Tree Adjoining and Head Wrapping

In this paper we discuss the formal relationship between the classes of languages generated by Tree Adjoining Grammars and Head Grammars. In particular, we show tha t Head Languages are included in Tree Adjoining Languages and tha t Tree Adjoining Grammars are equivalent to a modification of Head Grammars called Modified Head Grammars. The inclusion of MHL in HL, and thus the equivalence of HG's and TAG's, in the most general case remains to be established. 1. I n t r o d u c t i o n This paper discusses the relationstfip between Tree Adjoining Grammars (TAG's) and :Head Grammars (HG's). TAG's and HG's, introduced to capture certain s tructural properties of na tura l languages, were developed independently. TAG's deal with a set of elementary trees which are composed by means of an operat ion called a d j o i n i n g . HG's are like Context-free Grammars , except for the fact tha t besides concatenation of strings, s t r i n g w r a p p i n g operations are permitted. TAG's were first introduced in 1975 by Joshi, Levy and Wakahashi [3]. Joshi [2] investigated some formal and linguistic properties of TAG's with local constraints. The formulation of local constraints was then modified and formal properties were investigated by Vijay-Shanker and Joshi [9]. The linguistic properties were studied in detail by Kroeh and Joshi [5]. HG's were first introduced by Pollard [6] in 1983 and their formal properties were investigated by Roach [7]. It was observed tha t the two systems seemed to possess similar generative power and since they also appear to have the same closure properties [7,9] as well as similar parsing algori thn~ [6,9] a significant amount of indirect evidence existed to suggest tha t they were formally equivalent. In the present paper, we will a t tempt to provide a characterization of the formal relationship between HG's and TAG's. In [10] we consider various linguistic aspects of the relationship: in part icular what might be referred to as the strong equivalence of the two formalisms. Vijay-Shanker and Joshi [9] provided a brief description of the intuit ion behind the inclusion of Tree Adjoining Langages (TAL) in the class of languages generated by a variant of HG's called Modified Head Grammars (MHG's). In the present paper, we give a proof of this result as well as a proof for the inclusion of Modified Head Languages (MHL) in TAL: hence we show tha t MHG's and TAG's are equivalent. This result is presented in section 3. In section 2, we discuss the relationship between MHG's and HG's. J" This work was partially supported by NSF grants MCS82-19116-CER, MCS-82-07294 and DC1~-84-10413. We are grateful bo Tony Kroch and Carl Pollard for valuable discus~ sions. 1.1. '.£ree A d j o i n i n g G r a m m a r s D e f i n i t i o n 1. A T A G is a 54uple G -= ( V N , V . r , S , I , A ) where Vlv and V T are finite sets of nonterminals and terminals respectively, S is a distinguished nonterminal, I is a finite set of initial trees and A is a finite set of auxiliary trees. Initial trees and auxiliary trees have the following form: / \ ~ V. w~wz ~Vx "{