AbstractThere are (at least) three motivations to study the class of regular directed node-label controlled graph grammars (regular DNLC grammars for shor): (1) it fits very well into the hierarchy of subclasses of DNLC grammars, (2) it generalizes naturally right-linear string grammars and (3) it provides a useful framework for the theory of concurrent systems based on the theory of traces.The complexity of (the membership problem for) the class of regular DNLC grammars is investigated

Aalbersberg, IJ.J.

Ehrenfeucht, A.

Rozenberg, G.

Elsevier - Publisher Connector 

Discrete Applied Mathematics 13 (1986) 79-85 
North-Holland 
79 
NOTE 
ON THE MEMBERSHIP PROBLEM FOR REGULAR DNLC 
GRAMMARS 
IJ.J. AALBERSBERG, G. ROZENBERG 
Institute of  Applied Mathematics and Computer Science, University of  Leiden, 
Wassenaarseweg 80, 2333 AL Leiden, The Netherlands 
A. EHRENFEUCHT 
Department of Computer Science, University of Colorado at Boulder, Boulder, CO 80309, 
USA 
Received 4 April 1985 
There are (at least) three motivations to study the class of regular directed node-label controlled 
graph grammars (regular DNLC grammars for short): (l) it fits very well into the hierarchy of 
subclasses of DNLC grammars, (2) it generalizes naturally right-linear string grammars and (3) 
it provides auseful framework for the theory of concurrent systems based on the theory of traces. 
The complexity of (the membership roblem for) the class of regular DNLC grammars is 
investigated. 
Introduction 
The theory of graph grammars i a natural extension of the formal string language 
theory and it has become a well-established topic of research (see, e.g., Ehrig et al., 
1983). The potential applicability of graph grammars in many fields of computer 
science provides substantial motivation for theoretical studies. 
One of such fields is the theory of traces, initiated by Mazurkiewicz (1977), which 
has become quite popular as an approach to the theory of concurrent systems (see, 
e.g., Mazurkiewicz, 1984, Mazurkiewicz, 1985, Bertoni et al., 1981, and Aalbersberg 
and Rozenberg, 1985a). In Aalbersberg and Rozenberg (1985b) the theory of traces 
was related to the theory of graph grammars through regular directed node-label 
controlled graph grammars (abbreviated regular DNLC grammars) - a subclass of 
the directed node-label controlled graph grammars (see, e.g., Janssens and Rozen- 
berg, 1981). It also turns out that the class of regular DNLC grammars fits very well 
into the hierarchy of various subclasses of DNLC grammars - e.g., it is a very 
natural subclass of the (directed version of) boundary NLC grammars (see, Rozen- 
berg and Welzl, 1984). Moreover the notion of a regular DNLC grammar transfers 
very nicely the notion of a right-linear grammar into the framework of graph 
grammars. 
0166-218X/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland) 
80 IJ.J. Aalbersberg et al. 
In this note we investigate he complexity of regular DNLC grammars and in par- 
ticular we prove that the membership problem for regular DNLC grammars i NP- 
complete. 
O. Preliminaries 
We assume the reader to be familiar with both basic formal string language theory 
(see, e.g., Salomaa, 1973) and the theory of NP-Completeness (see, e.g., Garey and 
Johnson, 1979). 
For sets A and B, A -B  denotes their difference; 0 denotes the empty set. For a 
set A, #A denotes the cardinality of A. 
A directed node labeled graph, in the sequel simply called a graph, will be 
specified in the form y=(V ,E ,Z , I )  where V is its set of nodes, 
Ec_ (Vx V)-{(o, o) loe V} is its set of edges, Z is its label alphabet and 1: V-*Z 
is its (node) labeling function. If graphs y and y' are isomorphic (and we consider 
only the node-label preserving isomorphisms), then we write y_--y'. We will 
sometimes identify isomorphic graphs as equal (then we really consider the so-called 
abstract graphs). For an alphabet ,S, ~z denotes the set of all graphs with the label 
alphabet Z. 
For a graph y, # y denotes the number of nodes of y. For a symbol c, a graph 
y=(V,E ,Z , I )  such that ceZ,  V={ol,...,On} for some n>_l, l(oi)=c for every 
1 <i<_n, and E= {(oi, Oi+l)[1 <_i<_n-1} is called a c-path. A graph y is a k- 
element c-path, where k> 1, if y is a c-path and #y=k.  For graphs y=(V, E, Z, 1) 
and y'=(V',E' ,  Z', 1') where VN V'=O, their union is the graph (VtJ V', EU E', 
Z t3 Z', 1U l'). (Hence we consider unions of disjoint graphs only, or, if we consider 
abstract graphs, then disjoint representatives are chosen.) 
We recall now basic notions concerning regular DNLC grammars. 
Definition 1. (1) A directed node-label controlled graph grammar, abbreviated 
DNLC grammar, is a system G = (F, A, P, (?in, Coat, Z), where: (i) F is an alphabet, 
called the total alphabet of  G, (ii) d c_ F is called the terminal alphabet of G, (iii) 
P c_ (F -  A) x ~r is called the set of productions of G, (iv) Cin C_/"X/-' is called the 
in-connection relation of G and Con tc_ F× F is called the out-connection relation of  
G, and (v) Z, called the axiom of G, is a graph over Fconsisting of one node labeled 
by an element o f / " -A .  
(2) A DNLC grammar G = (F, A, P, Cin, (?out, Z) is called regular, if every pro- 
duction of G is 
a Y a 
either of the form (X, e--,---.) or of the form (X,*), 
with a e d and Y e F -  A . 
The membership problem for regular DNLC grammars 81 
Informally speaking, a DNLC grammar G = (F, A, P, Cin, Gout, Z) generates a set 
of graphs as follows. Given a graph y to be rewritten and a production of the form 
(X, fl), where XeF-A  and fie ~r, one chooses a node o of y labeled by X and 
replaces it by (a graph isomorphic to) ft. Then, in order to embed ,6 in 'the remainder 
of y' (i.e., the graph resulting from y by removing o), one uses relations Cin and 
Cout as follows. For every pair (b, c)~ Cin one establishes an (incoming) edge from 
each direct neighbour node of o labeled c to each node of fl labeled b. Analogously, 
for every pair (b, c)e Cout one establishes an (outgoing) edge from each node 
labeled b in `6 to each direct neighbour node of o labeled c. Every graph y' iso- 
morphic to the resulting raph is said to be directly derived from y in G. Iterating 
the direct derivation step (starting with the axiom graph Z of G) and choosing only 
those derived graphs that are labeled by labels from the terminal alphabet A one gets 
the (graph) language L(G) of  G. 
The notions are defined formally in Janssens and Rozenberg (1981). 
1. The complexity of regular DNLC grammars 
We consider now the complexity of the membership problem for regular DNLC 
grammars. 
Theorem 1. There exists a regular DNLC grammar G (with the empty out- 
connection relation) such that the membership problem for L(G) is NP-complete. 
Proof. Let G = (F, A, P, Gin, Gout, Z) be the regular DNLC grammar, such that: 
(i) F= {A o, A 1, Bo, BI, a, b}, 
(ii) A = { a, b}, 
(iii) P= {(Bo ' b_~Ao), (BI, b_+._~Ao), (Bl, bo_~A~), (Ao, a~Bo), 
a B 
(.Ao, a.~Bl~ (A 1, ~_~o), (A 1, a.~B,), (Ao, a a ,, *), (Al, *)}, 
(iv) Cin= {(a, a), (b, b), (B1, b), (AI, a)}, 
(v) Coat = O, and 
(vi) Z is a graph consisting of one node labeled by B0. 
Informally speaking G works as follows. G generates sets of b-paths and a-paths 
by generating alternatively a b-labeled node and an a-labeled node ('b-generating' 
nonterminals B0 and BI always introduce one of the 'a-generating' nonterminals A0 
and AI). In this way in each graph of L(G) the number of b-labeled nodes equals 
the number of a-labeled nodes. Moreover, at any moment G can 'decide' to break 
82 IJ.J. Aalbersberg et ai. 
the paths it is generating, by introducing one of the 'disconnecting' nonterminals 
A0 and B0. However, after breaking a b-path one has to break the 'associated' a- 
path: 
is the only production for the 'breaking' nonterminal B0. 
A 'typical' graph in L(G) looks as follows: 
b 
b ~a 
b a 
b oa 
b 1 
N 
b oa  
b • oa 
b oa 
b 
Fig. 1. 
The membership problem for regular DNLC grammars 83 
The following result follows directly from the construction of G. 
Lemma 1. Let Y be a graph which is the non-empty union of a-paths and b-paths. 
Let, for m>_O, Sb= {ill, ... ,tim} be the set of disjoint b-paths of y and let Sa be the 
collection of  a-paths of y. Then y ~ L(G) if and only if there exists a partition 
(S~,...,S m) of  So, such that, for every l <i<m, #fli = ~s~#6.  [] 
Lemma 2. The membership problem for L(G) is NP-complete. 
Proof. Obviously the membership roblem for L(G) is in NP. 
In order to show that the membership problem for L (G) is NP-hard, consider the 
following strong NP-complete problem (see Garey & Johnson, 1979, Problem SP 15, 
page 224). 
3-PARTITION 
Instance. A finite set S of 3n elements, where n > 1, a positive integer k and, for 
every s~S, a positive integer o(s), such that, for every seS,  k/4< o(s)<k/2 and 
Es~s o(s) = kn. 
Question. Can S be partitioned into n sets SI,... ,  S,, in such a way that, for every 
1 <i<_n, ~,seSi u(s)=k? 
Let f be the function which maps every instance I = (S, n, k, u) of 3-PARTITION 
into a graph f ( I )  in ~zi as follows, f ( I )  is the union of (i) n k-element b-paths, and 
(ii) for every s e S, a o(s)-element a-path (hence, every s e S corresponds uniquely 
to an a-path in f(I)). (Thus, for every instance I of 3-PARTITION, f ( I )  is the union 
of at least one a-path and at least one b-path.) 
It is easily seen that f has a pseudo-polynomial-bounded time-complexity. Fur- 
thermore, for every instance I of 3-PARTITION, I is a 'yes'-instance of 
3-PARTITION if and only i f f ( I )  is an element of L(G); this can be seen as follows. 
Assume that I=  (S, n, k, o) is a 'yes'-instance of 3-PARTITION. Hence, there ex- 
ists a partition (Sl,...,Sn) of S, such that, for every l<_i<_n, ~,s~sO(s)=k. Fur- 
thermore, since f(1) is the union of at least one a-path and at least one b-path, we 
can partition f ( I )  into Sb = {ill,--., tim} and Sa, in the way described in the first 
part of the statement of Lemma 1 (note that now m>_l). Let, for every 1 <_i<_n, 
S / be the set of a-paths in Sa corresponding to the elements of Si. It is easily seen 
that: (1) m=n, and (2) for every 1 <i<_n, 
#Bi=k = E v(s)= E 
s~S~ 6~S" 
Consequently, by Lemma 1, f ( I )  E L(G) which proves the 'only if'-part of the claim. 
Assume now that I=  (S, n, k, o) is an instance of 3-PARTITION, such that 
f ( I )  ~ L(G). Hence, because f ( I )  is the union of at least one a-path and at least one 
84 IJ.J. Aalbersberg et aL 
b-path, it follows from Lemma I that f(1) can be partitioned into Sb= {]~l,-.. ,tim} 
consisting of b-paths, where m_> I, and Sa consisting of a-paths in such a way that 
there exists a partition (Sla,...,S m) of S a, such that for every l<_i<_m, 
#fli = ~s$  #~" Let, for every l <_i<_m, Si be the set of elements in S correspon- 
ding to the a-paths of S/. It is easily seen that: (1) m =n, and (2) for every I <_i<_n, 
E o(s)= E #a= ##i =k. 
seS~ ~eS~ 
Consequently, I is a 'yes'-instance of 3-PARTITION, which proves the 'if '-part of 
the claim. 
Since 3-PARTITION is strong NP-complete, the membership roblem for L(G) 
is NP-complete. [] 
The theorem follows directly from Lemma 2. [] 
This is a rather surprising result. A regular DNLC grammar (with the empty out- 
connection relation) can be considered as a right-linear string grammar in which 
during the (left-to-right) generation of (the graph representation of) a string (with 
all edges resulted by 'transitivity' also added) some edges are removed (cutted) ac- 
cording to the in-connection relation. It is well-known that the membership problem 
for right-linear grammars i linear while the membership problem for regular DNLC 
grammars turns out to be so difficult!!! 
Acknowledgements 
The second and the third author gratefully acknowledge the support by NSF 
Grant MCS 83-05245. The authors are indebted to J. Engelfriet, H.J. Hoogeboom, 
H.C.M. Kleijn and E. Welzl for useful comments concerning the first version of this 
note. 
The authors are grateful to the referee who has pointed out that (modifying Lem- 
ma 1) we get the NP-completeness of the following number-theoretic problem of 
general interest: 
'Given two partitions n = nl + ... + nk, n = ml + ... + mt of an integer n, is the one 
a subpartition of the other?'. 
It seems that this is the first time that the NP-completeness of this subpartition 
problem is written out. 
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On the membership problem for regular DNLC grammars 

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