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10
AnnaPorębska
“1”cellisshiftedleftorrightorupordown,butalwaysitisashift.Eachruleis
connectedwiththerotationofgraphoritsmirrorreflection.Inconsequencethe
knowledgeofcellularautomatonbehaviourforoneoftheserulesenablesusto
inferenceaboutbehaviourwiththerestoftransitionfunctions.Thus,wecan
narrowsearchingofthespaceofcellularautomatatotheanalysisofrulesspace
representativespossiblefordifferentgraphs.
Theruleisf2ifthesum5=2.Itmeansthatform,m=0,..,10,from10such
configurations(TableI)thenewstateofcellis“1”(or“0”insymmetriccase).Thuseach
ofthoserulesbelongstothesubclasswhichcanbemarkas
reg
()
10
m
.Thetruthtable
contains“1”valuesonmpositionsingreyareaconnectedwithsum5=2(seeFig.2).
f2∈{
reg
()
10
0
,
reg
()
10
1
,
reg
()
10
2
,
reg
()
10
3
,
reg
()
10
4
,
reg
()
10
5
,
reg
()
10
6
,
reg
()
10
7
,
reg
()
10
8
,
reg
()
9
10
,
reg
()
10
10
}
(3)
Foreachof
reg
()
10
m
m=1,...,10subclasseswecancreateappropriategraphs.
Case
reg
()
10
1
Thereare10rulesinthissubclassonly.3differentgraphs-seeFig.3.-are
possiblefor3groupsof1configuration.
Fig.3.Graphsmadefor
reg
()
10
1
typerules
Fourrulesareconnectedwiththeleftgraph,fourwiththerightgraphandtwo
withthecentralgraph.Thusthereareonly3rulesforwhichthebehaviourof
cellularautomataisdifferentiated.Eachgraphhas2nodesand1branch.
Case
reg
()
10
2
Thereare45rulesinthissubclass.9differentgraphs-seeFig.4.-arepossible
for10groupsof2configurations.
Eightrulesareconnectedwitheachof(a)graphs,fourwitheachof(b)graphs,
twowiththeleftof(c)graphs,onewiththecentralof(c)graphsand6withthe
rightofthem.Thusthereare10groupsofrulesforwhichthebehaviourofcellular
automataisdifferentiated.Eachgraphhasupto4nodesand2branches.
