Partial descriptions are sets of constraints which do not necessarily determine a unique object. Two related issues arise when objects are specified by partial descriptions. The first asks whether a given description is satisfied by any object. The second seeks to produce, from a description, an appropriate representative of those objects which satisfy it. We explore these issues in the context of two recent proposals utilizing partial descriptions of trees in the area of Tree-Adjoining Grammars. In this context, what counts as an appropriate representative of the set of trees which satisfy a description is a tree in that set which is minimal in the sense that every other tree in the set can be derived from it by adjunction. We call a partial description for which there is a unique such minimal tree a quasi-tree. We formalize the notions of partial descriptions of trees and of quasi-trees, and, using these, provide mechanisms which resolve any description of trees into an equivalent se..
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