Restrictions on Monadic Context-Free Tree Grammars
Akio Fujiyoshi
Department of Computer and Information Sciences, Ibaraki University
4-12-1 Nakanarusawa, Hitachi, Ibaraki, 316-8511, Japan
fujiyoshi@cis.ibaraki.ac.jp
Abstract
In this paper, subclasses of monadic context-
free tree grammars (CFTGs) are compared.
Since linear, nondeleting, monadic CFTGs gen-
erate the same class of string languages as
tree adjoining grammars (TAGs), it is examined
whether the restrictions of linearity and non-
deletion on monadic CFTGs are necessary to
generate the same class of languages. Epsilon-
freeness on linear, nondeleting, monadic CFTG
is also examined.
1 Introduction
The context-free tree grammars (CFTGs) were in-
troduced by W. C. Rounds (1970) as tree gener-
ating systems, the definition of which is a direct
generalization of context-free grammars (CFGs)
from strings to rooted, ordered, labeled trees. For
the application of CFTGs to natural languages,
many kinds of restrictions on CFTGs have been
considered because the string languages generated
by CFTGs are exactly indexed languages, whose
emptiness problem and uniform membership prob-
lem are exponential time complete, i.e., nonre-
stricted CFTGs are formidable. One approach to
define subclasses of CFTGs is to restrict the ranks
of nonterminals. The rank of a nonterminal is a nat-
ural number assigned to each nonterminal by which
the number of children of the node labeled by the
nonterminal is fixed. Through this approach, the
simplest model of CFTGs is regular tree grammars
(RTGs) (Brainerd, 1969), where the ranks of non-
terminals are all 0. The string languages generated
by RTGs are the languages generated by context-
free grammars (CFGs). Since recent research on
natural languages has suggested that formalisms for
natural languages need to generate a slightly larger
class of languages than CFGs, this paper focuses on
monadic CFTGs, where the ranks of nonterminals
are either 0 or 1.
Another formalism of tree generating systems is
tree adjoining grammars (TAGs) (Joshi et al., 1975;
Joshi and Schabes, 1996; Abeill´e and Rambow,
2000). TAGs have been widely studied relating
them to natural languages, and it was shown that
TAGs have the same generative power of string lan-
guages as other formalisms for natural languages
developed independently such as head grammars,
combinatory categorial grammars and linear in-
dexed grammars (Vijay-Shanker and Weir, 1994).
It is also noteworthy that there are recognition
algorithms for the string languages generated by
TAGs that run in O(n6) and O(M(n2)) time (Ra-
jasekaran, 1996; Rajasekaran and Yooseph, 1998).
From the view point of CFTG, the languages gener-
ated by TAGs were examined (Fujiyoshi and Kasai,
2000; Fujiyoshi, 2004; M¨oennich, 1997), and it was
shown that linear, nondeleting, monadic CFTGs
generate the same class of string languages as TAGs
and a strictly larger class of tree languages than
TAGs. Linearity is a restriction on CFTGs that
requires the number of occurrences of every vari-
able in the right-hand side of a rule be no more
than 1, and nondeletion requires all variables in the
left-hand side of a rule occur at least once in the
right-hand side. In other words, linear, nondelet-
ing, monadic CFTGs are those with nonterminals of
rank 0 and 1 only and with exactly one occurrence
of a variable in every right-hand side of a rule for a
nonterminal of rank 1.
In this paper, the subclasses of monadic CFTGs
are compared to examine whether the restrictions
of linearity and nondeletion on monadic CFTGs are
necessary to generate the same class of string lan-
guages as TAGs. It is shown that nondeletion is un-
necessary since for any linear, monadic CFTG, there
exists an equivalent linear, nondeleting, monadic
CFTG. On the other hand, it is shown that linearity
is necessary since there exists a non-linear, monadic
CFTG which is not weakly equivalent to any linear,
monadic CFTG.
For the development of parsing algorithm, the
property of epsilon-freeness is very important, and
in this paper, epsilon-freeness on linear, monadic
CTFGs is also considered. Epsilon-freeness is a
restriction on grammars that requires no use of
epsilon-rules, that is, rules defined with the empty
string. It is shown that for any linear, monadic
CFTG, there exists an epsilon-free, linear, nondelet-
ing, monadic CFTG that generate the same string
language.
2 Preliminaries
In this section, some terms, definitions and former
results which will be used in the rest of this paper
are introduced.
2.1 Ranked Alphabets, Trees and Substitution
A ranked alphabet is a finite set of symbols in which
each symbol is associated with a natural number,
called the rank of a symbol. Let  be a ranked al-
phabet. For n  0, it is defined that  n = fa 2
 j the rank of a is ng.
The set T (trees over  ) is the smallest set of
strings over  , parentheses and commas such that
(1)  0  T and (2) if  1; 2;::: ; n 2 T and
a 2  n for some n  1, then a( 1; 2;::: ; n) 2
T .
Let  be the empty string. Let " be the special
symbol that may be contained in  0. The yield of
a tree is a function from T into   defined as fol-
lows. For  2 T , (1) if  = a 2 ( 0  f"g),
yield( ) = a, (1’) if  = ", yield( ) =  , and
(2) if  = a( 1; 2;::: ; n) for some a 2  n
and  1; 2;::: ; n 2 T , yield( ) = yield( 1)  
yield( 2)   yield( n).
Let X be the fixed countable set of variables
x1;x2;:::. It is defined that X0 = ; and for n  1,
Xn = fx1;x2;::: ;xng. x1 is situationally denoted
by x. T (Xn) is defined to be T [Xn taking the
ranks of elements in X are all 0. For  2 T (Xn)
and  1; 2;::: ; n 2 T (X),  [ 1; 2;::: ; n] is
defined to be the result of substituting each  i (1  
i  n) for the occurences of the variable xi in  .
A tree  2 T (Xn) is linear if no variable occurs
more than once in  , and nondeleting if all variables
in Xn occur at least once in  . The set of all linear
trees and all nondeleting trees in T (Xn) are de-
noted by T (dXne) and T (bXnc), respectively.
In this papaer, the conventional way of illus-
trating trees is used. See Figure 1. The tree
A(b(a);a;B(E;d)) is illustrated as (1). An arbi-
trary tree  2 T is illustrated as (2). When the
variables of a tree  2 T (X3) occur in the order of
x1;x2;x3;x1, the tree is illustrated as (3).
2.2 Context-Free Tree Grammars
The context-free tree grammars (CFTGs) were in-
troduced by W. C. Rounds (1970) as tree generating
systems. The definition of CFTGs is a direct gener-
alization of context-free grammars (CFGs).
$
E D %
(D G
α
β 
x1 x2 x3 x1 
(1) (2) 
(3) 
Figure 1: Trees
A context-free tree grammar (CFTG) is a four-
tuple G = (N; ;P;S), where:
 N and  are disjoint ranked alphabets of non-
terminals and terminals, respectively.
 P is a finite set of rules of the form
A(x1;x2;::: ;xn) !  
with n  0, A 2 Nn and  2 TN[ (Xn). For
A 2 N0, rules are written as A !  instead of
A() !  .
 S, the initial nonterminal, is a distinguished
symbol in N0.
For a CFTG G, the one-step derivation G) is
the relation on TN[  TN[ such that for a tree
 2 TN[ , if  =  0[A( 1; 2;::: ; n)] for some
 0 2 TN[ (dX1e) \ TN[ (bX1c), A 2 Nn and
 1; 2;::: ; n 2 TN[ , and A(x1;x2;::: ;xn) !
 is in P, then  G) 0[ [ 1; 2;::: ; n]]. Figure 2
is an example of a one-step derivation where the rule
A(x) !  is applied to the tree  =  0[A( 00)] and
the tree  0[ [ 00]] is obtained.
An (n-step) derivation is a finite sequence of
trees  0; 1;::: ; n 2 TN[ such that n  0 and
 0 G) 1 G)    G) n. When there exists a deriva-
tion  0; 1;::: ; n, it is writen that  0 Gn) n or
 0 G ) n.
The tree language generated by G is the set
L(G) = f 2 T j S G ) g. The string language
generated by G is LS(G) = fyield( ) j  2 L(G)g.
Note that LS(G)  ( 0  f"g) .
α•
$
α••
 α•
α•• α••
β
β
x x 
$
*
x
Figure 2: One-step derivation
Let G and G0 be CFTGs. G and G0 are equivalent
if L(G) = L(G0). G and G0 are weakly equivalent if
LS(G) = LS(G0).
2.3 Restrictions on CFTGs
A CFTG G = (N; ;P;S) is monadic if the
rank of any nonterminal is 0 or 1, i.e., N =
N0 [ N1 and Nn = ; for n  2. G is lin-
ear if for any rule A(x1;x2;::: ;xn) !  in P,
 2 TN[ (dXne), and nondeleting if for any rule
A(x1;x2;::: ;xn) !  in P,  2 TN[ (bXnc).
A CFTG G = (N; ;P;S) is epsilon-free if for
any rule A(x1;x2;::: ;xn) !  in P, the symbol "
doesn’t occur in  .
When G is monadic, all rules are either of the
form A(x) !  with A 2 N1 and  2 TN[ (X1)
or of the form B !  with B 2 N0 and  2 TN[ .
When G is monadic, linear and nondeleting, for any
rule A(x) !  with A 2 N1 in P, there exists
exactly one occurrence of x in  .
For linear, nondeleting, monadic CFTGs, the fol-
lowing results are known.
Theorem 2.1 (Fujiyoshi and Kasai, 2000) The
class of string languages generated by linear, non-
deleting, monadic CFTGs coincides with the class
of string languages generated by TAGs.
Theorem 2.2 (Fujiyoshi and Kasai, 2000) For any
linear, nondeleting, monadic CFTG, there exists
a weakly equivalent linear, nondeleting, monadic
CFTG G = (N; ;P;S) that satisfies the following
conditions:
 For any a 2  , the rank of a is either 0 or 2.
 For each A 2 N0, if A !  is in P, then either
 = a with a 2  0, or  = B(C) with B 2
N1 and C 2 N0. See (1) and (2) in Figure 3.
$ D $
%
&
$
x
$
x
%
&
x
$
x
E
x &
E
x&
(1) (2) 
(3) 
(4) (5) 
Figure 3: Strong normal form
 For each A 2 N1, if A(x) !  is in P, then  
is one of the following forms:
 = B(C(x)) with B;C 2 N1,
 = b(C;x) with b 2  2 and C 2 N0, or
 = b(x;C) with b 2  2 and C 2 N0.
See (3),(4) and (5) in Figure 3.
If a linear, nondeleting, monadic CFTG satisfies the
condition of Theorem 2.2, it is said that the grammar
is in strong normal form1.
3 Linearity and Nondeletion on Monadic
CFTGs
Because linear, nondeleting, monadic CFTGs gen-
erate the same class of string languages as TAGs,
the question is whether the restrictions of linearity
and nondeletion on monadic CFTGs are necessary
to generate the same class of languages. First, it
will be shown that nondeletion is unnecessary.
Theorem 3.1 For any linear, monadic CFTG G,
there exists an equivalent linear, nondeleting,
monadic CFTG G0.
Proof. Let G = (N; ;P;S) be a linear, monadic
CFTG. An equivalent linear, nondeleting, monadic
CFTG G0 = (N0; ;P0;S) can be constructed as
follows.
The set of nonterminal is N0 = N00 [ N01 such
that N00 = N0 [ fAjA 2 N1g and N01 = N1.
For the preparation of the definition of P 0, for  2
TN[ (X1) we define  ( )  TN0[ (X1) as the
smallest set satisfying the following conditions:
  2  ( )D
1We say “strong” because a grammar in this normal form
only preserves weak equivalence.
 
x 
*•
Π δ * β
(1) 
α•
α•
γ
x 
δ•
γ
Π  
*•
* β
(2) 
^ 
α•
γ
δ•
γΠ  
*•
* 
β
(3) 
^ 
α•• δ••
Figure 4: The three different cases
 If  2  ( ) and  =  0[B( 00)] for some B 2
N1,  0 2 TN0[ (dX1e) \ TN0[ (bX1c) and
 00 2 TN0[ , then  0[ B ] 2  ( ).
The set of rules is defined as follows.
P0 = fA ! ^ j A 2 N0, A !  2 P, ^ 2  ( )g
[ fA(x) ! ^ j A 2 N1, A(x) !  2 P,
^ 2  ( ) \ TN0[ (bX1c)g
[ fA ! ^ j A 2 N1, A(x) !  2 P,
^ 2  ( ) \ TN0[ g
Because of the construction of N0 and P0, G0 is
monadic and nondeleting.
To show the equivalence of G and G0, we prove
the following statement holds for any  2 TN[ 
and  2 T by induction on the length of deriva-
tions:
 G ) if and only if there exitsts ^ 2
 ( ) such that ^ G0 ) .
We start with proving “only-if” part. Let  Gk) .
If k = 0, then clearly  =  ,  2  ( ) and
 G0 ) . For k  1, assume that the statement
holds for any derivation of length less than k. If
a rule of the form A !  with A 2 N0 is
used at the first step, the proof is rather simple,
so we only prove the other case. Suppose that
a rule A(x) !  with A 2 N1 is used at the
first step and  =  0[A( 00)] G) 0[ [ 00]] G ) for
some  0 2 TN[ (dX1e) \ TN[ (bX1c) and  00 2
TN[ . By the induction hypothesis, there exist
 2  ( 0[ [ 00]]) such that  G0 ) . Here, we have to
think of the three different cases: (1)  2  ( 0), (2)
 can be written as  0[^ ] for some  0 2  ( 0) and
^ 2  ( ), and (3)  can be written as  0[^ [ 00]] for
some  0 2  ( 0), ^ 2  ( ) and  00 2  ( 00). See
Figure 4. In the case (1),  2  ( ) and  G0 ) .
In the case (2), A ! ^ is in P 0 and therefore,
 0[ A ] 2  ( ) and  0[ A ] G0) 0[^ ] G0 ) . And in
the case (3), A(x) ! ^ is in P 0 and therefore,
 0[A( 00)] 2  ( ) and  0[A( 00)] G0) 0[^ [ 00]] G0 ) .
The “if” part is proved as follows. Let ^ G0k) 
for some ^ 2  ( ). If k = 0, then clearly
^ =  ,  = ^ and  G )  . For k  1, as-
sume that the statement holds for any derivation of
length less than k. The rule used at the first step
is one of the following forms: (1) A ! ^ with
A 2 N0, (2) A(x) ! ^ with A 2 N1, or (3)
A ! ^ with A 2 N00  N0. The proof of the
case (1) is similar to the proofs of the other cases,
so we start proving the case (2). In the case (2),
^ = ^ 0[A(^ 00)] G0) ^ 0[^ [ ^ 00]] G0 )  for some ^ 0 2
TN0[ (dX1e)\TN0[ (bX1c) and ^ 00 2 TN0[ . By
the definition of P0, A(x) !  is in P such that
^ 2  ( ). By the induction hypothesis, for any
 2 TN[ such that ^ 0[^ [^ 00]] 2  ( ),  G ) . By
the definition of  , there exists  0 2 TN[ (X1) and
 00 2 TN[ such that  =  0[A( 00)] G) 0[ [ 00]],
and ^ 0[^ [^ 00]] 2  ( 0[ [ 00]]). Therefore,  G ) .
And in the case (3), ^ = ^ 0[ A ] G0) ^ 0[^ ] G0 ) 
for some ^ 0 2 TN0[ (dX1e) \ TN0[ (bX1c). By
the definition of P0, A(x) !  is in P such that
^ 2  ( ). By the induction hypothesis, for any
 2 TN[ such that ^ 0[^ ] 2  ( ),  G ) . By the
definition of  , there exists  0 2 TN[ (X1) and
 00 2 TN[ such that  =  0[A( 00)] G) 0[ [ 00]],
and ^ 0[^ ] 2  ( 0[ [ 00]]). Therefore,  G ) .
Because  (S) = fSg, L(G) = L(G0).
Next, consideration will be given to whether
the restriction of linearity can be removed from
monadic CFTGs to generate the same class of lan-
guages. The answer is negative. The following ex-
ample is a non-linear, monadic CFTG that generates
a string language that no linear, monadic CFTG can
generate.
Example 3.2 The following is an example of a
monadic CFTG that generates the string language
Lw4 = fwwww j w 2 fa;bg+g. G = (N; ;P;S)
where N = fS;Ag, the ranks of S and A are 0 and
1, respectively,  = fa;b;c;dg, the ranks of a, b, c
and d are 0, 0, 2 and 4, respectively, and P consists
of the following rules:
S ! A(a), S ! A(b), A(x) ! d(xxxx),
A(x) ! A(c(xa)), and A(x) ! A(c(xb)).
Because G has the rule A(x) ! d(xxxx), G is not
linear.
Theorem 3.3 There exists a monadic CFTG which
is not weakly equivalent to any linear, monadic
CFTG.
Proof. It is known that the string language Lw4
in Example 3.2 cannot be generated by any TAG. It
cannot be generated by any linear, monadic CFTG,
neither.
4 Epsilon-Freeness on Linear, Monadic
CFTGs
According to our definition of CFTGs, they are al-
lowed to generate trees with the special symbol ",
which is treated as the empty string while taking
the yields of trees. In this section, it will be seen
that for any linear, monadic CFTG, there exists a
weakly equivalent epsilon-free, linear, nondeleting,
monadic CFTG. Because any epsilon-free CFTG
cannot generate a tree with ", it is clear that for
a CFTG with epsilon-rules, there generally doesn’t
exist an equivalent epsilon-free CFTG.
Theorem 4.1 For any linear, monadic CFTG G =
(N; ;P;S), if  62 LS(G), then there exists a
weakly equivalent epsilon-free, linear, nondeleting,
monadic CFTG G0. If  2 LS(G), then there exists
G0 whose epsilon-rule is only S ! ".
Proof. Since it is enough to show the existence of
a weakly equivalent grammar, without loss of gen-
erality, we may assume that G is in strong normal
form. We may also assume that the initial nonter-
minal S doesn’t appear in the right-hand side of any
rule in P.
We first construct subsets of nonterminals E0 and
E1 as follows. For initial values, we set E0 = fA 2
N0jA ! " 2 Pg and E1 = ;. We repeat the fol-
lowing operations to E0 and E1 until no more oper-
ations are possible:
 If A ! B(C) with B 2 E1 and C 2 E0 is in
P, then add A 2 N0 to E0.
 If A(x) ! b(C;x) with C 2 E0 is in P, then
add A 2 N1 to E1.
 If A(x) ! b(x;C) with C 2 E0 is in P, then
add A 2 N1 to E1.
 If A(x) ! B(C(x)) with B;C 2 E1 is in P,
then add A 2 N1 to E1.
In the result, E0 satisfies the following.
E0 = fA 2 N0j9 2 T ;A G ) ; yield( ) =  g
We construct G0 = (N0; 0;P0;S) as follows.
The set of nonterminals is N0 = N00 [ N01 such that
N00 = N0 [ fAjA 2 N1g and N01 = N1. The set
of terminal is  0 =  [fcg, where c is a new sym-
bol of rank 1. The set of rules P 0 is the smallest set
satisfying following conditions:
 P0 contains all rules in P except rules of the
form A ! ".
 If S 2 E0, then S ! " is in P0.
 If A ! B(C) is in P and C 2 E0, then A !
B is in P0.
 If A(x) ! B(C(x)) is in P, then A ! B(C )
is in P0.
 If A(x) ! b(C;x) or A(x) ! b(x;C) is in P
and C 2 E0, then A(x) ! c(x) is in P 0.
 If A(x) ! b(C;x) or A(x) ! b(x;C) is in P,
then A ! c(C) is in P0.
To show LS(G0) = LS(G), we prove the follow-
ing (i), (ii) and (iii) hold by induction on the length
of derivations:
(i) For A 2 N0, A G0 ) 0 and  0 2 T if and
only if A G ) for some  2 T such that
yield( ) = yield( 0) 6=  .
(ii) For A 2 N1, A(x) G0 ) 0 and  0 2 T (X1)
if and only A(x) G ) for some  2 T (X1)
such that yield( ) = yield( 0).
(iii) For A 2 N00  N0, A G0 ) 0 and  0 2 T if and
only if A(x) G ) for some  2 T (X1) such
that yield( ["]) = yield( 0) 6=  .
We start with “only if” part. For 0-step deriva-
tions, (i), (ii) and (iii) clearly hold since there
doesn’t exists  0 2 T nor  0 2 T (X1) for each
statement.
We consider the cases for 1-step derivations.
[Proof of (i)] If A G0) 0 and  0 2 T , then  0 = a
for some a 2  0 and the rule A ! a in P0 has been
used. Therefore, A ! a is in P and A G)a.
[Proof of (ii)] If A(x) G0) 0 and  0 2 T (X1),
then  0 = c(x) and the rule A(x) ! c(x)
in P0 has been used. By the definition of P 0,
A(x) ! b(C;x) or A(x) ! b(x;C) is in P for
some C 2 E0. There exists  2 T such that
C G ) and yield( ) =  . Therefore, A(x) G)
b(C;x) G )b( ;x) or A(x) G)b(x;C) G )b(x; ),
and yield(b( ;x))= yield(b(x; ))= yield(c(x)).
[Proof of (iii)] There doesn’t exists  0 2 T such
that A G0) 0.
For k  2, assume that (i), (ii) and (iii) holds for
any derivation of length less than k.
[Proof of (i)] If A G0k)  0, then the rule used at
the first step is one of the follwoing form: (1)
A ! B(C) or (2) A ! B. In the case
(1), A G0)B(C) G0 ) 0[ 0] =  0 for some  0 2
T (X1) and  0 2 T such that B(x) G0 ) 0 and
C G0 ) 0. By the induction hypothesis of (ii), there
exists  2 T (X1) such that B(x) G ) and
yield( ) = yield( 0). By the induction hypoth-
esis of (i), there exists  2 T such that C G ) 
and yield( ) = yield( 0). By the definition of P 0,
A ! B(C) is in P. Therefore, A G)B(C) G ) [ ]
and yield( [ ]) = yield( 0[ 0]). In the case (2),
A G0)B G0 ) 0. By the definition of P0, A ! B(C)
is in P for some C 2 E0. There exists  2 T such
that C G ) and yield( ) =  . By the induction hy-
pothesis of (iii), there exists  2 T (X1) such that
B(x) G ) and yield( ["]) = yield( 0). Therefore,
A G)B(C) G ) [ ] and yield( [ ]) = yield( 0).
[Proof of (ii)] If A(x) G0k) 0, then the rule used
at the first step is one of the follwoing form: (1)
A(x) ! B(C(x)), (2) A(x) ! b(C;x) or (3)
A(x) ! b(x;C). Becasue these rule are in P, the
proofs are direct from the induction hypothesis like
the proof of the case (1) of (i).
[Proof of (iii)] If A G0k) 0, then the rule used at
the first step is one of the follwoing form: (1)
A ! B(C) or (2) A ! c(C). In the case (1),
A G0)B(C) G0 ) 0[ 0] =  0 for some  0 2 T (X1)
and  0 2 T such that B(x) G0 ) 0 and C G0 ) 0.
By the induction hypothesis of (ii), there exists
 2 T (X1) such that B(x) G ) and yield( ) =
yield( 0). By the induction hypothesis of (iii),
there exists  2 T (X1) such that C(x) G ) 
and yield( ["]) = yield( 0). By the definition
of P0, A(x) ! B(C(x)) is in P. Therefore,
A(x) G)B(C(x)) G )  [ ] and yield( [ ["]]) =
yield( 0[ 0]). In the case (2), A G0)c(C) G0 ) c( 0) =
 0 for some  0 2 T such that C G0 ) 0. By the in-
duction hypothesis of (i), there exists  2 T such
that C G ) and yield( ) = yield( 0). By the def-
inition of P0, A(x) ! b(C;x) or A(x) ! b(x;C)
is in P. Without loss of generality, we may as-
sume that A(x) ! b(C;x) is in P. Therefore,
A(x) G)b(C;x) G )b( ;x) and yield(b( ;x)["]) =
yield(c( 0)).
The “if” part is similarly proved as follows. For
0-step derivations, (i), (ii) and (iii) clearly hold since
there doesn’t exists  2 T nor  2 T (X1) for
each statement.
The cases for 1-step derivations are proved.
[Proof of (i)] If A G) and  2 T , then  = a
for some a 2  0 and the rule A ! a in P has been
used. Therefore, A ! a is in P 0 and A G0) a.
[Proof of (ii) and (iii)] There doesn’t exists  2 T 
such that A G) .
For k  2, assume that (i), (ii) and (iii) holds for
any derivation of length less than k.
[Proof of (i)] If A Gk) , then the rule used at the
first step must be of the form A ! B(C). Thus,
A G)B(C) G ) [ ] =  for some  2 T (X1)
and  2 T such that B(x) G ) and C G ) .
Here, we have to think of the two cases: (1)
yield( ) 6=  and (2) yield( ) =  . In the
case (1), by the induction hypothesis of (ii), there
exists  0 2 T (X1) such that B(x) G0 ) 0 and
yield( 0) = yield( ), and by the induction hypothe-
sis of (i), there exists  0 2 T such that C G0 ) 0 and
yield( 0) = yield( ). By the definition of P 0, A !
B(C) is in P. Therefore, A G0)B(C) G )  0[ 0]
and yield( 0[ 0]) = yield( [ ]). In the case (2),
C 2 E0. Thus, A ! B is in P0. By the induction
hypothesis of (iii), there exists  0 2 T (X1) such
that B G ) 0 and yield( 0) = yield( ["]). There-
fore, A G0)B G0 ) 0 and yield( 0) = yield( [ ]).
[Proof of (ii)] If A(x) Gk) , then the rule used at
the first step is one of the follwoing form: (1)
A(x) ! B(C(x)), (2) A(x) ! b(C;x) or (3)
A(x) ! b(x;C). The proof of the case (1) is di-
rect from the induction hypothesis. In the case (2),
A(x) G)b(C;x) G )b( ;x) =  for some  2 T 
such that C G ) . Here, we have to think of the two
cases: (a) yield( ) 6=  and (b) yield( ) =  .
(a) If yield( ) 6=  , then by the induction hy-
pothesis of (i), there exists  0 2 T such that
C G0 ) 0 and yield( 0) = yield( ). By the defini-
tion of P0, A(x) ! b(C;x) is in P0. Therefore,
A(x) G0)b(C;x) G0 )b( 0;x) and yield(b( 0;x)) =
yield(b( ;x)). (b) If yield( ) =  , then C 2 E0
and A(x) ! c(x) is in P0. Therefore, A(x) G0)c(x)
and yield(c(x)) = yield(b( ;x)). The proof of the
case (3) is similar to that of the case (2).
[Proof of (iii)] If A(x) Gk) , then the rule used
at the first step is one of the follwoing form:
(1) A(x) ! B(C(x)), (2) A(x) ! b(C;x)
or (3) A(x) ! b(x;C). In the case (1),
A(x) G)B(C(x)) G )  [ ] =  for some  ; 2
T (X1) such that B(x) G ) and C(x) G ) . By
the definition of P0, A ! B(C) is in P0. By
the induction hypothesis of (ii), there exists  0 2
T (X1) such that B(x) G0 ) 0 and yield( 0) =
yield( ). By the induction hypothesis of (iii), there
exists  0 2 T such that C G0 ) 0 and yield( 0) =
yield( ["]). Therefore, A G0) B(C) G0 ) 0[ 0] and
yield( 0[ 0]) = yield( [ ["]]). In the case (2),
A(x) G)b(C;x) G )b( ;x) =  for some  2
T such that C G ) and yield( ) 6=  . By
the definition of P0, A ! c(C) is in P0. By
the induction hypothesis of (i), there exists  0 2
T such that C G0 ) 0 and yield( 0) = yield( ).
Therefore, A G0)c(C) G0 )c( 0) and yield(c( 0)) =
yield(b( ;x)["]). The proof of the case (3) is simi-
lar to that of the case (2).
By (i), we have the result LS(G0) = LS(G).
5 Conclusions
In this paper, the desirable features of linear
monadic CFTGs have been discovered: the re-
striction of nondeletion doesn’t affect their gener-
ative power of tree languages, and the restriction of
epsilon-freeness can be assumed when their gener-
ation of string languages is considered. The key to
the proofs of this paper was the simplicity of the def-
inition of linear, monadic CFTGs and their normal
form.
Recently, the class of grammars called mildly
context-sensitive grammars has been studied very
actively, to which TAGs and other well-established
formalisms for natural languages belong. Since it
is not difficult to study formal properties of linear,
monadic CFTGs, they are helpful tools for the study
of mildly context-sensitive grammars.

References

Anne Abeill´e and Owen Rambow, editors. 2000.
Tree adjoining grammars: formalisms, linguis-
tic analysis and processing. CSLI Publications,
Stanford, California.

Walter S. Brainerd. 1969. Tree generating regular
systems. Information & Control, 14(2):217–231.

Akio Fujiyoshi and Takumi Kasai. 2000. Spinal-
formed context-free tree grammars. Theory of
Computing Systems, 33(1):59–83.

Akio Fujiyoshi. 2004. Epsilon-free grammars and
lexicalized grammars that generate the class of
the mildly context-sensitive languages. In 7th In-
ternational Workshop on Tree Adjoining Gram-
mar and Related Formalisms: Proceedings of the
Workshop, Vancouver, pages 16–23.

Aravind K. Joshi and Yves Schabes, 1996. Hand-
book of Formal Languages, volume 3, chap-
ter Tree-adjoining grammars, pages 69–124.
Springer, Berlin.

Aravind K. Joshi, Leon S. Levy, and Masako Taka-
hashi. 1975. Tree adjunct grammars. J. Com-
puter & System Sciences, 10(1):136–163.

Uwe M¨oennich. 1997. Adjunction as substitution:
an algebraic formulation of regular, context-free
and tree adjoining languages. In G. V. Morrill G-

J. Kruijff and R. T. Oehrle, editors, Formal Gram-
mars 1997: Proceedings of the Conference, Aix-
en-Provence, pages 169–178.

Sanguthevar Rajasekaran and Shibu Yooseph.
1998. TAL recognition in O(M(n2)) time. J.
Computer & System Sciences, 56(1):83–89.

Sanguthevar Rajasekaran. 1996. Tree-adjoining
language parsing in O(n6) time. SIAM J. Com-
put., 25(4):862–873.

William C. Rounds. 1970. Mapping and gram-
mars on trees. Mathematical Systems Theory,
4(3):257–287.

K. Vijay-Shanker and David J. Weir. 1994.
The equivalence of four extensions of context-
free grammars. Mathematical Systems Theory,
27(6):511–546.
