A Note on the Complexity of Optimality Systems

Optimality Theory (OT) has a growing importance in various disciplines of linguistics. The way in which correct linguistic expressions are generated according to OT can be captured formally in an Optimality System (OS). An OS de nes a relation between an input and an output using a binary relation, called gen, on the domain, and a set of constraints. Frank and Satta ([1]) have shown that an OS in which gen is a rational relation on strings and which has only regular constraints de nes again a rational relation. This result is of great importance for phonological applications of OT. For syntax, however, it cannot be assumed that gen is a rational relation on strings. Since OT syntax is a theory about trees, it seems natural to lift OSs for this purpose to the domain of trees and relations on trees. It is shown in this paper that the result concerning regular string languages can be extended to regular tree languages and it is sketched how such a system might work for natural language syntax.