The context-free tree grammars (CFTGs) were in­troduced by W. C. Rounds (1970) as tree gener­ating 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 prob­lem 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 nat­ural 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 non­terminals 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; Abeille 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 lan­guages as other formalisms for natural languages developed independently such as head grammars, combinatory categorial grammars and linear in­dexed 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(n) and O(M(n)) time (Ra-jasekaran, 1996; Rajasekaran and Yooseph, 1998). From the view point of CFTG, the languages gener­ated by TAGs were examined (Fujiyoshi and Kasai, 2000; Fujiyoshi, 2004; Moennich, 1997), and it was shown that linear, nondeleting, monadic CFTGs generate the same class ofstring 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 vari­able 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 lan­guages as TAGs. It is shown that nondeletion is un­necessary 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.

In this section, some terms, definitions and former results which will be used in the rest of this paper are introduced.

Because linear, nondeleting, monadic CFTGs gen­erate 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 G/.

Proof. Let G = (N, S, P, S) be a linear, monadic CFTG. An equivalent linear, nondeleting, monadic follows.

The set of nonterminal is N' = Nq U N[ such that Nd = N0 U {A\A G Ni} and N[ = Nx. For the preparation of the definition of P/, for a G Tvus(Xl) we define n(a) C TW'us(Xl) as the smallest set satisfying the following conditions:

definition of TT, there exists a' G Tnus (X1) and

Next, consideration will be given to whether the restriction of linearity can be removed from 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 — 77 with and a'[7] G ü(a'[7[a"]]). Therefore, a == ß.

Because U(S) = {S}, L(Q) = L(Q<).

According to our definition of CFTGs, they are al­lowed to generate trees with the special symbol e, 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 e, 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, S,P, S), if A £ Ls(G), then there exists a weakly equivalent epsilon-free, linear, nondeleting, monadic CFTG G'. If A £ Ls (G), then there exists G' whose epsilon-rule is only S — e.

Proof. Since it is enough to show the existence of a weakly equivalent grammar, without loss of gen­erality, we may assume that G is in strong normal form. We may also assume that the initial nonter­minal S doesn t appear in the right-hand side ofany rule in P.

We first construct subsets of nonterminals E0 and E1 as follows. For initial values, we set E0 = {A £

N0 |A — e £ P} and E1 = 0. We repeat the fol­lowing operations to E0 and E1 until no more oper­ations are possible:

P, then add A £ N0 to E0.

add A £ N1 to E1.

then add A £ N1 to E1.

In the result, E0 satisfies the following.

We construct G' = (N', S',P',S) as follows. The set of nonterminals is N' = Nq U N[ such that No" = N0 U (A\A G Nx} and N[ = Nx. The set of terminal is S' = S U {c}, where c is a new sym­bol of rank 1. The set of rules P' is the smallest set satisfying following conditions:

• P' contains all rules in P except rules of the form A — e.

• If S £ E0,then S — e is in P'.

• If A -»■ B(C) is in P and C G E0, then A -»■ B is in P'.

• lfA(x) -»■ B(C{x)) is in P, then A^ B(C) is in P'.

• If A(x) — 6(C, x) or A(x) — 6(x, C) is in P and C £ E0, then A(x) — c(x) is in P'.

• UA{x) -»■ b(C, x) or A(x) -»■ b(x, C) is in P, then A c(C) is in P'.

To show Ls(G') = Ls(G), we prove the follow­ing (i), (ii) and (iii) hold by induction on the length of derivations:

only if A G for some £ Ts such that yield( ) = yield( ') = A.

such that yield( ) = yield( ').

(iii) For A G - N0, A % a' and a' G Ts if and that yield( [e]) = yield( ') = A.

yield(c(7')).

The "if part is similarly proved as follows. For 0-step derivations, (i), (ii) and (iii) clearly hold since there doesn t exists £ Ts nor £ Ts(X1) for each statement.

The cases for 1-step derivations are proved.

for some a £ S0 and the rule A — a in P has been used. Therefore, A — a is in P' and A a.

A =B(C) =ft[7] andyield(ft[7]) = yield(a').

at the first step is one of the follwoing form: (1) A(x) — B(C(x)), (2) A(x) — 6(C,x) or (3) A(x) — 6(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).

the first step is one of the follwoing form: (1) such that A .

and 7 £ Ts such that B(x) t and C 7.

yield(t') = yield(t), and bythe inductionhypothe-sis of (i), there exists 7' £ Ts such that C ' 7' and and yield(ft/[Y]) = yield(/%]). In the case (2), C g Eq. Thus, A —> B is in P'. By the induction hypothesis of (iii), there exists ft' g TE (X\) such that B ft' and yield (ft') = yield (ft [e]). There­fore, A^B ^13' andyield(ft') = yield(/%]).

yield(t) = yield(t'). By the induction hypothesis of (i), there exists 7 £ Ts such that C the first step is one of the follwoing form: (1) [Proof of (ii)] If A(x) k , then the rule used at form.

and yield(c(x)) = yield(6(7, x)). The proofofthe case (3) is similar to that ofthe case (2).

at the first step is one of the follwoing form:

the definition of P', A -»■ B(C) is in P'. By the induction hypothesis of (ii), there exists t' £ yield (ft). By the induction hypothesis of (iii), there exists 7' € Ts such that C ^7' and yield(7') = the definition of P', A -»■ c(C) is in P'. By the induction hypothesis of (i), there exists 7' £

Ts such that C ' 7' and yield(7') = yield(7).

yield(6(7, x)[e]). The proofofthe case (3) is simi­lar to that of the case (2).

In this paper, the desirable features of linear monadic CFTGs have been discovered: the re­striction of nondeletion doesn't affect their gener­ative power of tree languages, and the restriction of epsilon-freeness can be assumed when their gener­ation of string languages is considered. The key to the proofs of this paper was the simplicity of the def­inition of linear, monadic CFTGs and their normal

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.