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8
AnnaPorębska
2.CELLULARAUTOMATARULES
Wecandescribeour2-Dbinarycellularautomatonasaquadruple:
A=(L
2,V
N,Q,f)
(1)
where:Q={0,1}-thesetofstates,LxL-thelatticedimension,VN-von
Neumannneighbourhoodf-atransitionfunction,rule.
Thetransitionfunctionisdefinedontheneighbouringcellsstates(markedby
geographicaldirectionsconnectedwiththeirpositioninthenet,CCmarksthe
centralcellstate-seeFig.1.).
signalconnections
neighbourhoodVN
cellCC(•)
forCC(•)
Fig.1.5-elementvonNeumannneighbourhoodofthecellCC(•),•-i,jfor2DCA,
markedbydotelinecontour
Weadditionallyassumethatarulefisasymmetricfunctionwithrespecttothe
0-1symmetry.
f
(
CC
()()()()()
t
,
N
t
,
W
t
,
S
t
,
E
t
,
)
=
CC
(
t
+
1
)
(2)
Thesequenceϑ(t)=(CC(t),E(t),N(t),W(t),S(t))iscalledaconfiguration.Inthe
caseofassumedsymmetryonlyahalfofallpossibleconfigurationsarevalidfor
furtheranalysis[4].Wecangroupthemaccordingtothevalueofsum5-the
numberof“1”cellsinconfiguration.For5-elementsconfigurationsthereare1
configurationwithsum5=0,5configurationswithsum5=1and10configurations
withsum5=2.Ifthesum5>2functionsaresymmetricin“0-1symmetry”tothe
previousones.Theareaswithdifferentnumberof“1”inconfigurationswithinthe
truetableofarulearepresentedinFig.2.
Thewholetransitionfunctionfcanbedecomposedtothetriplet(f0,f1,f2)where
f0,f1,f2arethesub-functionsdefinedforseparategroupsofconfigurations
mentionedabove.Itmeanstherulehasformulaf0ifthesum5=0,formulaf1ifthe
sum5=1andformulaf2ifthesum5=2.Becauseofassumedsymmetryforsum5
biggerthan2fisnegationofpreviousfunctions.Changingoneofthesefunctions
generatesthenewcellularautomaton.
