zone! But I think it is only a small mistake in definition I still can't explain what is said on forum, but I hope I helped a bit.

For the range 1 zone, we need only be able to reach a cell in 1 way. The definition can be amended to specify that we must be able to reach the cell in n or fewer steps in at least n different ways.

[Wojowu]

But in that way we can show that it is outside its range 1 zone! But I think it is only a small mistake in definition I still can't explain what is said on forum, but I hope I helped a bit.

For the range 1 zone, we need only be able to reach a cell in 1 way. The definition can be amended to specify that we must be able to reach the cell in n or fewer steps in at least n different ways.

Yes, you are right, but I said it in my post (and I later forgotten about that )

...we can get here in at least n possible ways making not more than n jumps.

[Nathaniel]

I guess I should clarify that I understand how the definition works and how to construct visually the zone of influence from the definition. What I don't understand is why the definition is important. When would the zone of influence ever be important rather than just the Moore neighborhood (i.e., the square with side length 2n+1)? Does it ever actually come up, and if so when?

[tod222]

Zone of Influence is useful when constructing stable aggregations of smaller stable components, like during a search for stable reflectors. The ZOI of the aggregation is calculated as the union of the ZOI of each cell in the component patterns.

I conducted for several months last year a search for stable reflectors. Candidate patterns were built using several basic stable patterns. As the candidate was built up, its ZOI was calculated to guide valid locations for additional components to ensure stability of the resulting aggregation.

The notch is there because Life is B3/S23, which allows stable components to stably nest together closer than the Moore neighborhood. ZOI != Moore neighborhood when B > 2.

An example using two blocks

Because it takes 3 neighboring cells to birth a new cell, two blocks can be so close that they can occupy each other's ZOI notches without causing instability:



1. A block
2. The zone of influence for a block
3. Two blocks, clear of each other's zones of influence, are stable
4. One block moved so that the blocks are within each other's ZOIs
5. Step one generation, pattern 3 is stable
6. Step one generation, pattern 4 is unstable

[Tropylium]

This seems to work fine for blocks, but not patterns with thinner peaks:

Code: Select all

x = 8, y = 5, rule = B3/S23
2o$obo$2bo3bo$2b2obobo$6bo!

These eater and tub are within each other's zone of influence as defined here, but yet quite uninteracting.

If the concept is motivated in finding out how close together patterns can be placed, I'd suggest a modification:
— Neighbor cells count for 1 influence point.
— Cells with a (Moore) distance of 2 count for ½ influence points.
— A pattern's influence points are the sum of its cells' influence points.
— A pattern is in another's zone of influence, if a cell of it that has 1 or more influence points from it, OR if the latter pattern is in the former's zone of influence.

Thus, in the eater/tub example, the opposing corners only gain ½ influence point from the other, and are not within each other's zone of influence. In this block/tub example

Code: Select all

x = 6, y = 4, rule = B3/S23
bo$obo$bo2b2o$4b2o!

the tub's corner gains ½ influence points from both of the block's left-side cells, so we can tell it is within the block's zone of influence (and thus the block also in the tub's.)

[137ben]

Interesting idea. I think the motivation for the zone of influence was to provide a sharper estimate for how close things could be than the moore neighborhood. Your idea is even closer, though still not perfect:

Code: Select all

x = 4, y = 7, rule = B3/S23
b2o$o2bo$b2o2$b2o$o2bo$b2o!

.

[Tropylium]

Interesting idea. I think the motivation for the zone of influence was to provide a sharper estimate for how close things could be than the moore neighborhood. Your idea is even closer, though still not perfect:

Code: Select all

x = 4, y = 7, rule = B3/S23
b2o$o2bo$b2o2$b2o$o2bo$b2o!

.

Yes, pseudo still lifes are certainly an issue that won't be easy to handle, even without going into things like the pseudo still lifes divisible in n but not n-1 components.

Code: Select all

x = 17, y = 10, rule = B3/S23
13b2o$14bo$13bo$b2obo8b2o$obob3o9bo$bo5bo4bob3o$2b3obobo3b2o$4bob2o7b
2o$12b3obo$12bo!

For that matter, various "tie" still lifes show that stability is not a monotonous function of range:

Code: Select all

x = 47, y = 36, rule = B3/S23
26b2o13bo$7b2o17bo13bobo$6bobo5bob2o9b3o9bo2bo$5bo8b2obo11bo10b2ob2o$
4bob2o9bob2o9b2o10bo2bo$3bo2bobo8b2obo10bo10bobo$3b2o2b2o22bob2o6b2obo
bo$b2o29bobo10b2o$obo$bo$14bob2o8b2o13bo$14b2obo8bo13bobo$7b2o18b3o9bo
2bo$6bobo8bob2o8bo10b2o$5bo11b2obo22b2o$4bo25b2o10bo2bo$3bo2b2o23bo10b
obo$3b2obobo22bob2o6b2obobo$7b2o23bobo10b2o$b2o$obo$bo2$14bob2o8b2o13b
o$14b2obo8bo13bobo$7b2o18b3o9bo2bo$6bobo20bo10b2o$5bo11bob2o$4bo12b2ob
o22b2o$3bo3b2o21b2o10bo2bo$3b2o2bobo21bo10bobo$8b2o21bob2o6b2obobo$32b
obo10b2o$b2o$obo$bo!

I would just subsume these cases under some other term than "zone of influence"; these are clearly objects that DO influence each other, it's just influence that doesn't lead to instability.

[wwei23]

For that matter, various "tie" still lifes show that stability is not a monotonous function of range

This is even less monotonous:

Code: Select all

x = 67, y = 4, rule = B3/S23
ob2o16bob2o16bob2o16bob2o$2obo16b2obo16b2obo16b2obo$6bob2o15bob2o15bob
2o15bob2o$6b2obo15b2obo15b2obo15b2obo!

ZOI for still lives up to 8 cells:

Code: Select all

x = 98, y = 49, rule = LifeHistory
.4B$6B5.5B$2B2A2B5.2BA2B$2B2A2B5.BABAB$6B5.2BA2B$.4B6.5B6$.5B$.2BA3B$
.BABA2B$.2B2A2B$.6B$2.4B4$2.4B5.4B16.4B6.5B$.6B3.6B5.5B4.7B3.8B$.2B2A
2B3.2B2A3B4.2BA3B3.2B2A4B2.2B2ABA2B$.BA2BAB3.2BABA2B4.BABA2B3.2BA2BA
2B2.2BAB2A2B$.2B2A2B3.3B2A2B4.2BABAB3.4B2A2B2.8B$.6B4.6B4.3BA2B4.7B4.
5B$2.4B6.4B6.5B6.4B4$2.4B15.4B6.4B$.6B4.5B4.7B3.6B$3B2A2B4.2BA3B3.2B
2A5B.2B2A3B$2BA2BAB4.BABA3B2.2BABABA2B.2BABA2B$2BABA2B4.2BABA2B2.5B2A
2B.4BA2B$3BA3B4.3B2A2B4.7B3.2B2AB$.5B6.6B6.4B4.5B$13.4B16.4B3$2.4B6.
4B5.4B16.4B6.4B26.4B7.3B$.6B4.6B3.6B14.6B4.6B5.5B5.5B4.6B4.7B$3B2A3B
3.2B2A3B2.2B2A4B12.2B2A3B3.2B2A3B4.2BA3B4.2BA3B3.2B2A3B3.2BABA3B$2BA
2BA2B3.BA2BA2B2.2BABA5B10.2BABA3B2.2BABA3B3.BABA3B3.BABA3B2.2BABA2B3.
2B2ABA2B$2BA2BA2B3.2BA2BAB2.5BABA2B10.5BA3B.3BABA2B3.2BABA2B3.2BABA2B
2.4BA2B3.5BA2B$3B2A3B3.3B2A2B4.4B2A2B12.4BA2B2.3B2A2B3.3BABAB3.4BA2B
3.2BA3B4.4B2AB$.6B5.6B6.6B13.2B2A2B3.6B4.3BA2B5.2B2AB3.2B2A2B6.5B$2.
4B7.4B8.4B15.5B4.4B6.5B5.5B3.5B8.4B$44.4B26.4B4.4B!

[fluffykitty]

Can you use a theme with lighter history? The contrast isn't very good.

[wwei23]

How?

[rowett]

Can you use a theme with lighter history?

Unfortunately you can't at the moment. There's a bug in LifeViewer which prevents LifeHistory patterns working with the [[ THEME ]] script command at generation 0. This has been fixed and will be in the next released build.

In the mean time try this:

Code: Select all

x = 98, y = 49, rule = LifeHistory
.4B$6B5.5B$2B2A2B5.2BA2B$2B2A2B5.BABAB$6B5.2BA2B$.4B6.5B6$.5B$.2BA3B$
.BABA2B$.2B2A2B$.6B$2.4B4$2.4B5.4B16.4B6.5B$.6B3.6B5.5B4.7B3.8B$.2B2A
2B3.2B2A3B4.2BA3B3.2B2A4B2.2B2ABA2B$.BA2BAB3.2BABA2B4.BABA2B3.2BA2BA
2B2.2BAB2A2B$.2B2A2B3.3B2A2B4.2BABAB3.4B2A2B2.8B$.6B4.6B4.3BA2B4.7B4.
5B$2.4B6.4B6.5B6.4B4$2.4B15.4B6.4B$.6B4.5B4.7B3.6B$3B2A2B4.2BA3B3.2B
2A5B.2B2A3B$2BA2BAB4.BABA3B2.2BABABA2B.2BABA2B$2BABA2B4.2BABA2B2.5B2A
2B.4BA2B$3BA3B4.3B2A2B4.7B3.2B2AB$.5B6.6B6.4B4.5B$13.4B16.4B3$2.4B6.
4B5.4B16.4B6.4B26.4B7.3B$.6B4.6B3.6B14.6B4.6B5.5B5.5B4.6B4.7B$3B2A3B
3.2B2A3B2.2B2A4B12.2B2A3B3.2B2A3B4.2BA3B4.2BA3B3.2B2A3B3.2BABA3B$2BA
2BA2B3.BA2BA2B2.2BABA5B10.2BABA3B2.2BABA3B3.BABA3B3.BABA3B2.2BABA2B3.
2B2ABA2B$2BA2BA2B3.2BA2BAB2.5BABA2B10.5BA3B.3BABA2B3.2BABA2B3.2BABA2B
2.4BA2B3.5BA2B$3B2A3B3.3B2A2B4.4B2A2B12.4BA2B2.3B2A2B3.3BABAB3.4BA2B
3.2BA3B4.4B2AB$.6B5.6B6.6B13.2B2A2B3.6B4.3BA2B5.2B2AB3.2B2A2B6.5B$2.
4B7.4B8.4B15.5B4.4B6.5B5.5B3.5B8.4B$44.4B26.4B4.4B!
[[ COLOR ALIVE WHITE COLOR BACKGROUND BLACK COLOR DEAD 0 128 255 ]]
[[ AUTOSTART STOP 1 NOREPORT ]]
[[ COLOR GRID 32 32 32 GRID GRIDMAJOR 0 ]]
[[ ZOOM 4 ]]

[fluffykitty]

Much better. Thanks!