A unifying approach to picture grammars

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by  Matteo Pradella

⋆
A unifying approach to picture grammars
MatteoPradella1,AlessandraCherubini2,andStefanoCrespiReghizzi2
1 CNRIEIIT-MI
2 PolitecnicodiMilano
P.zzaL.daVinci,32,20133Milano,Italy
1 {alessandra.cherubini, stefano.crespireghizzi,
1 matteo.pradella}@polimi.it
0
2

n
Abstract. Severaloldandrecentclassesofpicturegrammars,thatvariouslyex-
a
J tend context-free string grammars in two dimensions, are based on rules that
rewritearraysofpixels.Suchgrammarscanbeunifiedandextendedusingatiling
8
basedapproach,wherebytherightpartofaruleisformalizedbymeansofafinite

] setofpermittedtiles.Wefocusonasimpletypeoftiling,namedregional,andde-
L finethecorrespondingregionaltilegrammars.TheyincludebothSiromoney’s(or
F Matz’s)KolamgrammarsandtheirgeneralizationbyPru˚sˇa,aswellasDrewes’s
. gridgrammars.Regionallydefinedpicturescanberecognizedwithpolynomial-
s
c time complexity by an algorithm extending the CKY one for strings. Regional
[ tilegrammarsandlanguagesarestrictlyincludedintoourprevioustilegrammars

and languages, and are incomparable with Giammarresi-Restivo tiling systems
3
(orWangsystems).
v
9
2 Keywords: picture language, tiling, picture grammar, 2D language, CKY al-
8 gorithm,syntacticpatternrecognition.
2
.
0
1 Introduction
1
9
0 Since the early days of formallanguagetheory,considerableresearch efforthas been
:
v spent towardsthe objectiveof extendinggrammarbased approachesfrom one to two
i dimensions(2D),i.e.,fromstring languagesto picturelanguages.Severalapproaches
X
havebeenproposed(andsometimesre-proposed)in thecourseofthe years,whichin
r
a different ways take inspiration from regular expressions and from Chomsky’s string
grammars,but,tothebestofourknowledge,nogeneralclassificationordetailedcom-
parisonofpicturegrammarshasbeenattempted.Itisfairtosaythattheimmensesuc-
cessofgrammar-basedapproachesforstrings,e.g.incompilationandnaturallanguage
processing, is far from being matched by picture grammars. Several causes for this
may exist. First, the lack of broadly accepted reference models has caused a disper-
sion of research efforts. Second, the algorithmic complexity of parsing algorithm for
2Dlanguageshasrarelybeenconsidered,andveryfewefficientalgorithms,andfewer
⋆Apreliminaryversionis[2].WorkpartiallysupportedbyPRINProject“Mathematicalaspects
and emerging applications of automata and formal languages”, ESF Programme Automata:
from Mathematics toApplications (AutoMathA), and CNR RSTLProject 760 Grammatiche
2Dperladescrizionediimmagini.
implementations,exist.Last,butnotleast,mostgrammartypeshavebeeninventedby
theoreticiansandtheirapplicabilityinpictureorimageprocessingremainstobeseen.
Wetrytoremove,oratleasttopartiallyoffset,thefirsttwocauses,thushopingtoset
inthiswaythegroundforappliedresearchonpicturegrammars.First,weofferanew
simpleunifyingapproachencompassingmostexistinggrammarmodels,basedonthe
notionofpicturetiling.Then,weintroduceanewtypeofgrammar,calledregionalthat
is moreexpressivethanseveralexistingtypes,yetit offersa polynomial-timeparsing
algorithm.
Weoutlinehowseveralclassicalmodelsofpicturegrammarsbasedonarrayrewrit-
ingrulescanbeunifiedbyatilingbasedapproach.Atypicalrewritingrulereplacesa
pixelarray,occurringin somepositionin thepicture,bya rightpart,whichisa pixel
arrayofequalsize.Eachgrammartypeconsidersdifferentformsofrewritingrules,that
we showhowcanbeformalizedusingmoreorlessgeneralsets oftiles. Inparticular,
wefocusonasimpletypeoftilesets,thoseofregionaltilegrammars.Thisnewclass
generalizessomeclassicalmodels,yetitisprovedtopermitefficient,polynomial-time
recognitionofpicturesbyanapproachextendingtheclassicalCocke-Kasami-Younger
(CKY)algorithm[23]ofcontext-free(CF)stringlanguages.
From the standpoint of more powerful grammar models, regional tile grammars
correspondto a naturalrestriction of our previoustile (rewriting)grammars (TG) [4,
3]. For such grammars, a rule replaces a rectangular area filled with a nonterminal
symbolwithapicturebelongingtothelanguagedefinedbyaspecifiedsetoftilesover
terminalornonterminalsymbols.ItisknownthattheTGfamilydominatesthefamily
oflanguagesdefinedbythetilingsystems(TS)ofGiammarresiandRestivo[10](which
areequivalenttoWangsystems[1][6]),andthatthelatterareNP-completewithrespect
topicturerecognitiontimecomplexity.Thenewmodelenforcestheconstraintthatthe
local language used to specify the right part of a rule is made by assembling a finite
number of homogeneousrectangularpictures. Such tiling is related to Simplot’s [20]
interestingclosureoperationonpictures.
Regionaltilegrammarsarethenshowntodominateothergrammartypes.Thefirst
istheclassicalKolamgrammartypeofSiromoney[22](which,initscontext-freeform,
isequivalenttothegrammarsofMatz[15]);itislessgeneralbecausetherightpartsof
grammarrulesmustbetiledinwaysdecomposableasverticalandhorizontalconcate-
nations.Threeothergrammarfamiliesarethenshowntobelessgeneral:Pru˚sˇa’stype
[18],grid grammars[8],andcontext-freematrix grammars[21].Thelanguageinclu-
sionpropertiesforalltheabovefamiliesarethusclarified.
The presentationcontinuesin Section 2 with preliminarydefinitions, then in Sec-
tions 3 and 4 with the definitionof tile grammars,their regionalvariant, and relevant
examples.InSection4.1wepresenttheparsingalgorithmandproveitscorrectnessand
complexity.InSection5wecompareregionaltilegrammarsandlanguageswithother
picturelanguagefamilies.Thepaperconcludesbysummarizingthemainresults.
2 Basicdefinitions
Thefollowingnotationanddefinitionsaremostlyfrom[11]and[4].
2
Definition1. LetΣ beafinitealphabet.Atwo-dimensionalarrayofelementsofΣ is
apictureoverΣ.ThesetofallpicturesoverΣ isΣ++.Apicturelanguageisasubset
ofΣ++.
Forh,k ≥1,Σ(h,k) denotesthesetofpicturesofsize(h,k)(wewillusethenota-
tion|p| = (h,k),|p| = h,|p| = k).#∈/ Σ isusedwhenneededasaboundary
row col
symbol;pˆreferstotheborderedversionofpicturep.Thatis,forp∈Σ(h,k),itis
# # ... # #
p(1,1)... p(1,k) # p(1,1) ... p(1,k) #
p= ... ... ... pˆ= ... ... ... ... ...
p(h,1)...p(h,k) #p(h,1)...p(h,k)#
# # ... # #
A pixel is an element p(i,j) of p. If all pixels are identical to C ∈ Σ the picture is
calledC-homogeneousorC-picture.
Row and column concatenationsare denoted ⊖ and , respectively. p⊖q is de-
finediffpandq havethesamenumberofcolumns;theresultingpictureisthevertical
ȅ
juxtapositionofpoverq.pk⊖ istheverticaljuxtapositionofk copiesofp;p+⊖ isthe
correspondingclosure. ,kȅ,+ȅ arethecolumnanalogous.
ȅ
Definition2. LetpbeapictureoverΣ.Thedomainofapicturepisthesetdom(p)=
{1,2,...,|p| }×{1,2,...,|p| }. A subdomainof dom(p) is a set d of the form
row col
{x,x+1,...,x′}×{y,y+1,...,y′}where1≤x≤x′ ≤|p| , 1≤y ≤y′ ≤|p| .
row col
Wewilloftendenoteasubdomainbyusingitstop-leftandbottom-rightcoordinates,in
thepreviouscasethequadruple(x,y;x′,y′).
ThesetofsubdomainsofpisdenotedD(p).Letd = {x,...,x′}×{y,...,y′} ∈
D(p),thesubpicturespic(p,d)associatedtodisthepictureofsize(x′−x+1, y′−y+
1)suchthat∀i∈{1,...,x′−x+1}and∀j ∈{1,...,y′−y+1},spic(p,d)(i,j)=
p(x+i−1,y+j−1).
AsubdomainiscalledC-homogeneous(orhomogeneous)whenitsassociatedsub-
pictureisaC-picture.C iscalledthelabelofthesubdomain.
Twosubdomainsd =(i ,j ;k ,l )andd =(i ,j ;k ,l )arehorizontallyadja-
a a a a a b b b b b
cent(resp.verticallyadjacent)iffj =l +1,andk ≥i ,k ≥i (resp.i =k +1,
b a b a a b b a
andl ≥j ,l ≥j ).Wewillcalltwosubdomainsadjacent,iftheyareeithervertically
b a a b
orhorizontallyadjacent.
Thetranslationofasubdomaind=(x,y;x′,y′)bydisplacement(a,b)∈Z2 isthe
subdomaind′ =(x+a,y+b;x′+a,y′+b).Wewillwrited′ =d⊕(a,b).
Definition3. Ahomogeneouspartitionofapicturepisanypartitionπ ={d ,d ,...,d }
1 2 n
ofdom(p)intohomogeneoussubdomainsd ,d ,...,d .
1 2 n
The unit partition of p, written unit(p), is the homogeneous partition of dom(p)
definedbysinglepixels.
An homogeneous partition is called strong if adjacent subdomains have different
labels.
We observe that if a picture p admits a strong homogeneouspartition of dom(p)
intosubdomains,thenthepartitionisuniqueandwillbedenotedbyΠ(p).
3
To illustrate,allthepicturesin Figure2butthelasttwo admita stronghomogeneous
partition, which is depicted by outlining the borders of the subdomains. The marked
partitionsofthelasttwopicturesarehomogeneousbutnotstrong,becausesomeadja-
centsubdomainsholdthesameletter.
Wenowintroducethecentralconceptsoftile,andlocallanguage.
Definition4. Wecalltileasquarepictureofsize(2,2).Wedenoteby p thesetofall
tilescontainedinapicturep.
J K
Let Σ be a finite alphabet.A (two-dimensional)languageL ⊆ Σ++ is local if there
existsafinitesetθoftilesoverthealphabetΣ∪{#}suchthatL={p∈Σ++ | pˆ ⊆
θ}.WewillrefertosuchlanguageasLOC(θ).
J K
Locally testable languages (LT) are analogous to local languages, but are defined
throughsquare tiles with side size possibly bigger than2. In the rest of the paperwe
willcallthesevariantoftilesk-tiles,toavoidconfusionwithstandard2×2tiles.For
instance,3-tilesaresquarepicturesofsize(3,3).
Last, we define tiling systems (TS). Tiling systems define the closure w.r.t. alpha-
beticprojectionoflocallanguages,andarepresentedandstudiedextensivelyin[11].
Definition5. A tiling system (TS) is a 4-tupleT = (Σ,Γ,θ,π), where Σ andΓ are
twofinitealphabets,θisafinitesetoftilesoverthealphabetΓ∪{#},andπ :Γ →Σ
isaprojection.
ThelanguagedefinedbythetilingsystemT (intherestdenotedbyL(T))isthesetof
pictures{π(p)|pˆ∈LOC(θ)}.
3 Tile grammars
Wearegoingtointroduceandstudyaverygeneralgrammartypespecifiedbyasetof
rewritingrules(orproductions).Atypicalrulehasaleftandarightpart,bothpictures
of unspecified but equal (isometric) size. The left part is an A-homogeneouspicture,
whereA is a nonterminalsymbol.The rightpartis a pictureofa locallanguageover
nonterminalsymbols.Thusa ruleis a schemedefininga possiblyunboundednumber
ofisometricpairs:leftpicture,rightpicture.Inadditiontherearesimplerruleswhose
rightpartisasingleterminal.
The derivation process of a picture starts from a S(axiom)-homogeneouspicture.
At each step, an A-homogeneoussubpicture is replaced with an isometric picture of
thelocallanguage,definedbytherightpartofaruleA → ....Theprocessterminates
whenallnonterminalshavebeeneliminatedfromthecurrentpicture.
For simplicity, this presentation focuses on nonterminal rules, thus excluding for
instance that both terminal and nonterminal symbols are in the same right part. This
normalizationhasacostintermsofgrammardimensionandreadability,butdoesnot
losegenerality.Indeed,moregeneralkindsofrules(e.g.likethoseusedin[4]),canbe
easilysimplifiedbyintroducingsomeauxiliarynonterminalsandrules.Wewillpresent
and use analogous transformations when comparing with other grammar devices in
Section5,wherewewilltalkaboutnonterminalnormalforms.
4
Definition6. A tile grammar (TG) is a tuple (Σ,N,S,R), where Σ is the terminal
alphabet,N isasetofnonterminalsymbols,S ∈N isthestartingsymbol,Risasetof
rules.
LetA∈N.Therearetwokindsofrules:
Fixedsize: A→t, wheret∈Σ; (1)
Variablesize: A→ω, ωisasetofnon-concavetilesoverN ∪{#}. (2)
B B
Concave tiles are like or a rotation thereof, where B 6= # (so we use concave
C B
tilesonlyforborders).ItiseasytoseethatallpicturesinLOC(ω),whereωisasetof
non-concavetiles,admitastronghomogeneouspartition.
Picturederivationisnextdefinedasarelationbetweenpartitionedpictures.
Definition7. ConsideratilegrammarG=(Σ,N,S,R),letp,p′ ∈(Σ∪N)(h,k) be
picturesofidenticalsize.Letπ ={d ,...,d }beahomogeneouspartitionofdom(p).
1 n
Wesaythat(p′,π′)derivesinonestepfrom(p,π),written
(p,π)⇒ (p′,π′)
G
iff,forsomeA∈N,thereexistinπanA-homogeneoussubdomaind =(x,y;x′,y′),
i
called applicationarea, and a rule A → α ∈ R such that p′ is obtainedsubstituting
spic(p,d )inpwith:
i
– α∈Σ,ifA→αisoftype(1);3
– s∈LOC(α),ifA→αisoftype(2).
Moreover,π′ =(π\{d })∪(Π(s)⊕(x−1,y−1)).
i
Wesaythat(p′,π′)derivesfrom(p,π)innsteps,written(p,π)⇒n (p′,π′),iffp=p′
G
andπ = π′, whenn = 0, orthere are a picture p′′ and a homogeneouspartitionπ′′
such that (p,π) =n−⇒1 (p′′,π′′) and (p′′,π′′) ⇒ (p′,π′). We use the abbreviation
G G
(p,π)⇒∗ (p′,π′)foraderivationwithafinitenumberofsteps.
G
Roughlyspeaking,ateachstep ofthederivationanA-homogeneoussubpictureis
replacedwithanisometricpictureofthelocallanguage,definedbytherightpartofa
ruleA→α,thatadmitsastronghomogeneouspartition.Theprocessterminateswhen
allnonterminalshavebeeneliminatedfromthecurrentpicture.
Intherestofthepaper,andwhenconsideringalsoothergrammaticaldevices,we
willdroptheGsymbolwhenitisclearfromthecontext,writinge.g.(p,π)⇒∗ (p′,π′).
Definition8. ThepicturelanguagedefinedbyagrammarG(written L(G))istheset
ofp∈Σ++suchthat
S|p|,{dom(p)} ⇒∗ (p,unit(p))
G
(cid:16) (cid:17)
∗
ForshortwealsowriteS ⇒ p.
G
3Inthiscase,x=x′andy=y′.
5
Weemphasizethat,togenerateapictureofacertaindimension,onemuststartfrom
apictureofthesamedimension.
We also willusethe notationL(X)todenotetheclassoflanguagesgeneratedby
someformaldeviceX,e.g.L(TG)willdenotetheclassoflanguagesgeneratedbytile
grammars.
Thefollowingexampleswillbeusedlaterforcomparinglanguagefamilies.
Example1. Onerowandonecolumnofb’s.
The set ofpictureshavingonerow andone column(bothnotatthe border)thathold
b’s,andtheremainderofthepicturefilledwitha’sisdefinedbythetilegrammarG in
1
Figure 1, where the nonterminals are {A ,A ,A ,A ,V ,V ,H ,H ,X,A,B}. We
1 2 3 4 1 2 1 2
# # # # # # #
u }
# A A V A A #
1 1 1 2 2
ww# A1 A1 V1 A2 A2 #(cid:127)(cid:127)
G1 : S →ww#H1 H1 V1 H2 H2 #(cid:127)(cid:127)
ww# A3 A3 V2 A4 A4 #(cid:127)(cid:127)
w# A A V A A #(cid:127)
w 3 3 2 4 4 (cid:127)
v# # # # # # #~
# # # #
u }
# X X # # # # #
u }
Ai→w#Ai Ai #(cid:127)| #X X # , for1≤i≤4
w (cid:127)
ww#Ai Ai #(cid:127)(cid:127) v# # # #~
v# # # #~
## # # # ## # # #
u } u }
X → # A X X # |a; Hi → # B Hi Hi # |b, for1≤i≤2
v## # # #~ v## # # #~
# # #
u }
# B #
A→a; B→b; Vi→w#Vi #(cid:127)|b, for1≤i≤2.
w (cid:127)
w#Vi #(cid:127)
w (cid:127)
v# # #~
aabaa
b b b b b
p =
1 aabaa
aabaa
Fig.1.TilegrammarG (top)andapicturep (bottom)ofExample1.
1 1
recallthat denotesthesetoftilescontainedintheargumentpicture.Thisnotationis
preferabletothelistingofalltiles,shownnext:
JK
# # # # A V V A A A A #
S → , ,..., 1 1, 1 2,..., 4 4, 4 .
(cid:26)#A1 A1 A1 H1 V1 V1 H2 # # # #(cid:27)
6
AnexampleofderivationisshowninFigure2,wherepartitionsareoutlinedforread-
ability.
SSSSS A A V A A
1 1 1 2 2
SSSSS H H V H H
⇒ 1 1 1 2 2 ⇒
SSSSS A A V A A
3 3 2 4 4
SSSSS A A V A A
3 3 2 4 4
A A V A A A A V A A
1 1 1 2 2 1 1 1 2 2
H H V H H H H V H H
⇒ 1 1 1 2 2 ⇒ 1 1 1 2 2 ⇒
X X V A A A X V A A
2 4 4 2 4 4
A A V A A A A V A A
3 3 2 4 4 3 3 2 4 4
A A V A A A A V A A aabaa
1 1 1 2 2 1 1 1 2 2
H H V H H H H V H H b b b b b
⇒ 1 1 1 2 2 ⇒ 1 1 1 2 2 ⇒+
A a V A A a a V A A aabaa
2 4 4 2 4 4
A A V A A A A V A A aabaa
3 3 2 4 4 3 3 2 4 4
Fig.2.DerivationusinggrammarG ofExample1,Figure1,withoutlinedpartitions.
1
Example2. Pictureswithpalindromicrows.Eachrowisanevenpalindromeover{a,b}.
ThegrammarG isshowninFigure3.
2
# # # #
u }
# R R # ####
u }
G2 : SP →ww#SP SP #(cid:127)(cid:127)| # R R #
ww#SP SP #(cid:127)(cid:127) v####~
v# # # #~
#### # # #### # #
R→u# A R R A′ #}|u# B R R B′ #}
v#### # #~ v#### # #~
## # # ## # #
R→u# A A′ #}|u# B B′ #}
v## # #~ v## # #~
A→a; B→b; A′→a; B′ →b.
a b b a
p = b aa b
2
aaaa
Fig.3.TilegrammarG (top)andapicturep (bottom)ofExample2.
2 2
7
3.1 Propertiesoftilegrammars
First,westatealanguagefamilyinclusionbetweentilingsystems(Definition5)andtile
grammars,provedin[4].We willillustrateitwithanexample,bothtogivethereader
anintuitiveideaoftheresult,andtolaterre-usetheexample.
Proposition1. L(TS)⊂L(TG).
Considera TS T = (Σ,Γ,θ,π), where Σ is the terminalalphabet,θ is a tile-set,
Γ isthe tile-setalphabet,andπ : Γ → Σ isan alphabeticprojection.Itisquiteeasy
to definea TG T′ such thatL(T′) = L(T).Informally,the ideais to takethe tile-set
θ andaddtwomarkers,e.g.{b,w}ina“chessboard-like”fashiontobuildupatile-set
suitableforthe rightpartofthe variablesize startingrule;otherstraightforwardfixed
sizerulesareusedtoencodetheprojectionπ.
WenotehowbothL(TS)andL(TG)areclosedunderintersectionwiththeclassof
allheight-1pictures:theclassesresultinginthatintersectionarethewell-knownclasses
recognizable and context-free, respectively, string languages. The inclusion is hence
proper:anycontext-free,non-recognizablestringlanguageisalso(whenconsideredas
apicturelanguage)inL(TG),butnotinL(TS).
ThenextexampleillustratesthereductionfromaTStoaTG.
Example3. Squarepicturesofa’s.
The TS T is based on a local language over {0,1} such that all pixels of the main
3
diagonalare1andtheremainingonesare0,andontheprojectionπ(0) = π(1) = a.
T andtheequivalentTGG areshowninFigure4.
3 3
The“chessboard-like”constructionisusedtoensurethattheonlystronghomoge-
neous partition obtained in applying a rule is the one in which partitions correspond
to single pixels. This allows the application of terminal rules encoding projection π.
Note that in the first rule of grammarG we used tiles arising from the two possible
3
chessboardstructures,i.e.theonewitha“black”intop-leftposition,andtheonewith
a“white”inthesameplace.Indeed,tofillareasaboveandbelowthediagonalwith0’s
weneedbothtiles
0 0 0 0
b w and w b .
0 0 0 0
w b b w
ThefollowingcomplexitypropertywillbeusedtoseparatetheTGlanguagefamily
fromseveralsubfamiliestobeintroduced.
Inthispaperas“parsingproblem”weconsidertheproblemofdecidingifagiven
inputpictureisinL(G),forafixedgrammarG(i.e.thealsocallednon-uniformmem-
bershipproblem).Thecomplexityofparsingalgorithmsisthusexpressedintermofthe
sizeoftheinputstring,inthiscasethepicturesize.
Proposition2. TheparsingproblemforL(TG)isNP-complete.
Proof From Proposition 1 and the fact that the parsing problem for L(TS) is NP-
complete(see[14]wheretilingsystemsarecalledhomomorphismsoflocallatticelan-
guages,or[13])itfollowsthatparsingL(TG)isNP-hard.
8
######
u }
# 1 0 0 0 #
w# 0 1 0 0 #(cid:127)
T3 : θ=ww# 0 0 1 0 #(cid:127)(cid:127), π(0)=a, π(1)=a.
w (cid:127)
w# 0 0 0 1 #(cid:127)
w (cid:127)
v######~
# # # # # # # # # # # #
u } u }
# 1b 0w 0b 0w # #1w 0b 0w 0b #
G3 : S →wwww##00wb 01wb 01wb 00wb ##(cid:127)(cid:127)(cid:127)(cid:127)∪wwww##00wb 10wb 10wb 00wb ##(cid:127)(cid:127)(cid:127)(cid:127)
ww#0w 0b 0w 1b #(cid:127)(cid:127) ww# 0b 0w 0b 1w #(cid:127)(cid:127)
v# # # # # #~ v# # # # # #~
1w →a, 1b →a, 0w →a, 0b →a.
Fig.4.ForExample3theTSdefining{a(n,n) | n > 1}(top),andtheequivalentTGgrammar
(bottom).
ForNP-completeness,weshowthatparsingL(TG)isinNP.First,weassumewithout
lossofgeneralitythataTGGdoesnotcontainanychainrule,i.e.aruleoftheform
####
u }
#B B #
A→ , B ∈N
w#B B #(cid:127)
w (cid:127)
v####~
thatcorrespondstoarenamingruleofastringgrammar.
Ifthisisnotthecase,itispossibletodiscardchainrulesbydirectlyusingthewell-
known(e.g.[12])approachforcontext-freestringgrammars.
Wesupposetohaveacandidatederivation
S(h,k),dom(p) ⇒ (p ,π )⇒ (p ,π )⇒ ···⇒ (p ,π )⇒ (p,unit(p))
G 1 1 G 2 2 G G n−1 n−1 G
(cid:16) (cid:17)
andwe are goingto provethatcheckingits correctnesstakespolynomialtime in h,k
(sizeofthepicture),byconsideringthedominantparametersoftimecomplexity.
First, the length n of this derivation, since there are no chain rules, is at most h·k.
Infact,westartfromapartitionwithonlyoneelementcoincidingwithdom(p),andat
eachstepatleastoneelementisadded,arrivingatstepn,wherethenumberofelements
ish·k,eachcorrespondingtoapixel.
For each step, we must find the application area in (p ,π ), and the corresponding
i i
rewrittennonterminalA,bycomparing(p ,π )with(p ,π ).Thenumberofcom-
i i i+1 i+1
parisonstobeperformedisatmosth·k.
Then,wehavetofindaruleA → ω inRwhichiscompatiblewiththerewrittensub-
pictureofp correspondingtotheapplicationarea.So,atmostwemustcheckevery
i+1
ruleinR,andeverytileofitsrightpart,onasubpicture,givenbytheapplicationarea,
9
whichisatmosth·k.Hence,wehavetoconsiderforthisstepanumberofchecksthat
isatmost
h·k·|R|· max |ω|
A→ω∈R
Each of these considered steps can be done in polynomial time in every reasonable
machinemodel,hencetheresultingtimecomplexityisstillpolynomial. ⊓⊔
From [4] it is known that the family of TG languages is closed w.r.t. union, col-
umn/rowconcatenation,column/rowclosureoperations,rotation,andalphabeticmap-
ping.
We mention that all the families presented in this work, that exactly define the
context-freestring languagesif restricted to one dimension (i.e. all buttiling systems
and grid grammars, presented in Section 5.3), are not closed w.r.t. intersection and
complement.This is provedas for string context-freelanguages:it is straightforward
to see that they are all closed w.r.t. union. But it is well known that the language
{anbncn | n > 0} is not context-free, and can be expressed as intersection of two
contextfreelanguages,e.g.{anbmcn | m,n > 0}and{anbncm | m,n > 0}.Hence,
theyarenotclosedw.r.t.intersection,butthisalsomeansthattheyarenotclosedw.r.t.
complement.
4 Regional tilegrammars
We nowintroducethecentralconceptofregionallanguage,andacorrespondingspe-
cialization of tile grammars. The adjective “regional” is a metaphor of geographical
political maps, where different regions are filled with different colors; of course, re-
gionsarerectangles.
Regionaltilegrammarsarecentraltothiswork,becausetheyarethemostgeneral
amongthepolynomial-timeparsablegrammarmodelsconsideredinthispaper.Wewill
seethatitiseasytodefinetheotherkindsof2Dgrammarsbyrestrictingthetilesused
inregionaltilegrammars.
Definition9. A homogeneous partition is regional (HR) iff distinct (not necessarily
adjacent) subdomains have distinct labels. A picture p is regional if it admits a HR
partition.Alanguageisregionalifallitspicturesareso.
Forexample,considerFigure5:thepartitionsinsubdomainsofthepictureonthe
leftishomogeneousandstrong,butnotregional,sincefourdifferentsubdomainsbear
the same symbol A. On right, a variant of the same picture with regional partitions
outlinedisdepicted.
Another(negative)exampleisinFigure4:“chessboard-like”picturesadmitunique
homogeneouspartitions, i.e. those in which every subdomaincorrespondsto a single
pixel.Notethatingeneralthesepartitionsarestrong(adjacentsubdomainshavediffer-
entsymbols,likeinachessboard),butarenotregional(e.g.inthevariablesizeruleof
grammarG therearemultiple0 symbols).
3 b
Definition10. Aregionaltilegrammar(RTG)isatilegrammar(seeDefinition6),in
whicheveryvariablesizeruleA→ωissuchthatLOC(ω)isaregionallanguage.
10