Using data from Jèrriais, Dyirbal, Moroccan Arabic, Polish, and Catalan I develop a full theory of HS/LS. I explore how GEN is conceived of in HS, how allomorph selection functions within the theory, and how HS/LS can handle certain cases of opacity. I propose a revision of the constraint PRIORITY, which is the LS constraint responsible for ensuring respect of allomorph ordering, from a gradient faithfulness constraint to a categorical markedness constraint. The incorporation of LS into HS is important in that it allows HS to more fully account for phonologically conditioned allomorphy.

With this new characterization of ‘typology’ in hand, we can pose and answer fundamental questions about the structure imposed by OT on its grammars.

(1) Typological Status. Since a typology must have a well-formed MOAT, we can assess whether a given collection of grammars constitutes an OT typology. Simply dividing the set of all rankings into individually well-formed grammars is not guaranteed to produce a legitimate typology. Failures are detected by the appearance of cycles in the EPO graphs of the MOAT. Cycles indicate that it is impossible to realize the EPOs as OT constraints assigning violations in a consistent manner. Concomitantly, we can determine which VT representations are equivalent in the sense that they yield the same typology.

(2) Classification. Within a typology, MOAT structure determines whether a collection of grammars can be classified together as a kind of super-grammar, one that retains their shared linguistic patterns while abstracting away from their differences. This contributes to the foundations of the Classification Program of Alber & Prince (2015, in prep.).

(3) Representation. The MOAT arises from a notion of adjacency between constraint orders, which has a natural geometric interpretation. Each typology is associated with a geometric figure that represents the relations between its grammars: the typohedron. Super-grammars appear as regions on the typohedron. The MOAT brings out symmetries between constraints, and these appear on the typohedron as symmetries between super-grammar regions.

The argument proceeds in both concrete and abstract terms. We pursue the main line of analysis by examining the Elementary Syllable Theory (EST) of Prince & Smolensky, which presents the basic issues accessibly and allows for thorough application of the ideas and techniqes developed here. We also look at instructive typologies that are not as obviously rooted in language-based issues. Proceeding more abstractly, we provide formal analysis and proofs of assertions. In investigations of this nature, where broad claims are advanced, it is not possible to rest on examples, and we have introduced formal apparatus and methods of proof that allow us to state and establish claimed results. Not every reader will wish to work through every proof, but the leading ideas are built from the common vocabulary of linguistic analysis and worked out through concrete examples, so that they should be accessible to interested readers in essence and in detail.