Book Reviews Tree Adjoining Grammars: Formalisms, Linguistic Analysis and Processing

Take a finite set T of trees, and close it under the operation of substitution—replacing daughterless nonterminals by other trees in T whose root node label matches the nonterminal. The set of all terminal strings of trees in the resultant tree-set will be a context-free language (CFL), and for every CFL there will be such a finite set of trees that generates it under substitution. For an elegant formalization of CFLs in these terms, see Rogers (1999, pages 25–26). Now assume an additional operation: besides substituting trees for daughterless nonterminals, you can also squeeze new material into the middle of a tree, substituting it for a nonterminal node that has daughters. More precisely, assume a finite set A of insertable trees that share a special form: each A ∈ A has a single node on its frontier, known as A's foot node, that has the same label as A's root. Squeezing some A ∈ A into a tree T ∈ T means replacing a node n in T so that n's mother becomes the mother of A's root, and everything dominated by n comes to be dominated by A's foot node. Closing T under the operation of squeezing in new material from A in this manner (an operation called tree adjunction) yields a tree-set of which the set of all terminal strings will be a tree adjoining language (TAL), and for every TAL there will be an appropriate pair T , AA. Such a pair (called a tree adjunct grammar in Joshi, Levy, and Takahashi [1975]) is known today as a tree adjoining grammar (TAG). The research program on TAGs that Aravind Joshi has led since 1975 is perhaps the most interesting and significant research program in formal language theory of the last 40 years. General linguists have clearly underrated it, though computational linguists have in general kept more closely in touch with it. The TALs are a mathematically natural class with closure and decision properties very similar to those of the CFLs, including a polynomial-time recognition problem. Several independent but equivalent characterizations of the class have been discovered: Vijay-shanker and Weir (1994) present a weak equivalence result for head grammars gave a characterization in terms of embedded pushdown automata; and Rogers (1998) gives a new model-theoretic characterization in terms of linearized terminal strings of three-dimensional tree models of monadic second-order logic formulae.