Computing Optimal Descriptions for Optimality Theory Grammars with Context-Free Position Structures

This paper describes an algorithm for computing optimal structural descriptions for Optimality Theory grammars with context-free position structures. This algorithm extends Tesar's dynamic programming approach (Tesar, 1994) (Tesar, 1995a) to computing optimal structural descriptions from regular to context-free structures. The generalization to contextfree structures creates several complications, all of which are overcome without compromising the core dynamic programming approach. The resulting algorithm has a time complexity cubic in the length of the input, and is applicable to grammars with universal constraints that exhibit context-free locality. 1 Computing Optimal Descriptions in Optimality Theory In Optimality Theory (Prince and Smolensky, 1993), grammaticality is de ned in terms of optimization. For any given linguistic input, the grammatical structural description of that input is the description, selected from a set of candidate descriptions for that input, that best satis es a ranked set of universal constraints. The universal constraints often con ict: satisfying one constraint may only be possible at the expense of violating another one. These con icts are resolved by ranking the universal constraints in a strict dominance hierarchy: one violation of a given constraint is strictly worse than any number of violations of a lower-ranked constraint. When comparing two descriptions, the one which better satis es the ranked constraints has higher Harmony. Cross-linguistic variation is accounted for by di erences in the ranking of the same constraints. The term linguistic input should here be understood as something like an underlying form. In phonology, an input might be a string of segmental material; in syntax, it might be a verb's argument structure, along with the arguments. For expositional purposes, this paper will assume linguistic inputs to be ordered strings of segments. A candidate structural description for an input is a full linguistic description containing that input, and indicating what the (pronounced) surface realization is. An important property of Optimality Theory (OT) grammars is that they do not accept or reject inputs; every possible input is assigned a description by the grammar. The formal de nition of Optimality Theory posits a function, Gen, which maps an input to a large (often in nite) set of candidate structural descriptions, all of which are evaluated in parallel by the universal constraints. An OT grammar does not itself specify an algorithm, it simply assigns a grammatical structural description to each input. However, one can ask the computational question of whether e cient algorithms exist to compute the description assigned to a linguistic input by a grammar. The most apparent computational challenge is posed by the allowance of faithfulness violations: the surface form of a structural description may not be identical with the input. Structural positions not lled with input segments constitute overparsing (epenthesis). Input segments not parsed into structural positions do not appear in the surface pronunciation, and constitute underparsing (deletion). To the extent that underparsing and overparsing are avoided, the description is said to be faithful to the input. Crucial to Optimality Theory are faithfulness constraints, which are violated by underparsing and overparsing. The faithfulness constraints ensure that a grammar will only tolerate deviations of the surface form from the input form which are necessary to satisfy structural constraints dominating the faithfulness constraints. Computing an optimal description means considering a space of candidate descriptions that include structures with a variety of faithfulness violations, and evaluating those candidates with respect to a ranking in which structural and faithfulness constraints may be interleaved. This is parsing in the generic sense: a structural description is being assigned to an input. It is, however, distinct from what is traditionally thought of as parsing in computational linguistics. Traditional parsing attempts to construct a grammatical description with a surface form matching the given input string exactly; if a description cannot be t exactly, the input string is rejected as ungrammatical. Traditional parsing can be thought of as enforcing faithfulness absolutely, with no faithfulness violations are allowed. Partly for this reason, traditional parsing is usually understood as mapping a surface form to a description. In the computation of optimal descriptions considered here, a candidate that is fully faithful to the input may be tossed aside by the grammar in favor of a less faithful description better satisfying other (dominant) constraints. Computing an optimal description in Optimality Theory is more naturally thought of as mapping an underlying form to a description, perhaps as part of the process of language production. Tesar (Tesar, 1994) (Tesar, 1995a) has developed algorithms for computing optimal descriptions, based upon dynamic programming. The details laid out in (Tesar, 1995a) focused on the case where the set of structures underlying the Gen function are formally regular. In this paper, Tesar's basic approach is adopted, and extended to grammars with a Gen function employing fully context-free structures. Using such context-free structures introduces some complications not apparent with the regular case. This paper demonstrates that the complications can be dealt with, and that the dynamic programming case may be fully extended to grammars with context-free structures. 2 Context-Free Position Structure