The Learnability of Optimality Theory: An Algorithm and Some Basic Complexity Results

If Optimality Theory (Prince & Smolensky 1991, 1993) is correct, Universal Grammar provides a set of universal constraints which are highly general, inherently conflicting, and consequently rampantly violated in the surface forms of languages. A language’s grammar ranks the universal constraints in a dominance hierarchy, higher-ranked constraints taking absolute priority over lower-ranked constraints, so that violations of a constraint occur in well-formed structures when, and only when, they are necessary to prevent violation of higher-ranked constraints. Languages differ principally in how they rank the universal constraints in their language-specific dominance hierarchies. The surface forms of a given language are structural descriptions of inputs which are optimal in the following sense: they satisfy the universal constraints, or, when these constraints are brought into conflict by an input, they satisfy the highest-ranked constraints possible. This notion of optimality is partly language-specific, since the ranking of constraints is language-particular, and partly universal, since the constraints which evaluate well-formedness are (at least to a considerable extent) universal. In many respects, ranking of universal constraints in Optimality Theory plays a role analogous to parameter-setting in principles-and-parameters theory. Evidence in favor of this Optimality-Theoretic characterization of Universal Grammar is provided elsewhere; most work to date addresses phonology: see Prince & Smolensky 1993 (henceforth, ‘P&S’) and the several dozen works cited therein, notably McCarthy & Prince 1993; initial work addressing syntax includes Grimshaw 1993 and Legendre, Raymond & Smolensky 1993. Here, we investigate the learnability of grammars in Optimality Theory. Under the assumption of innate knowledge of the universal constraints, the primary task of the learner is the determination of the dominance ranking of these constraints which is particular to the target language. We will present a simple and efficient algorithm for solving this problem, assuming a given set of hypothesized underlying forms. (Concerning the problem of acquiring underlying forms, see the discussion of ‘optimality in the lexicon’ in P & S 1993:§9). The fact that surface forms are optimal means that every positive example entails a great number of implicit negative examples: for a given input, every candidate output other than the correct form is ill-formed.1 As a consequence, even a single positive example can greatly constrain the possible grammars for a target language, as we will see explicitly. In §1 we present the relevant principles of Optimality Theory and discuss the special nature of the learning problem in that theory. Readers familiar with the theory may wish to proceed directly to §1.3. In §2 we present the first version of our learning algorithm, initially, through a concrete example; we also consider its (low) computational complexity. Formal specification of the first version of the algorithm and proof of its correctness are taken up in the Appendix. In §3 we generalize the algorithm, identifying a more general core called Constraint Demotion (‘CD’) and then a family of CD algorithms which differ in how they apply this core to the acquisition data. We sketch a proof of the correctness and convergence of the CD algorithms, and of a bound on the number of examples needed to complete learning. In §4 we briefly consider the issue of ties in the ranking of constraints and the case of inconsistent data. Finally, we observe that the CD algorithm entails a Superset Principle for acquisition: as the learner refines the grammar, the set of well-formed structures shrinks.