0 0 -> 0
1 0 -> 0
0 1 -> 0
1 1 -> 1
I tried a few things.
First of all let me refer to the rule you have previously evolved
as Rule 1. It's graphic form is:
Also. P_Dens is the unbiased performance of a rule to the density task, while P_AND is the performance for the AND task. The P_AND is also an unbiased random generation of 10000 IC's. Each cell has 0.5 probability of being on or off. Thus, the probabilities of the each lattice being (0,0), (0,1), (1,0), and (1,1) in terms of the first and second bit halves is 0.25 for each.
P_Dens (rule 1) = 0.77
P_AND (rule 1) = 0.71
Avg = 0.74.
P_Dens (rule 4): 0.5
P_AND (rule 4): 0.79
Therefore the fitness function of the AND task is biased to a uniform distribution of lattices leading to (0, 0), (0,1), (1,0) 50% of the time, and (1,1) the other 50%. I set up a GA with this fitness function, and injected a few good rules on the first population, including rule 1 and a few others I have evolved. Two interesting rules evolved were:
Rule2: 0504058705000E77037755837AFFB77F, which is almost identical
to rule 1, but yields P_Dens(rule 2) = 0.70 and
(rule 2) = 0.79, average 0.75.
However, it does evolve a different, similar, set of particle
interactions, more suitable for the AND than the Density task:
But since this GA was evolving rules that solve the AND task alone, this rule did not stick around and was quickly over ruled by rules like Rule 3: 2D0008000000008F77FFBFFF27F7FFFF, with P_Dens(rule 3) = 0.58 and P_AND(Rule 3) = 0.80. This rule is a block expansion rule:
Rule 5: 0504058505000D77073755437BFFB77F,
5) = 0.75 and P_AND (rule 5) = 0.77, average 0.76. Which
is clearly related to rule 1 but evolves different particle interactions
that are suitable for both tasks. It can solve the AND task well wothou
sacrificing the density task.
Rule 8: 0504058F050025750337578B7BFFB77F , P_Dens(Rule
6) = 0.77 and P_AND (rule 5) = 0.75, average 0.76. Which
has results symmetrical to rule 6.
Now, very interestingly, in these simulations I evolved a rule that solves the density task better than rule 1. Rule 7: 05040587050025770377578B7BFFB77F with P_DEns(Rule 7) = 0.79 and P_AND (Rule 7) = 0.66. It did not stick around because its fitness for the AND task is not very good. But it is VERY good with the density task. Have you evolved this one before? It is clearly related to rule 1.
Other neat rules evolved include Rule 9: 0544058705000E770375554373FF977F, with P_Dens(Rule9) = 0.62 and P_AND (Rule 9) = 0.83, Average = 0.72. This rule has a set of particles that is excellent for the AND task, but not so good for the density task:
It is related to Rule 1, but with a few more differences:
Notice that when rules are evolved with a fitness function designed tol solve both the density and the AND task, it is not an option to come up with rules that tend to 0 a lot. Such rules would have good P_AND, but lousy P_Dens, besides, I defined a fitness function (see I) that avoids such rules.
A few other rules evolved were also interesting, but you can check those
out in the text file attached to this.
It is also interesting to note that the evolution of rules that can solve the two tasks evolved rule 7 which is excellent for the density task, and rule 9 which is excellent for the AND task.
Regarding what we discussed earlier, it is great to note that the particle computation system can evolutionarily adapt to perform more complicated computations that require the solution of two similar but distinct tasks. There is clearly evolutionary advantage to this adaptation potential. Furthermore, solving different tasks makes the case that these particles set up a proper syntax stronger.
Now I should try to evolve other logical gates. I want to incorporate these results into the paper I gave you eventually. Let me know what you think.
# Rule P_Dens P_AND Avg. File Notes
1 - 0504058705000F77037755837BFFB77F 0.77 0.71 0.74 rule1.txt Original Particle
2 - 0504058705000E77037755837AFFB77F 0.70 0.79 0.75 rule2.txt particle (popped up in the same run of 3 but did not stick because it was evolved with AND alone)
3 - 2D0008000000008F77FFBFFF27F7FFFF 0.58 0.80 0.69 rule3.txt block exp with dif boundaries. Evolved with AND alone with good rules from dens injected at start (tends to all 0)
4 - 0B42D6BF647FA7336EFA083C3C599A1F 0.50 0.79 0.65 rule4.txt Tends to 0 always. Evolved with AND only and random initial pop.
5 - 0504058505000D77073755437BFFB77F 0.75 0.77 0.76 rule5.txt Particle. Evolved with both AND/Density.
6 - 0500058705000777037757817BFFB77F 0.70 0.80 0.75 rule6.txt Particle. Evolved with both AND/Density. Much better for AND.
7 - 05040587050025770377578B7BFFB77F 0.79 0.66 0.73 rule7.txt Particle. Evolved with both AND/Density. Solves Density REALLY well, not so good for AND.
8 - 0504058F050025750337578B7BFFB77F 0.77 0.75 0.76 rule8.txt Particle. Evolved with both AND/Density. Good for both. Simetrical with 5.
9 - 0544058705000E770375554373FF977F 0.62 0.83 0.72 rule9.txt particle. Evolved with both AND/Density. Very Good for AND, VERY bad for Density.
10 - 05000587050007570373758B7BFFB77F 0.73 0.79 0.76 rule10.txt particle. Evolved with both AND/Density. Good average.
11 - 05000587250007770373158B7BFFB77F 0.74 0.77 0.76 rule11.txt Particle. Evolved with both AND/Density.
12 - 0504058301020F7F037755A37BFFB37F 0.74 0.77 0.75 rule12.txt Particle. Evolved with both AND/Density.
13 - 0504058705000F77037555837BFFA77F 0.71 0.80 0.75 rule13.txt Particle. Evolved with both AND/Density.
14 - 050405C705000F77017755837BBFB77F 0.74 0.79 0.76 rule14.txt Particle. Evolved with both AND/Density.
15 - 050405C705000F77077755C13BFFB77F 0.77 0.67 0.72 rule15.txt Particle. Evolved with both AND/Density. Very Good for Density Task.
16 - 0504058704000F73037755937BFFF77F 0.76 0.77 0.77 rule16.txt Particle. Evolved with both AND/Density. Very Good for Both.
17 - 0504258704000F73077755937BFFF77F 0.78 0.60 0.69 rule17.txt Particle. Evolved with both AND/Density. Very Good for Density.
18 - 0504059505001F7F037455833BFFF77F 0.78 0.72 0.75 rule18.txt Particle. Evolved with both AND/Density. Very Good for Density and Not bad for AND.