Complexity of the Acquisition of Phonotactics in Optimality Theory

The problem of the acquisition of Phonotactics in OT is shown to be not tractable in its strong formulation, whereby constraints and generating function vary arbitrarily as inputs of the problem. Tesar and Smolensky (1998) consider the basic ranking problem in Optimality Theory (OT). According to this problem, the learner needs to find a ranking consistent with a given set of data. They show that this problem is solvable even in its strong formulation, namely without any assumptions on the generating function or the constraint set. Yet, this basic ranking problem is too simple to realistically model any actual aspect of language acquisition. To make the problem more realistic, we might want, for instance, to require the learner to find not just any ranking consistent with the data, rather one that furthermore generates a smallest language (w.r.t. set inclusion). Prince and Tesar (2004) and Hayes (2004) note that this computational problem models the task of the acquisition of phonotactics within OT. This paper shows that, contrary to the basic ranking problem considered by Tesar and Smolensky, this more realistic problem of the acquisition of phonotactics is not solvable, at least not in its strong formulation. I conjecture that this complexity result has nothing to do with the choice of the OT framework, namely that an analogous result holds for the corresponding problem within alternative frameworks, such as Harmonic Grammar (Legendre et al., 1990b; Legendre et al., 1990a). Furthermore, I conjecture that the culprit lies with the fact that generating function and constraint set are completely unconstrained. From this perspective, this paper motivates the following research question: to find phonologically plausible assumptions on generating function and constraint set that make the problem of the acquisition of phonotactics tractable. 1 Statement of the main result Let the universal specifications of an OT typology be a 4-tuple (X ,Y, Gen, C), as illustrated in (1): X and Y are the sets of underlying and surface forms; Gen is the generating function; and C is the constraint set. X = Y = {ta, da, rat, rad} Gen = ˆ ta, da→ {ta, da} rat, rad→ {rat, rad} ̃ C = <:Fpos = IDNT[VCE]/ONSET, F = IDNT[VCE], M = ∗[+VCE, −SON] =; (1) Let , ′, . . . be rankings over the constraint set, as illustrated in (2) for the constraint set in (1). Fpos M F Fpos ′ F ′ M (2) Let OT be the OT-grammar corresponding to a ranking (Prince and Smolensky, 2004), as illustrated in (3) for the ranking in (2). OT (/ta/) = [ta] OT (/da/) = [da] OT (/rat/) = [rat] OT (/rad/) = [rat] (3) Let L( ) be the language corresponding to a ranking , illustrated in (4) for the rankings (2). L( ) = {ta, da, rat} L( ′) = {ta, da, rat, rad} (4) A data set D is a finite set of pairs (x, ŷ) of an underlying form x ∈ X and an intended winner surface form ŷ∈Gen(x)⊆Y , as illustrated in (5). D = {(/da/, [da]), (/rat/, [rat])} (5) A data set D is called OT-compatible with a ranking iff the corresponding OT-grammar accounts for all the pairs in D, namely OT (x)= ŷ for every pair (x, ŷ) ∈ D. A data set D is called OTcompatible iff it is OT-compatible with at least a ranking. Suppose that the actual universal specifications (X ,Y, Gen, C) are fixed and known. The