We investigate the state change behavior of one-dimensional cellular automata during the solution of the binary density-classification task. Update rules of high, low and un- known fitness are applied to cellular au- tomata, thereby providing examples of high and low rates of successful classification. A spread factor, ω, is introduced and investi- gated as a numerical marker of state change behavior. The nature of ω describes complex or particle-like behavior on the part of the cellular automata over the middle region of initial configuration density-distribution, but breaks down at the ends. Because of the lim- itation on ω, a related jump-out term, jot, is selected for incorporation into the finess func- tion for genetic algorithm evolution of update rules. The inclusion of jot in the fitness func- tion significantly reduces the number of gen- erations required to reach high rates of suc- cessful classification (≥90%)

Blank, Doug

Rolniak, Douglas

Thompson, Dale R.

Ward, Em

English

Scholarship, Research, and Creative Work at Bryn Mawr College | Bryn Mawr College Research

Bryn Mawr CollegeScholarship, Research, and Creative Work at Bryn MawrCollegeComputer Science Faculty Research andScholarship Computer Science2001Complexity as Fitness for Evolved CellularAutomata Update RulesEm WardDoug BlankBryn Mawr College, dblank@brynmawr.eduDouglas RolniakDale R. ThompsonLet us know how access to this document benefits you.Follow this and additional works at: http://repository.brynmawr.edu/compsci_pubsPart of the Computer Sciences CommonsThis paper is posted at Scholarship, Research, and Creative Work at Bryn Mawr College. http://repository.brynmawr.edu/compsci_pubs/33For more information, please contact repository@brynmawr.edu.Custom CitationWard, Em, Blank, Douglas S., Rolniak, Douglas, and Thompson, Dale R. (2001). Complexity as Fitness for Evolved Cellular AutomataUpdate Rules. In Late Breaking Papers of the 2001 Genetic and Evolutionary Computation Conference.Complexity as Fitness for Evolved Cellular Automata Update RulesEm WardEE DepartmentUniversity of ArkansasFayetteville, AR 72701eward@uark.eduDouglas S. BlankCSCE DepartmentUniversity of ArkansasFayetteville, AR 72701dblank@uark.eduDouglas RolniakCSCE DepartmentUniversity of ArkansasFayetteville, AR 72701drolnia@uark.eduDale R. ThompsonCSCE DepartmentUniversity of ArkansasFayetteville, AR 72701drt@uark.eduAbstractWe investigate the state change behavior ofone-dimensional cellular automata during thesolution of the binary density-classificationtask. Update rules of high, low and un-known fitness are applied to cellular au-tomata, thereby providing examples of highand low rates of successful classification. Aspread factor, ω, is introduced and investi-gated as a numerical marker of state changebehavior. The nature of ω describes complexor particle-like behavior on the part of thecellular automata over the middle region ofinitial configuration density-distribution, butbreaks down at the ends. Because of the lim-itation on ω, a related jump-out term, jot, isselected for incorporation into the finess func-tion for genetic algorithm evolution of updaterules. The inclusion of jot in the fitness func-tion significantly reduces the number of gen-erations required to reach high rates of suc-cessful classification (≥90%).1 INTRODUCTIONThis paper investigates state changes of one-dimensional, binary, cellular automata (CA) throughtime and space during attempts to solve the density-classification task. CAs are viewed as models of dy-namical systems and computation [10], [11], [4]. Thestate of each cell may then change over time. Thefrequency of state changes and the spatial distribu-tion of the state changes may reflect complex behaviorby the CA [12], [6]. Our study looks at these statechanges in CAs updated by rules evolved by a geneticalgorithm (GA) for good performance in the density-classification task and compares them to those of CAsupdated by evolved rules with poor performance. Wedescribe our attempt to find a simple measure that canrepresent complex, or particle-like, behavior by CA,its inclusion in the fitness function for evolving updaterules and its effect on evolution time.1.1 BACKGROUNDCellular automata are arrays, or lattices, of cells. Eachcell may be in a finite number of states. At discretetime steps, the states of the cells are changed accordingto update rules [10], [11]. The solution of the density-classification task may be viewed as nontrivial compu-tation that requires a CA to determine the majoritystate in its initial configuration (IC) and update itselfwithin a given number of time steps to achieve themajority state in every cell [9]. Figure 1 shows a time-space diagram of this computation. The IC contains amajority of 0 states (off or white). Correct classifica-tion would require all cells in the lattice to achieve a 0state. The CA has a solution when two successive timesteps have identical states for all cells. The maximumnumber of time steps allowed for a solution is prob-abilistically determined as a function of the numberof cells in the CA. Previous authors have examinedthis scheme and concluded that CAs performing thedensity-classification task exhibit a range of behaviors[12], [6], [9], [3]. When the behavior is one of self-organized cell state changes through time and space,it is called ”complex”. or particle-like [12], [9], [5].Genetic algorithms have been used to evolve theserules [9], [8]. Mitchell, Hraber and Crutchfield de-scribed a GA approach by Packard [9]. Previous ap-plication of GA-evolved update rules have had lowerperformance than the Gacs-Kurdyamov-Levin (GKL)rule, with approximately 95% best performance of cor-rect classification, with performance decreasing as lat-tice size increases [9], [8], [5]. The GKL, though notdesigned for the density-classification task, has prop-erties (all 1s or all 0s attraction states) that lend itFigure 1: Diagram of a cellular automaton (CA) solv-ing density-classification.to successful application to the problem [2], [9]. TheGKL rule consistently maintains approximately 98%average performance on increasingly large lattice sizesand number of trials [9], [8]. Investigators at the SantaFe Institute found an evolved rule, φpar, which inducedself-organized behavior and performed similarly to theGKL rule [8].1.2 RATIONALEThe rationale for examining cellular state changesthrough time is based on previous assertions that com-plex behavior, described by Wolfram as propagating,localized structures, sometimes long-lived, is requiredfor computation [12], [6]. Different parameters havebeen offered in the literature as markers of complexbehavior, including entropy, mutual-site informationand difference-pattern spreading factor [7], [6], [1].We investigate a spread factor, ω, and a jump outterm, jot, that represent simpler, more intuitive mea-sures of complex behavior. Incorporation of some mea-sure of complexity into the fitness function of the GArewards behavior. Presumably, this would speed evo-lution of high-performance update rules.2 METHODS2.1 UPDATE RULE GENERATIONWe used the following parameters for the evolution ofupdate rules: intital configuration (IC) length N =149, radius r = 3, population size p = 100, numberof generations g = 100, and the number of IC trialst = 100.Each set of randomly generated ICs had uniform dis-tribution of 1s and 0s. The radius is the number ofcells to either side of the index cell, which constitutedthe neighborhood of the index cell. The lattice shouldbe considered circular, as the neighborhoods wraparound. The neighborhood of 1s and 0s, read from leftto right, served as rule table input. The inputs wereordered in the rule table from 0 to 22r+1. A geneticalgorithm evolved the output column of the rule table.Fixed-point crossover, followed by two-point mutationwas the means of reproduction. Twenty percent ofthe rule table outputs with highest fitness were keptin the population for each generation. The remain-ing 80% of the population were replaced with the pro-duced children. Fitness was defined as the numberof ICs correctly classified by M time steps, where Mwas a Poisson distribution with mean twice the initialconfiguration length, 2N .2.2 STATE CHANGE INVESTIGATIONFrom the evolved population of rule table outputs(rules) five with the highest fitness and five with thelowest fitness were fitness-tested on a very large set oftrials, t = 10, 000. This was done to insure we selectedtruly good performers and truly poor performers touse for the state change investigation. The good ruleshad fitness ≥89% (mean = 90.52%), and the five badrules had fitness ≤51% (mean = 31.05%) on the 10,000trials. In addition to these ten, five random rules weregenerated. The random rules were created as a pop-ulation with no evolution. Five random rules and theGKL (fitness = 97.69%) rule were used as benchmarkcomparisons to the findings from the good and badrules.Each of the 16 total rules was then applied to ten trialICs. The state change characteristics of these ICs wereexamined. We defined the state change frequency, f ,as the number of state changes of a cell divided by thenumber of time steps before the automaton reacheda fixed point or the maximum number of time stepsallowed (M).We analyzed f with the following measures: mean andlattice range (the maximum f in the automaton latticeminus the minimum f in the automaton lattice). Ourmeasure of interest, ω, was intended to be a measure,a spread factor, of f across the lattice space and wasdefined asω = range/mean. (1)Here is an example calculation - For cell #14, therewere six state changes before a fixed point was reachedat the 12th time step: f14 = 6/12 = 0.5. Across thelattice (149 cells), the minimum f (of 149) was 0 (nostate change in at least one cell), and the maximum f(of 149) was 0.833 (ten state changes in at least onecell). Thus, the range is 0.833. Let the mean f acrossthe lattice be fave = 0.425:ω = 0.833/0.425 = 1.960. (2)Since each set of ICs contained a uniform distributionof 1s, ω could give a divide-by-zero error if the ICs wereall 0s. Also, at the ends of the spectrum, ω could bevery large, but not representative of complex behav-ior. Therefore, we also examined ω over the sectionof the distribution, where 1s density ranges from 10%to 90% (ωˆ). This includes the region of greatest dif-ficulty for classification, where 1s density approaches50%. Statistics were examined for each rule and eachrule group (good, bad, random, GKL).2.3 COMPLEXITY AS FITNESSFUNCTIONFor incorporation into the fitness function, ω was ex-amined and rejected because of the problem with highω values on the ends of the IC density spectrum. Thisposed a problem with online or realtime evolution, as acorrect classification also could be rewarded for a highω value for solving the almost-all-1s or almost-all-0stasks. Therefore, we introduced a jump-out term, jot,defined asjot = m/M, (3)where m is the number of time steps to a solutionand M is the maximum allowed time steps. jot cap-tures the idea that complex behavior by CA requiresmore computational time (long-transients, or long-lived propagating structures). The normalization wasrequired for jot to be less than one, since the term wasadded to the original performance fitness function, toreward complex behavior, only for a correct classifica-tion. Thus, the new fitness function was a real num-ber with integer and fractional components. In thisscheme, fitness could be greater than 100 (<101), butthe performance, equal to the integer component ofthe fitness divided by total number of trials, could notexceed 100%.For evaluation of a complexity measure incorporatedinto the fitness function, 30 runs with the same pa-rameters describe above, were performed with each ofTable 1: Spread Factors, ω and ωˆ.Parameter Rule Group ω ωˆ n nˆGood 6.620 6.493 50 40Bad 2.702 2.959 50 40Random 3.266 3.932 50 40GKL 7.514 7.294 10 8n and nˆ are number of trials for calculating ω and ωˆ,respectively.Table 2: Generations to High Performance (≥90%).Fitness Function mean s. d. varianceStandard 34.733 50.053 2505.306jot 12.133 5.894 34.740Random 26.067 41.460 1718.892s. d. = standard deviationthree different fitness functions: standard, standardwith jot and standard with a random number addedto the right of the decimal point. The test group of100 ICs was maintained for all functions.3 RESULTSTable 1 contains ω and ωˆ for each rule group. ω wascalculated replacing 0/0 (range = 0, mean = 0) termswith ones. One can clearly appreciate the difference inω for the good and bad rule groups (6.620 and 2.702,respectively). The GKL rule spread factor (ω = 7.514)was close in value to the good rule group. The randomrule group has ω similar to the bad rule group–3.266.The values for ωˆ are close to those for ω in each rulegroup.Figure 2 is a representative plot of state change fre-quency, f across the cell lattice for a computation froma good rule (ω = 5.787, averaged over ten ICs. Fig-ure 3 is f from a bad rule (ω = 0.152), averaged overten ICs. These two plots show the frequency rangeacross the lattice and will be discussed in the nextsection. There were instances in all groups where theminimum and maximum frequencies and the mean fre-quency were zero. In our samples, a bad rule rarelyproduced plots with state changes every single timestep.Table 2 shows the comparisons between the standard,standard with jot, and standard with random fitnessfunctions with regard to number of generations to highperformance (≥90%).00.050.10.150.20.250.30 20 40 60 80 100 120 140 160"good4.avg.vbd"Figure 2: State change frequency, f , across space, goodrule (ω = 5.787). The vertical axis is f ; the horizontalaxis is cell number of the lattice. f is averaged overten ICs.0.8720.8740.8760.8780.880.8820.8840.8860.8880.890.8920.8940 20 40 60 80 100 120 140 160"bad4.avg.vbd"Figure 3: State change frequency, f , across space, badrule (ω = 0.152). The vertical axis is f ; the horizontalaxis is cell number of the lattice. f is averaged overten ICs.Table 3: Data Analysis of ω.Comparison Result Confidenceωg vs. ωb ωg > ωb >99.9%ωg vs. ωr ωg > ωr >99.9%ωg vs. ωGKL ωg = ωGKL NRSubscripts indicate the following: g - good rule group,b - bad rule group, r - random rule group, and GKL -GKL rule group. NR = not rejected4 DISCUSSION4.1 DATA ANALYSISDifference of means tests for ω were applied to threegroup comparisons: good vs. bad, good vs. randomand good vs GKL. The hypotheses tested were thatthe average ω for good rules are significantly higherthan for bad or random rules and the same as theGKL rule. Table 3 contains the findings. With a highdegree of confidence (> 99.9%), the spread factor, ω,is significantly larger for a high rate of classificationsuccess than it is with a low rate of classification suc-cess. The analysis of ωˆ gave the same results, exceptthe confidence of ωˆg > ωˆr was 98.8%.4.2 COMPLEX BEHAVIORIn reports of computational models, the dynamical be-havior of the computing entity has been studied [12],[6], [9]. Wolfram described four classes of behavior:I. Homogeneous stateII. Separated simple stable or periodicstructuresIII. Chaotic patternIV. Complex localized structures [12].In our investigation, ω and ωˆ represent patterns ofstate changes of a CA through time. Our results in-dicate that successful computation, or correct classi-fication, occurs more reliably when lattice segmentsundergoing a high state change frequency are inter-spersed with segments of low state change frequency(see Figure 2). Classification is less successful when fis relatively uniform across the lattice (see Figure 3).This is consistent with the patterns of long-transientspreviously described [12], [6], [9], [8].Table 4: Data Analysis of Fitness Functions.Comparison Result Confidencestandard vs. jot standard > jot =99.3%standard vs. rnd standard = rnd NRrnd = random, NR = not rejected4.3 FITNESS FUNCTIONEvolved GA update rules have had best performanceranging from 90% to 95%, occurring, generally, bythe generation 40 [9]. When an evolved rule inducesparticle-like, or complex, behavior in the CA, the num-ber of generations to high performance decreases [8].We believe that our spread factor, ω, represents suchbehavior on the part of the CA. However, a limitationto ω is that it will be falsely high at the ends of the 1sdensity spectrum of the IC. This is a byproduct of theforced bias of uniform distribution of states across theinitial configuration–a move to reduce the likelihoodof having to classify difficult densities (near 0.5). Thisproblem can be reduced with a form of ωˆ as describedabove, though ωˆ is not easily incorporated into the fit-ness function during online or realtime evolution. Webelieve ω would speed evolution on ICs with randomdistribution, since the probability of almost-all-1s andalmost-all-0s would be very low. This is left as futurework. Another comparison to investigate is the rangeof f (the numerator of ω) as a fitness function, remov-ing the problem with the denominator being zero orvery small, giving very high ω.A simpler parameter, jot, represents computation timeby the CA, an indicator of complex behavior. Whenjot is incorporated into the fitness function, complex-ity, in addition to performance, is awarded. Whenthis happens, the number of generations to high per-formance is significantly reduced. Table 4 contains theanalysis results for the number of generations to highperformance for these comparisons: standard vs. jotand standard vs. random. Our samples did not pro-vide support for rejecting the hypothesis that speedof evolution is the same for the standard fitness andstandard with random number added.5 CONCLUSIONSThis paper describes an investigation into the statechange frequency of cellular automata during perfor-mance of the density classification task. We intro-duced a spread factor, ω, whose value was significantlyhigher when CAs had a high rates of successful clas-sification compared to low rates. ω is the range ofstate change frequencies of the cells in the CA latticedivided by the average state change frequency. It isa numerical representation of the activity of the CAand consistent with previous descriptions of complexbehavior required for computation.A jump out term, jot, captures the idea that com-plex behavior by CA requires more computational time(long-transients, or long-lived propagating structures).When jot is incorporated into the fitness function forevolving CA update rules with a GA, the number ofgenerations to high performance rules is significantlyreduced.AcknowledgmentsWe thank Ross Williamson for his significant program-ming contributions and Russell Deaton for his helpfulcommentary.References[1] R. Badii and A. Politi. Thermodynamics andcomplexity of cellular automata. Physical ReviewLetters, 78(3):444–447, 1997.[2] P. Gonzaga de Sa and C. Maes. The Gacs-Kurdyumov-Levin automaton revisited. Journalof Statistical Physics, 67(3/4):507–522, 1992.[3] M. Garzon. Models of Massive Parallelism: Anal-ysis of Cellular Automata and Neural Networks,chapter 8, Classification, pages 141–166. Springer,New York, 1995.[4] J.R. Koza. Genetic Programming, chapter 14,Entropy-Driven Evolution, pages 395–417. MITPress, 1992.[5] M. Land and R.K. Belew. No perfect two-statecellular automata for density classification exists.Physical Review Letters, 74(25):5148–5150, 1995.[6] C.D. Langton. Computation at the edge of chaos:Phase transisitons and emergent computation.Physica D, 42:12–37, 1990.[7] W. Li, N.H. Packard, and C.G. Langton. Transi-tion phenomena in cellular automata rule space.Physica D, 45:77–94, 1990.[8] M. Mitchell, J.P. Crutchfield, and R. Das. Evolv-ing cellular automata with genetic algorithms: Areview of recent work. In Proceedings of the FirstInternational Conference on Evolutionary Com-putation and its Applications, Moscow, 1996. Rus-sian Academy of Sciences.[9] M. Mitchell, P.T. Hraber, and J.P. Crutchfield.Revisiting the edge of chaos: Evolving cellular au-tomata to perform computations. Complex Sys-tems, 7:89–130, 1993.[10] J. von Neumann. John von Neumann CollectedWorks, chapter 9, The General and Logical The-ory of Automata, pages 288–328. PergamonPress, New York, 1961.[11] S. Wolfram. Statistical mechanics of cellular au-tomata. Reviews of Modern Physics, 55:601–644,1983.[12] S. Wolfram. Universality and complexity in cel-lular automata. Physica D, 10:1–35, 1984.

Complexity as Fitness for Evolved Cellular Automata Update Rules

Scholarship;  Research;  and Creative Work at Bryn Mawr College

Bryn Mawr College
Scholarship, Research, and Creative Work at Bryn Mawr
College
Computer Science Faculty Research and
Scholarship Computer Science
2001
Complexity as Fitness for Evolved Cellular
Automata Update Rules
Em Ward
Doug Blank
Bryn Mawr College, dblank@brynmawr.edu
Douglas Rolniak
Dale R. Thompson
Let us know how access to this document benefits you.
Follow this and additional works at: http://repository.brynmawr.edu/compsci_pubs
Part of the Computer Sciences Commons
This paper is posted at Scholarship, Research, and Creative Work at Bryn Mawr College. http://repository.brynmawr.edu/compsci_pubs/33
For more information, please contact repository@brynmawr.edu.
Custom Citation
Ward, Em, Blank, Douglas S., Rolniak, Douglas, and Thompson, Dale R. (2001). Complexity as Fitness for Evolved Cellular Automata
Update Rules. In Late Breaking Papers of the 2001 Genetic and Evolutionary Computation Conference.
Complexity as Fitness for Evolved Cellular Automata Update Rules
Em Ward
EE Department
University of Arkansas
Fayetteville, AR 72701
eward@uark.edu
Douglas S. Blank
CSCE Department
University of Arkansas
Fayetteville, AR 72701
dblank@uark.edu
Douglas Rolniak
CSCE Department
University of Arkansas
Fayetteville, AR 72701
drolnia@uark.edu
Dale R. Thompson
CSCE Department
University of Arkansas
Fayetteville, AR 72701
drt@uark.edu
Abstract
We investigate the state change behavior of
one-dimensional cellular automata during the
solution of the binary density-classification
task. Update rules of high, low and un-
known fitness are applied to cellular au-
tomata, thereby providing examples of high
and low rates of successful classification. A
spread factor, ω, is introduced and investi-
gated as a numerical marker of state change
behavior. The nature of ω describes complex
or particle-like behavior on the part of the
cellular automata over the middle region of
initial configuration density-distribution, but
breaks down at the ends. Because of the lim-
itation on ω, a related jump-out term, jot, is
selected for incorporation into the finess func-
tion for genetic algorithm evolution of update
rules. The inclusion of jot in the fitness func-
tion significantly reduces the number of gen-
erations required to reach high rates of suc-
cessful classification (≥90%).
1 INTRODUCTION
This paper investigates state changes of one-
dimensional, binary, cellular automata (CA) through
time and space during attempts to solve the density-
classification task. CAs are viewed as models of dy-
namical systems and computation [10], [11], [4]. The
state of each cell may then change over time. The
frequency of state changes and the spatial distribu-
tion of the state changes may reflect complex behavior
by the CA [12], [6]. Our study looks at these state
changes in CAs updated by rules evolved by a genetic
algorithm (GA) for good performance in the density-
classification task and compares them to those of CAs
updated by evolved rules with poor performance. We
describe our attempt to find a simple measure that can
represent complex, or particle-like, behavior by CA,
its inclusion in the fitness function for evolving update
rules and its effect on evolution time.
1.1 BACKGROUND
Cellular automata are arrays, or lattices, of cells. Each
cell may be in a finite number of states. At discrete
time steps, the states of the cells are changed according
to update rules [10], [11]. The solution of the density-
classification task may be viewed as nontrivial compu-
tation that requires a CA to determine the majority
state in its initial configuration (IC) and update itself
within a given number of time steps to achieve the
majority state in every cell [9]. Figure 1 shows a time-
space diagram of this computation. The IC contains a
majority of 0 states (off or white). Correct classifica-
tion would require all cells in the lattice to achieve a 0
state. The CA has a solution when two successive time
steps have identical states for all cells. The maximum
number of time steps allowed for a solution is prob-
abilistically determined as a function of the number
of cells in the CA. Previous authors have examined
this scheme and concluded that CAs performing the
density-classification task exhibit a range of behaviors
[12], [6], [9], [3]. When the behavior is one of self-
organized cell state changes through time and space,
it is called ”complex”. or particle-like [12], [9], [5].
Genetic algorithms have been used to evolve these
rules [9], [8]. Mitchell, Hraber and Crutchfield de-
scribed a GA approach by Packard [9]. Previous ap-
plication of GA-evolved update rules have had lower
performance than the Gacs-Kurdyamov-Levin (GKL)
rule, with approximately 95% best performance of cor-
rect classification, with performance decreasing as lat-
tice size increases [9], [8], [5]. The GKL, though not
designed for the density-classification task, has prop-
erties (all 1s or all 0s attraction states) that lend it
Figure 1: Diagram of a cellular automaton (CA) solv-
ing density-classification.
to successful application to the problem [2], [9]. The
GKL rule consistently maintains approximately 98%
average performance on increasingly large lattice sizes
and number of trials [9], [8]. Investigators at the Santa
Fe Institute found an evolved rule, φpar, which induced
self-organized behavior and performed similarly to the
GKL rule [8].
1.2 RATIONALE
The rationale for examining cellular state changes
through time is based on previous assertions that com-
plex behavior, described by Wolfram as propagating,
localized structures, sometimes long-lived, is required
for computation [12], [6]. Different parameters have
been offered in the literature as markers of complex
behavior, including entropy, mutual-site information
and difference-pattern spreading factor [7], [6], [1].
We investigate a spread factor, ω, and a jump out
term, jot, that represent simpler, more intuitive mea-
sures of complex behavior. Incorporation of some mea-
sure of complexity into the fitness function of the GA
rewards behavior. Presumably, this would speed evo-
lution of high-performance update rules.
2 METHODS
2.1 UPDATE RULE GENERATION
We used the following parameters for the evolution of
update rules: intital configuration (IC) length N =
149, radius r = 3, population size p = 100, number
of generations g = 100, and the number of IC trials
t = 100.
Each set of randomly generated ICs had uniform dis-
tribution of 1s and 0s. The radius is the number of
cells to either side of the index cell, which constituted
the neighborhood of the index cell. The lattice should
be considered circular, as the neighborhoods wrap
around. The neighborhood of 1s and 0s, read from left
to right, served as rule table input. The inputs were
ordered in the rule table from 0 to 22r+1. A genetic
algorithm evolved the output column of the rule table.
Fixed-point crossover, followed by two-point mutation
was the means of reproduction. Twenty percent of
the rule table outputs with highest fitness were kept
in the population for each generation. The remain-
ing 80% of the population were replaced with the pro-
duced children. Fitness was defined as the number
of ICs correctly classified by M time steps, where M
was a Poisson distribution with mean twice the initial
configuration length, 2N .
2.2 STATE CHANGE INVESTIGATION
From the evolved population of rule table outputs
(rules) five with the highest fitness and five with the
lowest fitness were fitness-tested on a very large set of
trials, t = 10, 000. This was done to insure we selected
truly good performers and truly poor performers to
use for the state change investigation. The good rules
had fitness ≥89% (mean = 90.52%), and the five bad
rules had fitness ≤51% (mean = 31.05%) on the 10,000
trials. In addition to these ten, five random rules were
generated. The random rules were created as a pop-
ulation with no evolution. Five random rules and the
GKL (fitness = 97.69%) rule were used as benchmark
comparisons to the findings from the good and bad
rules.
Each of the 16 total rules was then applied to ten trial
ICs. The state change characteristics of these ICs were
examined. We defined the state change frequency, f ,
as the number of state changes of a cell divided by the
number of time steps before the automaton reached
a fixed point or the maximum number of time steps
allowed (M).
We analyzed f with the following measures: mean and
lattice range (the maximum f in the automaton lattice
minus the minimum f in the automaton lattice). Our
measure of interest, ω, was intended to be a measure,
a spread factor, of f across the lattice space and was
defined as
ω = range/mean. (1)
Here is an example calculation - For cell #14, there
were six state changes before a fixed point was reached
at the 12th time step: f14 = 6/12 = 0.5. Across the
lattice (149 cells), the minimum f (of 149) was 0 (no
state change in at least one cell), and the maximum f
(of 149) was 0.833 (ten state changes in at least one
cell). Thus, the range is 0.833. Let the mean f across
the lattice be fave = 0.425:
ω = 0.833/0.425 = 1.960. (2)
Since each set of ICs contained a uniform distribution
of 1s, ω could give a divide-by-zero error if the ICs were
all 0s. Also, at the ends of the spectrum, ω could be
very large, but not representative of complex behav-
ior. Therefore, we also examined ω over the section
of the distribution, where 1s density ranges from 10%
to 90% (ωˆ). This includes the region of greatest dif-
ficulty for classification, where 1s density approaches
50%. Statistics were examined for each rule and each
rule group (good, bad, random, GKL).
2.3 COMPLEXITY AS FITNESS
FUNCTION
For incorporation into the fitness function, ω was ex-
amined and rejected because of the problem with high
ω values on the ends of the IC density spectrum. This
posed a problem with online or realtime evolution, as a
correct classification also could be rewarded for a high
ω value for solving the almost-all-1s or almost-all-0s
tasks. Therefore, we introduced a jump-out term, jot,
defined as
jot = m/M, (3)
where m is the number of time steps to a solution
and M is the maximum allowed time steps. jot cap-
tures the idea that complex behavior by CA requires
more computational time (long-transients, or long-
lived propagating structures). The normalization was
required for jot to be less than one, since the term was
added to the original performance fitness function, to
reward complex behavior, only for a correct classifica-
tion. Thus, the new fitness function was a real num-
ber with integer and fractional components. In this
scheme, fitness could be greater than 100 (<101), but
the performance, equal to the integer component of
the fitness divided by total number of trials, could not
exceed 100%.
For evaluation of a complexity measure incorporated
into the fitness function, 30 runs with the same pa-
rameters describe above, were performed with each of
Table 1: Spread Factors, ω and ωˆ.
Parameter Rule Group ω ωˆ n nˆ
Good 6.620 6.493 50 40
Bad 2.702 2.959 50 40
Random 3.266 3.932 50 40
GKL 7.514 7.294 10 8
n and nˆ are number of trials for calculating ω and ωˆ,
respectively.
Table 2: Generations to High Performance (≥90%).
Fitness Function mean s. d. variance
Standard 34.733 50.053 2505.306
jot 12.133 5.894 34.740
Random 26.067 41.460 1718.892
s. d. = standard deviation
three different fitness functions: standard, standard
with jot and standard with a random number added
to the right of the decimal point. The test group of
100 ICs was maintained for all functions.
3 RESULTS
Table 1 contains ω and ωˆ for each rule group. ω was
calculated replacing 0/0 (range = 0, mean = 0) terms
with ones. One can clearly appreciate the difference in
ω for the good and bad rule groups (6.620 and 2.702,
respectively). The GKL rule spread factor (ω = 7.514)
was close in value to the good rule group. The random
rule group has ω similar to the bad rule group–3.266.
The values for ωˆ are close to those for ω in each rule
group.
Figure 2 is a representative plot of state change fre-
quency, f across the cell lattice for a computation from
a good rule (ω = 5.787, averaged over ten ICs. Fig-
ure 3 is f from a bad rule (ω = 0.152), averaged over
ten ICs. These two plots show the frequency range
across the lattice and will be discussed in the next
section. There were instances in all groups where the
minimum and maximum frequencies and the mean fre-
quency were zero. In our samples, a bad rule rarely
produced plots with state changes every single time
step.
Table 2 shows the comparisons between the standard,
standard with jot, and standard with random fitness
functions with regard to number of generations to high
performance (≥90%).
00.05
0.1
0.15
0.2
0.25
0.3
0 20 40 60 80 100 120 140 160
"good4.avg.vbd"
Figure 2: State change frequency, f , across space, good
rule (ω = 5.787). The vertical axis is f ; the horizontal
axis is cell number of the lattice. f is averaged over
ten ICs.
0.872
0.874
0.876
0.878
0.88
0.882
0.884
0.886
0.888
0.89
0.892
0.894
0 20 40 60 80 100 120 140 160
"bad4.avg.vbd"
Figure 3: State change frequency, f , across space, bad
rule (ω = 0.152). The vertical axis is f ; the horizontal
axis is cell number of the lattice. f is averaged over
ten ICs.
Table 3: Data Analysis of ω.
Comparison Result Confidence
ωg vs. ωb ωg > ωb >99.9%
ωg vs. ωr ωg > ωr >99.9%
ωg vs. ωGKL ωg = ωGKL NR
Subscripts indicate the following: g - good rule group,
b - bad rule group, r - random rule group, and GKL -
GKL rule group. NR = not rejected
4 DISCUSSION
4.1 DATA ANALYSIS
Difference of means tests for ω were applied to three
group comparisons: good vs. bad, good vs. random
and good vs GKL. The hypotheses tested were that
the average ω for good rules are significantly higher
than for bad or random rules and the same as the
GKL rule. Table 3 contains the findings. With a high
degree of confidence (> 99.9%), the spread factor, ω,
is significantly larger for a high rate of classification
success than it is with a low rate of classification suc-
cess. The analysis of ωˆ gave the same results, except
the confidence of ωˆg > ωˆr was 98.8%.
4.2 COMPLEX BEHAVIOR
In reports of computational models, the dynamical be-
havior of the computing entity has been studied [12],
[6], [9]. Wolfram described four classes of behavior:
I. Homogeneous state
II. Separated simple stable or periodic
structures
III. Chaotic pattern
IV. Complex localized structures [12].
In our investigation, ω and ωˆ represent patterns of
state changes of a CA through time. Our results in-
dicate that successful computation, or correct classi-
fication, occurs more reliably when lattice segments
undergoing a high state change frequency are inter-
spersed with segments of low state change frequency
(see Figure 2). Classification is less successful when f
is relatively uniform across the lattice (see Figure 3).
This is consistent with the patterns of long-transients
previously described [12], [6], [9], [8].
Table 4: Data Analysis of Fitness Functions.
Comparison Result Confidence
standard vs. jot standard > jot =99.3%
standard vs. rnd standard = rnd NR
rnd = random, NR = not rejected
4.3 FITNESS FUNCTION
Evolved GA update rules have had best performance
ranging from 90% to 95%, occurring, generally, by
the generation 40 [9]. When an evolved rule induces
particle-like, or complex, behavior in the CA, the num-
ber of generations to high performance decreases [8].
We believe that our spread factor, ω, represents such
behavior on the part of the CA. However, a limitation
to ω is that it will be falsely high at the ends of the 1s
density spectrum of the IC. This is a byproduct of the
forced bias of uniform distribution of states across the
initial configuration–a move to reduce the likelihood
of having to classify difficult densities (near 0.5). This
problem can be reduced with a form of ωˆ as described
above, though ωˆ is not easily incorporated into the fit-
ness function during online or realtime evolution. We
believe ω would speed evolution on ICs with random
distribution, since the probability of almost-all-1s and
almost-all-0s would be very low. This is left as future
work. Another comparison to investigate is the range
of f (the numerator of ω) as a fitness function, remov-
ing the problem with the denominator being zero or
very small, giving very high ω.
A simpler parameter, jot, represents computation time
by the CA, an indicator of complex behavior. When
jot is incorporated into the fitness function, complex-
ity, in addition to performance, is awarded. When
this happens, the number of generations to high per-
formance is significantly reduced. Table 4 contains the
analysis results for the number of generations to high
performance for these comparisons: standard vs. jot
and standard vs. random. Our samples did not pro-
vide support for rejecting the hypothesis that speed
of evolution is the same for the standard fitness and
standard with random number added.
5 CONCLUSIONS
This paper describes an investigation into the state
change frequency of cellular automata during perfor-
mance of the density classification task. We intro-
duced a spread factor, ω, whose value was significantly
higher when CAs had a high rates of successful clas-
sification compared to low rates. ω is the range of
state change frequencies of the cells in the CA lattice
divided by the average state change frequency. It is
a numerical representation of the activity of the CA
and consistent with previous descriptions of complex
behavior required for computation.
A jump out term, jot, captures the idea that com-
plex behavior by CA requires more computational time
(long-transients, or long-lived propagating structures).
When jot is incorporated into the fitness function for
evolving CA update rules with a GA, the number of
generations to high performance rules is significantly
reduced.
Acknowledgments
We thank Ross Williamson for his significant program-
ming contributions and Russell Deaton for his helpful
commentary.
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