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<Paper uid="J98-2006">
  <Title>Optimality Theory and the Generative Complexity of Constraint Violability</Title>
  <Section position="3" start_page="0" end_page="308" type="metho">
    <SectionTitle>
2. Basics of OT
</SectionTitle>
    <Paragraph position="0"> As in derivational systems, the general form of phonological computation in OT proceeds from an underlying representation (UR). 3 Such a UR is fed as input to the function OEN, which produces as output the set of all possible surface realizations (SRs) for this UR, called the candidate set. The notion of a possible SR, as realized in Prince and Smolensky (1993), is governed by the containment condition, requiring any SR output by GEN to include a representation of the UR as a (not necessarily contiguous) subpart. Thus, an SR must at a minimum include all of the structure that is specified in the UR, but may also include extra structure absent from the UR, called epenthetic structure. This is not to say that all parts of the input are necessarily pronounced at the surface. Rather, the analogue of &amp;quot;deletion&amp;quot; may occur by marking that part of the SR corresponding to the deleted material as unparsed, meaning that it is not visible to the phonetic interface.</Paragraph>
    <Paragraph position="1"> The candidate set produced by GEN for any UR will in general be infinite, as there is no bound on the amount of epenthetic material that may be added to the UR to prowhether the constraint satisfaction problem for a specific input form can be compiled into a finite-state automaton. He provides an algorithm to produce a nondeterministic finite-state automaton that represents the set of winning candidates for any particular underlying form given finite-state representations of the input and the constraints. We are, however, interested in the more general question of whether the input-output mapping specified by OT for the class of inputs as a whole can be simulated with finite-state machinery. Another related study is that of Tesar (1995), who shows how the set of optimal output forms can be efficiently computed using a dynamic programming technique.</Paragraph>
    <Paragraph position="2"> Tesar does not, however, address the question of the generative complexity of the mappings his algorithm computes. 3 Length constraints prevent us from presenting a more comprehensive introduction to OT. For further discussion of the formal structure of the model and its empirical consequences, see Prince and Smolensky (1993) and references cited therein.</Paragraph>
    <Paragraph position="3">  Frank and Satta Constraint Violability duce the SR. The core of the OT machinery is devoted to choosing among the members of this candidate set to determine which is the actual SR. To do this, OT imposes a set of well-formedness constraints on the elements of the candidate set. Note, however, that these constraints are not imposed conjunctively, meaning that the &amp;quot;winning&amp;quot; SR need not, and most often will not, satisfy them all. Instead, OT allows for the specification of a language-particular ranking among the constraints, reflecting their relative importance. The candidate SRs are evaluated with respect to the constraints in a number of stages. At each stage, the entire candidate set is subjected to one of the constraints, the stage at which a constraint is applied being determined by the specified constraint ranking. 4 There are two possible outcomes of such an evaluation. The first arises when some members of the candidate set violate the constraint, but others do not. In this case, the constraint permits us to distinguish among the members of the candidate set: those that do not satisfy the constraint are eliminated from the candidate set and are not considered in subsequent constraint evaluation. (Alternatively, if a constraint can be violated multiple times by a single SR, the relevant evaluation compares the number of violations incurred by each of the SRs in the candidate set. Candidates with the fewest violations are preferred and those with more violations are eliminated.) The second possible outcome from a constraint evaluation ensues when all of the members of the candidate set violate the constraint to the same degree, perhaps massively or perhaps not at all. When this happens, the constraint does not help us in narrowing down the candidate set. Hence, no candidates are eliminated from the candidate set and violations of the constraint do not block any of them from being considered further to be the actual SR. At the end of the last stage, i.e., when all constraints have been applied, what remains is precisely the subset of the candidate set that are the optimal satisfiers of the constraints under their ranking. This set of candidates, which will often contain only a single member under the system of constraints suggested by Prince and Smolensky (1993), is taken as the set of actual SRs for the original UR.</Paragraph>
    <Paragraph position="4"> OT makes the strong assumption that the constraints used to evaluate the members of the candidate set are universal, and are therefore active in the phonology of every language. What varies from one language to another is the relative ranking of constraints. Thus, as soon as a commitment is made concerning the set of constraints, there is a concomitant commitment concerning the range of possible typological variation: every ordering of the constraints corresponds to a possible phonological system.</Paragraph>
  </Section>
  <Section position="4" start_page="308" end_page="309" type="metho">
    <SectionTitle>
3. Formal Preliminaries
</SectionTitle>
    <Paragraph position="0"> Before proceeding with our formalization of OT, it will be useful to review some formal notation. Given a finite alphabet ~ we denote by ~ the set of all strings over G, including the empty string ~, and we denote by 2 E the power set of ~*.</Paragraph>
    <Paragraph position="1"> We assume that the reader is familiar with the notions of finite-state automaton, regular language, finite-state transducer, and rational relation; definitions and basic properties can be found in Gurari (1989). To recap briefly, a finite-state transducer is a finite-state automaton whose transitions are defined over the cross-product set (~ U {~}) x (&amp; U {~}), with ~ and &amp; two (finite) alphabets. If we interpret ~ as the alphabet of input to the machine and &amp; as the alphabet of output, each accepting 4 We note that there is nothing about the OT system that requires that candidates be evaluated in this serial manner. Instead, all of the coostraints could be seen as being imposed in parallel, with the relative importance among violations being determined after the evaluation. From the perspective of specifying the abstract computation that is determined by the OT model, nothing hinges on this serial versus parallel distinction, so far as we can see.  Computational Linguistics Volume 24, Number 2 computation of the transducer can be viewed as defining a mapping between a string in E* and a string in A*. Of course, the finite-state transducer may be nondeterministic, in which case a single input string may give rise to multiple outputs. Thus, every finite-state transducer can be associated with what is called a rational relation, a relation over E* x A* containing all possible input-output pairs. A rational relation R can also be regarded as a function \[R\] from E* to 2 a*, by taking \[R\](u) = {v \] (u,v) E R} for each u E E*. We will use this latter representation of rational relations throughout our subsequent discussion.</Paragraph>
  </Section>
  <Section position="5" start_page="309" end_page="312" type="metho">
    <SectionTitle>
4. A Model of OT
</SectionTitle>
    <Paragraph position="0"> We are now in a position to present our formal model of the OT system. Let us denote as N the set of nonnegative integers.</Paragraph>
    <Paragraph position="1"> Definition An optimality system (OS) is a triple G = (E, tEN, C), where E is a finite alphabet, GEN is a relation over E* x E&amp;quot; and C = (cl ..... Cp), p &gt; 1, is an ordered sequence of total functions from E* to N.</Paragraph>
    <Paragraph position="2"> The basic idea underlying this definition is as follows: If w is a well-formed UR, \[GEN\](W) is the nonempty set of all associated SR, otherwise \[C/nN\](W) = 0. Each function c in C represents some constraint of the grammar. For a given SR w, the non-negative integer c(w) is the &amp;quot;degree of violation&amp;quot; that w incurs with respect to the represented constraint. Given a set of candidates S, we are interested in the subset of S that violates c to the least degree, i.e., whose value under the function c is lowest. To facilitate reference to this subset, we define argminc{S } = {w\] w E S, c(w) = min{c(w') \] w' E S}}.</Paragraph>
    <Paragraph position="3"> We can now define the map an OS induces. We do this in stages, each one representing the evaluation of the candidates according to one of the constraints. For each w E E* and for 0 &lt; i &lt; p we define a function from E* to 2~*:</Paragraph>
    <Paragraph position="5"> if i&gt; 1 and argminci{OT~-l(w)} = OT~-l(w); if i &gt; 1 and argminc,{OT~-l(w)} # OT~-l(w).</Paragraph>
    <Paragraph position="6"> Function OTPc is called the optimality function associated with G, and is simply denoted as OTc. We drop the subscript when there is no ambiguity. The question of the expressive power of OT can now be stated precisely: what is the generative capacity of the class of optimality functions? The answer to this question depends, of course, upon the character of the functions that serve as GRN and the constraints. Though we will not make any substantive empirical claims about these functions, we will make a number of specific assumptions concerning their formal nature. Regarding GEN, we assume that the mapping from the UR to the candidate set is specifiable in terms of a finite-state transducer, that is to say, we will consider only OSs for which GEN is a rational relation (viewing rational relations as functions, as specified in the previous section). Since the question that we focus on in this research is that of determining whether the class of mappings specifiable in OT is beyond the formal power of finite-state transducers, allowing ann to be beyond the power of a  Frank and Satta Constraint Violability finite-state transducer would decide the question byfiat, s In addition, we assume that each constraint c in C is regular in that it satisfies the following requirement: For each k E N, the set {w I w E ~*, c(w) = k} (i.e., the inverse image of k under c) is a regular language. In other words, this requires that the set of candidates that violate a given constraint to any particular level must be regular. The choice of regular constraints is for reasons essentially identical to those that motivated the use of rational relations for GEN.</Paragraph>
    <Paragraph position="7"> It turns out that nearly all of the constraints that have been proposed in the OT phonological literature are regular in this sense. The reason for this is that OT constraints have tended to take the form of local conditions on the well-formedness of phonological representations, where local means bounded in size. Because of this restriction, we can characterize all possible violations of a given constraint c through a finite set of configurations Vc. More precisely, a phonological representations w attests as many violations of c as the number of occurrences of strings in Vc appearing as sub-strings of w. Since Vc is finite, it can be represented through some regular expression. Under the standard assumption that phonological representations are not structurally recursive, but rather are combined using essentially iterated concatenation, we can use well-known algebraic properties of regular languages (see for instance Kaplan and Kay 1994) to show that c is regular. (See Tesar 1995 for further discussion of a related notion of locality in constraints.) 5. OT as a Rational Relation This section presents the main result of this paper. We show that OSs of the sort outlined in the last section can be implemented through finite-state transducers so long as each constraint of the system satisfies one additional restriction: that it have a finite codomain, meaning that it distinguishes among only a finite set of equivalence classes of candidates. We start with some properties of the class of rational relations that will be needed later (proofs of these properties can be found for instance in Gurari 1989). Let R be a rational relation. The left projection of R is the language Left(R) = {u I (u, v) E R}. Symmetrically, the fight projection is the language Right(R) = {v I (u, v) E R}. It is well known that Left(R) and Right(R) are both regular languages. If R' is a rational relation, the composition of R and R', defined as R o R' = {(u,v) I (u,w) E R, (w,v) E R', for some w}, is still a rational relation.</Paragraph>
    <Paragraph position="8"> Let L be a regular language. We define the left restriction of R to L as the relation</Paragraph>
    <Paragraph position="10"> relations. The idea underlying a proof of this fact is to compose R (to the left or to the right) with the identity relation {(w, w) I w E L}, which is rational.</Paragraph>
    <Paragraph position="11"> Let G = (G, GEN, C) be an OS. We start the presentation of our result by restricting our attention to constraints having codomain of size two, that is, each ci in C is a total function from ~* to {0,1} such that both the set L(ci) -- {w I w E ~*, Ci(W) -~ 0} and its complement are regular. Recall that L(ci) denotes the language of all strings in G* that satisfy the constraint of the grammar represented by ci, and its complement, the strings 5 We recognize that this assumption, while plausible for phonological representations, is perhaps less so for syntactic representations. Further, as a reviewer points out, recent developments of OT in the domain of reduplication phenomena (McCarthy and Prince 1995), which assume that GEN produces a correspondence relation between the UR and SR, might constitute a phonological case in which tEN is not a rational relation. If well-formedness conditions on this correspondence relation are guaranteed only by the constraints, however, GEN could remain rational, though the constraints would no doubt cease to be expressible as regular languages.  Computational Linguistics Volume 24, Number 2 mapped to I by ci, includes all strings that violate it. Thus, such cis correspond to constraints that can distinguish only between complete satisfaction and violation. Using the above restriction, we can reformulate the definition of OT i reported in Section 4:</Paragraph>
    <Paragraph position="13"> Note that the case where all candidates in OT i-1 (w) satisfy constraint ci falls under the second clause of the definition in Section 4, but under the third clause of (1).</Paragraph>
    <Paragraph position="14"> However, this case is treated in the same way in both definitions, since OTi-l(w) =</Paragraph>
    <Paragraph position="16"> Let G = (G, GEN, C) be an OS such that GEN is a rational relation and each constraint in C is regular and has co-domain of size two. Then OTc is a rational relation.</Paragraph>
    <Paragraph position="17"> Let us start with the basic idea underlying the proof of this lemma. Assume that for i _&gt; 1 we have already been able to represent OT i-1 by means of a rational relation R.</Paragraph>
    <Paragraph position="18"> Consider some UR w and the set of associated candidate SRs that are optimal with respect to OT i-1, that is, the set OTi-l(w) = \[R\](w). To compute the strings in this set that are optimal with respect to ci, we must perform what amounts to a &amp;quot;conditional intersection&amp;quot; with the regular language L(ci), as determined by (1). That is, we check if there are candidates from \[R\] (w) that are also compatible with ci, i.e., that are members of L(ci). If there are some some, we eliminate any nonsatisfying candidates by intersecting \[R\](w) with L(ci) (third condition in \[1\]). However, if no such candidates remain, we do nothing to the set of candidates from OT i-1 (second condition in \[1\]).</Paragraph>
    <Paragraph position="19"> As shown in the proof below, it turns out that this can be done by partitioning the left projection of relation R into two regular languages. This results in the &amp;quot;splitting&amp;quot; of R into two relations, one of which must be &amp;quot;refined&amp;quot; by taking its right restriction to language L(ci). The union of the two resulting relations is then the desired representation of OT i. Putting these ideas together, we are now ready to present a formal proof.</Paragraph>
    <Paragraph position="20"> Proof We show that OT i is a rational relation for 0 &lt; i &lt; p. We proceed by induction on i.</Paragraph>
    <Paragraph position="21"> For i = 0, the claim directly follows from our assumptions about OEN. Let 1 &lt; i &lt; p.</Paragraph>
    <Paragraph position="22"> From the inductive hypothesis, there exists a rational relation R such that \[R\] = OT i-1.</Paragraph>
    <Paragraph position="23"> Since L(ci) is a regular language, from an already mentioned property it follows that:</Paragraph>
    <Paragraph position="25"> is a rational relation as well. Function \[al\] associates a UR to the set of SRs that are optimal up to constraint ci-1 and that also satisfy ci, the latter being the effect of the right restriction operator. Since R1 is rational, we have that L1 = Left(R1), the set of URs for which function JR1\] results in some non-empty set, is a regular langua__ ge. By a well-known closure property of regular languages, the com_plement of L1, L1 = G* - L1, is a regular language as well. Note that, for each UR in L1, no associated SR is both optimal up to constraint ci-1 and satisfies ci. It then follows, by an already mentioned property, that:</Paragraph>
    <Paragraph position="27"> Frank and Satta Constraint Violability is a rational relation. Note that function \[R2\] computes optimality up to constraint ci-1, but only over those URs whose optimal satisfiers do not satisfy ci. It is not difficult to see from an inspection of (1) that OT i = \[R1 U a2\]. Then the statement of the lemma follows from the fact that the class of rational relations is closed under finite union (see for instance Gurari 1989). \[\] The result in the above lemma can be extended to regular constraints having arbitrarily large finite codomain, corresponding to constraints that rank candidates along some finite-valued scale. This is done using a construction, first suggested in Ellison (1994), which, expressed intuitively, replaces any such constraint function by a finite number of constraint functions having codomain of size two. More formally, assume constraint c has codomain {0,1,...,k}, k &gt; 1. We introduce new constraints (c,i), 1 &lt; i &lt; k, defined as follows: For each 1 &lt; i &lt; k and w C G*, we let (c,i)(w) = 0 if c(w) &lt; i, (c, i)(w) -- 1 if c(w) &gt;_ i. Each (c, i) has codomain of size two. Since the class of regular languages is closed under finite union, if c is regular then each (c, i) is regular.</Paragraph>
    <Paragraph position="28"> We can finally state our main result, which directly follows from the above discussion and from Lemma 1.</Paragraph>
    <Paragraph position="29"> Theorem 1 Let G = (G, GEN, C) be an OS such that ORN is a rational relation and each constraint in C is regular and has a finite codomain. Then OTG is a rational relation.</Paragraph>
  </Section>
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