Regular Description of Context Free
Graph Languages
Jo ost Engelfriet and Vincent van Oostrom
Department of Computer Science Leiden University
P O Box RA Leiden The Netherlands
email engelfri wi leidenuniv nl
Abstract A set of lab eled graphs can b e de ned by a regular tree
language and one regular string language for each p ossible edge lab el as
follows For each tree t from the regular tree language the graph g r t
has the same no des as t with the same lab els and there is an edge
with lab el from no de x to no de y if the string of lab els of the no des on
the shortest path from x to y in t b elongs to the regular string language
for Slightly generalizing this de nition scheme we allow g r t to have
only those no des of t that have certain lab els and we allow a relab eling
of these no des It is shown that in this way exactly the class of C edNCE
graph languages generated by C edNCE graph grammars is obtained
one of the largest known classes of context free graph languages
Introduction
There are many kinds of context free graph grammars see e g ENRR EKR
Some are no de rewriting and others are edge rewriting In b oth cases a pro duc
tion of the grammar is of the form X D C Application of such a pro duction
to a lab eled graph H consists of removing a no de or edge lab eled X from H
replacing it by the graph D and connecting D to the remainder of H accord
ing to the embedding pro cedure C Since these grammars are context free in
the sense that one no de or edge is replaced their derivations can b e mo d
eled by derivation trees as in the case of context free grammars for strings
However in particular for certain types of no de rewriting grammars the gram
mar may still b e context sensitive in the sense that the edges of the graph
generated according to the derivation tree may dep end on the order in which
the pro ductions are applied A graph grammar that do es not su er from this
quite disastrous context sensitivity is said to b e con uent or to have the
nite Church Rosser prop erty see Cou for a uniform treatment Thus for a
con uent graph grammar G each derivation tree of G yields a unique graph
?
The rst author was supp orted by ESPRIT BRWG No COMPUGRAPH I I
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The present address of the second author is Faculty of Mathematics and Computer
Science Vrije Universiteit de Bo elelaan a HV Amsterdam The Nether
lands email o ostrom cs vu nl
in the graph language generated by G Due to this close relationship to deriva
tion trees the generated graph language can b e describ ed in terms of a reg
ular tree language the set of derivation trees and a nite number of regular
string languages to simulate the embedding pro cedure We will show this for
the particular case of the no de rewriting edNCE graph grammars studied in
Kau Bra Bra ELR ELR Schu ELW EL EL ER CER Thus we
de ne the notion of a regular path description of a graph language mainly deter
mined by a regular tree language and a nite number of regular string languages
and prove that regular path descriptions have the same p ower as the con uent
edNCE grammars or C edNCE grammars The idea of using regular tree and
string languages for the description of graphs was introduced in Wel and in
vestigated in ELW for sp ecial cases of the C edNCE grammar
The structure of this pap er is as follows In Section we de ne the edNCE
grammar and in particular the con uent edNCE grammar In Section we in
tro duce the notion of a regular path description of a graph language generalizing
the regular path descriptions of Wel ELW In Sections and also some ex
amples and some easy lemmas can b e found Section contains the pro of of the
main result the characterization of the C edNCE graph languages by regular
path descriptions We use this result to show that the b oundary edNCE gram
mars or B edNCE grammars cf e g RW ELW have less generating p ower
than the C edNCE grammars In Section we consider a number of sp ecial cases
of the main result In particular we de ne sp ecial types of regular path descrip
tions that characterize the b oundary ap ex and linear edNCE graph languages
In Section we investigate the string generating p ower of C edNCE grammars
we view a graph grammar as a generator of all the strings that lab el directed
paths in the generated graphs We use the main result to show that the class of
string languages generated by C edNCE grammars in this way equals the class
of output languages of nondeterministic tree walking transducers This implies
that this string generating metho d is more p owerful than the one of EH that
gives the output languages of deterministic tree walking transducers
The main result of this pap er strengthens our b elief that the class of C edNCE
graph languages which seems to b e the largest known class of graph languages
that can b e generated by context free graph grammars where context free is
taken in the sense of Cou is a robust class of context free graph languages
it can b e characterized in several di erent ways Other characterizations can b e
found in CER by handle rewriting hypergraph grammars and in Oos Eng
by monadic second order logic
The results of this pap er were established in and presented in Oos
and in Eng The only added result is the characterization of ap ex edNCE
languages Theorem which uses EHL More recent work on the class of
C edNCE graph languages or its sub classes can b e found in e g Bra Cou
Cou Eng Eng SW SW KL For a survey see ER
The reader is assumed to b e familiar with the basic concepts of formal lan
guage theory see e g HU and of regular tree languages see e g GS
Con uent edNCE Graph Grammars
In this subsection we give formal de nitions for the edNCE graph grammars
and in particular for the con uent edNCE C edNCE graph grammars These
grammars generate directed graphs with lab eled no des and lab eled edges
Let b e an alphab et of no de lab els and an alphab et of edge lab els A
graph over and is a tuple H V E where V is the nite set of no des
E f v w j v w V v w g is the set of edges and V is
the no de lab eling function The comp onents of H are also denoted as V E
H H
and resp ectively Thus we consider directed graphs without lo ops multiple
H
edges b etween the same pair of no des are allowed but they must have di erent
lab els A graph is undirected if for every v w E also w v E Graphs
with unlab eled no des and or edges can b e mo deled by taking and or to b e
a singleton resp ectively
The set of all graphs over and is denoted GR A subset of GR is
called a graph language
As usual two graphs H and K are disjoint if V V Also as usual
H K
H and K are isomorphic if there is a bijection f V V such that E
H K K
f f v f w j v w E g and for all v V f v v The
H H K H
reader is assumed to b e familiar with the way in which concrete graphs are used
as representatives of abstract graphs which are equivalence classes of concrete
graphs with resp ect to isomorphism We are usually interested in abstract graphs
but mostly discuss concrete ones For instance whereas a graph language is
de ned to b e a set of concrete graphs we usually view it as a set of abstract
graphs
After these preliminaries we turn to the de nition of edNCE graph gram
mar The name of these grammars can b e explained as follows NCE stands for
neighbourhood controlled embedding the d stands for directed graphs and the
e means that not only the no des but also the edges of the graphs are lab eled in
particular the e stresses the fact that the edNCE grammar allows for dynamic
edge relabeling Thus edNCE grammars are graph grammars with neighbour
hood controlled embedding and dynamic edge relabeling They were introduced in
Nag Nag Nag as depth context free graph grammars and studied in
e g Kau Bra Bra Schu They were also investigated as generalizations of
NLC graph grammars in e g EL EL ELW
De nition An edNCE grammar is a tuple G P S where
is the alphab et of no de lab els is the alphab et of terminal no de lab els
is the alphab et of edge lab els is the alphab et of nal edge lab els
P is the nite set of pro ductions and S is the initial nonterminal
A pro duction is of the form X D C with X D GR and
C V fin outg ut
D
Elements of are called nonterminal no de lab els and elements of
non nal edge lab els A no de with a terminal or nonterminal lab el is said
to b e a terminal or nonterminal no de resp ectively and similarly for nal and
non nal edges For a pro duction p X D C X is the left hand side of
p D is the right hand side of p and C is its connection relation We write
lhs p X rhs p D and con p C Each element x d of C with
x V and d fin outg is a connection instruction of
H
p To improve readability a connection instruction x d will always b e
written as x d In the literature the elements of a connection instruction
are often listed in another order Two pro ductions X D C and X
D C are called isomorphic if X X and there is an isomorphism f from
D to D such that C f f x d j x d C g We will assume
that P do es not contain distinct isomorphic pro ductions By copy P we denote
the in nite set of all pro ductions that are isomorphic to a pro duction in P an
element of copy P will b e called a production copy of G
The pro cess of rewriting in an edNCE grammar is de ned through the appli
cation of pro ductions or rather pro duction copies in the usual way Informally
a rewriting step according to a pro duction p X D C consists of removing
a no de v lab eled X the mother no de from the given host graph H substi
tuting D the daughter graph in its place and connecting D to the remainder
of H in a way sp eci ed by the connection instructions in C Together with v all
edges incident with v are removed to o A connection instruction x out
of C means that if there was a lab eled edge from the mother no de v to a no de
w with lab el in H then the connecting pro cess will establish a lab eled edge
from x to w And similarly for in instead of out where in refers to incoming
edges of v and out to outgoing edges of v Note in particular that the edge
lab el is changed from into which explains the notation The formal
de nition is as follows
De nition Let G P S b e an edNCE grammar Let H and
0
H b e graphs in GR let v V and let p X D C b e a pro duction
H
0
copy of G such that D and H are disjoint Then we write H H or just
v p
0 0 0
H H or H H if v X and H is the graph V E in GR
p H
such that
V V fv g V
H D
E f x y E j x v y v g E
H D
f w x j w v E w x in C g
H H
f x w j v w E w x out C g
H H
x x if x V fv g and x x if x V
H H D D
0
H H is called a derivation step and a sequence of such derivation steps is
v p
called a derivation A derivation
H H H
n v p v p v p
n n 2 2 1 1
n is creative if the graphs H and rhs p i n are mutually disjoint
i
0
We wil l restrict ourselves to creative derivations Thus we write H H if
0
there is a creative derivation as ab ove with H H and H H Let sn S z
n
denote the graph with a single S lab eled no de z and no edges A sentential form
of G is a graph H such that sn S z H for some z The set of all sentential
forms of G is denoted SF G The graph language generated by G is
L G fH GR j sn S z H for some z g
ut
0
It is not di cult to show that if H and H are isomorphic and sn S z H
0 0 0
then sn S z H for some z Thus L G is closed under taking isomorphic
copies
An edNCE grammar is nonblocking if L G fH GR j sn S z
H for some z g This means that if a sentential form H has terminal no des only
i e cannot b e rewritten any more then all its edges are nal Note that we do
not assume that edNCE grammars are nonblocking as opp osed to Eng Eng
Example To draw a pro duction X D C of an edNCE grammar we draw
the graph D in the usual fashion with no des represented by dots and edges
by arrows and we add C to D in the following way a connection instruction
x in C is represented by a dashed arrow from a symbol to the dot
representing x with lab el for the connection instruction x out
the direction of the arrow is reversed
As a rst example consider the edNCE grammar G P S
with fS X i n f g fi n f g f g f g
and P consists of the three pro ductions drawn in Fig Thus pro duction p is
X D C with V fx y g E f x y g x n y f and
D D D D
C f n x in n y out g L G consists of all ladders of the form
shown in Fig with at least six no des Note that intuitively stand
for right left down and up resp ectively
As another example consider the edNCE grammar G P S
with fS X ng fng f g and P consists of the three
pro ductions p p p shown in Fig A dashed arrow from or to X n repre
a b c
sents two connection instructions one for X and one for n in the obvious way
G generates all ro oted binary trees with lab eled edges from each parent to
its children with additional lab eled edges from each leaf to the ro ot and with
additional lab eled edges that chain the leaves of the tree An example of such
a tree like graph is given in Fig except that the lab els a b b c c of all
l r l r
no des should b e replaced by n Note that in a derivation of G the lab eled
edges to the ro ot are created by pro duction p are passed from nonterminal
a
no de to nonterminal no de by pro duction p and are nally attached to the leaves
b
by pro duction p
c
0
Let G b e the edNCE grammar that is obtained from G by erasing all edges
0
and connection instructions that involve It should b e clear that L G consists
of all ro oted binary trees with additional edges from the leaves to the ro ot i e
all graphs of L G without their lab eled edges
As a third and nal example consider the edNCE grammar G
P S with fS ng fng f g and P consists
of the three pro ductions p p p shown in Fig An undirected edge b etween
a b n
i n
p S
X
n
n
n
n
p X
X
n
n
n
n
p X
n
f
Fig Pro ductions of G
i n n n
n n n f
Fig A ladder generated by G
x and y stands for two directed edges one from x to y and one from y to x
and similarly for connection instructions L G is the set of all cographs see
CLS It is the smallest set of unlab eled undirected graphs that contains the
one no de graph and is closed under the op erations of join and disjoint union
The one no de graph corresp onds to pro duction p the join op eration i e tak
n
ing the disjoint union of two graphs and joining every no de of the one graph
with every no de of the other graph corresp onds to pro duction p and the op
a
eration of disjoint union corresp onds to pro duction p An example of a cograph
b
is the square it is the join of two discrete graphs each of which is the disjoint
n
p S
a
X X
n
n
n n
p X
b
X n X n
X X
n n
X n p X X n
c
n
Fig Pro ductions of G
a
c b
r l
c b
l r
c c
l r
Fig A tree like graph generated by G where all no de lab els should b e n
S n S n
p S
a
S S
S n S n
p S
b
S S
S n
p S
n
n
Fig Pro ductions of G
union of two one no de graphs ut
The edNCE grammar has certain undesirable non context free prop erties This
is caused by the fact that it need not be con uent i e that the result of a
derivation may dep end on the order in which the pro ductions are applied This
problem turns up in sentential forms that have edges b etween two nonterminal
no des as in the following example
Example Consider an edNCE grammar G P S with
0 0
fS A B a bg fa bg f g f g and the following
three pro ductions X D C
p X S V fu v g E f u v g u A v B and C
S D D D D
p X A V fxg E x a and
A D D D
0
C f B x out b x out g
p X B V fy g E y b and
B D D D
C f A y in a y in g
The application of pro ductions p p p in that order gives a derivation
S A B
H where H is rhs p and H is the H H sn S z
S v p u p z p
B A S
graph with two no des x and y lab eled a and b resp ectively and one edge
x y However interchanging the application of p and p to no des u and v
A B
H where H H resp ectively gives a derivation sn S z
u p v p z p
A B S
0
H is the same as H except that its edge is x y Since the order of appli
cation of pro ductions results in two di erent graphs H and H the grammar
is not con uent ut
In the literature it is customary to give a dynamic de nition of con uence see
e g Kau Schu Bra Cou Eng Eng CER Here we prop ose a static one
that can easily b e checked on the embedding relations of the pro ductions of the
grammar cf De nition of EJKR and Lemma of Cou
De nition An edNCE grammar G P S is con uent or a
C edNCE grammar if for all pro ductions X D C and X D C in
and all edge lab els the following and x V P all no des x V
D D
2 1
equivalence holds
x x in C X x out C and
D
1
x x out C ut X x in C and
D
2
By C edNCE we denote the class of graph languages generated by C edNCE
grammars
All grammars discussed in Example are C edNCE grammars Many other
sets of graphs with tree like graph theoretic prop erties can b e de ned by C
edNCE grammars For example series parallel graphs transitive VSP graphs
complete bipartite graphs maximal outerplanar graphs edge complements of
trees and for xed k k trees graphs of treewidth k pathwidth k cutwidth
k bandwidth k cyclic bandwidth k and top ological bandwidth k see
e g RW EL
A symbolic picture of De nition is given in Fig Intuitively the de nition
means that if the two pro ductions are applied to a graph with a single edge
v v where v is lab eled X then the same edges x x are established
i i
b etween no des of their right hand sides indep endent of the order in which the
pro ductions are applied From this intuition the following characterization of
con uence easily follows the result of a derivation do es not dep end on the order
in which the pro ductions are applied
X X
0
x x
Fig Con uence
Prop osition An edNCE grammar G P S is con uent if and
only if the fol lowing holds for every graph H GR
H are creative deriva H H and H H if H
v p v p v p v p
1 1 2 2 2 2 1 1
tions of G with v v V and v v then H H
H
The de nition of con uence from the literature where it is also called the nite
Church Rosser or fCR prop erty is exactly the same as the previous prop osition
except that H is restricted to b e a sentential form of G let us call this dynamic
con uence Although there are more dynamically con uent than con uent ed
NCE grammars it can b e shown that they generate the same class C edNCE of
graph languages see ER Although dynamic con uence is a decidable prop
erty of edNCE grammars see Kau the advantage of our notion of con uence
is that it is completely static
It is shown in SW that for every C edNCE grammar an equivalent non
blo cking C edNCE grammar can b e constructed This fact will not b e used but
will b e a consequence of our pro ofs cf the discussion after Theorem
Several natural sub classes of the C edNCE grammars have b een investigated
in the literature We will consider three of them Whenever we de ne an X
edNCE grammar for some X X edNCE will denote the class of graph languages
generated by X edNCE grammars
An edNCE grammar G P S is boundary or a B edNCE gram
mar if for every pro duction X D C D do es not contain edges b etween
nonterminal no des and C do es not contain connection instructions x d
where is nonterminal Obviously the second condition implies that b ound
ary grammars are con uent A B edNCE grammar G P S is
apex or an A edNCE grammar if for every pro duction X D C and every
connection instruction x d C x and are terminal The b ound
ary restriction on graph grammars was introduced in RW the ap ex restriction
was rst considered in ELR An edNCE grammar G P S is
linear or a LIN edNCE grammar if for every pro duction X D C D has at
most one nonterminal no de It is easy to see that LIN edNCE B edNCE It is
shown in EL that A edNCE and LIN edNCE are incomparable sub classes of
B edNCE The class B edNCE also contains the B NLC languages of RW and
as shown e g in ER the hyperedge replacement HR graph languages of
BC HK Hab
0
Grammar G from Example is b oth linear and ap ex Grammar G is b ound
ary but not linear or ap ex Grammars G and G are not b oundary
The class A edNCE can b e characterized within the class C edNCE there
is a simple condition on a graph language L C edNCE that expresses mem
b ership of L in A edNCE viz that L is of b ounded degree i e that there is a
number k such that all no des in all graphs of L have degree at most k This
characterization was shown in EHL see also Eng
Prop osition For every graph language L C edNCE
L A edNCE if and only if L is of bounded degree
The class C edNCE of graph languages generated by C edNCE grammars has a
large number of nice closure prop erties Here we need a very simple one closure
0
under edge relab eling Let b e an edge relab eling i e a mapping
0
where and are edge lab el alphab ets For a graph H GR we de ne
0
to b e the graph V E with E f v w j v w H GR
H H
E g
H
Prop osition C edNCE is closed under edge relabelings i e if is an edge
relabeling and L C edNCE then L C edNCE The classes B edNCE
A edNCE and LIN edNCE are also closed under edge relabelings
0
Proof Let G P S b e a C edNCE grammar and let
0
b e an edge relab eling It can b e assumed that
0
is de ned to b e the graph For a graph D GR D GR
[
V E such that
D D
E f v w j v w E v and and w g
D D D
f v w E j v or or w g
D D D
Thus only the nal edges b etween terminal no des are relab eled Similarly for a
connection relation C V fin outg we de ne the connection
D
relation C to b e
f x d j x d C x and and g
D
f x d C j x or or g
D
0 0 0 0
We now construct the edNCE grammar G P S with
0
P fX D C j X D C is in P g Using the con uence of G it
0
is easy to verify that G is still con uent in De nition rst consider the case
that x and x are terminal and is nal and then the remaining case Clearly
the b oundary ap ex and linear prop erties are preserved It is also easy to see that
0
the derivations of G starting with sn S z are of the form sn S z H
H where sn S z H H is a derivation of G Formally
n n
0
this can b e proved by induction on n It shows that L G L G ut
Regular Path Descriptions
As observed b efore all graphs generated by a C edNCE grammar are tree
like An alternative grammar indep endent way of describing a set of tree like
graphs H is by taking a tree t from some regular tree language de ning the no des
of H as a subset of the vertices of t and de ning an edge b etween no des u and
v of H if the string of vertex lab els on the shortest undirected path b etween u
and v in t b elongs to some regular string language Such a description of a graph
language will b e called a regular path description This idea was introduced in
Wel for linear graph grammars and investigated for LIN eNCE and B eNCE
grammars in ELW where the missing d means that the generated graphs are
undirected Note that the no des of trees will also b e called vertices in order
not to confuse them with the no des of the graphs involved
Let us rst de ne the regular tree languages The following de nition of a
regular tree grammar as a sp ecial type of edNCE grammar is equivalent in
generating p ower with the usual one in tree language theory see GS The
ro oted ordered trees from tree language theory will b e viewed here as usual
as a sp ecial type of graph each vertex of the tree has a directed edge to each
of its k children k and the order of the children is indicated by using the
numbers k to lab el these edges
As usual a ranked alphabet is an alphab et together with a mapping rank
f g By m we denote the maximal number rank
De nition An edNCE grammar G P S is a regular tree gram
mar or a REGT grammar if is a ranked alphab et f m g
and for every pro duction X D C in P
V fx x x g for some k
D k
x is in and has rank k and for every i k
D
either x is in or x is in and has rank
D i D i
E f x i x j i k g and
D i
C f i i x in j i g
G is in normal form if x for all i k ut
D i
Since and are uniquely determined by we will sp ecify a REGT grammar
as P S
Thus for a pro duction p of a regular tree grammar G rhs p consists of a
terminal parent x and k terminal or nonterminal children x x Edges
k
lead from the parent to each child and the children are ordered by numbering
their edges from to k Such a pro duction X D C can mo dulo isomor
phism b e denoted uniquely as X with x and x
k D i D i
for i k Every sentential form of G is a tree of which all internal ver
tices are terminal thus only leaves are rewritten into an internal vertex with
k children
A graph language generated by a REGT grammar is a regular tree language
It is easy to see and well known that every regular tree language can b e
generated by a regular tree grammar in normal form For a ranked alphab et
we denote by T the set of all trees over i e the regular tree language
generated by the REGT grammar with one nonterminal S and all pro ductions
k
S S for every where k rank
It is obvious that every REGT grammar is a con uent edNCE grammar
b ecause it is even an A edNCE grammar Hence every regular tree language is
an A edNCE graph language
We now turn to the regular path description of graph languages An essential
concept to b e used is the string of lab els on the shortest undirected path from
one vertex u of a tree to another vertex v Clearly such a path rst ascends from
u to the least common ancestor of u and v and then descends to v In the string
we indicate this change of direction by barring the lab el of the least common
ancestor
De nition Let b e a ranked alphab et and let f j g For t T
as follows Let z V b e the least and u v V we de ne path u v
t t
t
common ancestor of u and v in t Let u u m and v v n
m n
b e the vertices on the directed paths in t from z to u and from z to v resp ectively
thus z u v u u and v v Then
m n
z v v path u v u u
t t t n t m t
t
ut
We are now ready for the de nition of a regular path description
De nition A regular path description is a tuple R T h W
where is a ranked alphab et and are alphab ets of no de and edge lab els
resp ectively T T is a regular tree language h is a partial function from
to and W is a mapping from to the class of regular string languages such
that for every W
The graph language described by R is L R fg r t j t T g where g r t
R R
is the graph H GR such that
V is the set of vertices v of t for which v is in the domain of h
H t
v h v for v V and
H t H
E is the set of all edges u v with path u v W ut
H
t
Note that L R GR Note that h is used b oth to determine which vertices
of the tree t are no des of the graph g r t and to de ne their lab els in that
R
graph on the basis of their lab els in the tree Note that for each edge lab el
W is the regular string language that de nes the graph edges with lab el
Note nally that L R is closed under taking isomorphic copies b ecause T is
Let RPD denote the class of graph languages that are describ ed by regular
path descriptions The main result of this pap er is that C edNCE RPD
We now de ne some natural sub classes X RPD of RPD by restricting the
regular path descriptions to b e of type X
Let B RPD b e the sub class of RPD obtained by restricting every W to
This means for a regular path description of type B b e a subset of
that graph edges are only established b etween tree vertices of which one is a
descendant of the other We will show that B edNCE B RPD essentially the
same result is shown in Theorem of ELW for undirected graphs
Let A RPD b e the sub class of RPD obtained by restricting every W to
b e nite Thus for a regular path description of type A graph edges can only
b e established b etween tree vertices that are at a b ounded distance from each
other We will show that A edNCE A RPD
Let LIN RPD b e the sub class of RPD obtained by restricting the symbols
of the ranked alphab et to have rank or This means that the trees in
the regular tree language are in fact strings Thus a regular path description of
type LIN uses regular string languages only Note that obviously LIN RPD
B RPD We will show that LIN edNCE LIN RPD as for B essentially the
same result for undirected graphs is shown in Theorem of ELW
Example We will give regular path descriptions of the graph languages gener
ated by the C edNCE grammars of Example
The language L G of all ladders can b e describ ed by the regular
path description R T h W of type LIN with fi n a f g
rank i rank n rank a rank f fi n f g f g
T iana na nf where we have written the trees as strings in pre x notation
h is the total function with h i i h n h a n and h f f and W
ian nang W fna nf g W ff na anag and is given by W f
ig W fa
a a a
i f
n n n
Fig Regular path description of a ladder
Note that R is not only of type LIN but also of type B and of type A To
see how R works see Fig Assuming that a n this gure represents the
t of the tree t ianananf which is in fact the ladder of Fig graph g r
R
1
The tree t itself is not drawn but is suggested as a horizontal chain with the
ro ot at the left and the leaf at the right
The graph language L G of all binary trees with additional edges is
describ ed by the regular path description R T h W with
fa b b c c g rank a rank b rank b rank c rank c
l r l r l r l r
fng f g T L G where G is the regular tree grammar with
pro ductions S aLR L b LR R b LR L c and R c with
l r l r
nonterminals S L R of which S is the initial one h is the total function with
a b b b b c c h n for all and W is de ned by W
l r l r l r
W c c b b a and W c b b c a b b b b c c
l r l r r l l l r r l r
r
l
To see how R works see Fig Taking all no de lab els to b e n the gure
t of the tree t ab c b c c c in pre x notation represents the graph g r
l l r l r r R
2
The tree t itself is not drawn but equals the tree spanned by the lab eled edges
apart from the edge lab els which should b e or
Note that R is not of type B A or LIN Removing from and W
0 0
is obtained that is of type B of L G from W a regular path description R
but not of type A or LIN
The graph language L G of all cographs is describ ed by the regular path
description R T h W with fa b ng rank a rank b
and rank n fng f g T T h n n h a and h b are
unde ned and W n a b a a b n
To see how R works see Fig Disregarding the dotted lines the gure
represents the tree t abnnbnn in pre x notation generated by G The no des
a
b b
n n n n
Fig Regular path description of a cograph
with lab el n together with the dotted undirected edges form the cograph
t which is the square By the de nition of W two leaves of t are g r
R
3
connected by an edge in g r t if their least common ancestor is lab eled a This
means that R formalizes the usual cotree representation of cographs see CLS
In fact the co trees in L G can b e viewed as expressions with a representing
the join op eration b disjoint union and n the one no de graph ut
We need the following easy closure prop erty of RPD
Lemma Let L be a graph language in RPD with L GR
0 0
0 0
is in RPD for al l and Then L GR
A similar statement holds for B RPD A RPD and LIN RPD
Proof Let L L R for the regular path description R T h W
0 0
0 0 0 0
L R where g Obviously L GR De ne T ft T j g r t GR
R
0 0
R T T h W Since the regular tree languages are closed under
0 0
intersection it su ces to show that T is regular Now T T T where T
0 0
g Clearly T is ft T j g r t GR g and T ft T j g r t GR
R R
0
0
where f j h is unde ned or h the regular tree language T
S
0
T where T is the set of all t T such that g Also T T
0
2
g r t has at least one edge with lab el Since the regular tree languages are
R
closed under union and complement it su ces to show that T is regular for
every Clearly T is the set of all t T such that there exist u v V
t
with path u v W Since W is a regular string language there is a
t
nite automaton A that accepts W It is not di cult to write a regular
tree grammar G that generates T G decides nondeterministically that it is
generating the least common ancestor z of some u and v guesses a state of A
and then simulates the b ehaviour of A on the path from z to v where it should
arrive in a nal state of A and simulates the b ehaviour of A backwards on the
path from z to u where it should arrive in an initial state of A We leave the
details of G as an exercise to the reader ut
The Main Result
In this section we prove the main result RPD C edNCE We start with the
inclusion of RPD in C edNCE We rst consider a di erent but strongly related
kind of graph description The edge lab els of the graphs are restricted to b e pairs
of states of a nite automaton
It is convenient to consider nite automata that are deterministic except that
they have an arbitrary number of initial states A nite automaton is a tuple
A Q I F where Q is the nite set of states is the input alphab et
Q Q is the transition function I Q is the set of initial states
and F Q is the set of nal states The transition function is extended in the
usual way to a mapping Q Q and the language recognized by A is
L A fw j i I i w F g
De nition An automaton path description is a tuple R T h A
where is a ranked alphab et is an alphab et T is a regular tree language
I F is a in T h is a partial function from to and A Q
nite automaton with L A The graph language described by R is
L R fg r t j t T g where g r t is the graph H GR such that
R R I F
V is the set of vertices v of t for which v is in the domain of h
H t
v h v for v V and
H t H
E is the set of all edges u hq q i v such that q path u v q q I
H
t
and q F ut
Lemma If R is an automaton path description then L R C edNCE
Proof Let R T h A and A Q I F Moreover let G
N P S b e a regular tree grammar in normal form generating T where
N disjoint with is the set of nonterminals of G We construct a C edNCE
0 0 0
grammar G N Q Q I F P S such that L G L R The idea
0
is that G simulates G generating the appropriate vertices of the tree t generated
0
by G To generate the edges of g r t G simulates the b ehaviour of A in its
R
edge lab els through the use of the dynamic edge relab eling feature of edNCE
0
grammars To this aim G also keeps edges b etween nonterminal vertices u
and v of a sentential form s generated by G viz all edges u hq q i v such
0 0
that q path u v q where path u v is obtained from path u v by
s s s
erasing the lab els of u and v note that all nonterminal vertices are leaves of s
0
Similarly G keeps edges from each nonterminal vertex u to each terminal vertex
v with q F and only the lab el of u erased and from each terminal vertex u
to each nonterminal vertex v with q I and only the lab el of v erased
0
The set of pro ductions P is de ned as follows Let p X X X
k
b e a pro duction of G with V fx x x g x and
k
rhs p rhs p
x X N for i k With this pro duction p P we asso ciate
i i
rhs p
0 0
one pro duction p P We rst consider the case that h is de ned Then
0
p X D C where
V fx x x g
D k
E f x hq q i x j i j q q g
D i j
f x hq q i x j i q I q q g
i
q g f x hq q i x j i q F q
i
x h and x X for i k
D D i i
0 0
C f hq q i hq q i x out j q q i g
i
0 0 0
f hq q i hq q i x out j q q q I g
0 0
f hq q i hq q i x in j q q i g
i
0 0 0
f hq q i hq q i x in j q q q F g
where we have used to denote an arbitrary element of N In the case that
0
h is unde ned the ab ove de nition of p should b e changed in the obvious
way by dropping x from V and restricting all other comp onents accordingly
D
such that only the rst part of E the second part of and the rst and
D D
third parts of C remain
0 0
This ends the de nition of G It is straightforward to verify that G is con
0
uent Intuitively the reason is that every edge in a sentential form s of G is of
the form u hq q i v Moreover if u is rewritten then the edge lab el changes
0 0
and the connection instructions do not insp ect q or q i for some q into hq
0 0
v Similarly if v is rewritten it changes into hq q i for some q and q and
s
u are not insp ected As a consequence if b oth u and v are rewritten the
s
0 0
edge lab el will b e hq q i indep endent of the order of rewriting In fact this idea
is formalized in Lemma of ER
To show the correctness of the construction we extend the de nition of g r
R
to sentential forms of G Let s b e a sentential form of G Note that s is a
tree with the nonterminals having rank We de ne g r s to b e the graph
R
H GR such that
N [ QQ
V is the union of the set of nonterminal vertices of s and the set of terminal
H
vertices v of s for which v is in the domain of h
s
v v for a nonterminal vertex v v h v for a terminal
H s H s
vertex v and
0
E is the set of all edges u hq q i v such that q path u v q and if
H
s
0
u is terminal then q I and if v is terminal then q F where path u v is
s
obtained from path u v by erasing all elements of N
s
It can now b e shown by induction on the length of the derivations that
for every graph H GR sn S z H if and only if there exists
0
N [ QQ
G
s T such that sn S z s and H g r s Since g r s GR
N [ R R I F
G
0
i s T this implies that L G L R This proves the lemma
Note that if g r s GR then s T and hence g r s GR
R QQ R I F
0
This shows that G is nonblocking cf the de nition of the nonblocking prop erty
just after De nition ut
Having done most of the work we now show that RPD is included in C edNCE
Lemma RPD C edNCE
Proof Let R T h W b e a regular path description Since for every
W is regular there is a nite automaton with one initial state that
recognizes W By putting all these automata disjointly together it should
I F such that b e clear that there exists a nite automaton A Q
I and for every W fw j w F g Now consider
0
the automaton path description R T h A and let I F b e
0
the edge relab eling such that q q q Obviously L R L R Hence
by Lemma and Prop osition L R is in C edNCE This proves the lemma
It is easy to see that the construction in the pro of of Prop osition preserves
the nonblocking prop erty of the edNCE grammars Together with the remark at
the end of the pro of of Lemma this shows that L R can b e generated by a
nonblocking C edNCE grammar ut
To prove that C edNCE RPD we will rst develop a few technical to ols First
of all in order to nd a regular path description for a given C edNCE grammar
it is convenient to assume that all no des of a right hand side of a pro duction have
distinct lab els It is easy to do this for the nonterminal no des for the terminal
no des a no de relab eling is needed to re establish the original lab els Let b e
0 0
a no de relab eling i e a mapping where and are no de lab el
0
alphab ets For a graph H GR we de ne H GR to b e the graph
V E where v v for every v V Let us say that a graph
H H H H
D is uniquely labeled if for all u v V if u v then u v And let
D D D
us say that a C edNCE grammar is uniquely lab eled if all right hand sides of its
pro ductions are uniquely lab eled
Lemma For every C edNCE grammar G one can construct a uniquely la
0 0
beled C edNCE grammar G and a node relabeling such that L G L G
The same is true for B edNCE A edNCE and LIN edNCE grammars
Proof Let G P S and let m b e an upp er b ound on the number
of no des with the same lab el in the right hand sides of the pro ductions in P
0 0
Let b e any alphab et and let b e any surjective mapping such that
m for every where A is the cardinality of a nite set A
0 0 0 0 0 0 0
Then we construct G P S where S is any
0
element of S and P is de ned as follows Let p X D C b e in P
0 0
0
Construct a uniquely lab eled graph D GR such that D D De ne
0 0 0 0 0
C f x d j x d C g Then corresp onding to p
0 0 0 0 0
P contains the pro ductions X D C for every X X
0
It can easily b e veri ed that G is con uent and that the construction pre
serves the b oundary ap ex and linear prop erties It should also b e clear that
0 0 0 0
L G L G where is the restriction of to ut
In the ab ove lemma the no de relab eling is not really needed for the case of
arbitrary C edNCE grammars In fact the grammar can rst b e transformed
into an equivalent one in which each right hand side of a pro duction contains at
most one terminal no de see e g Oos ER
The following lemma is obvious it su ces to comp ose the function h of the
regular path description with the no de relab eling
Lemma RPD is closed under node relabelings The same holds for B RPD
A RPD and LIN RPD
Thus in the remainder of this section it su ces to consider uniquely lab eled
C edNCE grammars
The main idea in the construction of a regular path description R for a
C edNCE grammar G is to use the set of derivation trees of G as the regular
tree language of R In our case it is convenient to de ne derivation trees in
such a way that the internal vertices are lab eled by pro ductions of G whereas
the leaves are lab eled by the no de lab els of G Intuitively if H is generated
by G according to a derivation tree t the pro ductions that are used in the
generation are on the lab els of the internal vertices of t whereas the leaves of t
represent the no des of H The edges b etween two no des u and v of H can then b e
determined by verifying that path u v b elongs to a certain regular language
t
Note that path u v consists of the two sequences of pro ductions that are used
t
to generate u and v and of the lab els of u and v which determine u and v in
the right hand sides of the pro ductions that are applied last It su ces to know
this information in order to simulate the connection instructions that build the
edges b etween u and v and this simulation can b e done by a nite automaton
We rst de ne a regular tree grammar that generates the derivation trees of
a C edNCE grammar
De nition Let G P S b e a uniquely lab eled C edNCE gram
mar We will view the elements of P and also as symbols of a ranked alphab et
where the rank of a pro duction p is the number of no des of rhs p and the rank of
every is The derivation tree grammar of G is the regular tree grammar
0 0 0 0 0 0 0 0 0
G P S where P P S S and P consists
of all pro ductions X p with p P X lhs p k V and
k
rhs p
f g f v j v V g The order of the no de lab els of rhs p
k
rhs p rhs p
0
in this pro duction of P is arbitrary but xed Note that all are distinct
i
b ecause rhs p is uniquely lab eled ut
We now de ne the set of all p ossible lab els of paths from a vertex to a leaf in a
derivation tree of G
De nition Let G P S b e a uniquely lab eled C edNCE gram
mar The set PS G of path strings of G is de ned to b e the set of all strings
p p P such that n p P for i n for every
n i
i n there is a unique no de in rhs p with lab el lhs p and if
i i
n there is a unique no de in rhs p with lab el
n
Let p p PS G Let y y b e the unique no des describ ed ab ove
n n
with y rhs p Then we de ne rst p p y where we assume that
i i n
n We also de ne conn p p f d j
n n
and y d con p for all i ng And we
n i i i i
de ne lhs p p lhs p if n and if n ut
n
Recall that con p is the connection relation of the pro duction p Intuitively
i i
p p PS G is the sequence of lab els of the vertices on a path from a
n
0
vertex to a leaf in a sentential form of the derivation tree grammar G of G the
internal vertices are lab eled by pro ductions p p and the leaf is lab eled by
n
The set conn p p formalizes the total e ect of the connection instructions
n
that are used when the pro ductions are applied and pro duce the leaf y Note
n
that for n conn f d j d fin outgg Note also
that conn p f d j y d con p g where y is the unique
no de of rhs p with lab el
It should b e clear that PS G is regular it is easy to de ne a nite automaton
that checks the requirements in its de nition The following fact is obvious from
the de nition of conn
Lemma Let w w PS G with w P and Then d
i
conn w w if and only if there exists such that d conn w
and d conn w where lhs w
In the next lemma we show that the prop erty of con uence can b e generalized
to two arbitrary sequences of pro ductions rather than just two pro ductions
Although this is a well known fact we need it here in the following technical
form
Lemma Let G P S be a uniquely labeled C edNCE gram
mar For al l path strings w w PS G with w P and and
i i
al l edge labels the fol lowing equivalence holds where X lhs w
i i i
X out conn w and in conn w
X in conn w and out conn w
Proof For w w P the ab ove equivalence is exactly the one in the de nition
of con uence De nition For w i e the empty string b oth conditions
say that in conn w and similarly for w
For the remaining cases we use induction on the sum of the lengths of w
and w The basis of the induction has already b een treated In the induction
step we may assume that w and w are nonempty Or viewed in another way
consider p w and p w in PS G with p P w P and Let
i i i
lhs w Intuitively it should now b e p ossible to change the sequence of
i i i
pro ductions p w p w into the sequence p p w w by interchanging w and p
then changing it into p p w w by interchanging b oth p p and w w and
nally changing it into p w p w by interchanging p and w Formally this is
proved as follows
X out conn p w and in conn p w
by Lemma twice
0
X out conn p
X out conn w
0
in conn p and
0
in conn w
by induction for the second and third line
0
X out conn p
in conn p
0
out conn w and
0
in conn w
by con uence for the rst two lines and by induction for the last two lines
0
X in conn p
out conn p
0
in conn w and
0
out conn w
by induction for the second and third line
0
X in conn p
X in conn w
0
out conn p and
0
out conn w
by Lemma twice
X in conn p w and out conn p w ut
We are now ready to show the inclusion of C edNCE in RPD
Lemma C edNCE RPD
Proof By Lemma s and it su ces to consider a uniquely lab eled C edNCE
grammar G P S Instead of constructing a regular path descrip
tion of L G we will construct one of SF G GR the set of sentential
forms of G This is su cient by Lemma b ecause L G SF G GR
0
We de ne the regular path description R P S F G h W where
0 0
G P P P S is the derivation tree grammar of G and h is the
identity mapping on It remains to de ne W
0
Thus we use the set of sentential forms of G as the regular tree language
of R and it should b e clear that it is indeed regular By the de nition of h
0
the no des of g r t are exactly the leaves of the sentential form t of G to b e
R
honest this is not entirely true if the right hand side of a pro duction p is the
empty graph then p is of rank and hence may b e the lab el of a leaf of t
For a string w P we denote by we the reverse of w i e if w p p p
n
then we p p p For we de ne
n
W f wf pw j p P w w P
pw pw PS G and for x rst pw
i i i
x x E
rhs p
x out conn w and
rhs p
in conn w g
It is straightforward to show that W is regular The main p oint is that
conn w can b e computed by a nite automaton
This ends the de nition of the regular path description R To prove that
L R SF G it can b e shown by induction on the length of the derivations
H if and only if there exists t T that for every H GR sn S z
P [
G
such that sn S z t and g r t H For the induction step it su ces to
0
R
G
show the following statement for every t T
P [
0 0 0 0 0
0
t in G then g r t H if g r t H H H in G and t
R R v p v p
0 0
where p is the pro duction of G of the form X p Note that since
k
g r t H every no de of H and in particular v is also a vertex of t Let
R
0
us sketch the pro of of the ab ove statement To b e more precise p and p are
0
pro duction copies and we may assume that the right hand side of p has ver
tices x x x with lab els p resp ectively where x x are
k k k
0
the no des of rhs p with the same lab els Thus t is obtained from t by replac
0
t t
u u
x p v X
x x x
i k i k
Fig A derivation step
ing the leaf v by the internal vertex x with children x x cf Fig and
k
0
H is obtained from H by replacing the no de v by the no des x x and of
k
course executing the connection instructions From this and the de nition of
0 0
h it should b e clear that g r t and H have the same no des with the same
R
lab els It remains to show that they have the same edges Note that since G is
uniquely lab eled the lab els of x x are all distinct Let u and u
k k
0
b e two leaves of t It should b e clear that if u and u are b oth leaves of t then
0
path u u path u u and hence they have the same edges in g r t
0
R
t t
0
and g r t since they also have the same edges in H and H by De nition
R
0 0
they have the same edges in g r t and H If u x and u x for some
R i j
p Since clearly p W if and i j k then path u u
0
j i j i
t
0 0
only if x x E u and u have the same edges in g r t and H
i j R
rhs p
The last and most imp ortant case to consider is that u is a leaf of t and u x
i
0 0
for some i k see Fig Let us rst show that g r t and H have the
R
0
p w p with same edges u x from u to u Clearly path u x wf
0
i i i
t
0
p w X recall that X is the lab el of v Now path u v wf
t
0
u x E
i
g r t
de nition of W
0
x x E
rhs p
0
x out conn w and
rhs p
in conn w p
i
0 0
where x rst p w and x rst p w p
i
Lemma
0
x x E
rhs p
0
x out conn w
rhs p
in conn w X and
in conn p
i
de nition of W and de nition of conn
u v E and x in con p
i
g r t
g r t H
u v E and x in con p
H i
De nition
0
u x E
i H
0 0
Finally it has to b e shown that g r t and H have the same edges x u
R i
0
from u to u In this case we have path x u pwf p w and path v u
0
i i
t t
0
X wf p w Trying the same pro of as ab ove would not work b ecause the de ni
tion of W forces the execution of w p b efore the execution of w which do es
not allow the reduction to the case of w and w However Lemma allows us
to interchange the order of these executions To b e more precise by Lemma
the last two lines of the de nition of W can b e changed into
x in conn w and
rhs p
out conn w
Using this alternative de nition of W the pro of is completely analogous
to the one ab ove syntactically change u x into x u x x into
i i
x x u v into v u and interchange in and out
0 0
This shows that g r t H and ends the pro of ut
R
Lemma s and give the main result
Theorem C edNCE RPD
As can b e seen from the remarks at the end of the pro ofs of Lemma and
Lemma we have also shown that for every C edNCE grammar there is an
equivalent nonblocking C edNCE grammar The essence of this fact is contained
in the pro of of Lemma
As an application of Theorem we show that B edNCE is a prop er sub
class of C edNCE For a graph H its edge complement is the graph com H
V E where E is the set of all v w v w that are not in E It is
H H H
easy to see that RPD is closed under edge complement i e if L RPD then
fcom H j H Lg RPD In fact it su ces to change each W into the
W Hence by Theorem C edNCE is regular string language
closed under edge complement The class of B edNCE languages is not closed
under edge complement see Theorem of ELW This proves that B edNCE
is a prop er sub class of C edNCE A concrete example of a graph language in
C edNCE that is not in B edNCE is the set of edge complements of binary trees
Theorem B edNCE is a proper subclass of C edNCE
Sp ecial Cases
In this section we consider a number of variations of the main result
It follows from the pro of of Theorem and in particular from the pro of of
Lemma that for a regular path description R T h W we may
always assume that h is only de ned for symbols in of rank This means
that for every tree t T all the no des of g r t are leaves of t In fact if
R
L R do es not contain the empty graph then we may even assume that the
domain of h is exactly the set of symbols in of rank b ecause it is easy to
show that a C edNCE language without the empty graph can b e generated by a
C edNCE grammar of which all right hand sides of pro ductions are non empty
That means that the no des of g r t are exactly the leaves of t It is an open
R
problem whether it may always b e assumed that h is a total function i e that
the no des of g r t are exactly all vertices of t again assuming that the empty
R
graph is not in L R In the following prop osition we treat a sp ecial case
Prop osition Let G be a C edNCE grammar such that every right hand side
of a production has exactly one terminal node Then there is a regular path
description R of L G such that h of R is a total function
Proof Note rst that the two prop erties are preserved by Lemmas and
resp ectively Now consider the construction in the pro of of Lemma followed
by the one in the pro of of Lemma This shows that L G has a regular path
description R which can b e obtained from R in the pro of of Lemma by
changing the regular tree language of R into some regular tree language T
0
L G Thus R P T h W where h is the identity on and
W is de ned as in the pro of of Lemma for every Clearly we
may assume that for every internal vertex v of every tree t T the last child
of v is a leaf whereas the other children are not leaves The no des of g r t
R
are exactly the leaves of t The idea is now to prune all leaves from t and to
let each former internal vertex v take over the role of its former last child
0 0 0
Formally we de ne the regular path description R P T h W such
that the rank of a pro duction p of G is the number of nonterminal no des of rhs p
0 0
T fpr t j t T g where pr t is obtained from t by removing all its leaves h
0
is the total function from P to such that for every p P h p x
p
rhs p
0
where x is the unique terminal no de of rhs p and W is the set of all strings
p
pw with w P and p P such that wf pw W where is the wf
i i
lab el of the terminal no de of the right hand side of the last pro duction of pw
i
0 0
Using the regularity of T and W it is easy to show that T and W are
regular It should b e clear that L R L G This proves the prop osition
As an example we observe that the regular path description R of Exam
ple is obtained from the C edNCE grammar G of Example by the ab ove
construction Note that G satis es the assumption of this prop osition The reg
ular tree grammar G that generates the regular tree language of R is obtained
from G as follows apply the construction of Lemma to G in order to make
it uniquely lab eled replacing X by L and R take the derivation tree grammar
of the resulting C edNCE grammar and prune the terminal leaves from the pro
ductions of the resulting regular tree grammar Note that a b b c c are the
l r l r
pro ductions of the uniquely lab eled C edNCE grammar ut
Let us now consider the class B edNCE of graph languages generated by b ound
ary edNCE grammars see Section for the de nition Recall from Section
the de nition of the class B RPD of regular path descriptions of type B every
W is a subset of
Theorem B edNCE B RPD
Proof First the inclusion B RPD B edNCE Let us say that an automaton
path description R T h A is of type B if L A Now
consider the pro of of Lemma for R of type B It should b e clear that the gram
0
mar G need not keep edges b etween nonterminal vertices any more Hence in
0 0
the de nition of the pro duction p X D C of G the edges x hq q i x
i j
can b e dropp ed from E Then the in the connection instructions of C can b e
D
0
restricted to This turns G into a B edNCE grammar It should b e clear that
the construction in the pro of of Lemma changes a regular path description of
type B into an automaton path description of type B
We now turn to the second inclusion B edNCE B RPD It is shown in
Theorem of ELW that every B edNCE language not containing the empty
graph can b e generated by a B edNCE grammar of which the right hand side of
every pro duction contains exactly one terminal no de the pro of is for undirected
graphs but can b e adapted in a straightforward way to directed graphs This
allows us to use the construction in the pro of of Prop osition To this aim
we rst have to check the construction in the pro of of Lemma Consider the
pw is in W then de nition of W in that pro of We claim that if wf
either w or w where is the empty string In fact if x is a
nonterminal no de then w b ecause x out conn w
rhs p
and G is a B edNCE grammar if on the other hand x is a terminal no de then
P PP w b ecause x lhs w This shows that W P
Hence for the regular path description R in the pro of of Prop osition we
0
obtain that W P P PP which shows that R is of type B ut
Next we consider the class LIN edNCE of graph languages generated by linear
edNCE grammars see again Section for the de nition Recall again from
Section the de nition of the class LIN RPD of regular path descriptions of
type LIN the symbols of have rank or
Theorem LIN edNCE LIN RPD
Proof It is easy to check the pro ofs to see that they preserve linearity The
inclusion LIN edNCE LIN RPD is based on the fact shown in Theorem of
ELW for undirected graphs that every LIN edNCE language can b e generated
by a LIN edNCE grammar of which the right hand side of every pro duction
contains exactly one terminal no de Then all elements of the ranked alphab et P
of R in the pro of of Prop osition have rank or ut
We nally consider the class A edNCE of graph languages generated by ap ex
edNCE grammars see again Section Recall again from Section the de n
ition of the class A RPD of regular path descriptions of type A every W is
nite The next result solves a conjecture on p of ELW
Theorem A edNCE A RPD
Proof To show that A RPD A edNCE we use Prop osition By this prop osi
tion and the inclusion of RPD in C edNCE Theorem it su ces to show that
every graph language in A RPD is of b ounded degree Let R T h W
b e a regular path description such that W is nite for every Let k
b e the maximal length of the strings in the W s Consider some t T and
u V In g r t if there is an edge b etween u and v then either path u v
t R
t
or path v u is in W for some Hence the distance b etween u and all
t
such vertices v is at most k Let m b e the maximal rank of the elements of
k
Then there are at most m vertices within distance k from u Hence u is
k
of degree at most m in g r t
R
To show that A edNCE A RPD consider the pro of of Lemma If G is
an A edNCE grammar then W is nite for every In fact it is easy
to see from De nition that in that case if n then conn p p
n
b ecause all y must b e terminal This implies that all path strings in W
i
have length at most and even b ecause at least one of the strings w in the
i
de nition of W is empty cf the pro of of Theorem ut
It can b e shown using a slightly more complicated construction than the one
in the pro of of Prop osition that every regular path description of type A is
The equivalent to one for which every W is a nite subset of
basic idea is to remove all children of an internal vertex v that are leaves and
replace them by a sequence of unary vertices ab ove v
String Languages
It is well known that an edge lab eled graph can b e used to de ne a regular string
language consisting of the strings of edge lab els along all directed paths in H
from certain initial no des of H to certain nal no des of H In fact this is just
another way of saying that the graph H is viewed as a nondeterministic nite
automaton To de ne nonregular string languages one might use a set of graphs
rather than just one graph Clearly allowing arbitrary sets of graphs would give
arbitrary string languages Thus it would b e more natural to use only graph lan
guages that can b e generated by certain graph grammars Here we investigate
the string generating p ower in the ab ove sense of C edNCE graph grammars
We will show that in this way they generate the class of output languages of non
deterministic tree walking transducers cf ERS The LIN edNCE grammars
generate the class of checking stack languages cf Gre To simplify the pro ofs
we will allow the edges of the graphs to b e lab eled by arbitrary strings including
the empty string This corresp onds to nite automata that can read an arbi
trary p ossibly empty string in one step It is formalized by applying a string
homomorphism to the language recognized by an ordinary nite automaton
De nition Let H GR and let i f Then path H is de ned
i f
to b e the set of all such that there is a directed path with no des
n
v v v n in H with v i v f and v v E
n H H n j j j H
for all j n
Let K b e a class of graph languages Then Path K is the class of all string
languages path L fpath H j H Lg where L K and i f are
i f i f
arbitrary no de lab els And HPath K is the class of string languages L where
L Path K and is an arbitrary string homomorphism ut
As an example for the graph language L G of ladders from Example
n n m n n
1 1
k k
with k path L G is the set of all strings
i f
m and n m for all j k Clearly this language is not regular
j
not even context free but it is a checking stack language guess the number
m on the checking stack walk up and down part of the stack k times and walk
up the whole stack
We now describ e the tree walking transducer in an informal way detailed
de nitions can b e found in e g ERS A tree walking transducer abbreviated
twt is a nondeterministic automaton with a nite control an input tree and an
output tap e The input trees are taken from a given regular tree language over
some ranked alphab et At any moment of time the automaton is at a certain
vertex of the input tree Dep ending on the state of its nite control and the
lab el of the vertex it changes state outputs a string to the output tap e and
either moves to the parent or to a sp eci c child of the vertex The automaton
starts in its initial state at the ro ot of the input tree and halts whenever it
reaches a nal state In this way it nondeterministically translates the input tree
into an output string The output language of the automaton is the set of all
output strings that are translations of input trees from the given regular tree
language By OUT TWT we denote the class of all output languages of tree
walking transducers From a slightly di erent p oint of view one could view a
twt as computing a relation b etween trees and strings OUT TWT is the class
of images of regular tree languages under such twt relations In the case that
the lab els of the input tree all have rank or the input tree can b e viewed
as a two way input tap e and the twt as a nondeterministic two way gsm i e a
nite state transducer with a two way input tap e gsm abbreviates generalized
sequential machine see e g HU By OUT GSM we denote the class of all
output languages of two way gsm s It is well known that OUT GSM equals
the class of one way checking stack languages in fact the input tap e and
output tap e of the two way gsm can b e viewed as the checking stack and the
one way input tap e of the checking stack automaton resp ectively For more
details on the ab ove see ERS where the twt is called checking tree transducer
or ct transducer and the two way gsm is called checking string transducer or
cs transducer
We now use our main result Theorem to show that C edNCE grammars
have the same string generating p ower as tree walking transducers For LIN
edNCE grammars we use Theorem to show that they have the same string
generating p ower as two way gsm s or checking stack automata The pro of will
b e as informal as the description of the twt ab ove
Theorem HPath C edNCE OUT TWT and
HPath LIN edNCE OUT GSM
Proof In this pro of whenever we consider a regular tree language T T
either of a regular path description or of a twt we will assume that there is a
mapping num f g such that for every vertex x of a tree t T if
num x j then either x is not the ro ot and j is the lab el of the incoming
t
edge of x i e x is the j th child of its parent or x is the ro ot of t and j
Clearly this assumption can b e made without loss of generality
We rst show that HPath RPD OUT TWT and HPath LIN RPD
OUT GSM Let R T h W b e a regular path description let
i f and let b e a string homomorphism with domain It is not di cult
to construct a twt M with input tree language T and with output language
path L R For a given input tree t T M rst nondeterministically
i f
walks to a vertex x of t such that h x i Then rep eatedly M chooses a
t
symbol outputs and nondeterministically walks to another vertex
y for which h y is de ned walking along the shortest undirected path from
t
x to y and using its nite control to check that path x y is in W Finally
t
M halts after checking that h x f for the current vertex x Note that
t
when walking from x to y along the shortest path from x to y M rst ascends
to the least common ancestor z of x and then in general descends to y to do
this M has to store the number num x of the child x of z of which x is a
t
descendant in its nite control in order to b e able to descend to another child
y of z of which y will b e a descendant
Next we show that OUT TWT HPath RPD Let M b e a twt with input
tree language T T and output alphab et Since a string homomorphism
is incorp orated into the de nition of HPath RPD it clearly su ces to assume
that M outputs exactly one symbol from at each move and to prove that
the output language of M is in Path RPD Also we may assume that M never
re enters its initial state and that it has exactly one nal state Let Q b e the
set of states of M and let i f Q i f b e its initial and nal state resp ec
0 0
tively We construct a regular path description R T h W such
0
that path L R is the output language of M The trees of T are obtained
i f
0 0
from those of T as follows for a tree t T we construct the tree t T by adding
Q new children to every vertex x of t lab eled distinctly with the ele
ments of Q fig and adding one more new child with lab el i if x is the ro ot of
t Such a new child with lab el q Q intuitively represents the fact that M is at
0
0
vertex x of t in state q Thus Q where rank rank Q
0
for every with num rank rank Q for every
0
with num and rank q for every q Q Clearly T is a regular tree
0 0
language We take and we take h to b e the identity on Finally
for every W is a nite language obtained from the nite control of
M as follows If M in state q and reading vertex lab el may go into state
p output and move to the parent then W contains the string q p for
every If M in state q and reading vertex lab el may go into state p
p for output and move to the j th child then W contains the string q
every such that num j From this construction of R it should b e
clear that path L R is the output language of M Intuitively for a tree t a
i f
0
directed path in the graph g r t represents a walk of M on t
R
Note that we have even shown that OUT TWT HPath A RPD
To prove that OUT GSM HPath LIN edNCE we rst construct R as
0
ab ove It is not di cult to see that the regular tree language T of R can b e
generated by a linear REGT grammar Consider the pro of of Lemma It
is left to the reader to show that it can b e adapted easily for an arbitrary
regular tree grammar G not necessarily in normal form the only thing that
0
changes is the de nition of E for the pro duction p X D C Since the
D
pro ofs of Lemma and Prop osition b oth preserve linearity cf the pro of of
Theorem this shows that L R can b e generated by a LIN edNCE grammar
ut
It can b e shown that HPath C edNCE Path C edNCE i e that the class
Path C edNCE is closed under homomorphisms but with the to ols we have
now the pro of is not easy to present and thus will not b e given here
Another way of generating string languages by graph grammars was investi
gated in EH Every string can b e viewed as a graph with n no des
n
that are connected into a directed chain by n edges lab elled In this
n
way every string language can b e viewed as a sp ecial type of graph language
For a class of graph languages K we denote by STR K the class of all string
languages in K The string languages generated by graph grammars in this way
were investigated in EH for another type of context free graph grammars
the hyperedge replacement grammars of Hab However it is known that these
grammars have the same string generating p ower in this sense as the C edNCE
grammars see e g Bra EH From this we nd that STR C edNCE
OUT DTWT the output languages of deterministic tree walking transducers
and STR LIN edNCE OUT DGSM the output languages of deterministic
two way gsm s Hence this string generation metho d is weaker than the one
m n
discussed in this section For instance the language f a j m n g is in
OUT GSM guess m on the checking stack and then walk up and down the
whole stack n times or there is a LIN edNCE grammar that generates the set
of all directed cycles This language is however not in OUT DTWT b ecause it
is not Parikh see ERS Note that in the other direction OUT DTWT con
tains all context free languages but OUT GSM do es not see Theorem
of Gre
References
BC M Bauderon B Courcelle Graph expressions and graph rewritings Math Sys
tems Theory
Bra F J Brandenburg On partially ordered graph grammars in ENRR
Bra F J Brandenburg On p olynomial time graph grammars Pro c STACS Lec
ture Notes in Computer Science Springer Verlag Berlin pp
Bra F J Brandenburg The equivalence of b oundary and con uent graph grammars
on graph languages of b ounded degree in Rewriting Technniques and Appli
cations R V Bo ok ed Lecture Notes in Computer Science Springer
Verlag Berlin pp
CER B Courcelle J Engelfriet G Rozenberg Handle rewriting hypergraph gram
mars J of Comp Syst Sci
CLS D G Corneil H Lerchs L Stewart Burlingham Complement reducible graphs
Discr Appl Math
Cou B Courcelle An axiomatic de nition of context free rewriting and its applica
tion to NLC graph grammars Theor Comput Sci
Cou B Courcelle The monadic second order logic of graphs VI I Graphs as rela
tional structures Theor Comput Sci
Cou B Courcelle Structural prop erties of context free sets of graphs generated by
vertex replacement Inform and Comput
EH J Engelfriet L M Heyker The string generating p ower of context free hyper
graph grammars J of Comp Syst Sci
EH J Engelfriet L M Heyker Hyp ergraph languages of b ounded degree J of
Comp Syst Sci
EHL J Engelfriet L M Heyker G Leih Context free graph languages of b ounded
degree are generated by ap ex graph grammars Acta Informatica
EJKR H Ehrig D Janssens H J Kreowski G Rozenberg Concurrency of no de lab el
controlled graph transformations Rep ort University of Antwerp U I A
EKR H Ehrig H J Kreowski G Rozenberg eds Graph Grammars and their Ap
plication to Computer Science Lecture Notes in Computer Science
Springer Verlag Berlin
EL J Engelfriet G Leih Linear graph grammars p ower and complexity Inform
and Comput
EL J Engelfriet G Leih Complexity of b oundary graph languages RAIRO The
oretical Informatics and Applications
ELR J Engelfriet G Leih G Rozenberg Ap ex graph grammars in ENRR
ELR J Engelfriet G Leih G Rozenberg Nonterminal separation in graph gram
mars Theor Comput Sci
ELW J Engelfriet G Leih E Welzl Boundary graph grammars with dynamic edge
relab eling J of Comp Syst Sci
Eng J Engelfriet Context free NCE graph grammars Pro c FCT Lecture Notes
in Computer Science Springer Verlag Berlin pp
Eng J Engelfriet A characterization of context free NCE graph languages by
monadic second order logic on trees in EKR
Eng J Engelfriet A Greibach normal form for context free graph grammars Pro c
ICALP W Kuich ed Lecture Notes in Computer Science Springer
Verlag Berlin pp
Eng J Engelfriet Graph grammars and tree transducers Pro c CAAP S Tison
ed Lecture Notes in Computer Science Springer Verlag Berlin
pp
ENRR H Ehrig M Nagl G Rozenberg A Rosenfeld eds Graph Grammars and
their Application to Computer Science Lecture Notes in Computer Science
Springer Verlag Berlin
ER J Engelfriet G Rozenberg A comparison of b oundary graph grammars and
context free hypergraph grammars Inform and Comput
ER J Engelfriet G Rozenberg No de replacement graph grammars Chap
ter for the Handbook of Graph Transformations Volume I Foundations
G Rozenberg ed World Scienti c to app ear
ERS J Engelfriet G Rozenberg G Slutzki Tree transducers L systems and two
way machines J of Comp Syst Sci
Gre S A Greibach Checking automata and one way stack languages J of Comp
Syst Sci
Gre S A Greibach One way nite visit automata Theor Comput Sci
GS F G ecseg M Steinby Tree automata Akad emiai Kiad o Budap est
Hab A Hab el Hyperedge Replacement Grammars and Languages Lecture Notes in
Computer Science Springer Verlag
HK A Hab el H J Kreowski May we introduce to you hyperedge replacement in
ENRR pp
HU J E Hop croft J D Ullman Introduction to Automata Theory Languages and
Computation Addison Wesley Reading Mass
Kau M Kaul Syntaxanalyse von Graphen bei Pr azedenz Graph Grammatiken Dis
sertation Universit at Osnabr uck
KL C Kim D H Lee Separating k separated eNCE graph languages Theor Com
put Sci
Nag M Nagl Formal languages of lab elled graphs Computing
Nag M Nagl A tutorial and bibliographical survey of graph grammars in Graph
Grammars V Claus H Ehrig G Rozenberg eds Lecture Notes in Computer
Science Springer Verlag Berlin pp
Nag M Nagl Graph Grammatiken Vieweg Braunschweig
Oos V van Oostrom Graph grammars and nd order logic in Dutch M Sc The
sis Leiden University January
RW G Rozenberg E Welzl Boundary NLC graph grammars basic de nitions
normal forms and complexity Inform and Control
Schu R Schuster Graphgrammatiken und Grapheinbettungen Algorithmen und
Komplexit at Technical Rep ort MIP Universit at Passau
SW K Skodinis E Wanke Exp onential time analysis of con uent and b oundary
eNCE graph languages in Graph Theoretic Concepts in Computer Science
WG E W Mayr G Schmidt G Tinhofer eds Lecture Notes in Computer
Science Springer Verlag Berlin pp
SW K Skodinis E Wanke The b ounded degree problem for eNCE graph gram
mars th International Workshop on Graph Grammars and their Application
to Computer Science Williamsburg Virginia USA November
Wel E Welzl Boundary NLC and partition controlled graph grammars in ENRR
A
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