|Title:||The Mother of All Tableaux: Order, equivalence, and geometry in the large-scale structure of OT|
|Authors:||Nazarre Merchant, Alan Prince|
|Comment:||Supersedes ROA-1285. Revision includes replacement of Ch. 3 (Analysis of the Moat) and Ch. 6 (Geometry)|
|Abstract:||[This manuscript thoroughly revises Merchant & Prince (2016), with complete replacement of Chapters 3 and 6, embodying new approaches to the central formal issues.]
OT grammars arise from the comparison of candidates over a set of constraints. An OT typology, we show, implicitly compares entire grammars over the same set of constraints. From the details of this comparison, each constraint can be seen in its essential form as an order and equivalence structure on grammars. At this level, a constraint is no longer a function penalizing concrete linguistic structures and mappings, but a more abstract order and equivalence structure that we call an EPO, an 'Equivalence-augmented Privileged Order'. The collection of the EPOs, each one representing a single constraint, forms the MOAT, the 'Mother of All Tableaux'. The unique MOAT of a typology is instantiated in every violation tableau that gives rise to that typology.
With this new characterization of 'typology' in hand, we can pose and answer fundamental questions about the structure imposed by OT on its typologies.
(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 merged together as a kind of super-grammar within a valid, coarser typology. Such a class retains the shared linguistic patterns across its elements while abstracting away from their differences. This contributes to the foundations of the broader analysis of OT typologies developing under the name of Property Theory (Alber & Prince 2015, 2021, in prep.).
(3) Geometry. The relations in the MOAT arise from a notion of adjacency between constraint orders. Rendered graphically, this notion leads to the permutohedron, an object familiar from combinatorics, in which each vertex corresponds to an order. Grammars are spherically convex regions on permutohedron (Riggle 2012), an important result that MOAT theory is used to prove. Further, if each grammar is shrunk a single point, then a typology is associated with a geometric figure that represents the relations between its grammars: the typohedron. The super-grammars that coarsen typologies appear as regions on the typohedron.
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 techniques 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 that are demonstrably true or demonstrably false, it is not possible to rest either on examples or on unexplored principles, and we have introduced the 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.
|Area/Keywords:||typology, formal analysis|