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A Constructive Solution to the Ranking Problem in Partial Order Optimality Theory

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Abstract

Partial order optimality theory (PoOT) (Anttila and Cho in Lingua 104:31–56, 1998) is a conservative generalization of classical optimality theory (COT) (Prince and Smolensky in Optimality theory: constraint interaction in generative grammar, Blackwell Publishers, Malden, 1993/2004) that makes possible the modeling of free variation and quantitative regularities without any numerical parameters. Solving the ranking problem for PoOT has so far remained an outstanding problem: allowing for free variation, given a finite set of input/output pairs, i.e., a dataset, \(\Delta \) that a speaker S knows to be part of some language L, how can S learn the set of all grammars G under some constraint set C compatible with \(\Delta \)?. Here, allowing for free variation, given the set of all PoOT grammars GPoOT over a constraint set C , for an arbitrary \(\Delta \), I provide set-theoretic means for constructing the actual set G compatible with \(\Delta \). Specifically, I determine the set of all STRICT ORDERS of C that are compatible with \(\Delta \). As every strict total order is a strict order, our solution is applicable in both PoOT and COT, showing that the ranking problem in COT is a special instance of a more general one in PoOT.

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Notes

  1. The standard notational convention in the literature for the dominance relation ‘\(>>\)’ and ‘\(C>> C'\)’ is read as, ‘\(C\) dominates \(C'\)’. I break with standard tradition to maintain internal conceptual clarity, particularly with respect to Theorem 3.

  2. Each node in Fig. 3 should be understood as a set of grammars. For example, ‘\(C_{1}C_{2}, C_1C_3\)’ is an abbreviation for ‘\(\{C_{1} \prec C_{2}, C_{1} \prec C_3 \}\)’, and ‘\(C_{1}C_{2}C_{3}\)’ would be shorthand for ‘\(\{C_{1} \prec C_{2}, C_{1} \prec C_{3}, C_{2} \prec C_{3}\}\)’.

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    Alex J. Djalali

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Djalali, A.J. A Constructive Solution to the Ranking Problem in Partial Order Optimality Theory. J of Log Lang and Inf 26, 89–108 (2017). https://doi.org/10.1007/s10849-017-9248-0

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