Welch sets for random generation and representation of reversible one-dimensional cellular automata !

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Information Sciences

journalhomepage:www.elsevier.com/locate/ins

Welch sets for random generation and representation of reversible one-dimensional cellular automata ^!

Juan Carlos Seck-Tuoh-Mora

^a,∗

, Joselito Medina-Marin

^a

,

Norberto Hernandez-Romero

^a

, Genaro J. Martinez

^b,c

, Irving Barragan-Vite

^a

a AAI-ICBI-UAEH. Carr Pachuca-Tulancingo Km 4.5. Pachuca 42184 Hidalgo, Mexico

b Unconventional Computing Centre, University of the West of England, BS16 1QY Bristol, United Kingdom

c Escuela Superior de Computo, Instituto Politecnico Nacional, Mexico

a r t i c l e i n f o

Article history:

Received 28 June 2016 Revised 7 November 2016 Accepted 4 December 2016 Available online 7 December 2016 Keywords:

Cellular automata Reversibility Welch indices Bipartite graph Block mapping

a b s t r a c t

Reversibleone-dimensionalcellularautomataarestudiedfromtheperspectiveofWelch Sets.ThispaperpresentsanalgorithmtogeneraterandomWelch setsthatdefineare- versiblecellularautomaton.Then,propertiesofWelchsetsareusedinordertoestablish twobipartitegraphsdescribingtheevolutionruleofreversiblecellularautomata.Thefirst graphgivesanalternativerepresentationforthedynamicsofthesesystemsasblockmap- pingsandshifts.Thesecondgraphoffersacompactrepresentationfortheevolutionrule ofreversiblecellularautomata.Bothgraphsandtheirmatrixrepresentationsareillustrated bythegenerationofrandomreversiblecellularautomatawith6and18states.

© 2016ElsevierInc.Allrightsreserved.

1. Introduction

Cellularautomata arediscrete dynamical systemsableto generatecomplexbehaviorbased onsimplelocalinteraction amongtheir components.Reversiblecellularautomata arecharacterizedtoproducean invertibleglobalbehaviorprovoked byasetoflocalmappingswhicharenotreversible.

Inthe one-dimensional case, reversible cellular automata havebeen widely investigated in orderto understand their topological,combinatorialanddynamicalproperties.

Due to their property of keeping the original information of the system, concepts and alternative definitions of re- versible cellularautomata have beenemployed inthe specificationof differentcyphering systems [2,8,9,11,19,20,22],and error-correctingcodes[13].

Recentworkshavepresenteddifferentprocedures tocharacterizereversiblebehaviorincellularautomata.Forinstance:

thegeneralization ofreversibilitydefinedby linearrulesover thebinaryfield caseundernullboundary conditionsisex- plained in[21].Reversibility of elementarycellular automata rulenumber 150is describedby circulant matricesin[14]. Reversibilityincellularautomatawithmemoryhasbeenstudiedin[1]and[17].

! Dedicated to the memory of Harold V. McIntosh (1929–2015), a leader in the development of cellular automata theory, whose teaching and knowledge were essential to this work.

∗ Corresponding author:

E-mail addresses: [email protected] (J.C. Seck-Tuoh-Mora), [email protected] (J. Medina-Marin), [email protected] (N. Hernandez-Romero), [email protected] (G.J. Martinez),[email protected] (I. Barragan-Vite).

http://dx.doi.org/10.1016/j.ins.2016.12.009 0020-0255/© 2016 Elsevier Inc. All rights reserved.

Fig. 1. Evolution of sequence v with four cells for r = 1 , in this case ϕ( v ) has two cells..

Thus,thereisacontinuousinvestigationtocharacterizethepropertiesofreversiblecellularautomata,bothfortheoretical researchandpracticalapplication.

The firsttopologicalstudyofreversiblecellular automataisdeveloped byHedlundin[10]establishingimportantcon- cepts for the one-dimensional case such asthe uniformmultiplicity of ancestors and Welch sets. Several investigations followedthispaperusingWelchsetstodefinenewgraphsrepresentingthelocalbehaviorinducingglobalreversibility[15], andusingtheirvectorrepresentationtogiveamaximumboundfortheminimuminformationneededtoobtainglobalin- vertibility[7].Anothertopicofinteresthasbeentoenumerateallthepossiblecasesofreversibleone-dimensionalcellular automata[3,4,18],thelasttwo referencestakingreversible automataasgroupoidswithalgebraicproperties.Inthissense, references[6]and[5]establishthepropertiesofthesegroupoidsandtheirrelationwithisotopyandCayleygraphs.

Based onthis previous work,the aimof thispaperis topresentan algorithm to createrandomWelchsets to define reversibleone-dimensionalcellularautomata.FromtheseWelchsets,wedefinetwobipartite graphs,bothforrepresenting theforwardandreverselocalmappingsoftheassociatedreversiblecellularautomaton.Wegiveexamplesshowingthatthe formergraphprovidesanalternativerepresentationforthedynamicsofreversibleautomatabyblockmappings andshifts firstlydemonstratedbyKariin[12],andthelatterneedslessspacethantheoriginalevolutionrule.

The paper is organized as follows: Section 2 explains the basic concepts of reversible one-dimensional cellular au- tomataandWelchsets. Section 3definestheproperties ofWelchSetsforreversible automatawithneighborhood size2.

Section4presentsthealgorithmgeneratingrandomWelchsetsandarbitraryreversiblecellularautomata.Section5exposes theconstructionofabipartitegraphbasedonWelchelementstocalculatetheevolutionofreversiblecellularautomataand representtheirdynamicsbyblockmappings andshifts.Section6specifiesashorterversion ofthisgraphinordertorep- resenttheevolutionofreversibleautomatawithareducednumberofelements.Thelastsectiongivesthefinal remarksof thestudy.

2. BasicconceptsofWelchsets

One-dimensionalcellularautomataaredefinedbyadiscretesetofstatesS,aneighborhoodradiusr∈N,aninitialcon- figuration c₀:

{

^1,2,...,m

}

→S for some m∈N, andan evolution rule

ϕ

:S^2r+1→S. The dynamicsis givenapplying the evolution rule over every cell in c₀, thus c₁(i)₌

ϕ

(c₀(i− r)...c₀(i+r)) taking periodic boundary conditions. In general ct+1(i)₌

ϕ

(ct(i− r)...ct(i+r)).

Forsimplicityweshallrepresent

ϕ

(c(i− r)...c(i+r))justas

ϕ

(c(i)).Thus,forastringv∈S^mwithm≥ 2r+1,wedefine

ϕ

(v)asfollows:

ϕ ( v )

=

ϕ ( v (

r+1

)) ϕ ( v (

r+2

))

. . .

ϕ ( v (

m− r

))

=w

(

1

)

w

(

2

)

. . .w

(

m− 2r

)

₌w (1)

wherewistheevolutionofv.Eq.(1)showsthat

ϕ

(v)has2rcellslessthan v.Fig.1illustrates thecaseforasequenceof fourcellsandr=1;inthisworkweareusingcenteredevolutions.

Inordertosimplifyourstudy,wetakethesimulationofeverycellularautomatonbyanotherwithneighborhoodsize2.

ByEq.(1),for

v

=c_t(i− 2r+1)...c_t(i+2r)wehavethat

ϕ

(

v

⁾=w=c_t+1(i− r+1)...c_t+1(i+r).Then

| v |

=4r and

|

^w

|

=2r.

LetusdefineanewsetofstatesS^′suchthat

|

^S′

|

=

|

^S

|

^{2r;thus,wecandefineabijectionfromS′toS2r.Withthis,thereare}

bijectionsv→s₁s₂andw→s₃ fors_i∈S^′suchthat:

ϕ

^′

(

s1s2

)

₌s3 (2)

Eq.(2) showsthat theoriginal evolution of acellular automaton canbe simulated by anotherevolution rule

ϕ

^′ with neighborhoodsize2(orradius1/2)andanextendedsetofstates.Fig.2describesthissimulationforacellularautomaton withr=1;wheretindicatestimestep.

A cellular automatonis reversible ifthere exists an integerr^′∈N andan inverse mapping

ϕ

⁻¹:S^2r′+1→S such that ct(i)₌

ϕ

⁻¹(ct+1(i)).Thisimpliesthat

ϕ

⁻¹(

ϕ

(ct(i)))₌ct(i).

Areversiblecellularautomatoncanbesimulatedbyanotherwithr=1/2forbothinvertiblerules,takingr^⋆=max(r,r^′) andusinganextendedsetofstatesS^⋆with

|

^S⋆

|

=

|

^S

|

^{2r⋆.Therefore,wehavetostudyonlyreversiblecellularautomatawith}

neighborhoodsize2inbothinvertiblerulesinordertounderstandtheothercases.

Fig. 2. Evolution of a cellular automaton with r = 1 represented by another with r = 1 / 2 according to Eq. (2) .

ThemainpropertiesofreversiblecellularautomatawerefirstlydefinedintheseminalworksofHedlund[10]andNasu [15].Adetailedexplanationofhowthesepropertiesgeneratereversiblebehaviorforreversibleautomatawithneighborhood size2canbeconsultedin[18].

Thefeaturesofreversibleone-dimensionalcellularautomataaresummarisedinthreemainproperties:

1.Uniformmultiplicityofancestors.Foreverystringw∈SⁿthereisasetofstringsA⊂ Sⁿ⁺¹suchthat

|

^A

|

=

|

^S

|

^andeachv∈A holdsthat

ϕ

(

v

)₌w.Eachvisanancestorofw.

2.Welchindices.Thereexisto_L,o_R∈Nsuchthatforeverystates∈StherearesetsLs,Rs⊂ Swith

|

^Ls

|

=o_L and

|

^Rs

|

=o_R such that

ϕ

(ab)₌sforeacha∈L_sandeveryb∈R_s,andtheproducto_Lo_R=

|

^S

|

^.TheseoL,o_RarecalledWelchindices.

3.Uniqueintersection.Everypairofstatesa,b∈Sholdsthat

|

^Ra∩L_b

|

=

|

^Rb∩La

|

=1.

Properties2and3definethattheancestorstringsofanysequenceof2ormorestatesshareacommoncentralpartand theirdifferencesareattheends.Thesepropertiesassuretheexistenceofaninverseevolutionruleabletoreverttheglobal dynamicsoftheautomaton.

ThesetsLsandRsare theWelchsets associatedwithstate s∈S.Noticethat indiceso_L ando_R areindependentofthe statedefiningLsandRs.Thatis,

|

^Ls

|

=o_L and

|

^Rs

|

=o_Rforalls∈S.

Since

ϕ

and

ϕ

⁻¹ aredefinedwithaneighborhoodsize 2,they canberepresentedbymatricesM_ϕ andM_ϕ−1 such that entriesM_ϕ(a,b)₌

ϕ

(ab)andM_ϕ−1(a,b)₌

ϕ

⁻¹(ab)foreverypairofstatesa,b∈S.

3. PropertiesofWelchsets

The previous propertiesgive a particularstructure for M_ϕ andM_ϕ−1. Property1 guarantees that there are |S| entries equaltosinbothmatricesforeverys∈S.Property2establishesthatifM(s₁,s₂)₌M(s₃,s₄)thenM(s₁,s₄)₌M(s₃,s₂)for s_i∈S.Finally,Property3specifiesthatforeverystates∈Sthereisauniquestatea∈SsuchthatM_ϕ(a,a)₌s.Thus,inM_ϕ andM_ϕ−1 everystatehasarectangulararrangement,andbothmatricesdefinearectangulargroupoid[5].

For every reversible cellular automaton, we can define a state permutation such that M_ϕ(s,s)₌s. Due to

ϕ

(ss)₌s, itishold that

ϕ

(sss)₌ss.Therefore,

ϕ

⁻¹(ss)₌sandthesame state permutationimpliesthat M_ϕ−1(s,s)₌s.Thisfeature simplifiesevenmoreourstudy,wejustneedtoanalyzereversiblecellularautomatawithneighborhoodsize2anddiagonal elementsM_ϕ(s,s)=M_ϕ−1(s,s)=sforeverys∈Sbecausealltheothercasescanberepresentedbythisone.Since

|

^Ls

|

=o_L and

|

^Rs

|

=o_R,statesappearsino_L rowsofM_ϕ,atthesameo_R columns.

Fora∈S,let M_ϕ(a, ^∗) andM_ϕ(^∗, a) bethe a-throwandthea-thcolumnrespectivelyofmatrix M_ϕ.LetD_ϕ(a) be the setofstatesinrowMϕ(a,^∗) andlet Iϕ(a)be thesetofstatesincolumnMϕ(^∗,a). Withthis, thefollowingremarkcanbe proved.

Remark1.

|

^Dϕ(a)

|

=o_L foreacha∈S.

Proof.SinceM_ϕ(a,^∗)has|S|columnsandeverystatesholdsthat

|

^Rs

|

=o_R,thenthereare

|

^S

|

^/oR=o_L statesinM_ϕ(a,^∗). ! AstraightforwardresultfromRemark1isthefollowingone.

Corollary1.

|

^Iϕ(a)

|

=o_R foreacha∈S.

ArelevantpropertyofD_ϕ(a)andI_ϕ(a)isthenextone.

Remark2. D_ϕ(a)_∩I_ϕ(a)₌aforeverya∈S.

Proof.SinceM_ϕ(a,a)₌awehavethata∈D_ϕ(a)anda∈I_ϕ(a).Ifthereisanotherstateb̸=asuchthatb∈D_ϕ(a)andb∈ I_ϕ(a),thena∈L_banda∈R_b.Thisimpliesthat

ϕ

(aa)₌b andthereforeM_ϕ(a,a)₌b,whichisacontradiction. !

Remark2saysthatD_ϕ(a)andI_ϕ(a)agreeonlyinthediagonal state.Foreachs∈S,let L⁻¹_s andR⁻¹_s betheWelchsets ofsassociatedwiththeinverseevolutionrule

ϕ

⁻¹.Accordingly,leto_L−1 ando_R−1 betheWelchindicesrelatedto

ϕ

⁻¹.The followingremarkestablishestherelationshipbetweentheWelchindicesof

ϕ

and

ϕ

⁻¹.

Remark3. o_L−1=o_R ando_R−1=o_L.

Proof. Forevery state s∈ S, we havethat a∈L⁻¹_i andb∈R⁻¹_i iff

ϕ

⁻¹(ab)₌s. Thisimpliesthat a ∈I_ϕ(s) andb ∈D_ϕ(s), henceL⁻¹_s =I_ϕ(s)andR⁻¹_s =D_ϕ(s).Remark1andCorollary1indicatethat

|

^Iϕ(s)

|

=o_R and

|

^Dϕ(s)

|

=o_L.Thereforeo_L−1=o_R ando_R−1=o_L. !

4. AlgorithmtogeneraterandomWelchsets

ForU⊆S,we canrepresenttheelementsofUbyabinaryvectorg_U:S→0,1ofsize1× |S|suchthatgU(a)₌1iff a∈ U;otherwise,g_U(a)₌0.Withthis,wedefinetwomappings

α

(g_U)₌U and

β

(U)₌g_U.

Thenext algorithmcomputesvalidWelchsets forevery states∈StofillupthematrixM_ϕ representingtheevolution ruleofareversiblecellularautomaton.ThealgorithmappliesthepropertiesofWelchsetsasvectorandmatrixoperations in orderto facilitatetheir computational implementation.The vectorsandmatricesused inthe algorithm aredefined as follows:

• G_L:|S|× |S|matrixrepresentingleftWelchvectorsasrowbinaryvectors.

• G_R:|S|× |S|matrixrepresentingrightWelchvectorsasrowbinaryvectors.

• G_D:|S|× |S|matrixwhereeverybinaryrowrepresentsthevalidstatesthatcanbeplacedatthesamerowinM_ϕ.

• G_I:|S|× |S|matrixwhereeverybinaryrowrepresentsthevalidstatesthatcanbeplacedatthecorrespondingcolumn inM_ϕ.

• h_D:1× |S|integervectorrepresentingthenumberofstatesusedineachrowofM_ϕ.

• h_I:1× |S|integervectorrepresentingthenumberofstatesusedineachcolumnofM_ϕ.

• E: |S| × |S| matrix representing the sumof Kronecker products to validate that the Welch vectors calculated so far composeavalidevolutionrule.

LetusrepresenttheidentitymatrixasI,thezeromatrixas0,andtherowvectorofonesas1.ForabinarymatrixB,let usdefineitsnegationby¬BanditstransposebyB^T.Withtheseelements,thealgorithmisdefinedasfollows:

1. Initializelimit_l,limitr∈N(loopvariables).

2. Initialize s=1,G_L=G_R=I, G_D=G_I=¬I, h_D=h_I=1, andM_ϕ=E=0. This stepspecifies that atthe beginning each states∈Sbelongsto L_s andR_s.Equally,theinitializationdetermines thatsbelongstoD_s andI_s andtherestofstates areavailabletofillupeachrowM_ϕ(s,^∗)andeverycolumnM_ϕ(^∗,s)respectively.

3. Setthreshold_l=0and f lag_l=1(controlvariables).

4. Calculateacandidateleft WelchsetL_s takingstatesando_L− 1randomstatesfrom{

α

(h_D <o_L)∩

α

(G_D(s,^∗)> 0)}.Set G_L(s,∗)₌

β

(Ls).ValidateProperty3forcandidateLs:for1≤ i<s,ifanydotproductG_R(i,^∗)· GL(s,^∗)̸=1then f lag_l=0. 5. If f lag_l=1,restrictthepossiblestatesforcandidateRs.ByRemark2,foreverya∈Ls,a̸=s,setG_I(s,a)₌0.For1≤ i

<s,takeeveryisuch that|

α

(GL(i,^∗))∩

α

(GL(s,^∗))|>0.Thenthestatesinthecorresponding

α

(GR(i,^∗))cannotbepart ofRstoavoidthat thesameneighborhoodhasmultipleevolutions.Therefore,setG_I(s,a)₌0foreverya ∈

α

(G_R(i,^∗)).

EnumeratethesetofpossiblestatesforthecandidaterightWelchsetRs:if

|{ α

(h_I<o_R)_∩

α

(G_I(s,∗)>0)

}|

^<oR− 1then f lag_l=0.

6. If f lag_l=0,setthreshold_l=threshold_l+1.Ifthreshold_l <limit_l thenset f lag_l=1andreturntostep4,otherwisereturn tostep2.

7. Setthresholdr=0and f lagr=1(controlvariables).

8. Calculate a candidate right Welchset R_s takingstate sand o_R− 1 random states from{

α

(h_I < o_R)∩

α

(G_I(s, ^∗) > 0)}.

SetG_R(s,∗)₌

β

(Rs).Validate Property3forcandidateRs: for1≤ i≤ s,ifanydot productG_R(s,^∗) · GL(i, ^∗) ̸= 1then f lagr=0.

9. If f lag_r=1,calculatetheKroneckerproductK_s=G_L(s,∗)^T_{! G}R(s,∗)andtakeK^′=K_s+E.IfK^′(i,j)>1for1≤ i≤ j≤ S then f lagr=0.Thismeansthatthereisaneighborhoodwithmultipleevolutions.

10. If f lagr=0setthresholdr=thresholdr+1.Ifthresholdr<limitrthenset f lagr=1andreturntostep8,otherwisegoto step4.

11. ValidWelchsetshavebeengeneratedforstates.Updateh_D(a)₌h_D(a)₊1foreacha∈

{ α

(G_L(s,∗))_{− s}

}

^,andanalogously h_I(b)₌h_I(b)₊1foreachb∈

{ α

(G_R(s,∗))_{− s}

}

^.SetE=E+KsandM_ϕ(a,b)₌s.Ifs<|S|thens=s+1andgotostep3.

OtherwisereturnmatricesGL,GRandMϕ.

TheassociatedpseudocodeispresentedinAlgorithm1.

In Algorithm1, loopparameters limit_l andlimitr define the numberofiterationsto search valid Welchsets forevery states.IncasethatavalidRshasnotbeencalculatedafterlimitrattempts,anewLsiscalculatedtorestartthesearchofan appropriateRs.IfavalidLshasnotbeencalculatedafterlimit_lattempts,avalidRshasnotbeencalculatedafterlimit_l^∗limitr

iterations.ThisimpliesthatthepreviouscalculatedWelchsetscouldnotbesoadequatetospecifyareversibleautomaton, especially forlarge valuesof limit_l andlimitr. Therefore,the search restartsfroms=1.Notice that parameters limit_l and limitrarenotdirectlyrelatedtothenumberofcellsintheconfigurationofthereversibleautomaton.

Theaimofthealgorithmproposedinthispaperisnottoenumeratereversibleautomatabutonlytogeneratearandom instancefora larger numberofstates.Infact, thealgorithm hasbeenableto calculaterandomWelchsets forreversible automataupto30states,withanycombinationofWelchindices.

Algorithm1 GeneratereversibleautomatabyrandomWelchsets.

Require:Numberofstates|^S|^,s=1,andloopparameterslimitl,limitr∈N; 1: whiles<=|^S|^do

2: ifs=1then ◃Initializedatastructures

3: GL=GR=I,GD=GI=¬I,hD=hI=1,andMϕ=E=0; 4: endif

5: thresholdl=0; ◃SetloopvariabletofindavalidLs

6: whilethreshold_l<=limit_ldo

7: Ls←o_L− 1randomstatesin{α(h_D<o_L)∩α(G_D(^s,∗)>0)}^{; ◃ValidateRemark1}

8: G_L(s,∗)₌β (Ls); f lag_l=1; ◃ValidateProperty3forLs

9: ifG_R(i,∗)_{· G}L(s,∗)_̸=1for1≤ i<sthen f lag_l=0; 10: endif

11: if f lag_l=1then

12: GI(s,a)₌0foreverya∈Ls,a̸=s; ◃RestrictstatesforRstoholdRemark2 13: foralli∈{^1...s− 1}^suchthat|α(GL(i,∗))_∩α(GL(s,∗))|^>0do

14: GI(s,a)₌0fora∈α(GR(i,∗)); ◃Toavoidneighborhoodswithmultipleevolutions

15: endfor

16: if|{α(hI<oR)_∩α(GI(s,∗)>0)}|^<oR− 1then f lag_l=0

17: else ◃EnoughavailablestatesforRs

18: thresholdr=0; ◃SetloopvariabletofindavalidRs

19: whilethresholdr<=limitrdo

20: Rs←oR− 1randomstatesin{α(hI<oR)_∩α(GI(s,∗)>0)}^{; ◃ValidateCor.1}

21: GR(s,∗)₌β (Rs);f lagr=1; ◃ValidateProperty3forRs

22: ifGR(s,∗)_{· G}L(i,∗)_̸=1for1≤ i<=sthen f lagr=0;

23: endif

24: if f lagr=1then ◃Kroneckerproducttocheckneighborhoodsalreadytaken

25: Ks=GL(s,∗)^T! GR(s,∗);K^′=Ks+E;

26: ifK^′(i,j)>1for1≤ i≤ j≤ Sthen f lagr=0;

27: else ◃ValidWelchsetshavebeengeneratedforstates

28: h_D(^a)=h_D(^a)+1fora∈{α(G_L(^s,∗))− s}^{; ◃UpdatestatesusedinrowsofMϕ}

29: h_I(^b)=h_I(^b)+1forb∈{α(G_R(^s,∗))− s}^{; ◃UpdatestatesusedincolumnsofMϕ}

30: E=E+Ks; ◃UpdatesumofKroneckerproducts

31: foralla∈{α(G_L(s,∗))}^andb∈{α(G_R(s,∗))}^{do ◃Updateevolutionrule}

32: M_ϕ(a,b)₌s

33: endfor

34: ifs<|^S|^{then ◃Therearemorestatesremaining}

35: s=s+1; ◃Advancetothenextstate

36: thresholdr=limitr+1;threshold_l=limit_l+1; ◃Breakwhileloops

37: else returnGL,GRandM_ϕ; ◃Reversibleruleiscomplete

38: endif

39: endif

40: endif

41: if f lagr=0thenthresholdr=thresholdr+1; ◃IncorrectRs,searchagain

42: endif

43: endwhile

44: endif

45: endif

46: if f lagl=0thenthresholdl=thresholdl+1; ◃IncorrectLs,searchagain

47: ifthresholdl>limitlthens=1; ◃PreviousWelchsetsareinadequate,startagain

48: endif

49: endif

50: endwhile 51: endwhile

Thisalgorithmcanbeclassifiedasaconstrainedrandomlocalsearch[16],inthesamecategoryasrandomsearchheuris- ticsorevolutionaryalgorithms.Randomlocalsearchareappliedtoproblemswhosestructureisnotwell-understood,aswell astohardcombinatorialproblems,liketheenumerationofreversibleautomata.

Thisenumerationisstillan openproblem. Infact,asetofformulasorageneralalgorithmto enumerateall reversible automatawithr=1/2andanynumberofstateshavenotbeendefinedyet.Themaximumnumberofstatesreportedfor anyproceduresofaris12[3,18].

Propertiesofreversibleautomata areimplicitinthegenerationoftheWelchvectorsG_L(s,^∗)andG_R(s,^∗)inlines8and 21foreachs∈S.WehavethattheWelchsets

| α

(G_L(s,∗))

|

=o_L duetolines3and7,and

| α

(G_R(s,∗))

|

=o_R duetolines3 and20,holdingProperty2(Welchindices).Property1(Uniformmultiplicityofancestors)isfulfilledinlines31–33where

|S|ancestorsarespecifiedforeverys.Lines9and22validateProperty3foralltheproductsbetweenleft andrightWelch sets.Thus,theobtainedmatrixM_ϕ satisfiesProperties1,2and3andthereforegeneratesareversibleautomaton.

Oneimportantpointinthealgorithmisthesetofrestrictionsimplementedtovalidateandreducethesearch space.In particular,inthealgorithm,Remark1inline7,Remark2inline12,discardmultipleevolutionsforthesameneighborhood inline14,enoughavailablestatesforRsinline16andCorollary1inline20,areimplementedtorestricttherandomsearch ofWelchsets,makingpossiblethegenerationofreversibleautomatawithlargernumberofstates.Property3andKronecker productvalidatetheobtainedWelchsetsinlines9,22and26.

Start

Initialize s=1 and arrays

Initialize control variables for Ls

Calculate candidate Ls

Property 3

Restrict states for Rs

Enough states

Initialize control variables for Rs

Calculate candidate Rs

Property 3 Kronecker

s<|S|

s=s+1 Return

arrays Stop

Yes Loop limit No

No Yes

Yes No

Yes

Loop limit No

No Yes

Yes

Yes No

No

Valid Welch sets, update arrays

Fig. 3. Flow chart to generate random Welch sets.

The associatedflow chart is presentedinFig. 3.Severalexamples ofreversible automata generatedwiththe previous algorithm are displayedin Fig. 5 using limit_l=limitr=100. The corresponding Welch indices are presented below each image.

5. Firstrepresentation:bipartitegraphwithWelchelements

Fors ∈S andits left Welch set L_s,a left Welch element isa pair (a, s) such that a ∈L_s. Analogously, a rightWelch elementisapair(s,a)suchthata∈Rs.

GiventhematricesG_L,G_RandM_ϕ,wecandefineabipartitegraphP=N_L× NRwhereN_ListhesetofleftWelchelements andNRisthesetofrightWelchelementsforeverystateinS.

EveryleftWelchelement(a,b) ∈N_L holdsthata ∈L_b fora,b ∈S.Analogously, everyrightWelchelement (d, e) ∈N_R holdsthate∈R_dford,e∈S.Thereisadirectededgefrom(a,b)to(d,e)labeledby

ϕ

⁻¹(bd).Similarly,thedirectededge from(d, e) to(a,b) islabeled by

ϕ

(ea). Fig.4presentstwo examplesofthe bipartitegraphs associatedwithareversible automatonof3states,o_L=1ando_R=3(leftside);andareversibleautomatonof4states,o_L=2ando_R=2.Edgesarecol- oredaccordingto

ϕ

⁻¹fromN_LtoN_R,andusing

ϕ

fromN_RtoN_L.Sincethegraphicalrepresentationcanbeverycomplicated forlargervaluesofnumberofstatesandWelchindices,amatrixdefinitionwillbeusedforfurtheranalysis.

Fig. 4. Digraphs P related to reversible automata of 3 and 4 states respectively..

Forp_L=(a,b)_∈N_L letusdefine

γ

(p_L)₌aand

δ

(p_L)₌b.Inthesameway,for p_R=(d,e)_∈N_R letusdefine

γ

(p_R)₌e and

δ

(p_R)₌d.Thus,thedirectededge(p_L,p_R)islabeledby

µ

(p_L,p_R)₌

ϕ

⁻¹(

δ

(p_L)

δ

(p_R))andthedirectededge(p_R,p_L) is labeledby

η

(p_R,p_L)₌

ϕ

(

γ

(p_R)

γ

(p_L)).

P can be representedby a matrix M_P withrowindices N_L andcolumnindices N_R. Everyposition M_P(p_L, p_R) has two entries,

µ

(p_L,p_R)and

η

(p_R,p_L).Withthis,thefollowingresultcanbeestablished.

Theorem1. Foreverys∈Sandfixedstatesa,e∈S,thereisauniquepairofnodes(pL,pR)inPwith

γ

(pL)₌aand

γ

(pR)₌e suchthat

µ

(p_L,p_R)₌s.

Proof.Every string in S³ is represented by a directed edge from a node in N_L to other node in N_R. By Property 2 and Remark1,for everya ∈S thereisa subset N_L^′⊂ NL with

|

^NL^′

|

=o_L such that

γ

(p_L)₌a foreach p_L∈N^′_L.Analogously, for everye∈SthereisasubsetN_R^′⊂ NRwith

|

^N′R

|

=o_Rsuchthat

γ

(p_R)₌eforeachp_R∈N_R^′.Thusthereareo_Lo_R=

|

^S

|

^directed

edgesinPtorepresentallthesequencesaseforfixedstatesa,eandeverys∈S.Sincethereareexactly|S|stringsase,there isauniquepair(p_L,p_R)withp_L∈N_L^′andp_R∈N_R^′ suchthat

µ

(p_L,p_R)₌s. !

DirectresultsfromTheorem1isthenextone.

Corollary2. Foreverys∈Sandfixedstatesd,b∈S,thereisauniquepairofnodes(p_R,p_L)inPwith

δ

(p_R)=d and

δ

(p_L)=b suchthat

η

(p_R,p_L)₌s.

Corollary3. Foreverys∈Sandfixedstatese,a∈S,thereare|S| pairsofnodes(p_R,p_L)inPwith

γ

(p_R)₌e,

γ

(p_L)₌asuch that

η

(p_R,p_L)₌s.

Proof.For every e∈Sthere isa subset N_R^′⊂ NR with

|

^NR^′

|

=o_R nodes suchthat

γ

(p_R)₌eforeach p_R∈N^′_R.Besides, for everya∈SthereisasubsetN_L^′⊂ NL with

|

^NL^′

|

=o_L suchthat

γ

(p_L)₌aforeach p_L∈N_L^′.Thusthereareo_Ro_L=

|

^S

|

^directed

edgesinPtorepresenttheneighborhoodeasuchthat

η

(p_R,p_L)=s. !

Asan illustrative example,Table1 presentsa reversible cellularautomata with

|

^S

|

=6, oL=3andoR=2withWelch setsgeneratedatrandomapplyingthealgorithmintheprevioussection.

Fig. 6 displayssome evolution examples of thisautomaton. The matrix M_P associated with this rule is presented in Table2.

1 2 3 4 5 6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

|S|=6, L=2, R=3 |S|=16, L=4, R=4

|S|=24, L=3, R=8 |S|=30, L=3, R=10

Fig. 5. Different reversible cellular automata with 6, 16, 24 and 30 cell-states generated by the algorithm. Every example is specified by a distinct combi- nation of Welch indices with 100 cells evolving in 120 steps. Cell-state colors are presented on the right side of every evolution.

Table 1

Random reversible cellular automata with 6 cell-states. Indexes of M ϕare implicitly defined, thus M ϕ( a, b ) indicates the evolution of the neighborhood ab .

Evolution rule M ϕ State s L s R s

4 1 4 1 6 6 1 1 2 6 2 4

3 1 5 1 3 5 2 3 4 5 2 4

3 2 5 2 3 5 3 2 3 5 1 5

4 2 4 2 6 6 4 1 4 6 1 3

3 2 5 2 3 5 5 2 3 5 3 6

4 1 4 1 6 6 6 1 4 6 5 6

1 2 3 4 5 6

Fig. 6. Evolution examples of the reversible automaton described in Table 1 for 100 cells in 100 time steps.

Table 2

Matrix M P for the evolution rule and Welch elements presented in Table 1 . For row and column indices p ∈ N L and q ∈ N R respectively, γ( p ) and γ( q ) are in bold. In the same way, every µ( p, q ) is in bold too.

N L / N R (1 ,2) (1 ,4) (2 ,2) (2 ,4) (3 ,1) (3 ,5) (4 ,1) (4 ,3) (5 ,3) (5 ,6) (6 ,5) (6 ,6) ( 1 ,1) ( 2 ,3) ( 2 ,4) ( 4 ,3) ( 4 ,4) ( 2 ,4) ( 2 ,3) ( 4 ,4) ( 4 ,3) ( 2 ,3) ( 2 ,4) ( 4 ,3) ( 4 ,4) ( 2 ,1) ( 2 ,1) ( 2 ,2) ( 4 ,1) ( 4 ,2) ( 2 ,1) ( 2 ,2) ( 4 ,1) ( 4 ,2) ( 2 ,2) ( 2 ,1) ( 4 ,2) ( 4 ,1) ( 6 ,1) ( 2 ,5) ( 2 ,6) ( 4 ,5) ( 4 ,6) ( 2 ,6) ( 2 ,5) ( 4 ,6) ( 4 ,5) ( 2 ,5) ( 2 ,6) ( 4 ,5) ( 4 ,6) ( 3 ,2) ( 2 ,5) ( 2 ,4) ( 4 ,5) ( 4 ,4) ( 2 ,4) ( 2 ,5) ( 4 ,4) ( 4 ,5) ( 2 ,5) ( 2 ,4) ( 4 ,5) ( 4 ,4) ( 4 ,2) ( 2 ,1) ( 2 ,2) ( 4 ,1) ( 4 ,2) ( 2 ,1) ( 2 ,2) ( 4 ,1) ( 4 ,2) ( 2 ,2) ( 2 ,1) ( 4 ,2) ( 4 ,1) ( 5 ,2) ( 2 ,3) ( 2 ,6) ( 4 ,3) ( 4 ,6) ( 2 ,6) ( 2 ,3) ( 4 ,6) ( 4 ,3) ( 2 ,3) ( 2 ,6) ( 4 ,3) ( 4 ,6) ( 2 ,3) ( 1 ,1) ( 1 ,2) ( 5 ,1) ( 5 ,2) ( 5 ,1) ( 5 ,2) ( 1 ,1) ( 1 ,2) ( 5 ,2) ( 5 ,1) ( 1 ,2) ( 1 ,1) ( 3 ,3) ( 1 ,5) ( 1 ,4) ( 5 ,5) ( 5 ,4) ( 5 ,4) ( 5 ,5) ( 1 ,4) ( 1 ,5) ( 5 ,5) ( 5 ,4) ( 1 ,5) ( 1 ,4) ( 5 ,3) ( 1 ,3) ( 1 ,6) ( 5 ,3) ( 5 ,5) ( 5 ,6) ( 5 ,3) ( 1 ,6) ( 1 ,3) ( 5 ,3) ( 5 ,6) ( 1 ,3) ( 1 ,6) ( 1 ,4) ( 1 ,3) ( 1 ,4) ( 3 ,3) ( 3 ,4) ( 3 ,4) ( 3 ,3) ( 1 ,4) ( 1 ,3) ( 3 ,3) ( 3 ,4) ( 1 ,3) ( 1 ,4) ( 4 ,4) ( 1 ,1) ( 1 ,2) ( 3 ,1) ( 3 ,2) ( 3 ,1) ( 3 ,2) ( 1 ,1) ( 1 ,2) ( 3 ,2) ( 3 ,1) ( 1 ,2) ( 1 ,1) ( 6 ,4) ( 1 ,5) ( 1 ,6) ( 3 ,5) ( 3 ,6) ( 3 ,6) ( 3 ,5) ( 1 ,6) ( 1 ,5) ( 3 ,5) ( 3 ,6) ( 1 ,5) ( 1 ,6) ( 2 ,5) ( 6 ,1) ( 6 ,2) ( 3 ,1) ( 3 ,2) ( 3 ,1) ( 3 ,2) ( 6 ,1) ( 6 ,2) ( 3 ,2) ( 3 ,1) ( 6 ,2) ( 6 ,1) ( 3 ,5) ( 6 ,5) ( 6 ,4) ( 3 ,5) ( 3 ,4) ( 3 ,4) ( 3 ,5) ( 6 ,4) ( 6 ,5) ( 3 ,5) ( 3 ,4) ( 6 ,5) ( 6 ,4) ( 5 ,5) ( 6 ,3) ( 6 ,6) ( 3 ,3) ( 3 ,6) ( 3 ,6) ( 3 ,3) ( 6 ,6) ( 6 ,3) ( 3 ,3) ( 3 ,6) ( 6 ,3) ( 6 ,6) ( 1 ,6) ( 6 ,3) ( 6 ,4) ( 5 ,3) ( 5 ,4) ( 5 ,4) ( 5 ,3) ( 6 ,4) ( 6 ,3) ( 5 ,3) ( 5 ,4) ( 6 ,3) ( 6 ,4) ( 4 ,6) ( 6 ,1) ( 6 ,2) ( 5 ,1) ( 5 ,2) ( 5 ,1) ( 5 ,2) ( 6 ,1) ( 6 ,2) ( 5 ,2) ( 5 ,1) ( 6 ,2) ( 6 ,1) ( 6 ,6) ( 6 ,5) ( 6 ,6) ( 5 ,5) ( 5 ,6) ( 5 ,6) ( 5 ,5) ( 6 ,6) ( 6 ,5) ( 5 ,5) ( 5 ,6) ( 6 ,5) ( 6 ,6)

Basedontheproof ofTheorem1,matrixM_Pcan beusedto obtaintheevolutionofa reversiblecellularautomatonin bothsenses.Foraninitialconfigurationcwith

|

^c

|

^{=mand1≤ i≤ m,theforwardevolutionisobtainedfollowingthenext}

steps:

1.Foreveryiwithmodulusmod(i,3)₌1,takeeverystatec(i) andthesetofrowsN^′_i⊆ NL suchthat

γ

(p)₌c(i)foreach p∈N_i^′.

2.Foreveryiwithmod(i,3)₌0,takeeverystatec(i)andthesetofcolumnsN_i^′⊆ NRsuchthat

γ

(q)₌c(i)foreachq∈N_i^′. 3.For every i with mod(i,3)₌2 and from the previous row and column sets, take the unique pair holding that

µ

(p_i−1,q_i+1)₌c(i)wherep_i−1∈N^′_i−1 andq_i+1∈N_i^′₊₁.Theevolutionofc(i− 1)c(i)c(i+1)isgivenby

δ

(p_i−1)

δ

(q_i+1). 4.Foreveryiwithmod(i,3)₌0andjwithmod(i+1,3)₌1,theevolutionofc(i)c(j)isobtainedtakingthedirectededge

η

(q_i,p_j).

Thebackwardsevolutioniscalculatedasfollows:

1.Foreveryiwithmod(i,3)₌1,takeeverystatec(i)andthesetofcolumnsN_i^′⊆ NR suchthat

δ

(q)₌c(i)foreachq∈N_i^′. 2.Foreveryiwithmod(i,3)₌0,takeeverystatec(i)andthesetofcolumnsN^′_i⊆ NLsuchthat

δ

(p)₌c(i)foreach p∈N_i^′. 3.Forevery i=mod(i,3)₌2andfromtheprevious rowsandcolumns,takethe uniquepairwith

η

(qi− 1,pi+1)₌c(i).

Theinverseevolutionofc(i− 1)c(i)c(i+1)isgivenby

γ

(q_i−1)

γ

(p_i+1).

4.Foreveryiwithmod(i,3)=0andjwithmod(i+1,3)=1,theinverseevolutionofc(i)c(j)isobtainedtakingthedirected edge

µ

(p_i,q_j).

Fig.7 describeshowthe previous algorithms are ableto calculatethe forwardandbackwardsevolution froma given randomconfiguration.

Inthisfigure,fortheforwarddirection,everysequencew∈S³ismappedintoa pairofnodes(p, q) suchthat

γ

(p)₌ w(1),

γ

(q)₌w(3)and

µ

(p,q)₌w(2)(boldfont).Thentheevolutionofwisgivenby

δ

(p)

δ

(q) (italicfont).Inthesecond partoftheprocess,themissingcell-statesareobtainedtakingeverydirectededge

η

(q,p)(italicfont)inordertocalculate thecompleteevolution.Thebackwardevolutioncanbeanalogouslydescribed.Noticethatthelast stateintheevolutionis repeatedtorepresentperiodicboundaryconditions.

6 1 3 2 4 5 5 3 4 6 2 2

6 1 3 2 4 5 5 3 4 6 2 2

4 4 1 6 5 2 1 1

6 5 5 4 6 2

4 4 2 1 6 5 2 1 1

2

3 2

3 6 5

5

5 4 4 2 1 6 3 5 2 6 1 1 5

Fig. 7. Forward and backward evolution obtained by means of the bipartite graph P . Table 3

Bijections from Welch elements of L s and R s into X and Y respectively.

L s Symbol L s Symbol L s Symbol R s Symbol R s Symbol ( 1 ,1) X 1 ( 2 ,3) X 7 ( 2 ,5) X 13 (1, 2 ) Y 1 (4, 1 ) Y 7

( 2 ,1) X 2 ( 3 ,3) X 8 ( 3 ,5) X 14 (1, 4 ) Y 2 (4, 3 ) Y 8

( 6 ,1) X 3 ( 5 ,3) X 9 ( 5 ,5) X 15 (2, 2 ) Y 3 (5, 3 ) Y 9

( 3 ,2) X 4 ( 1 ,4) X 10 ( 1 ,6) X 16 (2, 4 ) Y 4 (5, 6 ) Y 10

( 4 ,2) X 5 ( 4 ,4) X 11 ( 4 ,6) X 17 (3, 1 ) Y 5 (6, 5 ) Y 11

( 5 ,2) X 6 ( 6 ,4) X 12 ( 6 ,6) X 18 (3, 5 ) Y 6 (6, 6 ) Y 12

AlthoughthepreviousalgorithmsaremoretechnicalandcomplicatedthanusingthematrixM_ϕ toobtaintheevolution ofareversibleautomata,their relevanceistoshow thatinvertibleevolutionrules canberepresentedbyabipartite graph, whichopensafuturedirectionofresearch.

Withthe nodesof P we can obtaina reinterpretation ofKari’sproof about therepresentation ofthe dynamicsofre- versible cellularautomata by block permutationsandshifts [12].The original proof demonstrates thatthe evolution ofa reversiblecellularautomataisequivalenttomappingblocksof6rstatesintootherblockswiththesamelength,andfrom theseblocks,applyashiftof3rplacesandasecondmappingtootherblocksof6rstates.Thisdemonstrationiscompletely basedonthepropertiesofWelchindicesandonlytakesthecasewheno_L=1.

Supposethatwehaveareversibleautomatonwithr=1/2and

|

^S

|

=4;ifo_L=1then

|

^NL

|

=4and

|

^NR

|

=16.Thisimplies that we candefine a bijectionfromN_L to SandfromN_R toS².Thus, a permutationfromblocks w∈S³ toblocksof the samelengthiseasilydefined.

However, ifo_L ̸= 1a permutation cannot be defined. Forthe samecase assume that o_L=o_R=2, hence

|

^NL

|

=8 and

|

^NR

|

=8. In order to define a permutation, we should map blocks w into other blocks of the same size,such that they shouldhave8differentleftpartsand8distinctrightparts,whichisnotpossible.Thisfactdoesnotimplythattheoriginal resultiswrong,butitisrequiredtogeneralizetheideathattheevolutionofareversibleautomatonisequivalenttoperform twomappingsofblockswithashiftbetweenthem.

Inordertoobtainthisgeneralization, theelementsofN_L andN_R (orthenodesofP)canbe mappedintotwo different sets ofsymbolsXandYrespectively.Instead ofdefiningthat every blockwpermutesintoanotherblock of3states,itis onlynecessarythattheblockmapsintoablockxywherex∈Xandy∈Y.

Thus,itisnotmandatorythattheblockxyhasthesamelengththatw,thereforethesetsizes|X|and|Y|canbearbitrarily controlled to generalize the mappings ofblocks with a shiftbetween them forany combinationof o_L ando_R such that o_Lo_R=

|

^S

|

^.

Noticethateverysequencew∈S³ mapsintoauniquepair(p_L,p_R)inP suchthat

µ

(p_L,p_R)₌w(2).Thesamehappens intheinversedirection,there isauniquepair(p_R,p_L)inP suchthat

η

(p_R,p_L)=w(2).Thisimpliesthatevery leftWelch elementdefinedby

ϕ

isarightWelchelementdeterminedby

ϕ

⁻¹.

Thus, we can define two sets ofsymbols X andYsuch that

|

^X

|

=

|

^NL

|

=

|

^S

|

^oL and

|

^Y

|

=

|

^NR

|

=

|

^S

|

^oR. Everyleft Welch set definedby

ϕ

canbe mappedbijectively intoan element ofX andconsequentlyeach rightWelchset canbe mapped bijectivelyintoanelementofY.

Then,everyconfigurationcanbepartitionedinblocksofthreecellsandeveryblockisrepresentedbyapairofsymbols xyforx∈Xandy∈Y.Analogously, everypairyxmapsintoaunique sequencew∈S³ applyingnowthebijectionfromY intoN_RandfromXintoN_L.

Forthe reversiblecellular automatoninTable 1,let usdefine aset ofbijectionsfrom Ls intoX andfromRs intoY in Table3.