Information, unification and locality

In a unification framework we deal with a domain of descriptions P tha t are used to classify objects from some class under analysis 0 , u t terances and their f ragments in the case tha t concerns us here. Classification is given by a description relation ~ between objects in 0 and elements of P. If d is a [partial] description of e, we write e ~ d, e satisfies (or is described by) d. The set of all objects tha t satisfy a description d is wri t ten [dl. Descriptions are in general partial, that is, a description d may in general be extended to a more specific (more informative) description d I. With suitable technical assumptions, this gives a partial order d ~ d' on descriptions. In terms of the description relation ~ , d E d' iff for every object e, e ~ d whenever e ~ d ~. Two descriptions d and d' are compatible if there is a description d" such tha t d _C d" and d ~ _C d", tha t is if d and d' can both be extended to a single description more informative than both. If two descriptions d and d ~ are compatible, it is reasonable to assume tha t there is a least specific description d U d' more specific tha t both d and d'. In other words, d U d' contains all the information in d and d', bu t no more. For historical reasons, d t.J d' is called the unification of d and d'. In more s tandard mathemat ica l terminology, d u d' is the join of d and d'. In terms of the description relation, if e ~ d it d', e ~ d and e ~ d'. Fur thermore , we want unification to behave like logical conjunction: if e ~ d and e ~ d', e ~ d U d'. Thus lid U d'~ = [[d] n [d'~ holds for any compatible descriptions d and d'. The domains of objects, descriptions and the description relation are usually infinite, even though there may be some way of finitely characterizing the description relation. Such a characterization is a grammar . To write grammars, we need to be able to constrain entities to satisfy arbitrari ly complex descriptions without giving the descriptions in full. Our main ins t ruments for this are parameterized descriptions and rules. A parameter ized description d(pl,. . . ,pk) is not a description itself, but ra ther an encoding of a function from k-tuples of descriptions to descriptions. An object e satisfies such a parameter ized description iff there are descriptions f l , . . . , fk such tha t e satisfies d( f l , . . . , f~). Given a family of parameter ized descriptions (di)iel with parameters (Pi)i~s and a set C of constraints involving the parameters , a family of objects (ei)ie I satisfies the parameter ized descriptions relative to the constraints iff there are descriptions (fi)ieJ tha t can be uniformly replaced for the parameters in such a