Proof-Theoretic Cellular Automata as Logic of Unconventional Computing

Abstract

Conventional logic assumes that all basic logical processes (deduction, valuation) are sequential. Almost all modern logical systems are conventional in this meaning. However, everywhere in the nature we can observe massively parallel processes that might be simulated by cellular automata. In unconventional computing there have been proposed different computational models in which massive parallelism is taken into account. Nevertheless, the question what the logic of unconventional computing is should be considered as open still. In this chapter proof-theoretic automata are proposed, where all basic logical processes (deduction, valuation) are regarded as massive-parallel ones. In these automata cell states are examined as well-formed formulas of a logical language. As a result, in deduction we can obtain not only derivation trees, but also a linear evolution of each singular premise. Thereby some derivation traces may be circular, i.e. some premises could be derivable from themselves, and hence some derivation traces may be infinite.

Full text

Proof-Theoretic Cellular Automata as Logic of Unconventionl
Computing
Andrew Schumann
UniversityofInformationTechnologyandManagementinRzeszow,Poland.Email:
Andrew.Schumann@gmail.com
Abstract
Conventionallogicassumesthatallbasiclogicalprocesses(deduction,valuation)aresequential.Almostallmodern
logicalsystemsareconventionalinthismeaning.However,everywhereinthenaturewecanobservemassively
parallelprocessesthatmightbesimulatedbycellularautomata.Inunconventionalcomputingtherehavebeen
proposeddifferentcomputationalmodelsinwhichmassiveparallelismistakenintoaccount.Nevertheless,the
questionwhatthelogicofunconventionalcomputingisshouldbeconsideredasopenstill.Inthischapterproof‐
theoreticautomataareproposed,whereallbasiclogicalprocesses(deduction,valuation)areregardedasmassive‐
parallelones.Intheseautomatacellstatesareexaminedaswell‐formedformulasofalogicallanguage.Asaresult,in
deductionwecanobtainnotonlyderivationtrees,butalsoalinearevolutionofeachsingularpremise.Therebysome
derivationtracesmaybecircular,i.e.somepremisescouldbederivablefromthemselves,andhencesomederivation
tracesmaybeinfinite.
1. Introduction
TheBiblebeginswiththeverse:“InthebeginningGodcreatedtheheavenandtheearth[ץראה תאו םימשה תאםיהלא ארב
תישארב].”Judaichermeneuticsemphasizesthatinthepassagetheword‘et’isusedthatmeansnotonlyanaccusative
case,butalsoaconnectionwhichhasasimilarmeaningastheword‘with.’Thus,accordingtoJudaichermeneutics
‘theheaven’(‘etha‐shamaim’)shouldbeunderstoodastheskyincludingeverythingthatisrelatedtothegiven
notion,i.e.anythingthatisoutoftheearth:theSun,planets,stars,comets,galaxies,atomsofhydrogen,oxygenetc.,
and‘theearth’(‘etha‐arets’)astheearthandeverythingthattakesplaceontheearth:mountains,rivers,seas,oceans
andsoon.Inotherwords,accordingtoJudaictraditiontheworldhasnotarisenfromasingularpointbymeansof
deploymentofanysequentialprocesses.Theworldhasbeencreatedatonceasthemassive‐parallel.
Whileitisdifficulttosay,howtheUniverseexistsinfact,neverthelessthisexamplewithtraditionalhermeneutics
showsthattheideaofmassiveparallelismofnaturalprocessesexistsforalongtime.Theelementarywayof
simulatingmassiveparallelisminthenatureispresentedbycellularautomata.StephenWolframclaimsinhis
famousbookANewKindofScience[1]thatallbasicphysicalandbiologicalprocessesmayberegardedascomputer’s
programs,namelyascellular‐automaticsimulations.Inthischapterwediscussthepossibilitytoestablishthe
unconventionallogic,i.e.thelogicofunconventionalcomputingthatfirstcanbeusedforsimulatinganynatural
processandsecondallitsbasiclogicalprocessessuchasdeductionandvaluationaremassivelyparallel.
Recallthatanycellularautomatonconsistsofcellsbelongingtothesetd
Z,therebyeachofcelltakesitsvalueinS,
afiniteorinfinitesetofelementscalledthestatesofanautomaton.Usually,cellsareconsideredasunchangeable,but
theirstateschangepermanently.ThisdynamicsdependsonlocaltransitionruleSSn→
+1
:
δ
thattransformsstates
ofcellstakingintoaccountstatesofnneighborcells.Theorderedsetof nelements,N,issaidtobea
neighborhood.Eachstepofdynamicsisfixedbydiscretetime K0,1,2,=t
Atthemomentt,theconfigurationofthewholesystem(ortheglobalstate)isgivenbythemappingt
xfromd
Z
intoS,andtheevolutionisthesequenceK,,, 210 xxx definedasfollows:
))(,),(),((=)( 1
1n
tttt zxzxzxzx
ααδ
++
+K,where1
α
〈
,...,N
n
∈
〉
α
.Here0
xistheinitialconfiguration,andit
fullydeterminesthefuturebehavioroftheautomaton.
Letusconsideranexampleofcellularautomataforamassive‐parallelpresentationoflogicalconjunction(AND).
Assumethatitscellsbelongto2
Zandeachcellcanhavejustoneoftwostates:tobeeithertrueorfalse.Ifthecellis
black,thismeansitistrue,andifitiswhite,itisfalse.Theneighborhoodofourcellularautomatonconsistsoftwo
members:theleftandrightneighborforeachcell.Thelocaltransitionrulethatembodiesconjunctionsaysthatthe
cellpreservestheblackcolorifandonlyifbothitsneighborsareblackandthewhitecelldoesnotchangeitscolor.In
caseofinitialconfigurationpresentedinFigure1,wehaveonlytwostepsofevolution.Thentheautomatonstops
becauseallcellsbecomewhite.


t=0





Figur
e

This
e
oflo
g
massi
possi
b
differ
e
claim
possi
b
aprio
r
cause
s
circu
m
confi
g
Neve
r
One
o
paral
l
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y
activ
e
grey
c
activ
e
theg
r
swer
v
grey
a


0





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Figur
e

Also
w
move
area
c


0

 
e1.Anexam
p
e
xampleofce
l
g
icalnotionsd
i
velyparallel
b
ilitiesarec
o
e
ntvaluesof
thatalltheb
a
b
lecases.Th
e
r
istic.Onthe
c
sallfurther
d
m
stancesalll
g
uration,i.e.
o
r
theless,ther
e
o
ftheseauto
m
l
el.Themove
m
y
onlytothea
c
e
remainthes
a
c
ells.Thenei
g
e
cellmaybe
l
r
eyneighbors
;
v
esrightupp
e
a
ndhasgrey
n
123


e2.Anexam
p
w
ecouldus
e
inparallelin
c
tive.Theirbe
h
123
t=

p
leofcellular
l
lular‐automa
t
iffersfromth
e
thinkingth
e
o
nsideredap
r
atomicprop
o
a
sicnotionso
f
e
reisnoinit
i
c
ontrary,inu
n
d
ynamics.H
e
ogicalnotion
s
o
nourexperie
n
e
couldbepr
o
m
ataiscalle
d
m
entofsingl
e
c
tivecell,and
a
mefromon
e
g
hborhoodc
o
l
ocatedonlyi
n
;
itswervesif
e
riftheright
u
n
eighbors;the
45

p
leofmobile
a
e
so‐calledco
n
thiskindof
a
h
aviorisexe
m
45
1 

automatonfo
t
icconsiderat
i
e
convention
a
e
dynamics
o
r
iori,e.g.in
s
o
sitionsinflue
f
conventiona
i
alconfigurat
i
n
conventiona
l
e
rebyeachst
e
s
inunconve
n
n
ce.
o
posedalsoh
y
d
mobileauto
m
e
celldepend
s
alsospecify
h
e
generationt
o
o
nsistsoftwo
n
thegreycel
l
r
ight(atthe
n
u
ppercellis
g
activecellst
o

a
utomatonth
a
n
currentauto
m
utomata.Let
u
m
plifiedinFi
g
678
t=2
rconjunction
i
onofconjun
c
a
lone.First,i
n
o
fthewhol
e
s
emanticval
u
e
ncesonthe
v
a
llogicarea
p
t
ionofthew
h
llogicwefoc
u
e
pofdynami
c
n
tionallogic
y
bridcellular
m
ata[1]that
h
s
onitsenvir
o
h
owtheactiv
e
o
thenext.As
members:u
p
l
withthegre
y
n
extstep)the
c
g
reyandhast
h
o
psifitcanno
t

a
tstopsatste
p
m
ata
,
whenm
o
u
sconsidert
h
g
ure3.


thathaltsats
c
tionshowsh
o
n
standardse
q
e
system.Se
c
u
ationofa
w
v
alueofthe
w
p
riori,theydo
h
olethatsho
u
soninitialc
o
c
sisunique
a
couldbesai
d
automata,w
h
h
aveasingle
o
nment.Ina
e
cellmovesfr
anexample,l
p
perandlow
e
y
neighbors;i
t
c
ellisgrey,b
u
h
egreyneigh
b
t
moveorsw
e
p
4
o
rethanone
h
epreviouse
x
t=3
t
ep3
o
wdeepthe
m
q
uentialthink
i
c
ond,incon
v
w
ell‐formedf
o
w
holeproposi
t
notdepend
o
u
ldbetaken
o
nfiguration
o
a
ndevolutio
n
d
tobeapos
t
h
ereweaccen
activecelli
n
mobileauto
m
o
monestept
o
e
tusconside
r
e
rcell.Thetr
a
t
movesright
u
tonlyoneof
i
b
orsandrigh
t
e
rve(seeFigu
r
cellisactive.
x
ampleofmo
b
m
assive‐paral
l
i
ngIanalyses
v
entionallo
g
o
rmulaIrega
t
ion.Duetot
h
o
nourexperi
e
intoaccoun
t
o
fthewhole
s
n
maybeinfi
n
teriori
,
they
d
n
tondynamic
n
steadofup
d
m
aton,theloc
a
othenext.Al
l
r
anautomat
o
ansitionrule
iftherightc
e
itsneighbors
t
lowerifthe
r
r
e2).
Hence,som
e
b
ileautomato
n
l
elunderstan
d
sequencesan
g
icallseque
n
rdallcases
h
h
isfactwec
o
e
nceandcove
r
t
.Itwouldb
e
s
ystem,becau
s
n
ite.Undert
h
d
ependonin
sofsinglece
l
d
atingallcell
a
ltransitionr
u
l
cellsthatare
o
nwithwhite
isasfollows:
e
llisgreyand
i
sgrey,there
b
r
ightlowerc
e
e
butnotall
c
n
wheretwo
c
d
ing
din
n
tial
h
ow
o
uld
r
all
e
so
s
eit
h
ese
itial
l
l(s).
sin
u
les
not
and
the
has
b
yit
e
llis
c
ells
c
ells











Figur
e
cells.
A

Thel
a
sever
a
cells
(
thetr
a
same
rema
i
theg
r
auto
m
Mobi
l
proce
theor
e
massi
embo
Inthi
s
three
Phys
a
0









Figur
e
2. Pr
o
Fora
n
syste
m
auto
m

e3.Anexam
p
A
tstep7the
a
a
stkindofh
y
a
linitialcells
a
(
atstep0)ass
u
a
nsitionrule
a
andweaddt
h
i
nsthesame
a
r
eycellhast
h
m
atonispictu
r
l
e,concurrent
sseswhicha
r
e
ticcellulara
i
velyparallel
dyinguncon
v
s
chapterwe
s
simulations
a
rumpolycep
h
123
 

e4.
o
of-theoretic
n
ylogicalla
n
m
s)simulati
n
m
aton
A
isc
o
p
leofconcurr
e
a
utomatonha
l
y
bridautom
a
a
ndaddsom
e
u
mingthatt
h
a
sfollows:if
t
h
enewgrey
c
a
ndweaddt
h
h
ewhiteneig
h
r
edinFigure
4
andspatiala
u
r
eobservede
v
utomatawhe
abstractphe
n
v
entionallogi
c
s
ketchsome
b
onthebasis
h
alum.Allth
e


cellular aut
o
n
guage
L
we
n
gmassive‐p
a
o
llectedfrom
w
e
ntautomato
n
l
ts
a
taweareg
o
e
additionalc
e
h
eneighborh
o
t
hegreycell
h
c
ellupper;ift
h
h
enewgrey
c
h
bor,itbeco
m
4
.
u
tomataares
v
erywherein
realllogical
n
omenasot
h
c
ofproof‐the
o
asicdefinitio
n
ofthislogi
c
e
seexamples
a

o
mata
canconstruc
a
rallelproofs
.
w
ell‐formedf
o
n
withtwoac
t
o
ingtoconsi
d
e
llsbytheloc
a
o
odconsistso
f
h
asonlyoneg
h
egreycellh
a
c
elllower;ift
h
m
eswhite;th
e
s
implederiva
t
thenature.
O
processessu
c
h
atrealmass
i
o
reticcellular
n
sofunconve
c
:epidemic
s
a
ssumefeedb
a
c
taproof‐theo
r
.Inaproof
‐
o
rmulasofal
t
ivecells.Ats
d
erispresent
e
a
ltransitionr
u
f
twomembe
r
reyneighbor
a
sonlyonegr
h
egreycell
h
e
whitecelle
v
t
ionsofcellul
a
O
urclaimist
h
c
hasdeducti
i
veparallelis
m
automata.
ntionallogic
a
s
preading,B
e
a
ckrelations
w
r
eticcellulara
u
‐
theoreticcell
anguage
L
a
n
t
ep4weobse
r
e
dbyspatial
u
le.Forinsta
n
r
s:upperand
c
ellanditisl
o
e
yneighborc
e
asbothgrey
n
v
erremainst
h
a
rautomatai
n
h
atwecould
o
n,derivatio
n
m
inthewo
r
a
ndexemplif
y
lousov‐Zhab
o
w
hicharemo
d
u
tomaton(inst
e
u
larautomat
n
dtheroleof
rvethefusio
n
automata
,
w
h
n
ce,letustak
e
lowercell.
T
ower,thenth
e
ellanditisu
p
n
eighbors,it
b
h
esame.The
n
simulating
m
introduceth
e
n
,valuation
a
r
ldpresents
d
y
howtheno
v
o
tinskyreact
i
d
eledbycirc
u
e
adofconve
n
t
ontheset
S
localtransiti
o
n
oftwoactiv
e
h
erewehave
e
threeinitial
g
T
henletusde
e
cellremain
s
p
per,thenthe
b
ecomeswhi
t
dynamicsof
m
assivelypar
a
e
notionofpr
o
a
reconsidere
d
d
ifferentway
s
v
ellogicwork
s
on,dynamic
s
u
larproofs.
n
tionaldedu
c
S
ofstateso
f
o
nfunction,
δ
e

just
g
rey
fine
s
the
cell
t
e;if
this
a
llel
o
of‐
d
as
s
of
s
by
s
of
c
tive
f
an
δ
,is

playedbytheinferenceruleofalanguageL.Theinitialconfigurationof
A
isthesetofallpremises(notaxioms)
anditfullydeterminesthefuturebehavioroftheautomaton.Weassumethat
δ
isaninferencerule,i.e.amapping
fromthesetofpremises(theirnumbercannotexceed|=| Nn )toaconclusion.Foranyd
zZ∈thesequence
)(
0zx ,)(
1zx ,…,)( zxt,…iscalledaderivationtracefromastate)(
0zx .Ifthereexiststsuchthat
)(=)( zxzx lt foralltl >,thenaderivationtraceisfinite.Itiscircular/cyclicifthereexistslsuchthat
)(=)( zxzx ltt +forallt.Incaseallderivationtracesofaproof‐theoreticcellularautomatonAarecircular,this
automaton
A
issaidtobereversible.
1.Cellular‐automaticpresentationofmodusponens.ConsiderapropositionallanguageLthatisbuiltinthestandard
waywiththeonlybinaryoperationofimplication
⊃
.Letussupposethatwell‐formedformulasofthatlanguageareusedasthe
setofstatesforaproof‐theoreticcellularautomaton
A
.Further,assumethatmodusponensisatransitionruleofthis
automaton
A
anditisformulatedforany
ϕ
,L
∈
ψ
asfollows:

⎪
⎩
⎪
⎨
⎧+∈⊃
+
.),(
);(=)(,
=)(
1
otherwisezx
Nzandzxif
zx
t
t
t
ϕψϕψ

ThefurtherdynamicscanbeexemplifiedbytheevolutionofcellstatesinFig.5–Fig.7.

r
q
p
⊃⊃ )
(

)
(qpp ⊃
⊃
 qp
⊃

)
(
)
(qpqp
⊃
⊃
⊃

r
p
r
⊃
⊃
)
(
)
(
)
(
r
q
r
p⊃⊃⊃  q
p
⊃
 
p
 )
(
q
p
p
⊃
⊃

p
r
⊃

r
p⊃ p
)
)(( qpqp
⊃
⊃
⊃
p
r

)
(
r
q
p
⊃⊃  pp
⊃
 qp
⊃

)
(
)
(pq
r
p
⊃
⊃
⊃

r
p
⊃

qp ⊃ )( pqp ⊃
⊃
 q
r
p
⊃
 p
Figure5.Aninitialconfigurationofaproof‐theoreticcellularautomatonAwiththeneighborhoodconsistingof8
membersinthe2‐dimensionalspace,itsstatesrunoverformulassetupinapropositionallanguageLwiththeonly
binaryoperation⊃,0=t.Noticethat
r
q
p
,, arepropositionalvariables


r
 qp ⊃ q qp
⊃

r

r
q⊃ q p qp
⊃
 p
r

p
 )
(
q
p
q
⊃
⊃

p

r

r
q⊃ p q pq
⊃

r

qp ⊃ )
(
p
q
p
⊃⊃  q
r
 p
Figure6.Anevolutionof
A
describedinFig.5atthetimestep1=t




r
 q q q
r

r
 q p q p
r

p
 q
p

r

r

p
 q
p

r

q
p
 q
r

p

Figure7.AnevolutionofAdescribedinFig.5atthetimestep3=t.Itsconfigurationcannotvaryfurther
Thisexampleshowsthatfirstwecompletelyavoidaxiomsandsecondlywetakepremisesfromthecellstatesofthe
neighborhoodaccordingtoatransitionfunction.Asaresult,wedonotcomeacrossprooftreesinournovel
approachtodeductiontakingintoaccountthatacellstatehasjustalineardynamics(thenumberofcellsandtheir
locationdonotchange).Thisallowsusevidentlytosimplifydeductivesystems.
2.Concurrent‐automaticpresentationofmodusponens.Letusconsideraconcurrentmodificationofcellularautomaton
describedinthepreviousexample.Inthenewautomatonwehavetwoactivecells:orangeandpurple.Theymoveconcurrently.If
theneighborcellisthesecondpremiseofmodusponenswhiletheactivecellisthefirstpremise,thenatthenextsteptheactive
cellislocatedontheplaceofthisneighbor.Theneighborhoodisorderedclockwiseandifseveralneighborcellsarethesecond
premiseofmodusponensatthesametime,thenatthenextsteptheactivecelltakesplaceofthefirstcelloftheseneighbors.The
concurrentautomatonofmodusponensmaybeillustratedbytheevolutionofcellstatesinFig.8–Fig.10.
3.ExampleofHilbertʹsinferencerules.SupposeapropositionallanguageLcontainstwobasicpropositionaloperations:
negationanddisjunction.Asusual,thesetofallformulasofLisregardedasthesetofstatesofanappropriateproof‐theoretic
cellularautomata.InthatwewillusetheexclusivedisjunctionofthefollowingfiveinferencerulesconvertedfromJosephR.
Shoenfieldʹsdeductivesystem:

⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎨
⎧
+∈∨∨¬∨
+∈∨¬∨∨
∨∨∨∨
∨
∨
+
).()(=)(,
);()(=)(,
);(=)(,)(
;=)(,
;=)(,
=)(
1
Nzandzxif
Nzandzxif
zxif
zxif
zxif
zx
t
t
t
t
t
t
χϕψϕψχ
ψϕχϕψχ
ϕψχϕψχ
ϕϕϕ
ϕϕψ

r
q
p
⊃⊃ )
(

)
(qpp ⊃
⊃
 q
p
⊃

)
(
)
(qpqp
⊃
⊃
⊃

r
p
r
⊃
⊃
)
(
)
(
)
(
r
q
r
p⊃⊃⊃  qp
⊃
 p )
(
q
p
p
⊃
⊃
 p
r
⊃

r
p
⊃
p

)
)(( qpqp
⊃
⊃
⊃

p

r

)
(
r
q
p
⊃⊃  pp
⊃
 qp
⊃

)
(
)
(pq
r
p
⊃
⊃
⊃

r
p
⊃

q
p
⊃ )( pqp ⊃
⊃
 q
r
p
⊃

p

Figure8.Aninitialconfigurationofaproof‐theoreticconcurrentautomaton
A
withtheneighborhoodconsistingof
8membersinthe2‐dimensionalspace.CellsofautomatonarethesameasinFig.5–7.Inadditionthisautomaton
hastwoconcurrentactivecells:orangeandpurple





r
q
p
⊃⊃ )
(

)
(qpp ⊃
⊃
 qp
⊃

)
(
)
(qpqp
⊃
⊃
⊃

r
p
r
⊃⊃
)
(
)
(
)
(
r
q
r
p⊃⊃⊃  qp
⊃
 p )
(
q
p
p
⊃
⊃
 p
r
⊃
r
p
⊃
p

)
)(( qpqp
⊃
⊃
⊃

p

r

)
(
r
q
p
⊃⊃ 
p
p
⊃
 q
p
⊃

)
(
)
(pq
r
p
⊃
⊃
⊃

r
p
⊃
qp ⊃ )( pqp ⊃
⊃
 q
r
p
⊃
 p
Figure9.AnevolutionofAdescribedinFig.8atthetimestep1=t

r
q
p
⊃⊃ )
(

)
(qpp ⊃
⊃
 qp
⊃

)
(
)
(qpqp
⊃
⊃
⊃

r
p
r
⊃
⊃
)
(
)
(
)
(
r
q
r
p⊃⊃⊃  q
p
⊃
 
p
 )
(
q
p
p
⊃
⊃

p
r
⊃

r
p⊃ p
)
)(( qpqp
⊃
⊃
⊃
p
r

)
(
r
q
p
⊃⊃ 
p
p
⊃
 q
p
⊃

)
(
)
(pq
r
p
⊃
⊃
⊃

r
p
⊃

qp ⊃ )( pqp ⊃
⊃
 q
r
p
⊃
 p
Figure10.AnevolutionofAdescribedinFig.8atthetimestep2=t.Wehaveafusionoftwoactivecellsintothe
oneandhaltingoftheautomaton
Wehavejustconstructedthetheoryofproofsinwhichthereisnoplacefortheoremsorprovableformulas(apriori
propositions),formoredetailssee[2].Wecangofurtherandconstructaformallanguageinwhichtherearenobasic
notionsofclassicalsemantics,e.g.therearenovalidformulas.
Thenovellanguagewillbecalledunconventionalornon‐well‐funded[3].Itsformulasaredefinedasfollows:
• Eachatomicformulaisaformula.
• IfФ1∧Ф2orФ1∨Ф2orФ1⇒Ф2or¬Ф1,¬Ф2areformulas,thenФ1,Ф2areformulas.
Theabovedefinitionassumesthattheformulaisasyntacticobjectwhichdoesnotsatisfytheset‐theoreticaxiomof
foundation.Itmeansthatthegivensyntacticobjectscanalreadybeofaninfinitelengthandcomprisecycles.For
example,cyclicexpressions(Ф1⇒(Ф2⇒(Ф1⇒(Ф2⇒(…)))))or(Ф1∧(Ф1∧(Ф1∧(Ф1∧(…)))))arewellformed
formulas.Theseexpressionscanbedefinedbyrecursion,too:
- Theformulaofinfinitelength(Ф1⇒(Ф2⇒(Ф1⇒(Ф2⇒(…)))))isequivalenttothecyclicdefinitionФ=(Ф1⇒
(Ф2⇒Ф)).
- Theformulaofinfinitelength(Ф1∧(Ф1∧(Ф1∧(Ф1∧(…)))))isequivalenttothecyclicdefinitionФ=(Ф1∧Ф).
Meaningsofunconventional‐logicalconnectives∧,∨,⇒,¬aredefinednotbytruetablesasthisproceedsusually,
butbyhybridcellularautomata.Letustakenowacellularautomaton
A
,wherethesetSofcellstatesiscollected
fromtwotruthvalues1and0forwell‐formedformulasofalanguageL.ThelocaltransitionruleSSn→
+1
:
δ

thatis
{}
¬
⇒∨∧∈ ,,,
δ
isthetruthvaluationforeachlogicaloperationofalanguageL.
Definition1.Atruthvaluationofconjunctionisatransitionruleoftheautomaton
A
,where
{}
0,1=S,anditis
formulatedforanyformulaФ1∧Ф2L∈asfollows:

⎪
⎩
⎪
⎨
⎧+∉
+
.0,
);(01=)(1,
=)(
1
otherwise
Nzandzxif
zx
t
t

Definition2.AtruthvaluationofdisjunctionisatransitionruleoftheautomatonA,where
{}
0,1=S,anditis
formulatedforanyformulaФ1∨Ф2L∈asfollows:


⎪
⎩
⎪
⎨
⎧+∈
+
.0,
);(11=)(1,
=)(
1
otherwise
Nzorzxif
zx
t
t

Definition3.AtruthvaluationofimplicationisatransitionruleoftheautomatonA,where
{}
0,1=S,anditis
formulatedforanyformulaФ1⇒Ф2L∈asfollows:

⎪
⎩
⎪
⎨
⎧+∉
+
.1,
);(00=)(0,
=)(
1
otherwise
Nzandzxif
zx
t
t

Definition4.AtruthvaluationofnegationisatransitionruleoftheautomatonA,where
{}
0,1=S,anditis
formulatedforanyformula¬ФL∈asfollows:

⎪
⎩
⎪
⎨
⎧
+
.1,
1;=)(0,
=)(
1
otherwise
zxif
zx
t
t

Algorithmoftruthvaluation.LetФbeaformula.Ifitisatomic,thenitstruthvalueisaninitialconfigurationofacellular
automatonwiththesetofstates{1,0}.Ifitisnotatomic,thenwestarttheevaluationwiththemostoutsideconnectivek1.By
usingoneofdefinitions1–4thatcorrespondstok1,wetransformaninitialconfigurationofacellularautomatonwiththesetof
states{1,0}.Thistransformationisfixedbystepstk1=1,2,3,…Thenwemovetoamoreinsideconnectivek2.Foreachsteptk1=
1,2,3,…wetransformaconfigurationatsteptk1inaccordancewithoneofdefinitions1–4thatcorrespondstothenew
connectivek2.Thismeansthatforeachsteptk1weobtainanewcellularautomatonforthetruthevaluationofk2.Each
transformationofthatcellularautomatonisfixedbynewstepstk2=1,2,3,…Noticethatitispossiblethatsimultaneouslywe
havetwoinsideconnectivesk2andk2’ofthesamerange.Inthiscasewedoallthesame,butinparallelmanner.Further,wecan
movetoamoreinsideconnectivek3.Foreachsteptk2=1,2,3,…wetransformaconfigurationatthesteptk2inaccordancewith
oneofdefinitions1–4thatcorrespondstotheconnectivek3andsoon.
Toprovideanexample,letusevaluatetheformula(Ф1⇒(Ф2∧Ф1))inacellularautomaton
A
withthe
neighborhoodconsistingof8membersinthe2‐dimensionalspace,wheretheblackcelldesignatestrueandthewhite
false:
(I)Initialconfiguration,0=t



Webeginourevaluationwiththeconnective⇒asmostoutside.Therebyweareusingdefinition3:
(II) 1=t



So,wehaveonlyonetransformationfixedbystep1.Thenwemovetotheinsideconnective∧andbydefinition1we
obtainthefollowingdataover(II):
(III) 2=t   (IV) 3=t



Theautomaton(IV)cannotvaryfurther.Ifwehadanadditionalmoreinsideconnective,thenwewouldhavetwo
automata(III)and(IV)forfurthermodificationinaccordancewiththismoreinsideconnective.

3. T
h
Now
w
epide
m
susce
p
R
:
N
four
p
R
(‘
h
Thes
conta
c
infect
e
popu
l
indiv
i
transi
thetr
a
acycl
Whe
n
popu
l
susce
p
small
enou
g
thetr
a
andt
h
transi
again
Asa
n
auto
m
indiv
i
healt
h
t=0











h
e proof-theo
r
weconsider
s
m
icspreadin
g
p
tible
S
,
ac
l
SIN
N
∪=
p
ermittedsta
t
h
ealthyandu
n
tateofthei
n
c
tswithothe
r
e
dandnext
t
l
ation.Each
s
i
dualisinfec
t
i
tionthatanil
a
nsitionthat
a
einstatetran
n
thetransiti
o
l
ationisquic
k
p
tibleindivi
d
numberofst
a
g
h.Inthecas
e
a
nsition
S
→
h
espreading
i
tion
R
IL →
diesout.
n
exampleo
f
m
atonwhose
n
i
dual,thegre
y
h
yandsuscep
  



r
etic cellular
s
omenatural
e
g
.Letafixed
h
l
assofill
IL
,
RIL∪∪
.
T
t
es:
IN
(‘infe
n
susceptiblei
n

n
dividualsev
o
r
individuals.
t
heinfection
s
tatecanbet
r
t
ed
b
yanilli
n
lindividual
w
a
nunsuscepti
b
sitions:

o
n
I
L
IN →
k
lyinfected,t
h
d
ualscanbei
n
a
tes,theepid
e
e
when
R→
IN
→
.Ifthetr
a
processisst
o
R
coversthe
f
proof‐theor
e
n
eighborhoo
d
y
cellinfecte
d
tibleindivid
u



automaton
f
e
xamplesofp
r
h
umanpopul
a
andaclass
o
T
heneachcel
l
ctedindivid
u
n
dividual’)
w
+
=)(
1
zx
t
o
lvesintime
Whenoneil
l
spreadsouts
i
r
ansformedi
n
n
dividual,
I
N
w
illrecoveror
b
leindividua
l
L
coversthe
h
eninfectedi
n
n
fected,there
f
e
micspread
m
S
coversaf
e
a
nsition
S
→
o
pped,becau
s
moststates,
t
e
ticcellular
consistsof8
m
d
individual
,

t
u
al:



f
or epidemic
r
oof‐theoretic
ation
N

b
e
d
o
fhealthyan
d
l
ofourabstr
a
u
al’),
S
(‘hea
l
w
hichinteract
u
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
∨
),(
ILIN
R
=)(S
=
)(IN
=)(R
=
)(IL
other
w
zx xif
zxif
zxif
zxif zxif
t
t
t
t
t
t
anddepends
l
individuala
i
dethesecell
n
oneoffour
IL
N
→
atr
a
r
beisolatedf
r
l
loseitsimm
u
moststates
,
n
dividuals(st
a
foreepidemi
c
m
uchslower
a
e
wstates,the
i
IN
→
issprea
d
s
ethenumbe
r
t
henthenum
b
automatons
i
membersint
h
t
hepurplece
l

s
preadin
g
cellularauto
m
d
ividedintoa
d
unsusceptibl
a
ctmachinesi
m
thyandsusc
e
u
nderthefoll
o
∈
+∈
+
∈
.
IL,S=)(
(S,
R
(IN,S
=
IL;
IN;
=
w
ise
z
N
z
z
t
ontheirpre
v
ppearsinac
e
s
.Thisslows
ways:
I
S→
nsitionthati
n
r
omtheresto
f
u
nityandbec
INS→→
i.e.theepi
d
a
tes)become
u
c
diesout.O
n
a
ndthenum
b
i
nfectioncan
o
d
always,al
m
r
ofsuscepti
b
b
erofinfect
e
i
mulatingep
i
h
e2‐dimensi
o
l
lunsuscepti
b
m
ataandour
f
classofinfec
t
e
orisolated
f
m
ulatingepi
d
e
ptibleindivi
d
o
wingconditi
+
∈
∉
;)(
(ILIN,,)
;)
Nz
z
N
N
v
iousstatea
n
e
ll,firstindiv
downthes
p
I
N
meanst
h
n
fectedindivi
f
thepopulati
o
amehealthy
a
RIL →→
d
emicsprea
d
u
nsusceptible
n
theotherh
a
erofsuscepti
b
o
ccuronlyfo
r
ostthewhole
leindividual
s
dindividual
s
i
demicsprea
d
o
nalspace
,
w
h
b
leorisolated
firstillustrati
o
t
ed
IN
,
acl
a
f
romtherest
o
d
emicspread
i
d
ual’),
IL
(‘i
i
on:
+;)N
z

n
dtheconne
c
v
idualsfrom
n
p
readingpro
c
h
etransition
t
i
dualbecome
on(e.g.ina
h
a
ndsusceptib
l
.S

d
srapidlya
n
andthismea
n
a
nd,when
I
N
i
bleindividu
a
r
asmallnu
m
e
populationi
s
s
istoolow.
F
s
becomess
m
dingwec
o
h
eretheblack
individual,
a
o
nisdedicate
a
ssofhealthy
o
fthepopula
t
nghasoneo
f
l
lindividual’
)
c
tionsorran
d
n
eighborcells
c
essinthew
h
t
hatasuscep
t
ill,
RIL →
ospital),
R
→
l
e.Thus,we
h
n
dalmostw
h
n
sthatonlya
IL
N
→
cove
a
lsisalwaysl
a
b
erofstates
w
s
quicklyinfe
c
F
inally,when
m
allandepid
e
o
uldconside
r
celldesignat
e
a
ndthewhite
dto
and
t
ion
f
the
)
,or
d
om
are
h
ole
t
ible
the
S
→

h
ave
h
ole
few
rsa
a
rge
w
ith
c
ted
the
e
mic
r
an
e
sill
cell

t=1






Figur
e
4. T
h
The
m
comp
l
simpl
react
a
2
Br
,

massi
andt
h
color
syste
m
minu
t
in
w
+3
Ce
Whe
n
andt
h
b
rom
a
incre
a
B
r
O
proce

Figur
e
atlea
s
prem
i
ifthe
y
The
p
chem
i



e11.Thepro
o
h
e proof-theo
r
m
echanismof
licated:itsre
c
ificationof
B
a
nts:
+3
Ce
,
H
2
(CO
O
CH
i
ve‐parallels
y
h
ebehavioro
fromyellow
m
s(suchasa
t
eandarerep
w
hichceriu
m
→
→
+4
Ce
n

−
Br
hasb
e
h
eoxidizedf
o
a
te(
−
3
BrO
)
a
a
sebromide
(
−
3
O
)and
H
O
B
ssbeginsaga
i
2
HBrO
HOBr
e12.The
b
asi
c
s
tonedisjunc
t
i
seforfurther
y
areneighbo
r
p
roof‐theoreti
c
i
calmachine
[

o
f‐theoreticce
r
etic concur
r
theBelousov
‐
c
entmodelco
B
elousov‐Zha
b
2
H
BrO
,
B
r
O
2
)
OH
which
y
stemevident
l
fthewholes
y
tocolorless,
simplepetri
d
eatedoveral
o
m
changes
K→
→
+3
Ce
e
ensignifican
t
o
rmoftheme
t
a
nd
HOBr
.
(
−
Br
)conce
n
Br
toform
B
i
n.Thus,para
l
&&
&
&
2
3
2
3
4
3
Br
Br
HOBr
H
Br
H
BrO
HBrO CeBr
B
Ce C
e
Ce
∨
∨∨
∨
−
−
−
−
+
+
c
reactionsof
t
ivestatetake
deducing),a
n
r
s
c
simulation
o
[
4]orreaction
‐

l
lularautoma
t
r
ent automat
o
‐
Zhabotinsky
r
ntains80ele
m
b
otinskyrea
c
−
3
O
,
+
H
,
C
e
interactacc
o
l
yismuchm
o
y
stemismuc
h
allowingth
e
d
ish),theosci
l
o
ngperiodof
itsoxidati
.
t
lylowered,t
h
t
alioncatalys
Meanwhile,t
h
n
tration.Onc
e
2
Br
,further
B
l
lelprocesses
h
(&
(5)
(3)
(
&
&
22
3
2
4
2
B
rC
H
HC
O
CH
r
H
H
HCOO
H
CO
O
B
rC
H
HBrO
e
B
HBrO
∨
∨
∨
+
+
+
+
+
Belousov‐Zh
a
splace(inco
n
n
dthesign&
o
fBelousov‐Z
h
‐
diffusionco
m

tonforepide
m
o
n for Belou
s
reaction(na
m
m
entarysteps
c
tionassumi
n
+4
e
,
OH
2
,
B
o
rdingtobas
i
o
redifficultt
h
h
morecomp
l
e
oscillations
l
lationsprop
a
f
time.Thecol
i
onstate
f
h
ereactionc
a
s
tandindicat
o
t
henextstep
e
thebromid
e
2
Br
reactswit
h
haveseveral
c
)(
)
&
&
&)
&
2
2
2
3
2
22
2
2
3
COOH
H
O
O
H
Br
HOBr
HOB
r
BrO
H
CO
H
O
H
OH
OH H
B
rO
∨∨
∨
−
+−
a
botinskyrea
c
n
currentinfer
e
isafusionth
a
h
abotinskyr
e
m
puting[5],[
6
m
icspreadin
g
s
ov-Zhaboti
n
m
elycerium(II
I
and26varia
b
n
gthatthes
e
B
rCH

(CO
i
creactions
o
h
anthatinthe
l
icated.Inthi
s
tobeobser
v
a
gateasspira
l
o
rchangesar
e
romceriu
m
a
usesanexpo
n
o
r,cerium(IV)
reducesthec
e
e
concentrati
o
h

2
(CO
O
CH
c
ycleswhich
a
(7)
&
&
&
&
(2)
(1)
2
2
OH HBr
HBrO
r
HBr
H
O
∨
∨
+
−
+−
+
c
tion
,
wheret
h
e
nceruleswe
a
tmeansinco
e
actioncanbe
6
],[7]):
atstep0and
n
sk
y
reaction
I
)
↔
cerium
b
lespeciesco
n
e
tofstates
c
2
)OH
,
−
Br
o
fFig.12.T
h
previoussec
t
s
reactionwe
v
edvisually.
I
wavefronts.
e
causedbya
l
m
(III)toce
n
entialincrea
.Bromousaci
e
rium(IV)to
c
o
nishighen
o
2
)
OH
tofor
m
a
reperformed
(6)
(4)
+
h
esign∨isa
d
canuseany
m
n
currentauto
definedasf
o
1
m
(IV)catalyze
d
n
centrations.
c
onsistsjust
,
HCOOH
h
isexample
o
t
ion:therewe
observesud
d
Inspatially
n
Theoscillati
o
l
ternatingoxi
d
e
rium(IV)a
n
a
seinbromo
u
i
dissubsequ
e
c
erium(III)an
o
ugh,itreact
m

(C
O
BrCH
d
synchronou
s

d
isjunction,
w
m
emberofdi
s
o
matathefusi
o
o
llows(itisa
d
reaction)is
v
Letusconsid
ofthefollo
w
,

2
CO
,
H
O
B
o
fproof‐theo
r
havemorest
d
enoscillatio
n
n
onhomogen
e
o
nslastabout
d
ation‐reduct
i
n
dvicev
e
sacid(
HBr
O
e
ntlyconverte
d
dsimultaneo
u
s
withbrom
a
2
)
O
OH
an
d
ly.
w
hichmeanst
h
s
junctionas
o
nofactivec
e
generalizatio
n
v
ery
era
w
ing
Br
,
r
etic
ates
n
sin
e
ous
one
i
ons
e
rsa:
2
O
)
d
to
u
sly
a
te(
d
the
h
at
e
lls
n
of

Definition5.1Let+3
{= CeL ,2
HBrO ,−
3
BrO ,+
H
,+4
Ce ,OH2,BrCH 2
)(COOH ,−
Br
,HCOOH ,
2
CO ,HOBr,2
Br ,})( 22 COOHCH bethesetofstates.Thetwobasicoperations∨(disjunction)and&(fusion)are
definedbyinferencerulesinFig.12,whichdescribegeneralpropertiesoftransitions.Thentheproof‐theoreticsimulationof
Belousov‐Zhabotinskyreactionisaconcurrentautomaton.
Inthissystem,transitionsbetweenstatesareidentifiedwithderivationsdeterminedbyinitialvaluesofstates.Here
weobtainamassive‐parallelprocessofderivationtoo.Eachstepofderivationmeansatransition.Asaresult,the
circulartraceofstate+3
Ce (resp.+4
Ce )hasameaningofcircularproof,wherethestate+3
Ce (resp.+4
Ce )is
unfoldedinfinitelyoftenamongpremisesandatthesametimeamongderivableexpressions.
5. The proof-theoretic spatial automaton for dynamics of Plasmodium of Physarum polycephalum
ThedynamicsofplasmodiumofPhysarumpolycephalumcouldberegardedasanothersimpleexampleofthenatural
proof‐theoreticautomata.Thepointisthatwhentheplasmodiumiscultivatedonanutrient‐richsubstrate(agargel
containingcrushedoatflakes)itexhibitsuniformcirculargrowthsimilartotheexcitationwavesintheexcitable
Belousov‐Zhabotinskymedium.Ifthegrowthsubstratelacksnutrients,e.g.theplasmodiumiscultivatedonanon‐
nutrientandrepellentcontaininggel,awetfilterpaperorevenglasssurfacelocalizationsemergeandbranching
patternsbecomeclearlyvisible.
Theplasmodiumcontinuesitsspreading,reconfigurationanddevelopmentaslongasthereareenoughnutrients.
Whenthesupplyofnutrientsisover,theplasmodiumeitherswitchestofructificationstate(iflevelofilluminationis
highenough),whensporangiaareproduced,orformssclerotium(encapsulatesitselfinhardmembrane),ifin
darkness.
Thepseudopodiumpropagatesinamanneranalogoustotheformationofwave‐fragmentsinsub‐excitable
Belousov‐Zhabotinskysystems.Startingintheinitialconditionstheplasmodiumexhibitsforagingbehavior,
searchingforsourcesofnutrients.Whensuchsourcesarelocatedandtakenover,theplasmodiumforms
characteristicveinsofprotoplasm,whichcontractsperiodically.Belousov‐Zhabotinskyreactionandplasmodiumare
light‐sensitive,whichgivesusthemeanstoprogramthem.Physarumexhibitsarticulatednegativephototaxis,
Belousov‐Zhabotinskyreactionisinhibitedbylight.Thereforebyusingmasksofilluminationonecancontrolthe
dynamicsoflocalizationsinthesemedia.
ExperimentswithPhysarumpolycephalumwerecarriedoutbyProf.Adamatzky([8],[9])asfollows.Theplasmodiaof
Physarumpolycephalumwereculturedonwetpapertowels,fedwithoatflakes,andmoistenedregularly.He
subculturedtheplasmodiumevery5–7days.
ExperimentswereperformedinstandardPetridishes,9cmindiameter.Dependingonparticularexperimentshe
used2%agargelormoistenfilterpaper,nutrient‐poorsubstrates,and2%oatmealagar,nutrient‐richsubstrate
(Sigma‐Aldrich).Allexperimentswereconductedinaroomwithdiffusivelightof3–5cd/m,22°Ctemperature.In
eachexperimentanoatflakecolonizedbytheplasmodiumwasplacedonasubstrateinaPetridish,andfewintact
oatflakesdistributedonthesubstrate.Theintactoatflakesactedassourceofnutrients,attractantsforthe
plasmodium.PetridisheswithplasmodiumwerescannedonastandardHPscanner.Theonlyeditingdoneto
scannedimagesiscolorenhancement:increaseofsaturationandcontrast.
Resultsofexperimentsmaybedescribedintermsofproof‐theoreticspatialautomata.Letusassumethatitssetof
statesconsistsoftheentitiesfromthefollowingsets.
• Thesetofgrowingpseudopodia,1
{P,},
2KP,localizedinactivezones.Onanutrient‐richsubstrateplasmodium
propagatesasatypicalcircular,targetwave,whileonthenutrient‐poorsubstrateslocalizedwave‐fragmentsare
formed.
• Thesetofattractants},,{ 21 KAA ,theyaresourcesofnutrients,onwhichtheplasmodiumfeeds.Itisstill
subjectofdiscussionhowexactlyplasmodiumfeelspresenceofattracts,indeeddiffusionofsomekindis
involved.BasedonpreviousexperimentsbyProf.Adamatzkywecanassumethatifthewholeexperimental
areaisabout8—10cmindiameterthentheplasmodiumcanlocateandcolonizenearbysourcesofnutrients.
• Thesetofrepellents},,{ 21 KRR .PlasmodiumofPhysarumavoidslight.Thus,domainsofhighilluminationare
repellentssuchthateachrepellent
R
ischaracterizedbyitspositionandintensityofillumination,orforceof
repelling.
• Thesetofprotoplasmictubes},,{ 21 KCC .Typicallyplasmodiumspanssourcesofnutrientswithprotoplasmic
tubes/veins.Theplasmodiumbuildsaplanargraph,wherenodesaresourcesofnutrients,e.g.oatflakes,and
edgesareprotoplasmictubes.

Henc
e
polyc
e
Thep
Defi
n
setof
v
auto
m
Then
Defi
n
deduc
i
form
close
d
wefi
n
i
p
,
p
Ane
x
dime
n
(I)In
i






(III)






A
R
A
∨
R
e
,thesetof
e
phalumisequ
p
roof‐theoreti
c
n
ition6.Cons
v
ariables
=S
m
aton
A
.
A
ss
u

A
simulates
n
ition7.Let
i
ng
1
+
i
s

f
rom
Xp ∨
,
wh
e
d
undertheo
p
n
doutanexp
r
j
p
donotoc
c
x
ampleofthe
n
sionalspace
f
i
tialconfigur
a
2=t

A

R

∨
C
R

C
statesint
h
alto
21
,,{ PP
c
simulationo
f
ideraproposit
i
1
{P,,,
2
PK
u
methat0isth
e
+
x
t
thedynamic
s
1
{
,, ssp
ii
∈
+
i
s
byinferen
c
e
re
X
isa
p
p
eration
∨
.
T
r
ession
p
i
∨
c
urinothern
e
evolutionof
f
orPhysarum
p
a
tion,
0=t


A
P
A∨C
P
C
h
eproof‐theo
r
1
{} A∪
K
,
A
f
Physarumpo
l
i
onallanguage
,
1
A

,,
2
R
AK
e
emptycell.T
h
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎧
∨
∨
∨
∨
∨
∨
),(
0
0
(
(
=)(
1
o
zx
C
C
C
X
P
X
PX
PX
z
t
i
i
j
j
s
ofPlasmodi
u
1
{
P
,,,
2
A
PK
c
erules(1)iff
p
ropositional
m
T
hus,weassu
m
Xp
j
∨
ina
e
ighborcells,
t
aproof‐theo
r
p
olycephalum
i


r
eticspatial
∪},
2
K
A
{
1
R
l
ycephalumis
d
L
withtheo
n
,...,,
21
R
R
1
C
h
einferenceru
l
∨∨
∨∨
.
0=)(
0 =)(
)
()
)()
=)(
=)(
otherwise
P
premises
a
n
zxif
C
premises
a
n
zxif
CX
C
i
PX
P
Xzxif
P
Xzxif
P
t
tji
ji
t
j
t
j
u
mofPhysaru
,
1
A
,,
1
2
RA K
either
p
is
s
m
etavariable
,

m
ethateach
p
a
neighborcel
l
t
hen
i
p
,
j
p
r
eticspatiala
u
i
sasfollows:
A
R
P
R
automatonf
o
{
},,
2
1
KR∪
d
efinedasfol
l
n
lybinaryoper
a
},,
2
KC
.Le
t
l
eoftheautom
a
+∈
+∈
∨
∨
∨∨
∨∨
)
(,
(,
=)(
=)(
NzA
P
R
premises
n
d
NzA
C
R
premises
n
d
Xzxif Xzx
if
AP
AC
ij
ij
t
tij
in
mpolycephalu
m
,...,,
2
1
R,
1
C
C
i
s
orinane
i
i.e.itrunso
v
p
remiseshou
l
l
andboth
i
p
couldnotbe
c
u
tomatonwit
h
(II)
1=t
(IV)
3
=t
C
C
o
rdynamics
},,
{
21
KCC
.
ows:
a
tion
∨
,
itis
b
t

S
betheset
a
tonisasfollo
w
+∉
+∉
∨
∨
∨
∨
)
;
),(
)
;
),(
Nz
R
Nz
R
CC PP
i
i
ji
ji
m
.
},
2
K
C
.As
t
ghborcellw
e
v
ereitherthe
l
doccurina
s
and
j
p
are
c
onsideredas
h
theneighb
o
3

A
C
P
P
C
P
ofPlasmodi
u
b
uiltinthesta
n
ofstatesofpr
o
w
s.

tate
p
isca
e
findoutan
e
emptyseto
r
s
eparatecell.
T
neededford
e
premises.
o
rhoodof8
m
u
mofPhysa
n
dardwayove
r
o
of‐theoreticsp
a
(1)
l
ledapremis
e
expressionof
r
thesetofst
T
hismeansth
e
ducing,whe
r
m
embersint
h
rum
r
the
a
tial
e
for
the
ates
atif
r
eas
h
e2‐

6. Conclusion
Self‐organizationphenomenainnatureassumecircularityandcause‐and‐effectfeedbackrelations:eachcomponent
affectstheothercomponents,butthesecomponentsinturnaffectthefirstcomponent([10],[11]).Ourexamplesof
suchself‐organizationwerepresentedbyepidemicspreading,Belousov‐Zhabotinskyreaction,andPhysarum
polycephalum.Forinstance,inepidemicspreadingweobservecircularityasauto‐wavesofinfectingandrecovering
andinBelousov‐Zhabotinskyreactionweobservecircularityintheinterchangeofsolutioncolor:inthebeginningthe
solutioniscolorless,thenitbecomesyellow,thenitbecomescolorless,etc.Inlogicalsimulationofepidemic
spreading,Belousov‐ZhabotinskyreactionanddynamicsofPhysarumpolycephalumweobtaincircularproofs.This
showsthatunconventionalcomputingdealswithlogicalcircularitylikecyclicproofsandfeedbackrelationsinstate
transitions.Wecansupposethatlogicalcircularityshouldbeakeynotionof“lifecomputer,”i.e.ofeachself‐
organizedsystem.Unconventionallogicofproof‐theoreticautomataallowsustoconsidersuchnotions.
References
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