Descriptive complexity for pictures languages (extended abstract)

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by  Etienne Grandjean

Descriptive complexity for pictures languages
(Extended abstract)
EtienneGrandjeana,FrédéricOliveb,GaétanRicharda
aUniversitédeCaen/ENSICAEN/CNRS-GREYC-Caen,France
bAix-MarseilleUniversité-LIF-Marseille,France
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1 Abstract
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2 Thispaperdealswithdescriptivecomplexityofpicturelanguagesofanydimensionbysyntacticalfragments
ofexistentialsecond-orderlogic.
n
a
• We uniformly generalize to any dimension the characterization by Giammarresi et al. [GRST96] of
J
theclassofrecognizablepicturelanguagesinexistentialmonadicsecond-orderlogic.
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2
• We state several logical characterizations of the class of picture languages recognized in linear time

onnondeterministiccellularautomataofanydimension.Theyarethefirstmachine-independentchar-
]
O acterizationsofcomplexityclassesofcellularautomata.
L
Ourcharacterizationsareessentiallydeducedfromnormalizationresultsweproveforfirst-orderandexis-
.
s tentialsecond-orderlogicsoverpictures. Theyareobtainedinageneralanduniformframeworkthatallows
c
[ to extend them to other "regular" structures. Finally, we describe some hierarchy results that show the
optimalityofourlogicalcharacterizationsanddelineatetheirlimits.
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v Keywords: Picturelanguages,locality,recognizability,lineartime,cellularautomata,logical
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characterizations,existentialsecond-orderlogic.
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5
. Introduction
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Onegoalofdescriptivecomplexityistoestablishlogicalcharacterizationsofnaturalclassesofproblems
2
1 infinitemodeltheory.Manyresultsinthisareainvolvesecond-orderlogic(SO)anditsrestrictions,monadic
: second-orderlogic(MSO)andexistentialsecond-orderlogic(ESO). Indeed,therearetwolinesofresearch
v
thatroughlycorrespondtoeitheroftheserestrictions:
i
X
r (a)Theformallanguagecurrent. ItstartsfromthepioneeringresultbyBüchi,ElgotandTrahtenbrot[Büc60],
a
which states that the class of regular languages equals the class of MSO-definable languages; in short,
REG =MSO. This line of research aims at characterizing in logic the natural classes of algebraically de-
finedlanguages(setsofwords)orsetsofstructures(trees,graphs,etc.)definedbyfinitestaterecognizability
orlocalpropertiessuchastilings.
Emailaddresses:[email protected](EtienneGrandjean),[email protected](Frédéric
Olive),[email protected](GaétanRichard)
PreprintsubmittedtoElsevier January30,2012
(b)Thecomputationalcomplexitycurrent. Itoriginatesfromanotherfamousresult,Fagin’sTheorem[Fag74],
whichcharacterizestheclassNPastheclassofproblemsdefinableinESO.
Formanyyears,bothdirectionsofresearchhaveproducedplentyofresults: seee.g. [EF95,Lib04]for
+
descriptivecomplexityofformallanguagesand[EF95,Imm99,GKL 07,Lib04]fortheoneofcomplexity
classes. However,andthismaybesurprising,onlyfewconnectionsareknownbetweenthosetwoareasof
descriptivecomplexity. Ofcourse,anexplanationisthatformallanguagetheoryhasitsownpurposesthat
have little to do with complexity theory. In our opinion, the main reason is that while MSO logic exactly
fits the fundamental notion of recognizability, as exemplified in the work of Courcelle [CE12], this logic
seemstransversaltocomputationalcomplexity. WearguethisisduetotheintrinsiclocalitythatMSOlogic
inheritsfromfirst-orderlogic[Han65,Gai82]. Typically,whereasMSO,orevenexistentialMSO(EMSO),
expressessomeNP-completegraphproblemssuchas3-colourability,itcannotexpresssomeotheronessuch
asHamiltonicity(see[Tur84,dR87,Lib04])orevensometractablegraphproperties,suchastheexistence
ofaperfectmatching. Incontrast,thesituationisveryclearontreesasonwords: MSOonlycapturesthe
class of "easiest" problems; an extension of Büchi’s Theorem [TW68] states that a tree language is MSO
definableiffitisrecognizablebyafinitetreeautomaton
Thus,items(a)and(b)aboveseemquiteseparateforproblemsonwords,treesorgraphs,inonecase(for
wordsandtrees)becauseMSOonlyexpresseseasyproblems(regularlanguages),intheothercasebecause
MSOandEMSOdonotcorrespondtoanycomplexityclassovergraphs.
Whataboutpicturelanguages,thatissetsofd-pictures,i.e., d-dimensionalwords(orcolouredgrids)?
First,noticethefollowingresults:
1. In a series of papers culminating in [GRST96], Giammarresi and al. have proved that a 2-picture
language is recognizable, i.e. is the projection of a local 2-picture language, iff it is definable in
EMSO. Inshort: REC2=EMSO.
2. In fact, the class REC2 contains some NP-complete problems. More generally, one observes that
for each dimension d ≥1, RECd can be defined as the class of d-picture languages recognized by
nondeterministicd-dimensionalcellularautomatainconstanttime1.
Insomesense,thepresentpaperisanattempttobridgethegapbetweentheformallanguagecurrentof
descriptivecomplexityinvolvingMSOandthecomputationalcomplexitycurrentthatinvolvesESO. This
paperoriginatesfromtwoquestionsaboutword/picturelanguages:
1. How can we generalize the proof of the above-mentioned theorem of Giammarresi and al. to any
dimension? Thatis,canweestablishtheequalityRECd=EMSOforanyd≥1?
2. Canweobtainlogicalcharacterizationsoftimecomplexityclassesofcellularautomata2?
Thepaperaddressesbothquestions;italsocompares,inacommonframework,thepointofviewofformal
language theory with that of computational complexity. A d-picture language over an alphabet Σ is a set
1Thatmeans:forsuchapicturelanguageL,thereissomeconstantintegercsuchthateachcomputationstopsatinstantc,andp∈L
iffithasatleastonecomputationwhosefinalconfigurationisacceptinginthefollowingsense:allthecellsareinanacceptingstate.
2ThisoriginatesfromaquestionthatJ.Mazoyeraskedusin2000(personalcommunication):givealogicalcharacterizationofthe
lineartimecomplexityclassofnondeterministiccellularautomata.
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of d-pictures p:[1,n]d →Σ, i.e., d-dimensional Σ-words3. There are two natural manners to represent a
d-picture pasafirst-orderstructure:
• asapixelstructure: onthepixeldomain[1,n]d;
• asacoordinatestructure: onthecoordinatedomain[1,n].
Significantly, these two representations respectively correspond to the two above-mentioned points of
viewasshownbyourresults.
Ourresults:
Weestablishtwokindsoflogicalcharacterizationsofd-picturelanguages,foralldimensionsd≥1:
1. On pixel structures: RECd =ESO(arity1)=ESO(var1)=ESO(∀1,arity1). That means a d-picture
languageisrecognizableiffitisdefinableinmonadicESO (resp. inESO with1first-ordervariable,
orinmonadicESOwith1universallyquantifiedfirst-ordervariable).
2. Oncoordinatestructures: NLINd =ESO(vard+1)=ESO(∀d+1,arityd+1); thatmeansad-picture
ca
languageisrecognizedbyanondeterministicd-dimensionalcellularautomatoninlineartimeiffitis
definable in ESO with d+1 distinct first-order variables (resp. ESO with second-order variables of
arityatmostd+1andaprenexfirst-orderpartofprefix∀d+1).
Both items (1) and (2) are easy consequences of normalization results of, respectively, first-order and
ESO logics we prove over picture languages. In particular, the "normalization" equality ESO(arity1)=
ESO(∀1,arity1)isaconsequenceofthefactthatonpixelstructures(andmoregenerally,onstructuresthat
consistofbijectivefunctionsandunaryrelations)anyfirst-orderformulaisequivalenttoabooleancombi-
nationof"cardinality"formulasoftheform: "thereexistskdistinctelementsxsuchthatψ(x)",whereψisa
quantifier-freeformulawithonlyonevariable. The"normalization"equalityexplicitlyexpressesthe"local"
featureofMSOonpicturesandcanbegeneralizedtoother"regular"structures.
Thirdly,incontrastwith(1)and(2),weestablishseveralstricthierarchyresultsforanyfixedd≥2and
ford-picturelanguagesrepresentedbycoordinatestructures: inparticular,weprove
ESO(vard−1)(cid:40)RECd(cid:40)ESO(vard)=NLINd
ca
andESO(vard)(cid:40)ESO(arityd). Here,thefactthatnonaturalrestrictionofESO (inparticular,ESO(vard))
exactlycapturestheclassRECdforthecoordinaterepresentationofpicturesseemstousthesymptomofthe
largeexpressivenessofthislogic:significantly,thestrictinclusionESO(vard)(cid:40)ESO(arityd)ford-pictures
inthecoordinaterepresentation,ifd≥2,stronglycontrastswiththeequalityESO(arity1)=ESO(var1)that
holdsinthepixelrepresentation.
In this document, we sketch many proofs and omit other ones, in particular, the proofs of hierarchy
theorems,inSection4. However,theverytechnicalproofofthenormalizationofthelogicESO(∀d,arityd)
oncoordinateencodingsof(d−1)-pictures(ford≥2)iscompletelydescribedintheappendix.
3Moregenerally, thedomainofad-pictureisofthe"rectangular"form[1,n1]×...×[1,nd]. Forsimplicityanduniformityof
presentation,wehavechosentopresenttheresultsofthispaperintheparticularcaseof"square"picturesofdomain[n]d.Fortunately,
ourresultsalsoholdwiththesameproofsforgeneraldomains[1,n1]×...×[1,nd].
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1. Preliminaries
In the definitions below and all along the paper, we denote by Σ, Γ some finite alphabets and by d a
positiveinteger. Foranypositiveintegern, weset[n]:={1,...,n}. Weareinterestedinsetsofpicturesof
anyfixeddimensiond.
Definition1.1. A d-dimensional picture or d-picture on Σ is a function p:[n]d →Σ where n is a positive
integer. Thesetdom(p)=[n]d iscalledthedomainofpicture panditselementsarecalledpoints,pixelsor
cellsofthepicture. Asetofd-picturesonΣiscalledad-dimensionallanguage,ord-language,onΣ.
Noticethat1-picturesonΣarenothingbutnonemptywordsonΣ.
1.1. Picturesasmodeltheoreticstructures
Alongthepaper,wewilloftendescribed-languagesassetsofmodelsoflogicalformulas. Toallowthis
pointofview,wemustsettleonanencodingofd-picturesasmodeltheoreticstructures.
Forlogicalaspectsofthispaper,werefertotheusualdefinitionsandnotationsinlogicandfinitemodel
theory (see [EF95] or [Lib04], for instance). A signature (or vocabulary) σ is a finite set of relation and
function symbols each of which has a fixed arity. A (finite) structure S of vocabulary σ, or σ-structure,
consists of a finite domain D of cardinality n≥1, and, for any symbol s∈σ, an interpretation of s over
D,oftendenotedby sforsimplicity. Thetupleoftheinterpretationsoftheσ-symbolsover Discalledthe
interpretationofσoverDand,whennoconfusionresults,itisalsodenotedσ.Thecardinalityofastructure
isthecardinalityofitsdomain.Foranysignatureσ,wedenotebystruc(σ)theclassof(finite)σ-structures.
We write models(Φ) the set of σ-structures which satisfy some fixed formula Φ. We will often deal with
tuplesofobjects. Wedenotethembyboldletters.
There are two natural manners to represent a picture by some logical structure: on the domain of its
pixels,oronthedomainofitscoordinates. Thisgivesrisetothefollowingdefinitions:
Definition1.2. Given p:[n]d→Σ,wedenotebypixeld(p)thestructure
pixeld(p)=([n]d,(Qs)s∈Σ,(succi)i∈[d],(mini)i∈[d],(maxi)i∈[d]).
Here:
• succ is the (cyclic) successor function according to the jth dimension of [n]d, that is: for each a=
j
(a) ∈[n]d,succ (a)=a(j) wherea(j),theith componentofa(j),equalsa,theith componenta of
i i∈[d] j i i i
a,exceptthe jthonewhichequalsthecyclicsuccessorofthe jthcomponenta ofa. Moreformally:
j
– a(j)=a +1ifa <n,anda(j)=1otherwise;
j j j j
– a(j)=a foreachi(cid:44) j.
i i
• themin’s,max’sandQ ’sarethefollowingunary(monadic)relations:
i i s
– min ={a∈[n]d:a =1};
i i
– max ={a∈[n]d:a =n};
i i i
– Q ={a∈[n]d:p(a)=s}.
s
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Definition1.3. Given p:[n]d→Σ,wedenotebycoordd(p)thestructure
coordd(p)=(cid:104)[n],(Qs)s∈Σ,<,succ,min,max(cid:105). (1)
Here:
• EachQ isad-aryrelationsymbolinterpretedasthesetofcellsof plabelledbyans. Inotherwords:
s
Q ={a∈[n]d:p(a)=s}.
s
• <,min,maxarepredefinedrelationsymbolsofrespectivearities2,1,1,thatareinterpreted,respec-
tively,asthesets{(i,j):1≤i< j≤n},{1}and{n}.
• succisaunaryfunctionsymbolinterpretedasthecyclicsuccessor. (Thatis: succ(i)=i+1fori<n
andsucc(n)=1.)
Forad-languageL,wesetpixeld(L)={pixeld(p):p∈L}andcoordd(L)={coordd(p):p∈L}.
1.2. Logicsunderconsideration
Letusnowcometothelogicsinvolvedinthepaper. Allformulasconsideredhereafterbelongtorela-
tionalExistentialSecond-Orderlogic. Givenasignatureσ,indifferentlymadeofrelationalandfunctional
symbols, arelationalexistentialsecond-orderformulaofsignatureσhastheshapeΦ≡∃Rϕ(σ,R), where
R=(R ,...,R )isatupleofrelationalsymbolsandϕisafirst-orderformulaofsignatureσ∪{R}.Wedenote
1 k
byESOσ theclassthusdefined. Wewilloftenomittomentionσforconsiderationsontheselogicsthatdo
notdependonthesignature. Hence,ESO standsfortheclassofallformulasbelongingtoESOσ forsome
σ.
WewillpaygreatattentiontoseveralvariantsofESO.Inparticular,wewilldistinguishformulasoftype
Φ≡∃Rϕ(σ,R)accordingto:
- thenumberofdistinctfirst-ordervariablesinvolvedinϕ,
- thearityofthesecond-ordersymbolsR∈R,and
- thequantifierprefixofsomeprenexformofϕ.
With the logic ESOσ(∀d,arity(cid:96)), we control these three parameters: it is made of formulas of which
first-order part is prenex with a universal quantifier prefix of length d, and where existentially quantified
relationsymbolsareofarityatmost(cid:96). Inotherwords,ESOσ(∀d,arity(cid:96))collectsformulasofshape:
∃R∀xθ(σ,R,x)
whereθisquantifierfree,xisad-tupleoffirst-ordervariables,andRisatupleofrelationsymbolsofarity
smallerthan(cid:96). Relaxingsomeconstraintsoftheabovedefinition,weset:
(cid:91) (cid:91)
ESOσ(∀d)= ESOσ(∀d,arity(cid:96))andESOσ(arity(cid:96))= ESOσ(∀d,arity(cid:96)).
(cid:96)>0 d>0
Finally,wewriteESOσ(vard)fortheclassofformulasthatinvolveatmostdfirst-ordervariables,thus
focusingonthesolenumberofdistinctfirst-ordervariables(possiblyquantifiedseveraltimes).
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2. AlogicalcharacterizationofREC
Inordertodefineanotionoflocalitybasedonsub-picturesweneedtomarktheborderofeachpicture.
Definition2.1. ByΓ(cid:93) wedenotethealphabetΓ∪{(cid:93)}where(cid:93)isaspecialsymbolnotinΓ. Let pbeanyd-
pictureofdomain[n]d onΓ. Theborderedd-pictureof p,denotedby p(cid:93),isthed-picture p(cid:93):[0,n+1]d→Γ(cid:93)
definedby
(cid:40)
p(a)ifa∈dom(p);
p(cid:93)(a)=
(cid:93)otherwise.
Here,"otherwise"meansthataisontheborderof p(cid:93),thatis,somecomponenta ofais0orn+1.
i
Let us now define our notion of local picture language. It is based on some sets of allowed patterns
(calledtiles)oftheborderedpictures.
Definition2.2. 1. Givenad-picture pandaninteger j∈[d],twocellsa=(a) andb=(b) of p
i i∈[d] i i∈[d]
are j-adjacentiftheyhavethesamecoordinates,exceptthe jthoneforwhich|a −b |=1.
j j
2. Atileforad-languageLonΓisacouplein(Γ(cid:93))2.
3. A picture p is j-tiled by a set of tiles ∆ ⊆ (Γ(cid:93))2 if for any two j-adjacent points a,b ∈ dom(p(cid:93)):
(p(cid:93)(a),p(cid:93)(b))∈∆.
4. Given d sets of tiles ∆ ,...,∆ ⊆(Γ(cid:93))2, a d-picture p is tiled by (∆ ,...,∆ ) if p is j-tiled by ∆ for
1 d 1 d j
each j∈[d].
5. WedenotebyL(∆ ,...,∆ )thesetofd-picturesonΓthataretiledby(∆ ,...,∆ ).
1 d 1 d
6. Ad-languageLonΓislocalifthereexist∆ ,...,∆ ⊆(Γ(cid:93))2 suchthatL=L(∆ ,...,∆ ). Wethensay
1 d 1 d
thatLis(∆ ,...,∆ )-local,or(∆ ,...,∆ )-tiled.
1 d 1 d
Definition2.3. A d-language L on Σ is recognizable if it is the projection of a local d-language over an
alphabetΓ. Itmeansthereexistasurjectivefunctionπ:Γ→Σandalocald-language L onΓsuchthat
loc
L={π◦p: p∈L }.
loc
BecauseofDefinition2.2,italsomeansthereexistasurjectivefunctionπ:Γ→Σanddsubsets∆ ,...,∆
1 d
of(Γ(cid:93))2suchthat
L={π◦p: p∈L(∆ ,...,∆ )}.
1 d
WewriteRECd fortheclassofrecognizabled-languages.
Remark2.4. Ournotionof localityisweakerthantheonegivenbyGiammarresiandal.[GRST96]. But
thisdoesn’taffectthemeaningof recognizability,whichcoincideswiththatusedin[GRST96].Thisconfirms
therobustnessofthislatternotion.
Acharacterizationofrecognizablelanguagesofdimension2byexistentialmonadicsecond-orderlogic
wasprovedbybyGiammarresietal.[GRST96]. Theyestablished:
Theorem2.5([GRST96]). Forany2-dimensionallanguageL,L∈REC2⇔pixel2(L)∈ESO(arity1).
Inthissection,wecomebacktothisresult. Wesimplifyitsproof,refinethelogicitinvolves,andgeneralize
itsscopetoanydimension.
6
Theorem2.6. Foranyd>0andanyd-languageL,thefollowingassertionsareequivalent:
1. L∈RECd;
2. pixeld(L)∈ESO(∀1,arity1);
3. pixeld(L)∈ESO(arity1).
Proposition2.7statestheequivalence1⇔2.InProposition3.9,weestablishthenormalizationESO(arity1)=
ESO(∀1,arity1)onpixelstructures,fromwhichtheequivalence2⇔3immediatelyfollows.
Proposition2.7. Foranyd>0andanyd-languageLonΣ: L∈RECd⇔pixeld(L)∈ESO(∀1,arity1).
Sketch of Proof. ⇒ A picture belongs to L if there exists a tiling of its domain whose projection
coincideswithitscontent. Inthelogicinvolvedintheproposition,the“arity1”correspondstoformulating
theexistenceofthetiling,whilethe∀1 isthesyntacticresourceneededtoexpressthatthetilingbehavesas
expected. Letusdetailtheseconsiderations.
ByDefinition2.2,thereexistanalphabetΓ(whichcanbeassumeddisjointfromΣ),asurjectivefunction
π:Γ→Σanddsubsets∆ ,...,∆ ⊆(Γ(cid:93))2suchthatListheset{π◦p(cid:48):p(cid:48)∈L(∆ ,...,∆ )}.
1 d 1 d
Thebelongingofapicturep(cid:48):[n]d→ΓtoL(∆1,...,∆d)iseasilyexpressedonpixeld(p(cid:48))=(cid:104)[n]d,(Qs)s∈Γ,...(cid:105)
with a first-order formula which asserts, for each dimension i∈[d], that for any pixel x of p(cid:48), the couple
(x,succ(x))canbetiledwithsomeelementof∆. Becauseitdealswitheachcell xseparately,thisformula
i i
hastheform∀xΨ(x,(Qs)s∈Γ),whereΨisquantifier-free.
Now,apicturep:[n]d→ΣbelongstoLiffitresultsfromaπ-renamingofapicturep(cid:48)∈L(∆ ,...,∆ ). It
1 d
meansthereexistsaΓ-labelingof p(thatis,atuple(Qs)s∈Γofsubsetsof[n]d)correspondingtoapictureof
L(∆1,...,∆d)(i.e.fulfilling∀xΨ(x,(Qs)s∈Γ))andfromwhichtheactualΣ-labelingof p(thatis,thesubsets
(Qs)s∈Σ)isobtainedviaπ(easilyexpressedbyaformulaoftheform∀xΨ(cid:48)(x,(Qs)s∈Σ,(Qs)s∈Γ)).
Finally,theformula(∃Qs)s∈Γ∀x:Ψ∧Ψ(cid:48)conveysthedesiredpropertyandfitstherequiredform.
⇐ In order to prove the converse implication, it is convenient to first normalize the sentences of
ESO(∀1,arity1). Thisistheroleofthetechnicalresultbelow, whichassertsthatonpixelencodings, each
suchsentencecanberewritteninaverylocalformwherethefirst-orderpartalludesonlypairsofadjacent
pixelsoftheborderedpicture. Westateitwithoutproof:
Fact2.8. Onpixelstructures,anyϕ∈ESO(∀1,arity1)isequivalenttoasentenceoftheform:
 
∃U∀x(cid:94) mmainxi((xx)) →→ mMi((xx)) ∧∧ . (2)
i∈[d] ¬maxii(x) → Ψii(x) 
Here,Uisalistofmonadicrelationvariablesandm,M,Ψ arequantifier-freeformulassuchthat
i i i
• atomsofm andM havealltheformQ(x);
i i
• atomsofΨ havealltheformQ(x)orQ(succ(x)),
i i
where,inbothcases,Q∈{(Qs)s∈Σ,U}.
7
Now, consider Lsuchthatpixeld(L)∈ESO(∀1,arity1). Fact2.8ensuresthatpixeld(L)ischaracterized
byasentenceoftheform(2)above. WehavetoprovethatListheprojectionofsomelocald-languageL
loc
on some alphabet Γ, that is a (∆ ,..., ∆ )-tiled language for some ∆ ,..., ∆ ⊆Γ2. Let U ,...,U denote
1 d 1 d 1 k
the list of (distinct) elements of the set {(Qs)s∈Σ,U} of unary relation symbols of ϕ so that the first ones
U ,...,U are the Q ’s (here, min and max symbols are excluded). The trick is to put each subformula
1 m s i i
m(x),M(x)andΨ(x)ofϕintoitscompletedisjunctivenormalformwithrespecttoU ,...,U . Typically,
i i i 1 k
eachsubformulaΨ(x)whoseatomsareoftheformU (x)orU (succ(x)),forsome j∈[k],istransformed
i j j i
intothefollowing"completedisjunctivenormalform":
 
(cid:95) (cid:94)(cid:15)jUj(x)∧(cid:94)(cid:15)(cid:48)jUj(succi(x)). (3)
((cid:15),(cid:15)(cid:48))∈∆i j∈[k] j∈[k]
Here,thefollowingconventionsareadopted:
• (cid:15)=((cid:15) ,...,(cid:15) )∈{0,1}k andsimilarlyfor(cid:15)(cid:48);
1 k
• foranyatomαandanybit(cid:15) ∈{0,1},(cid:15) αdenotestheliteralαif(cid:15) =1,theliteral¬αotherwise.
j j j
For(cid:15)∈{0,1}k,wedenotebyΘ (x)the"completeconjunction"(cid:86) (cid:15) U (x).Intuitively,Θ (x)isacomplete
(cid:15) j∈[k] j j (cid:15)
descriptionof xandthesetΓ=(cid:83) {0i−110m−i}×{0,1}k−m isthesetofpossiblecolors(rememberthatthe
i∈[m]
Q ’sthataretheU ’sfor j∈[m]formapartitionofthedomain). Thecompletedisjunctivenormalform (3)
s j
ofΨ(x)canbewrittenintothesuggestiveform
i
(cid:95)
(cid:0)Θ(cid:15)(x)∧Θ(cid:15)(cid:48)(succi(x))(cid:1).
((cid:15),(cid:15)(cid:48))∈∆i
If each subformula m(x) and M(x) of ϕ is similarly put into complete disjunctive normal form, that is
i i
(cid:87)((cid:93),(cid:15))∈∆iΘ(cid:15)(x)and(cid:87)((cid:15),(cid:93))∈∆iΘ(cid:15)(x), respectively(thereisnoambiguityinourimplicitdefinitionofthe∆i’s,
since(cid:93)(cid:60)Γ),thentheabovesentence(2)equivalenttoϕbecomesthefollowingequivalentsentence:
 (cid:95) 
ϕ(cid:48)=∃U∀x(cid:94) mmaixnii((xx)) →→ ((cid:93),(cid:95)(cid:15))∈∆iΘΘ(cid:15)(cid:15)((xx)) ∧∧ 
i∈[d] ¬maxi(x) → ((cid:15),(cid:95)(cid:93))∈∆i (Θ(cid:15)(x)∧Θ(cid:15)(cid:48)(succi(x))) 
((cid:15),(cid:15)(cid:48))∈∆i
Finally,let L denotethed-languageoverΓdefinedbythefirst-ordersentenceϕ obtainedbyreplacing
loc loc
each Θ by the new unary relation symbol Q in the first-order part of ϕ(cid:48). In other words, pixeld(L ) is
(cid:15) (cid:15) loc
definedbythefollowingfirst-ordersentence:
 (cid:95) 
ϕ =∀x(cid:86)  mmaixnii((xx)) →→ ((cid:93),(cid:95)(cid:15))∈∆iQQ(cid:15)(cid:15)((xx)) ∧∧ 
loc i∈[d] ¬maxi(x) → ((cid:15),(cid:95)(cid:93))∈∆i (Q(cid:15)(x)∧Q(cid:15)(cid:48)(succi(x))) 
((cid:15),(cid:15)(cid:48))∈∆i
Hence,L =L(∆ ,...,∆ ). Thatis,L isindeedlocalandthecorrespondingsetsoftilesarethe∆’softhe
loc 1 d loc i
previousformula. Itisnoweasytoseethatourinitiald-language Listheprojectionofthelocallanguage
8
L bytheprojectionπ:Γ→Σdefinedasfollows:π((cid:15))=siff(cid:15) =1fori∈[m]andU isQ . Thiscompletes
loc i i s
theproof. (cid:3)
Proposition2.9. ESO(arity1)⊆ESO(∀1,arity1)onpixelstructures,foranyd>0.
Sketch of Proof. In a pixel structure, each function symbol succ is interpreted as a cyclic successor,
i
thatisabijectivefunction. So, apixelstructureisabijectivestructure, thatisafirst-orderunarystructure
whose (unary) functions f are bijective and that explicitly includes all their inverse bijections f−1. It has
been proved in [DG06] that any first-order formula on a bijective structure can be rewritten as a so-called
cardinalityformula, thatisasabooleancombinationofsentencesoftheformψ≥k =∃≥kxψ(x)(fork≥1)
whereψ(x)isaquantifier-freeformulawiththesinglevariablexandwherethequantifier∃≥kxmeans"there
exist at least k elements x". Therefore, it is easily seen that proving the proposition amounts to show that
eachsentenceoftheformψ≥k or¬ψ≥k canbetranslatedinESO(∀1,arity1)onpixelstructures.
Thisisdoneasfollows:foragivensentence∃≥kxψ(x),weintroducenewunaryrelationsU=0,U=1,...,U=k−1
andU≥k,withtheintendedmeaning:
Apixela∈[n]dbelongstoU=j(resp.U≥k)iffthereareexactly j(resp.atleastk)pixelsb∈[n]d
lexicographicallysmallerthanorequaltoasuchthatpixeld(p)|=ψ(b).
Thenwehavetocompeltheserelationsymbolstofittheirexpectedinterpretations,bymeansofafirst-order
formulawithasingleuniversallyquantifiedvariable. First,wedemandtherelationstobepairwisedisjoint:
(cid:94) (cid:16) (cid:17) (cid:94)(cid:16) (cid:17)
(1) ¬U=i(x)∨¬U=j(x) ∧ ¬U=i(x)∨¬U≥k(x) .
i<j<k i<k
Then, we temporarily denote by ≤ the lexicographic order on [n]d inherited from the natural order
lex
on [n], and by succ , min , max its associated successor function and unary relations corresponding
lex lex lex
to extremal elements. Then the sets described above can be defined inductively by the conjunction of the
followingsixformulas:
(2) (min (x)∧¬ψ(x))→U=0(x)
lex
(3) (min (x)∧ψ(x))→U=1(x)
lex
(cid:94)(cid:16) (cid:17)
(4) ¬max (x)∧U=i(x)∧¬ψ(succ (x)) →U=i(succ (x))
lex lex lex
i<k
(cid:94) (cid:16) (cid:17)
(5) ¬max (x)∧U=i(x)∧ψ(succ (x)) →U=i+1(succ (x))
lex lex lex
i<k−1
(cid:16) (cid:17)
(6) ¬max (x)∧U=k−1(x)∧ψ(succ (x)) →U≥k(succ (x))
lex lex lex
(cid:16) (cid:17)
(7) ¬max (x)∧U≥k(x) →U≥k(succ (x)).
lex lex
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Hence, under the hypothesis (1)∧...∧(7), the sentences ψ≥k and ¬ψ≥k are equivalent, respectively, to
∀x(max (x)→U≥k(x))and∀x(max (x)→¬U≥k(x)).
lex lex
Tocompletetheproof,itremainstogetridofthesymbolssucc ,min andmax thatarenotallowed
lex lex lex
inourlanguage. Itisdonebyreferringtothesesymbolsimplicitlyratherthanexplicitly. Forinstance,since
succlex(x)=succisucci+1...succd(x), for each non maximal x∈[n]d, i.e., distinct from (n,...,n), and for
(cid:86)
thesmallesti∈[d]suchthat max (x),eachformulaϕinvolvingsucc (x)actuallycorrespondstothe
j>i j lex
conjunction:
 
(cid:94)(¬maxi(x)∧ (cid:94) maxj(x))→ϕi,
i∈[d] i<j≤d
where ϕi is obtained from ϕ by the substitution succlex(x)(cid:32)succisucci+1...succd(x). Similar arguments
allowtogetridofmin andmax . (cid:3)
lex lex
Remark2.10. Inthisproof,twocrucialfeaturesofastructureoftypepixeld(p)areinvolved:
• its"bijective”nature,thatallowstorewritefirst-orderformulasascardinalityformulaswithasingle
first-ordervariable;
• the "regularity" of its predefined arithmetics (the functions succ defined on each dimension), that
i
endowspixeld(p)withagridstructure: itenablesustoimplicitlydefinealinearorderofthewhole
domaindom(p)bymeansoffirst-orderformulaswithasinglevariable,whichinturnallowstoexpress
cardinalityformulasby“cumulative”arguments,viathesetsU=i.
Proposition 3.9 straightforwardly generalizes to all structures – and there are a lot – that fulfill these two
properties.
Toconcludethissection,letusmentionthatwecouldrathereasilyderivefromTheorem2.6thefollowing
additionalcharacterizationofRECd:
Corollary2.11. Foranyd>0andanyd-languageL,thefollowingassertionsareequivalent:
1. L∈RECd;
2. pixeld(L)∈ESO(var1).
3. AlogicalcharacterizationofNLIN
ca
Besidesthenotionofrecognizablepicturelanguage,themainconceptstudiedinthispaperisthelinear
timecomplexityonnondeterministiccellularautomatonofanydimension.
Definition3.1. Aone-wayd-dimensionalcellularautomaton(d-automaton,forshort)overanalphabetΣis
atupleA=(Σ,Γ,δ,F),where
• the finite alphabet Γ called the set of states of A includes the input alphabet Σ and the set F of
acceptingstates: Σ,F⊆Γ;
• δisthe(nondeterministic)transitionfunctionofA: δ:Γ×(Γ(cid:93))d→P(Γ).
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