An approximation approach to the problem of the acquisition of phonotactics in Optimality Theory

The problem of the acquisition of phonotactics in Optimality Theory is intractable. This paper offers a way to cope with this hardness result: the problem is reformulated as a well known integer program (the Assignment problem with linear side constraints) paving the way for the application to phonotactics of approximation algorithms recently developed for integer programming. Knowledge of the phonotactics of a language is knowledge of its distinction between licit and illicit forms. The acquisition of phonotactics represents a distinguished and important stage of language acquisition. In fact, in carefully controlled experimental conditions, nine-month-old infants already react differently to licit and illicit sound combinations (Jusczyk et al., 1993). They thus display knowledge of phonotactics already at an early stage of language development. Usually, the problem of the acquisition of the phonotactics of a language given a finite set of linguistic data is formalized as the problem of finding a smallest language in the typology that is consistent with the data (Berwick, 1985; Manzini and Wexler, 1987; Prince and Tesar, 2004; Hayes, 2004; Fodor and Sakas, 2005). Section 1 formulates the problem of the acquisition of phonotactics along these lines within the mainstream phonological framework of Optimality Theory (Prince and Smolensky, 2004; Kager, 1999). Unfortunately, (such a formulation of) the problem of the acquisition of phonotactics in OT turns out to be intractable (NP-complete): for any attempted efficient solution algorithm, there are some instances of the problem where the algorithm fails (Magri, 2010; Magri, 2012b). This hardness result holds for the universal formulation of the problem, in the sense of Heinz et al. (2009): there are no restrictions on the constraint set that defines the OT typology and indeed the OT typology itself figures as an input to the problem. There are two strategies to cope with this hardness result. One approach weakens the formulation of the problem through proper restrictions on the constraint set: certain constraint sets are implausible from a phonological perspective, and should therefore be ignored in the proper formulation of the problem (Magri, 2011; Magri, 2012c). This approach raises interesting challenges, as it requires a through investigation of the algorithmic implications of various generalizations developed by phonologists on what counts as a “plausible” OT constraint set. Another approach is to bypass this difficulty, and weaken the formulation of the problem by lowering the standard for success: we settle on an approximate solution, namely a “small” language rather than a smallest language. This paper paves the way for the latter approach. I focus on the specific formulation of the problem of the acquisition of OT phonotactics developed in Prince and Tesar (2004). In Sections 2 and 3, I show that this formulation of the problem can be restated as a classical integer program, namely the Assignment problem with liner side constraints (AssignLSCsPbm). The theory of approximation algorithms for integer programing is a blooming field of Computer Science (Bertsimas and Weismantel, 2005). In particular, powerful approximation algorithms have been recently developed for the AssignLSCsPbm. A state-of-the-art algorithm is due to Arora et al. (2002). The integer programming formulation developed in this paper thus paves the way for a new approximation approach to the problem of modeling the acquisition of phonotactics within OT. In Magri (2012a), I report simulation results with Arora’s et. al. (2002) algorithm on various instances of the problem of the acquisition of phonotactics.