Efficient Generation in Primitive Optimality Theory

This paper introduces primitive Optimality Theory (OTP), a linguistically motivated formalization of OT. OTP specifies the class of autosegmental representations, the universal generator Gen, and the two simple families of permissible constraints. In contrast to less restricted theories using Generalized Alignment, OTP's optimal surface forms can be generated with finite-state methods adapted from (Ellison, 1994). Unfortunately these methods take time exponential on the size of the grammar. Indeed the generation problem is shown NP-complete in this sense. However, techniques are discussed for making Ellison's approach fast in the typical case, including a simple trick that alone provides a 100-fold speedup on a grammar fragment of moderate size. One avenue for future improvements is a new finite-state notion, "factored automata," where regular languages are represented compactly via formal intersections N~=IAi of FSAs. 1 W h y f o r m a l i z e O T ? Phonology has recently undergone a paradigm shift. Since the seminal work of (Prince & Smolensky, 1993), phonologists have published literally hundreds of analyses in the new constraint-based framework of Optimality Th.eory, or OT. Old-style derivational analyses have all but vanished from the linguistics conferences. The price of this creative ferment has been a certain lack of rigor. The claim for O.T as Universal Grammar is not substantive or falsifiable without formal definitions of the putative Universal Grammar objects Repns , Con, and Gen (see below). Formalizing OT is necessary not only to flesh it out as a linguistic theory, but also for the sake of computational phonology. Without knowing what classes of constraints may appear in grammars, we can say only so much about the properties of the system, or about algorithms for generation, comprehension, and learning. The central claim of OT is that the phonology of any language can be naturally described as successive filtering. In OT, a phonological grammar for a language consists of ~ vector C1, C2, • .. C, of soft c o n s t r a i n t s drawn from a universal fixed set Con. Each constraint in the vector is a function that scores possible output representations (surface forms): (1) Ci : R e p n s --* {0, 1, 2, . . .} (Ci E Con) If C~(R) = 0, the output representation R is said to sa t i s fy the ith constraint of the language. Otherwise it is said to v io la te that constraint, where the value of C~(R) specifies the degree of violation. Each constraint yields a filter that permits only minimal violation of the constraint: (2) Filteri(Set)= {R E Set : Ci(R) is minimal} Given an underlying phonological input, its set of legal surface forms under the grammar--typically of size 1--is just (3) Filter, (...Filter,. (Filter 1 (Gen(input)))) where the function G en is fixed across languages and Gen(input) C_ Rep n s is a potentially infinite set of candidate surface forms. In practice, each surface form in Gen(input) must contain a silent copy of input, so the constraints can score it on how closely its pronounced material matches input. The constraints also score other criteria, such as how easy the material is to pronounce. If C1 in a given language is violated by just the forms with coda consonants, then Filterl(Gen(input)) includes only coda-free candidates--regardless of their other demerits, such as discrepancies from input or unusual syllable structure. The remaining constraints are satisfied only as well as they can be given this set of survivors. Thus, when it is impossible to satisfy all constraints at once, successive filtering means early constraints take priority. Questions under the new paradigm include these: • Generation. How to implement the inputoutput mapping in (3)? A brute-force approach