Learnability in Optimality Theory (Short Version)

A central claim of Optimality Theory is that grammars may differ only in how conflicts among universal well-formedness constraints are resolved: a grammar is precisely a means of resolving such conflicts via a strict priority ranking of constraints. It is shown here how this theory of Universal Grammar yields a highly general Constraint Demotion principle for grammar learning. The resulting learning procedure specifically exploits the grammatical structure of Optimality Theory, independent of the content of substantive constraints defining any given grammatical module. The learning problem is decomposed and formal results are presented for a central subproblem, deducing the constraint ranking particular to a target language, given structural descriptions of positive examples and knowledge of universal grammatical elements. Despite the potentially large size of the space of possible grammars, the structure imposed on this space by Optimality Theory allows efficient convergence to a correct grammar. Implications are discussed for learning from overt data only, learnability of partially-ranked constraint hierarchies, and the initial state. It is argued that Optimality Theory promotes a goal which, while generally desired, has been surprising elusive: confluence of the demands of more effective learnability and deeper linguistic explanation. How exactly does a theory of grammar bear on questions of learnability? Restrictions on what counts as a possible human language can restrict the search space of the learner. But this is a coarse observation: alone it says nothing about how data may be brought to bear on the problem, and further, the number of possible languages predicted by most linguistic theories is extremely large. It would clearly be a desirable result if the nature of the restrictions 1 imposed by a theory of grammar could contribute further to language learnability. The central claim of this paper is that the character of the restrictions imposed by Optimality Theory (Prince and Smolensky 1991, 1993) have demonstrable and significant consequences for central questions of learnability. Optimality Theory explains linguistic phenomena through the complex interaction of violable constraints. The main results of this paper demonstrate that those constraint interactions are nevertheless restricted in a way that permits the correct grammar to be inferred from grammatical structural descriptions. These results are theorems, based on a formal analysis of the Optimality Theory framework; proofs of the theorems are contained in an appendix. The results have two important properties. First, they derive from central principles of the Optimality Theory framework. Second, they are nevertheless independent of the details of any substantive analysis of particular phenomena. The results apply equally to phonology, syntax, and any other domain admitting an Optimality Theoretic analysis. Thus, these theorems provide a learnability measure of the restrictiveness inherent in Optimality Theory’s account of cross-linguistic variation per se: constraint reranking. The structure of the paper is as follows. Section 1 formulates the Optimality Theoretic learning problem we address. Section 2 addresses this problem by developing the principle of Constraint Demotion, which is incorporated into an error-driven learning procedure in section 3. Section 4 takes up some issues and open questions raised by Constraint Demotion, and section 5 concludes. Section 6 is an appendix containing the formal definitions, theorems, and proofs. 2 Tesar & Smolensky Learnability in Optimality Theory 1. Learnability and Optimality Theory Optimality Theory (henceforth, ‘OT’) defines grammaticality by optimization over violable constraints. The defining reference is Prince and Smolensky 1993 (abbreviated ‘P&S’ here). Section 1.1 provides the necessary OT background, while section 1.2 outlines the approach to language learnability proposed here, including a decomposition of the overall problem; the results of this paper solve the subproblem involving direct modification of the grammar. 1.1 Optimality Theory In this section, we present the basics of OT as a series of general principles, each exemplified within the Basic CV Syllable Theory of P&S. 1.1.1 Constraints and Their Violation (1) Grammars specify functions. A grammar is a specification of a function which assigns to each input a unique structural description or output. (A grammar per se does not provide an algorithm for computing this function, e.g., by sequential derivation.) In Basic CV Syllable Theory (henceforth, ‘CVT’), an input is a string of Cs and Vs, e.g., /VCVC/. An output is a parse of the string into syllables, denoted as follows: [ F [ F (2) a. .V.CVC. = V] CVC] b. +V,.CV.+C, = V CV] C [ F c. +V,.CV.C~. = V CV] C~] [ F [ F ́ ́ d. .~V.CV.+C, = ~V] CV] C [ F [ F (These four forms will be referred to frequently in the paper, and will be consistently labeled a–d.) 3 Tesar & Smolensky Learnability in Optimality Theory Output a is an onsetless open syllable followed by a closed syllable; periods denote the boundaries of syllables (F). Output b contains only one, open, syllable. The initial V and final C of the input are not parsed into syllable structure, as notated by the angle brackets +,. These segments exemplify underparsing, and are not phonetically realized, so b is ‘pronounced’ simply as .CV. The form .CV. is the overt form contained in b. Parse c consists of a pair of open syllables, in which the nucleus of the second syllable is not filled by an input segment. This empty nucleus is notated ~, and exemplifies overparsing. The ́ phonetic interpretation of this empty nucleus is an epenthetic vowel. Thus c has .CV.CV. as its overt form. As in b, the initial V of the input is unparsed in c. Parse d is also a pair of open syllables (phonetically, .CV.CV.), but this time it is the onset of the first syllable which is unfilled (notated ~; phonetically, an epenthetic consonant), while the final C is unparsed. (3) Gen: Universal Grammar provides a function Gen which, given any input I, generates Gen(I), the set of candidate structural descriptions for I. The input I is an identified substructure contained within each of its candidate outputs in Gen(I). The domain of Gen implicitly defines the space of possible inputs. In CVT, for any input I, the candidate outputs in Gen(I) consist in all possible parsings of the string into syllables, including the possible overand underparsing structures exemplified above in (b–d). All syllables are assumed to contain a nucleus position, with optional preceding onset and following coda positions. CVT adopts the simplifying assumption (true of many languages) that the syllable positions onset and coda may each contain at most one C, and the nucleus position may contain at most one V. The four candidates of /VCVC/ in (2) are only illustrative of the full set Gen(/VCVC/). Since the possibilities of overparsing are unlimited, Gen(/VCVC/) in fact contains an infinite number of candidates. 4 Tesar & Smolensky Learnability in Optimality Theory The next principle identifies the formal character of substantive grammatical principles. (4) Con: Universal Grammar provides a set Con of universal well-formedness constraints. The constraints in Con evaluate the candidate outputs for a given input in parallel (i.e., simultaneously). Given a candidate output, each constraint assesses a multi-set of marks, where each mark corresponds to one violation of the constraint. The collection of all marks assessed a candidate parse p is denoted marks(p). A mark assessed by a constraint ÷ is denoted *÷. A parse a is more marked than a parse b with respect to ÷ iff ÷ assesses more marks to a than to b. (The theory recognizes the notions moreand less-marked, but not absolute numerical levels of markedness.) The CVT constraints are given in (5). (5) Basic CV Syllable Theory Constraints ONSET Syllables have onsets. NOCODA Syllables do not have codas. PARSE Underlying (input) material is parsed into syllable structure. Nucleus positions are filled with underlying material. FILLNuc Onset positions (when present) are filled with underlying material. FILLOns These constraints can be illustrated with the candidate outputs in (a–d). The marks incurred by these candidates are summarized in table (6). L1 5 Tesar & Smolensky Learnability in Optimality Theory (6) Constraint Tableau for Candidates ONSET NOCODA PARSE FILLNuc FILLOns