!-Graphs with Trivial Overlap are Context-Free

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by  Aleks Kissinger

!-Graphs with Trivial Overlap are Context-Free
AleksKissinger VladimirZamdzhiev
DepartmentofComputerScience DepartmentofComputerScience
UniversityofOxford UniversityofOxford
Oxford,UnitedKingdom Oxford,UnitedKingdom
[email protected] [email protected]
Stringdiagramsareapowerfultoolforreasoningaboutcompositestructuresinsymmetricmonoidal
categories. Byrepresenting stringdiagrams asgraphs, equationalreasoning can bedone automati-
callybydouble-pushoutrewriting. !-graphsgiveusthemeansofexpressingandprovingproperties
aboutwholefamiliesofthesegraphssimultaneously. While!-graphsprovideelegantproofsofsur-
prisinglypowerfultheorems,littleisknownabouttheformalpropertiesofthegraphlanguagesthey
define. Thispapertakesthefirststepincharacterisingtheselanguagesbyshowingthatanimportant
subclassof!-graphs—thosewhoserepeatedstructuresonlyoverlaptrivially—canbeencodedusing
a(context-free)vertexreplacementgrammar.
1 Introduction
String diagrams are essentially directed graphs, but instead of edges they have a more flexible notion
of wires. Wires can be left open at one or both ends to form inputs and outputs to the diagram, or
can be connected to themselves to form a circle. They have a formal semantics given by monoidal
categorytheory[9],andhavefoundapplicationsinmanyfields. Forexample: inmodelsofconcurrency,
they give an elegant presentation of Petri nets with boundary [15], in computational linguistics, they
areusedtocomputecompositionalsemanticsforsentences[4], incontroltheory, theyrepresentsignal-
flowdiagrams[1,2],andintheoreticalphysics,theyprovidetheformallanguageofcategoricalquantum
mechanics[3]. Astringgraphencodesastringdiagramasatyped,directedgraph,byreplacingthewires
withchainsofedgescontainingspecialdummyverticescalledwire-vertices(seeFig.1). Bycontrast,the
“real” vertices, labelled here f,g and h are called node-vertices. String graphs—originally introduced
underthename“opengraphs”in[7]—havetheadvantagethattheyarepurelycombinatoric(asopposed
togeometric)objects. Equationalreasoningonstringdiagramsisdonebyreplacingsub-diagrams. The
F
F
h
h F
E
F
E D
g D (cid:55)→
g
D
C
C
f f
A B
A B
Figure1: Astringdiagram,anditsencodingasastringgraph
©A.Kissinger&V.Zamdzhiev
A.Rensink&E.Zambon(Eds.):GraphsasModels2015(GaM’15)
Thisworkislicensedunderthe
EPTCS181,2015,pp.16–31,doi:10.4204/EPTCS.181.2
CreativeCommonsAttributionLicense.
A.Kissinger&V.Zamdzhiev 17
presence of dummy vertices in string graphs allows this to be done with double-pushout (DPO) graph
rewriting. Forexample,in:
(cid:55)→ (cid:55)→ (1.1)
theLHS(showningreenabove)isreplacedbytheRHS(blue),usingthecommonboundary(red). This
typeofrewritingforstringgraphsisimplementedinthegraphicalproofassistantQuantomatic[12].
Often, one wants to reason not just about single string graphs, but entire families of them. However,
informalnotionsofgraphswithrepetitionareill-suitedforautomatedtools. Toaddressthisproblem,!-
graphs(pronounced“bang-graphs”)wereintroducedin[6]andformalisedforstringgraphsin[10]. The
ideabehind!-graphsisthatcertainmarkedsubgraphs(alongwiththeiradjacentedges)canberepeated
anynumberoftimes,inamannersomewhatanalogoustotheKleenestar. Thesemarkedsubgraphsare
called!-boxes. Forexample:
(cid:117) (cid:125)
(cid:119) b (cid:127)
(cid:119) (cid:127)
(cid:119) (cid:127)
(cid:119) (cid:127)= , , , , ···
(cid:119) (cid:127)
(cid:119) (cid:127)
(cid:119) (cid:127)
(cid:119) (cid:127)
(cid:118) (cid:126)
The !-graph in the semantic brackets represents the depicted set of string graphs. More precisely, an
instance of a !-graph is obtained by repeatedly applying of the two operations EXPAND and KILL on
the!-graph:
b b b
EX−PA→NDb K−IL→Lb (1.2)
untilall!-boxeshavebeeneliminated. Thesetofconcretegraphs(i.e. thosenotcontainingany!-boxes)
obtainablefroma!-graphviatheseoperationsiscalledthelanguageofthe!-graph.
We call the set of languages expressible by !-graphs BG. In this paper, we will compare the expres-
siveness of this language to context-free vertex replacement grammars. We will focus on confluent
neighbourhood-controlled embedding grammars, with directions and edge-labels, i.e. C-edNCE gram-
mars[14].
Inarbitrary!-graphs,the!-boxesareallowedtonestinsideofeachother,butalsooverlap. Thelattercan
generatepatentlynon-context-freebehaviour. Thus,itisnaturaltoconsiderarestrictedsetoflanguages,
18 !-GraphswithTrivialOverlapareContext-Free
LIN-edNCE BG
BGTO
Figure2: RelationshipbetweenLIN-edNCE,BGandBGTO
BGTO of !-graphs with trivial overlap. In this paper, we will give an algorithm for producing a C-
edNCEgrammarthatisfurthermorelinear(LIN-edNCE)whichreproducesthelanguageofany!-graph
withtrivialoverlap. WethereforeshowtherelationshipbetweengraphlanguagesillustratedinFig.2.
After introducing some preliminaries on string graphs, !-graphs, and C-edNCE grammars in Section 2,
we will give an encoding of !-graphs into LIN-edNCE grammars in Section 3. In that section, we also
demonstrate that LIN-edNCE contains languages not in BG (and hence is strictly larger than BGTO).
Finally, we give one conjecture and discuss the prospects for reasoning with context-free string graph
grammarsinamanneranalogoustothe!-graphcaseinSection4.
2 Preliminaries
Definition2.1(Graph[14]). AgraphoveranalphabetofnodelabelsΣandanalphabetofedgelabelsΓ
is a tuple H =(V,E,λ), whereV is a finite set of nodes, E ⊆{(v,γ,w)|v,w∈V,v(cid:54)=w,γ ∈Γ} is the set
of edges and λ :V →Σ is the node labelling function. The set of all graphs with labels Σ,Γ is denoted
GR .
Σ,Γ
Remark 2.2. Note, that this definition of graphs does not allow for self-loops. This is a standard (and
convenient) assumption in the C-edNCE literature. By using this notion of graphs, we do not lose any
expressivity for string graphs, because any string graph with a self-loop is wire-homeomorphic (see
Definition2.6)toastringgraphwithnoself-loops.
Definition 2.3 (String Graph). A string graph is a graph labelled by the set {N,W}, where vertices
labelled N are called node-vertices and vertices labelledW are called wire-vertices, and the following
conditionshold: (1)therearenoedgesdirectlyconnectingtwonode-vertices,(2)thein-degreeofevery
wire-vertexisatmostoneand(3)theout-degreeofeverywire-vertexisatmostone.
We will depict wire vertices as small black dots and node-vertices as larger white circles, as we have
alreadydoneinSection1. Forsimplicity,weassumethereisonlyonetypeofnode-vertex,butwecould
easilycapturestringgraphslike1bytakingthelabelstobe{N ,N ,N ,W},forexample.
f g h
Wealsoidentifythosewire-verticeswhichformtheboundaryofastringgraph:
Definition 2.4 (Inputs, Outputs and Boundary Vertices). A wire-vertex of a string graph G is called an
input if it has no incoming edges. A wire-vertex with no outging edges is called an output. We denote
with In(G) and Out(G) the string graphs with no edges whose vertices are respectively the inputs and
outputsofG. TheboundaryofGisBound(G):=In(G)∪Out(G).
A.Kissinger&V.Zamdzhiev 19
Justasnode-verticesrepresenttheboxesinastringdiagram,chainsofwire-verticesrepresentthewires.
Definition 2.5 (Wires). A connected chain of vertices where each endpoint is either a boundary or a
node-vertex, and all other vertices are wire-vertices is called a closed wire. A closed wire minus its
endpoints is called the interior of a closed wire. A cycle consisting only of wire-vertices is called a
circle.
Often,itismorenaturaltoconsideranequivalenceclassofstringgraphs,uptowirehomeomorphism.
Definition2.6(Wire-homeomorphicstringgraphs). TwostringgraphsGandG(cid:48) arecalledwire-homeo-
morphic,writtenG∼G(cid:48)ifG(cid:48)canbeobtainedfromGbyeithermergingtwoadjacentwire-vertices(left)
orbysplittingawire-vertexintotwoadjacentwire-vertices(right)anynumberoftimes:
∗ ∗ (cid:55)→ ∗ ∗ ∗ ∗ (cid:55)→ ∗ ∗
That is, two graphs are wire-homeomorphic if the only dif-
ferencebetweenthemisthenumberofwire-verticesusedto
representawireinthediagram(seeFig.3).
∼
Next,weprovidedefinitionsfor!-graphsandrelatednotions.
First,weneedtoidentifywhichsubgraphsarelegaltoputin
!-boxes.
Definition 2.7 (Open Subgraph). A subgraph O of a string Figure3: Wirehomeomorphism
graphH issaidtobeopenifitisafullsubgraphandfurther-
moreIn(H\O)⊆In(H)andOut(H\O)⊆Out(H).
Definition 2.8 (!-graph). A !-graph H is a string graph, a partially ordered set !(H) of !-boxes, and an
opensubgraphB(b)⊆H foreveryb∈!(H)suchthatb≤b(cid:48) =⇒ B(b)⊆B(b(cid:48)).
Thenotionofopennessensuresthatweneverputonlypartofawireinsidea!-box,whichwouldcause
wiresto“split”inthemiddleasweexpandthe!-box,whichviolatesthearityconditionsforwire-vertices
fromDefinition2.3.
Ifb ≤b ,thismeansthatb isnestedinsideofb ,whichweindicatebydrawingalineconnectingtheir
2 1 2 1
corners. Thisistodistinguishfromthecasewhereb andb merelyoverlap.
1 2
b1 b1
vs
b2 b2
Definition 2.9 (Concrete Graph). A !-graph with no !-vertices is called a concrete graph or simply a
stringgraph.
A !-graph represents an infinite set of concrete string graphs, each of which is obtained after a finite
numberofapplicationsoftheEXPANDandKILLoperationspresentedbelow.
20 !-GraphswithTrivialOverlapareContext-Free
Definition2.10(!-boxOperations). Thetwoprimitive!-boxoperationsareasfollows:
• KILL (G)=G(cid:48)\B(b)
b
• EXPAND (G)=G (cid:116) G(cid:48)
b KILLb(G)
whereG(cid:48)hasthesameunderlyingstringgraphasG,butwithbandallofitschildrenremovedfrom!(G).
The expression for EXPAND is the disjoint union of G and G(cid:48), with the vertices, edges, and !-boxes in
thecommonsubgraphKILL (G)identified.
b
As seen in (1.2), KILL removes a !-box b and its contents, whereas EXPAND inserts a new copy of
b b
thecontentsofb. See[10]foramoredetaileddescriptionoftheseoperations.
Definition2.11(!-graphLanguage). Thelanguageofa!-graphGisthesetofallconcretestringgraphs
obtainedbyapplyingasequenceof!-boxoperationsonG.
Notethatoverlapping!-boxesarenotexplicitlyruledout. However,theycancreatesomeoddnon-local
behaviour. So,naïvely,onemightwanttoruleoutoverlapentirely,butitturnsoutthatcertaininstances
ofoverlaparemoreinnocentthanothers.
Definition2.12(OverlapandTrivialOverlap). Givenapairofnon-nested!-boxesb andb ,wesaythat
1 2
b and b are overlapping if B(b )∩B(b )(cid:54)=0/. b and b overlap trivially if B(b )∩B(b ) consists of
1 2 1 2 1 2 1 2
onlytheinteriorofzeroormoreclosedwires,whereoneendpointisanode-vertexonlyinB(b )andthe
1
otherisanode-vertexonlyinB(b ).
2
In particular, B(b )∩B(b ) can be empty, so trivial overlap
1 2
generalisesno-overlap. InFig.4b1 andb2 overlaptrivially, b1 b3
siningcneotdhee-yveorntilcyesovinertlhaeptownotbhoexienst.erbioraonfdabwoivreerclaopnnneocnt-- b2 b4 b5
3 4
trivially,sincetheyoverlaponanode-vertex. Themostsub- b6
tlecaseisb andb ,whichoverlapnon-triviallybecausethe
5 6
shared wire-vertices are not part of a wire whose endpoints
are in distinct !-boxes. Of course, all other pairs of !-boxes
overlaptrivially,astheirintersectionisjusttheemptyset. Figure4: Trivialvsnon-trivialoverlap
Definition 2.13 (BGTO). A !-graph where any two non-
nested!-boxesoverlaptriviallyiscalleda!-graphwithtrivialoverlap. Thesetofalllanguagesinduced
bythese!-graphsiscalledBGTO.
Next, we introduce C-edNCE graph grammars. Graph grammars are a generalization of context-free
grammars for strings. A graph grammar consists of a finite collection of productions which specify
instructions on how to generate a family of graphs. Like their context-free string counterparts, context-
free graph grammars have better structural, decidability and complexity properties compared to other
moreexpressivemechanismsforgraphtransformation. Weusethesamedefinitionsandconventionsfor
graph grammars as presented in [14], which also describes C-edNCE grammars and their properties in
muchgreaterdetail.
C-edNCE grammars are built using graphs with embedding. These provide the information needed to
replaceanon-terminalnodewithanewgraph,andupdateconnectionsaccordingly.
A.Kissinger&V.Zamdzhiev 21
(1) (2) (3)
b
b
δ δ
δ δ
a α X β a a X a
α β
a X a
α α β β
α γ α γ
γ
α γ α
a a a
Figure5: (1)Graphwithembedding(H,C )(2)Graphwithembedding(D,C )(3)Resultofsubstitution
H D
(H,C )[v/(D,C )]wherevisthevertexwithnon-terminallabelinH.
H D
Definition2.14(Graphwithembedding[14]). AGraphwithembeddingoverlabelsΣ,Γisapair(H,C),
whereH isagraphoverΣ,ΓandC⊆Σ×Γ×Γ×V ×{in,out}.Ciscalledaconnectionrelationandits
H
elements are called connection instructions. The set of all graphs with embedding over Σ,Γ is denoted
byGRE .
Σ,Γ
Graph grammars operate by substituting a graph (with embedding) for a non-terminal node of another
graph. Connection instructions are used to introduce edges connected to the new graph based on edges
connectedtothenon-terminal. Aconnectioninstructionas(σ,β/γ,x,d)saystoaddanedgelabelledγ
connected to the vertex x in the new graph, for every β-labelled edge connecting a σ-labelled vertex to
thenon-terminal. d thenindicateswhetherthisruleappliestoin-edgesorout-edgesofthenon-terminal.
Moreformally:
Definition 2.15 (Graph Substitution [14]). Let (H,C ),(D,C )∈GRE be two graphs with embed-
H D Σ,Γ
ding,whereH andDaredisjoint. Letv∈V beanodeofH. Thesubstitutionof(D,C )forvin(H,C )
H D H
isdenotedby(H,C )[v/(D,C )]andisgivenbythegraphwithembeddingwhosecomponentsare:
H D
V = (V −{v})∪V
H D
E = {(x,γ,y)∈E |x(cid:54)=v,y(cid:54)=v}∪E
H D
∪{(w,γ,x)|∃β ∈Γ:(w,β,v)∈E ,(λ (w),β/γ,x,in)∈C }
H H D
∪{(x,γ,w)|∃β ∈Γ:(v,β,w)∈E ,(λ (w),β/γ,x,out)∈C }
H H D
(cid:40)
λ (x) ifx∈(V −{v})
H H
λ(x) =
λ (x) ifx∈V
D D
C = {(σ,β/γ,x,d)∈C |x(cid:54)=v}
H
∪{(σ,β/δ,x,d)|∃γ ∈Γ:(σ,β/γ,v,d)∈C ,(σ,γ/δ,x,d)∈C }
H D
Thankfully, graphs with embedding allow for a simple graphical presentation (see Fig. 5). We draw
graphsasintheprevioussections,butwiththeadditionalconventionthatnodeswithnon-terminallabels
aredrawnasboxesinsteadofcircles. Asingleconnectioninstructionisdrawnasapairofedgescrossing
the outer frame. The edge and node types outside of the frame specify neighbouring node(s) which are
connectedtothenon-terminalinthehostgraphandtheedge(s)insidetheframeindicatewhatnewedges
willbeconstructedtoandfromthosenodes.
22 !-GraphswithTrivialOverlapareContext-Free
Next,wedefinetheconceptofanedNCEGraphGrammar. edNCEisanabbreviationforNeighbourhood
ControlledEmbeddingfordirectedgraphswithdynamicedgerelabeling.
Definition 2.16 (edNCE Graph Grammar [14]). An edNCE Graph Grammar is given by a tuple G=
(Σ,∆,Γ,Ω,P,S),whereΣisthealphabetofnodelabels,∆⊆Σisthealphabetofterminalnodelabels,Γ
is the alphabet of edge labels, Ω⊆Γ is the alphabet of final edge labels, P is a finite set of productions
and S∈Σ−∆ is the initial nonterminal. Productions are of the form X →(D,C), where X ∈Σ−∆ is a
non-terminalnodeand(D,C)∈GRE isagraphwithembedding.
Σ,Γ
WecangraphicallydepictanedNCEgrammarbydepictingallofitsproductions. Aproductionisdrawn
exactly like a graph with embedding with the addition that we depict its associated non-terminal in the
top-leftpartoftheframe.
Example2.17. Thefollowinggrammargeneratesthesetofallchainsofnodeverticeswithaninputand
nooutputs:
S X X
X X
Definition 2.18 (Derivation Step [14]). For a graph grammar G = (Σ,∆,Γ,Ω,P,S) and graphs (with
embedding) H,H(cid:48) ∈GRE consider a vertex v∈V and an isomorphic copy p:X →(D,C) of some
Σ,Γ H
production p(cid:48)∈P. WesayH ⇒ H(cid:48) isaderivationstepifλ (v)=X andH(cid:48)=H[v/(D,C)]. Ifvand p
v,p H
are clear from the context, then we write H ⇒H(cid:48). A sequence of derivation steps H ⇒ H ⇒
0 v1,p1 2 v2,p2
···⇒ H is called a derivation. A derivation is called creative if H and rhs(p) are all mutually
vn,pn n 0 i
disjoint. WewriteH ⇒ H(cid:48) ifthereexistsacreativederivationfromH toH(cid:48).
∗
Example2.19. Aderivationintheabovegrammarofthestringgraphwiththreenodevertices:
S ⇒ X ⇒ X ⇒ X ⇒
whereagainwecolorthenewlyestablishededgesinred.
Definition2.20(StartingGraph[14]). Wesaythatsn(S,z)∈GR isastartinggraphifithasonlyone
Σ,Γ
nodegivenbyz,itslabelisS,thegraphhasnoedgesandnoconnectioninstructions.
Definition 2.21 (Graph Grammar Language [14]). The graph language induced by a graph grammar
G=(Σ,∆,Γ,Ω,P,S) is given by L(G)={[H] | H ∈GR andsn(S,z)⇒ H forsomez}, where [H]
∆,Ω ∗
denotestheequivalenceclassofallgraphswhichareisomorphictoH.
Definition2.22(Confluence[14]). WesaythatagraphgrammarG=(Σ,∆,Γ,Ω,P,S)isconfluentifthe
followingholdsforeverygraphH∈GRE : ifH⇒ H ⇒ H andH⇒ H ⇒ H are
Σ,Γ u1,p1 1 u2,p2 12 u2,p2 2 u1,p1 21
creativederivationsofGwithu (cid:54)=u ,thenH =H .
1 2 12 21
TherearesimpleandeasilydecidableconditionsfordeterminingifanedNCEgrammarisconfluent. The
classofconfluentedNCEgrammarsisdenotedC-edNCE.
Definition 2.23 (LIN-edNCE grammar [14]). An edNCE grammar G is linear, or a LIN-edNCE gram-
mar,ifforeveryproductionX →(D,C),Dhasatmostonenonterminalnode.
Atanygivenpoint, onlyonenon-terminalcanbereplaced, soLIN-edNCEgrammarsareautomatically
confluent. Hence,theyareasubclassofC-edNCEgrammars.
A.Kissinger&V.Zamdzhiev 23
3 !-graphs vs C-edNCE grammars
In this section we outline the relationship between !-graph languages and context-free languages de-
scribedbyC-edNCEgrammarsandsomeoftheirrespectivesubclasses. Insubsection3.1,weshowthat
thelanguageinducedbyany!-graphH withnooverlapping!-boxescanbedescribedbyaLIN-edNCE
grammar, which can moreover be constructed effectively from H. In subsection 3.2, we first show that
thereexistsa!-graphwithtrivialoverlapof!-boxeswhoselanguagecannotbedirectlyrepresentedusing
anyC-edNCEgrammar. Next,weshowthatthelanguageofany!-graphH withtrivialoverlapbetween
!-boxescanberepresentedbyaLIN-edNCEgrammaruptowire-encodingandthegrammarcanbeef-
fectively constructed from H as well. In subsection 3.3 we show that the classes of languages induced
by(unrestricted)!-graphsandC-edNCEgrammarsrespectivelyareincomparable.
3.1 !-graphswithoutoverlap
Themainresultofthissubsectionisatheoremstatingthatthelanguageinducedbyany!-graphwithno
overlapping of !-boxes can be directly represented by a LIN-edNCE grammar, which can moreover be
generatedeffectively.
Before we present the main results in this subsection and the next one, we prove a series of lemmas
whichareusedintheproofofthetwomaintheorems. EachlemmadescribeshowtobuildbiggerLIN-
edNCEgrammarsoutofsmalleroneswhichinturndescribethelanguageinducedbycertainsubgraphs
of a given !-graph. All of our constructions are effective in the sense that they can be performed by a
computer.
Weintroducesomeconventionsthatareusedthroughoutourproofs. Givena!-graphH,itsverticeswill
be{v ,v ,...,v }. Tothesevertices,wewillassociatenon-finaledgelabels{α ,α ,...,α }whichwillbe
1 2 n 1 2 n
used in the productions of our grammars. Informally, an edge labeled with α will have as its source or
i
targeteithertheoriginalvertexv oroneofitscopies. Thisisusedinsomeproductionsofourgrammars,
i
sothatwecaneasilyrefertoallcopiesofsuchavertexatonceandconnectthemtoothervertices.
Throughouttheproof,wewillusegrammarssatisfyingthefollowingconditions:
1. ThegrammarisinLIN-edNCE,thuseveryproductionhasatmostonenon-terminalnodeinit
2. Thereisasinglefinalproductionwhichistheemptygraph
3. Every sentential form with a non-terminal node is such that all terminal vertices are connected
withanedgetothenon-terminalnode(inbothdirections)andeverysuchedgehasaslabelsome
α
i
4. Everyproductionexceptthefinalonehasconnectioninstructionswhichspecifythatbothincoming
andoutgoingedgesoftypeα areconnectedtothenon-terminal. Graphically,wewilldepictthat
i
usingbidirectionaledgesasashorthandnotationforanedgeineachdirection.
5. Foreveryi,thereisatmostoneterminalvertexintheproductionsofagrammarwhichisincident
toanedgewithlabelα
i
Forconvenience,wewillrefertoagrammarsatisfying1-5asbeingin!-linearform.
Lemma 3.1. Given a concrete string graph H, there exists a LIN-edNCE grammar G which generates
thelanguage{H}. Moreover,thisgrammarcanbeeffectivelygeneratedandisin!-linearform.
24 !-GraphswithTrivialOverlapareContext-Free
Proof. Thisisdonebythefollowinggrammar:
S
αα21 α1α2 αα12 F
H F
αn αn αn
wherethesetofverticesofH isV ={v ,v ,...,v }andeachedgewithlabelα hasassourceortarget
H 1 2 n i
v andthenon-terminalF.
i
Lemma3.2. Givena!-graphH anda!-linearformgrammarG whichgeneratesthesamelanguageas
H,thereexistsagrammarG(cid:48) whichgeneratesthesamelanguageasthefollowing!-graph:
H
Moreover,G(cid:48) canbeeffectivelygeneratedandisin!-linearform.
Proof. Let’sassumethatSisthestartingproductionofG,F isthefinalproductionofG andthatS(cid:48) and
F(cid:48) arenotproductionsofG.
First,wemodifytheproductionF tobethefollowing:
F
α1
α1α2 α2
S(cid:48)
αn αn
Finally,totheproductionsofG weaddthefollowingproductions,whereS(cid:48) isthestartingone:
S(cid:48) S(cid:48)
α1 α1
α1α2 α2 α1α2 α2 F(cid:48)
S F(cid:48)
αn αn αn αn
AderivationS(cid:48)⇒S⇒···⇒F createsaconcretestringgraphfromthelanguageofH,soitissimulating
a single EXPAND operation applied to the top-level !-box, together with a concrete instantiation of the
!-boxesinH. Byconstruction,wecaniteratethis,thusallowingustogeneratemultipledisjointconcrete
A.Kissinger&V.Zamdzhiev 25
graphs, all of which are in the language of H. A derivation S(cid:48) ⇒F(cid:48) simulates the final KILL operation
appliedtothetop-level!-box.
Lemma3.3. Givendisjoint!-graphsH,KandgrammarsG ,G whichgeneratethesamelanguagesasH
1 2
andK respectively,thenthereexistsagrammarG(cid:48) whichgeneratesthesamelanguageasthefollowing
!-graph:
H K
Moreover,G(cid:48) canbeeffectivelygeneratedandisin!-linearform.
Proof. LettheverticesofH be{v ,v ,...,v }andlettheverticesofK be{v ,v ,...,v }. Also,let
1 2 k k+1 k+2 n
S andF bethestartingandfinalproductionsrespectivelyofG. First,wemodifyeachproductionX of
i i i
G ,exceptF ,byaddingconnectioninstructionsforedgelabelsα ,...,α inthefollowingway(left):
1 1 k+1 n
X
α1 α1 F1 α1
αk αk α1α2 α2
Y
αk+1αk+1 S2
αn αn αn
αn
where the new additions are colored in red. This doesn’t change the language of G and is done so that
1
we can put the grammar in the required form. Similiarly, modify all productions of G , except F , by
2 2
adding to their connection instructions the missing edge labels α ,α ,...,α . Finally, modify F to be
1 2 k 1
theproductiondepictedabove(right),sothatwecanchaintogetherthetwogrammars.
The required grammar G(cid:48) has as its productions the modified productions of G and G with starting
1 2
production S . A derivation S ⇒ ··· ⇒ F creates a concrete graph from the language of H and a
1 1 1
derivation S ⇒···⇒F creates a graph from the language of K. By chaining the two grammars, we
2 2
simplygeneratetwodisjointconcretegraphs,onefromthelanguageofH andonefromthelanguageof
K,asrequired.
Lemma3.4. Given!-graphH,whereH containsa!-boxbandgivena!-linearformgrammarG which
generatesthesamelanguagesasH,thereexistgrammarsG(cid:48)andG(cid:48)(cid:48)whichgeneratesthesamelanguages
as:
and
respectively, where in both cases, the newly depicted edge (colored in red) is incident to the !-box b
in H and the edge is also incident to a node-vertex in H which is not in any !-boxes. Moreover, these
grammarscanbeeffectivelygeneratedandarein!-linearform.