|Title:||A Comparison of Lexicographic and Linear Numeric Optimization Using Violation Difference Ratios|
|Abstract:||Optimality Theory (henceforth OT) (Prince and Smolensky 1993/2004) is based upon lexicographic optimization. It differs in this respect from Harmonic Grammar (henceforth HG) (Legendre et al. 1990a, Legendre et al. 1990b), which is based upon linear numeric optimization. Differences between the two have been discussed in several places, including (Legendre et al. 2006, Pater et al. 2007a, Prince and Smolensky 1993/2004).
Patterns which are achievable in HG but not in OT can be called cumulative interactions. Extensive discussion and exemplification of cumulative interactions is provided in (Pater et al. 2007a) (henceforth PBP). An important point of PBP is that HG cannot realize every possible pattern: only some linguistic patterns which are not achievable in OT can be achieved via cumulative interactions. In part of their paper, PBP examine some undesirable cumulative interactions, in particular ones that occur when linear numeric optimization is used in the context of 'global' optimization models, in which the optimal candidate(s) in a competition is decided via a single optimization in which all candidates are simultaneously competing. PBP ultimately propose a variant of Harmonic Serialism (Prince and Smolensky 1993/2004), called Local Harmonic Serialism, to prevent the undesired interactions. Local Harmonic Serialism draws in part on recent work by McCarthy (McCarthy 2006b, McCarthy 2007). Harmonic Serialism uses 'local' optimization, in which the optimal candidate is determined via a series of optimizations, each of which consists only of candidates that differ in a specific localized property (see PBP for further discussion).
In this paper, I focus exclusively on 'global' optimization models, providing further investigation into the similarities and differences between OT and HG in that context. The relation is viewed with respect to violation differences: the difference (as a quantity) in the number of violations assessed to two candidates in a comparison. When two constraints conflict on a candidate comparison, we can examine the violation difference ratio of those constraints: the relative size of the differences in the number of violations for the constraints. The relation between violation difference ratios for different candidate comparisons can predict potential for cumulative interactions.
The major points of the present work are listed in (1).
(1) Major points of this paper.
a. Many cases permitting OT and HG to diverge can be understood in terms of disproportional constraint violation differences: the violation difference ratios for conflicting constraints differs across candidate comparisons.
b. Failure to exhibit cumulative interaction is not limited to instances of 'Anything Goes' competitions; there are competitions which are not 'Anything Goes', but do exhibit proportional violation differences, and do not permit cumulative interactions.
c. Efforts to avoid cumulative interactions in some comparisons can introduce cumulative interactions into other comparisons.
d. Imposing a fixed bound on the number of violations a constraint can assess does not eliminate typologically problematic patterns in HG, it only limits the number of variations.
Many parts of this paper are summary and synthesis of insights originally presented elsewhere, especially in (Legendre et al. 2006, Pater et al. 2007a, Prince and Smolensky 1993/2004, Prince 2002), with a few new observations along the way. I make particularly extensive use of the investigation in PBP; many of the examples are either taken from or adapted from PBP. The construct of violation difference ratios makes it possible to tie together many of these prior insights in a particularly understandable way.