A Structural Theory of Derivations

We propose a category Der of generalizations of partial orders which we use to model syntactic derivations. The objects of Der are ‘variable’ partial orders which keep track of derived syntactic objects (SOs), parameterized over a sequence of steps. This gives information relating the structure of each derived SO to each subsequent SO, i.e. the ‘structural changes’. Der characterizes isomorphisms between derivations (correspondence between steps, derived objects, and dependencies introduced), embeddings of derivations (with inclusions of constituents as a special case), and the relationship between derivations and their derived objects. We show that this category has many good properties, including the existence of disjoint unions of derivations (used to model workspaces and structural changes induced by syntactic operations) and products of derivations (which can be used to model relations on derivations). We also give a form of pushout methods for describing syntactic operations and contexts of application, which we use to recursively construct derivations. We formalize ‘overlap’ or ‘degree of connectivity’ introduced by operations, and we give interpretations of this data which characterize grammatical relations. Finally, we construct toy grammars as collections of derivations closed under some set of operations, and we define equivalence and isomorphism of grammars in terms of the structure of the lexical items they have and derivations they can construct. We then briefly discuss difficulties other mainstream computational methods have in capturing similar properties, and ways that derivations allow for more general constructions like feature geometry and feature sharing.