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<Paper uid="J98-2006">
  <Title>Optimality Theory and the Generative Complexity of Constraint Violability</Title>
  <Section position="2" start_page="0" end_page="0" type="abstr">
    <SectionTitle>
1. Introduction
</SectionTitle>
    <Paragraph position="0"> Analyses within generative phonology have traditionally been stated in terms of systems of rewrite rules, which, when applied in the appropriate sequence, produce a surface form from an underlying representation. As first pointed out by Johnson (1972), the effects of phonological rewrite rules can be simulated using only finite-state machinery, with iterative application accomplished by sending the output from one transducer to the input of the next, a process that can be compiled out into a single transducer (Kaplan and Kay 1994). 1 Using this insight, a vast majority of computational implementations of phonological rule systems have been done using finite-state transducers or extensions thereof (Sproat 1992).</Paragraph>
    <Paragraph position="1"> Recently, there has been a shift in much of the work on phonological theory, from systems of rules to sets of well-formedness constraints (Paradis 1988, Scobbie 1991, Prince and Smolensky 1993, Burzio 1994). This shift has, however, had relatively little impact upon computational work (but see Bird and Ellison 1994). In this paper, we begin an examination of the effects of the move from rule-based to constraint-based theories upon the generative properties of phonological theories. Specifically, we will focus our efforts on the issue of whether the widely adopted constraint-based view  Computational Linguistics Volume 24, Number 2 raises a particularly interesting theoretical question in this context: it allows the specification of a ranking among the constraints and allows lower-ranked constraints to be violated in order for higher-ranked constraints to be satisfied. This violability property means that certain well-known computational techniques for imposing constraints are not directly applicable. Our study can be seen, therefore, as the beginnings of an investigation of the generative complexity of constraint ranking and violability. In this paper, we present a general formalization of OT that directly embodies that theory's notion of constraint violability. We then study the formal properties of one particular case of this general formalization in which the mapping from input to possible output forms, GEN, is representable as a finite-state transducer, and where each constraint is represented by means of some total function from strings to non-negative integers, with the requirement that the inverse image of every integer be a regular set. These two formal assumptions are sufficiently generous to allow us to capture most of the current phonological analyses within the OT framework that have been presented in the literature. We prove that the generative capacity of the resulting system does not exceed that of the class of finite-state transducers precisely when each constraint has a finite codomain, i.e., constraints may distinguish among only a finite set of equivalence classes of candidates. As will be discussed in Section 6, this result is optimal with respect to the finite codomain assumption, in the sense that dropping this assumption allows the representation of relations that cannot be implemented by means of a finite-state transducer (the latter fact has been shown to us by Markus Hiller, and will be discussed here). Before proceeding with the discussion of our result, however, we describe the rudiments of OT and introduce some technical notions.</Paragraph>
  </Section>
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