On defining TALs with logical constraints

·rn Rogers (1997b) we introduced a new dass of models, three-dimensional tree manifolds (3-TM), that can serve as both thc derived and clerivation structures for TAGs in the same way that trees serve as both clerived and derivation structures for CFGs. The1->e tree-manifolds are higher-dimensional analogs of trees; in a 3-Tl'v! the children of a node form an o~·dinary (two-dimensional) tree just as in ordinary tree1-> the children of a node form a string. From t.his point of view the elementary struct.ures of a TAG can bc interpretcd as labeled local 3-Tl\faa root node and it.s set of children (a pyramidal structure)analogous to the interpretation of the rewrite rules of a CFG as local trees. Adjunction in TAGs and substitntion in CFGs both rcduce to a form of concatcnation, of local trees in CFGs, of local 3-TMs in TAGs. In Figure 1, for examplc, the local 3-Tl\'1s corresponding to the elementary trees o 1 aud ß1 are concatenated to form the 3-TM corresponding to the result of adjoining ß1 into 0:1. The two-dimcnsional yicld of this structure is the corresponding derivcd tree and its onc-dimensional yield is the derived string.