In some grammars, the string operations that are defined, are not only monotonic with respect to the words they dominate, but also with respect to the order constraints, that are defined between these words (`order-monotonic'). For example, Reape's sequence union operation preserves the linear precedence constraints, that are defined. TAGs do not allow to change the order of two elements once this order has been established. Thus, TAGs are examples of `order-monotonic' grammars. For that reason, a subsequence check can be used to good effect for the head-driven parser for TAGs. The analysis of verb second in the foregoing section, on the other hand, uses a string operation that does not satisfy this restriction.
For grammars that do satisfy the `order-monotony' restriction it is possible to extend the top-down prediction possibilities by the incorporation of extra clauses in the `head_corner' and `parse' predicates, which check that the phrase that has been analyzed up to that point, can become a sub-string of the input string. To this purpose, the input string is percolated through the parser as an extra argument. Each time a rule has applied the parser checks whether the string derived up to that point can be a subsequence of the input string. The head-corner parser for TAGs defined in the previous section constitutes an example of this technique.