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<Paper uid="P96-1014">
  <Title>Computing Optimal Descriptions for Optimality Theory Grammars with Context-Free Position Structures</Title>
  <Section position="3" start_page="101" end_page="102" type="intro">
    <SectionTitle>
2 Context-Free Position Structure
Grammars
</SectionTitle>
    <Paragraph position="0"> Tesar (Tesar, 1995a) formalizes Gen as a set of matchings between an ordered string of input segments and the terminals of each of a set of position structures. The set of possible position structures is defined by a formal grammar, the position structure grammar. A position structure has as terminals structural positions. In a valid structural description, each structural position may be filled with at most one input segment, and each input segment may be parsed into at most one position. The linear order of the input must be preserved in all candidate structural descriptions.</Paragraph>
    <Paragraph position="1"> This paper considers Optimality Theory grammars where the position structure grammar is context-free; that is, the space of position structures can be described by a formal context-free grammar.</Paragraph>
    <Paragraph position="2"> As an illustration, consider the grammar in Examples 1 and 2 (this illustration is not intended to represent any plausible natural language theory, but does use the &amp;quot;peak/margin&amp;quot; terminology sometimes employed in syllable theories). The set of inputs is {C,V} +. The candidate descriptions of an input consist of a sequence of pieces, each of which has a peak (p) surrounded by one or more pairs of margin positions (m). These structures exhibit prototypical context-free behavior, in that margin positions to the left of a peak are balanced with margin positions to the right. 'e' is the empty string, and 'S' the start symbol.</Paragraph>
    <Paragraph position="3"> Example 1 The Position Structure Grammar</Paragraph>
    <Paragraph position="5"> -(m/V) Do not parse V into a margin position -(p/C) Do not parse C into a peak position PARSE Input segments must be parsed FILL m A margin position must be filled FILL p A peak position must be filled  The first two constraints are structurM, and mandate that V not be parsed into a margin position, and that C not be parsed into a peak position. The other three constraints are faithfulness constraints. The two structural constraints are satisfied by descriptions with each V in a peak position surrounded by matched C's in margin positions: CCVCC, V, CVCCCVCC, etc. If the input string permits such an analysis, it will be given this completely faithful description, with no resulting constraint violations (ensuring that it will be optimal with respect to any ranking).</Paragraph>
    <Paragraph position="6"> Consider the constraint hierarchy in Example 3. Example 3 A Constraint Hierarchy {-(m/V),-(p/C), PARSE} ~&gt; {FILL p} &gt; {FILL m} This ranking ensures that in optimal descriptions, a V will only be parsed as a peak, while a C will only be parsed as a margin. Further, all input segments will be parsed, and unfilled positions will be included only as necessary to produce a sequence of balanced structures. For example, the input /VC/ receives the description 1 shown in Example 4.</Paragraph>
    <Paragraph position="7"> Example 4 The Optimal Description for/VC/ S(F(Y(M(C),P(V),M(C)))) The surface string for this description is CVC: the first C was &amp;quot;epenthesized&amp;quot; to balance with the one following the peak V. This candidate is optimal because it only violates FILL m, the lowest-ranked constraint. null Tesar identifies locality as a sufficient condition on the universal constraints for the success of his l In this paper, tree structures will be denoted with parentheses: a parent node X with child nodes Y and Z is denoted X(Y,Z).</Paragraph>
    <Paragraph position="8">  approach. For formally regular position structure grammars, he defines a local constraint as one which can be evaluated strictly on the basis of two consecutive positions (and any input segments filling those positions) in the linear position structure. That idea can be extended to the context-free case as follows. A local constraint is one which can be evaluated strictly on the basis of the information contained within a local region. A local region of a description is either of the following: * a non4erminal and the child non-terminals that it immediately dominates; * a non-terminal which dominates a terminal symbol (position), along with the terminal and the input segment (if present) filling the terminal position.</Paragraph>
    <Paragraph position="9"> It is important to keep clear the role of the position structure grammar. It does not define the set of grammatical structures, it defines the Space of candidate structures. Thus, the computation of descriptions addressed in this paper should be distinguished from robust, or error-correcting, parsing (Anderson and Backhouse, 1981, for example). There, the input string is mapped to the grammatical structure that is 'closest'; if the input completely matches a structure generated by the grammar, that structure is automatically selected. In the OT case presented here, the full grammar is the entire OT system, of which the position structure grammar is only a part. Error-correcting parsing uses optimization only with respect to the faithfulness of pre-defined grammatical structures to the input. OT uses optimization to define grammaticality.</Paragraph>
  </Section>
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