Gradient constraints in finite state OT: The unidirectional and the bidirectional case

Optimality Theory in the sense of Prince and Smolensky 1993 is computationally very expensive in the general case. It can be shown that the set of optimal candidates for a given generator GEN and a set of constraints CON may be undecidable even if both GEN and all constraints in CON are recursive. Under certain general conditions, however, optimization does not even exceed the limits of finite state techniques. Frank and Satta (1998) show that the optimal input-output relation is rational if (a) GEN is a rational relation and (b) all constraints in CON are binary markedness constraints that have a regular language as extension. The proof of this fact makes crucial use of an operation called “conditional intersection” (Karttunen 1998 calls it “lenient composition”), which, in the finite state calculus from Kaplan and Kay (1994), can be defined as