The TAG formalism can be motivated because it provides a larger domain of locality in which to state linguistic dependencies. In most formalisms, dependencies can be defined between the elements of a rule (i.e. between the nodes of a local tree). In TAG, it is possible to state dependencies between nodes of trees which are further apart, because the basic building blocks of the formalism are trees. For example, the relation between a topicalized constituent and its governor can be stated locally in a TAG, whereas in most other formalisms this relation must be stated with some global mechanism. Furthermore, the adjunction operation provides for the analysis of (at least some) kinds of discontinuous constituency constructions. This is reflected in the formal power of the formalism: some TAGs recognize context-sensitive languages.
An initial tree is a tree of which the interior nodes are all labelled with non-terminal symbols, and the nodes on the frontier are either labelled with terminal symbols, or with non-terminal symbols, which are marked with the substitution marker ( ).
An auxiliary tree is defined as an initial tree, except that exactly one of its frontier nodes must be marked as foot node (`*'). The foot node must be labelled with a non-terminal symbol which is the same as the label of the root node.
As an example, consider the following initial and auxiliary trees in figure 4.5.
Derived trees are built from initial and auxiliary trees by substitution and adjunction. Substituting a tree in a tree simply replaces a substitution node in with , under the convention that the non-terminal symbol of the substitution node is the same as the root node of . For example, substituting in gives the following tree:
Only initial trees, and derived trees, can be substituted in another tree.
Adjunction is a more complex operation. Adjoining an auxiliary tree at some node n of a derived tree proceeds as follows. Firstly, the non-terminal symbol of the root node (and hence the non-terminal symbol of the foot node) of should be the same as the non-terminal symbol associated with n. The sub-tree t of rooted by n is removed from , and is substituted for it instead; where t is substituted in the foot node of . An illustration of the adjunction operation is presented in figure 4.6. In (1) the sub-tree at the node n where adjunction takes place, is identified. In (2) the root node of the auxiliary tree is substituted at node n, and the sub-tree of n is substituted in the foot node of the auxiliary tree, resulting in (3).
As an example, consider the adjunction of at the `n' node of the derived tree:
This yields the derived tree in figure 4.7.
A way to look at this problem from a more general perspective is presented in . In a way, an elementary tree partially describes possible trees. For example, the elementary tree of figure 4.5 can be interpreted as in figure 4.9.
Thus, we might view such an initial tree as licensing any tree which can be built by adjoining at its interior nodes. For that reason, there is always a sense in which we view such nodes `from the bottom' or `from the top'. This intuition is formalized in constraint-based versions as follows. Each of the nodes of an elementary tree is associated with two variables (called top and bot) representing the view `from the top' and `from the bottom' on which constraints can be defined.
In the case of the substitution of a tree , at a substitution node n of a tree , the result is as before, under the convention that the top variable of the root node of is constrained to be equal to the top variable of n, and similarly the bot variable of the root of is equal to the bot variable of n.
Adjunction is slightly more complex. Adjoining an auxiliary tree at some node n of a derived tree is defined as before, under the convention that the top variable of n is constrained to be equal to the top variable of the root node of ; furthermore the bot variable of n is constrained to be equal to the bot variable of the foot node of . See figure 4.10 for an illustration. The top features of the adjunction node are unified with the top features of the root node of the auxiliary tree. The bottom features of the adjunction node are unified with the bottom features of the foot node of the auxiliary tree.
Finally, at the end of a derivation the top and bot variables of each node are constrained to be equal as well. A derived tree with a node of which the top and bot variables cannot be unified is not regarded as a tree derived by the grammar.
Examples of the use of constraints for TAG are the usual number agreement, case marking, etc. However, it is also possible to use the constraint machinery to define `sign-based' TAGs in which each node is associated with a value for a `phonology' and a `semantics' attribute. We shall see that this will crucially illustrate the need for two distinct variables for each node.
The figures 4.11 and 4.12 illustrate how this might work.
Thus, the argument structure of the bottom part of the verb `saw' is defined as a binary predicate, of which the arguments are the argument structures of the subject and the object. The argument structure of the top part of this verb, however, is not yet known. For example, modifiers may adjoin later at this node to construct a more complex argument structure, built from the simple one associated with the bottom part of the node. Once no more adjunctions are applied, the argument structure is percolated to the verb phrase node: hence the top part of the verb is unified with the bottom part of the verb-phrase. Again, the bottom-part of the verb phrase may be modified, because of adjunctions. Its top-part is unified with the bottom part of the root node, because the argument structure should be percolated, if no more adjunctions apply.