One Tableau Suffices

The grammars of any OT typology can be derived from a single tableau in which each row, when asserted as optimal, delivers the grammar of a language in the typology. In the single-tableau representation, each candidate represents an entire language. When, as is typically the case, a typology is constructed from more than one candidate set, a single tableau representation may be built from the Minkowski sum of the whole collection. Since all typologies are representable as a finite collection of finite candidate sets, it follows that a single tableau is always sufficient to represent the grammars of a typology. The study of formal typologies therefore reduces to the study of single tableaux, which are just matrices of nonnegative integers. The result shown here thus gives us a new point of entry into the study of typologies as abstract objects. It also allows us to move around easily in the set of all typologies on n constraints, shown here to be a lattice, because the meet of two typologies is the Minkowski sum of single tableaux that represent them. This is significant because, as a generalization structure, the lattice of typologies plays a role in classifying the grammars of a typology according to shared and contrasting ranking properties (Alber and Prince 2015, in prep.). Perhaps surprisingly, it also follows that an OT system imposes orders and equivalences not just on individual candidates but on the grammars of its typology, a matter explored in detail in Merchant & Prince 2016. Acknowledgment. The original work on this project was supported by grants from the University of Verona, Italy, in academic year 2010-2011. Thanks to Birgit Alber, Nazarré Merchant, and Natalie DelBusso for discussion that has improved the paper throughout. The anxieties expressed by unidentified commenters (2016) have been addressed in a few places, though concerns about the dangers of the internet have not. Version history. Posted to ROA November 10, 2015. Minor mods. Nov. 19, Nov. 26, 2015; Mar. 30, June 14, 2016.