Easy and Hard Constraint Ranking in Optimality Theory: Algorithms and Complexity

We consider the problem of ranking a set of OT constraints in a manner consistent with data. (1) We speed up Tesar and Smolensky’s RCD algorithm to be linear on the number of constraints. This finds a ranking so each attested form xi beats or ties a particular competitor yi. (2) We also generalize RCD so each xi beats or ties all possible competitors. Alas, neither ranking as in (2) nor even generation has any polynomial algorithm unless P = NP—i.e., one cannot improve qualitatively upon brute force: (3) Merely checking that a single (given) ranking is consistent with given forms is coNP-complete if the surface forms are fully observed and ∆ 2 -complete if not. Indeed, OT generation is OptP-complete. (4) As for ranking, determining whether any consistent ranking exists is coNP-hard (but in ∆ 2 ) if the forms are fully observed, and Σ 2 -complete if not. Finally, we show (5) generation and ranking are easier in derivational theories: P, and NP-complete.