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<Paper uid="P87-1015">
  <Title>CHARACTERIZING STRUCTURAL DESCRIPTIONS PRODUCED BY VARIOUS. GRAMMATICAL FORMALISMS*</Title>
  <Section position="7" start_page="109" end_page="110" type="concl">
    <SectionTitle>
5 Discussion
</SectionTitle>
    <Paragraph position="0"> We have studied the structural descriptions (trce sets) that can be assigned by various gr-mr-at;cal systems, and classified these formalisms on the basis of two fentures: path complexity; and path independence. We contrasted formalisms such as CFG's, HG's, TAG's and MCTAG's, with formalisms such as IG's and unificational systems such as LFG's and FUG's.</Paragraph>
    <Paragraph position="1"> We address the question of whether or not a formalism can generate only slructural descriptions with independent paths. This property reflects an important aspect of the underlying linguistic theory associated with the formalism. In a grammar which generates independent paths the derivations of sibling constituents can not share an unbounded amount of information.</Paragraph>
    <Paragraph position="2"> The importance of this property becomes clear in contrasting theories underlying GPSG (Gazdar, Klein, Pullum, and Sag, 1985), and GB (as described by Berwick, 1984) with those underlying LFG and FUG. It is interesting to note, however, that the ability to produce a bounded number of dependent paths (where two dependent paths can share an unbounded amount of information) does not require machinery as powerful as that used in LFG, FUG and IG's. As illustrated by MCTAG's, it is possible for a formalism to give tree sets with bounded dependent paths while still sharing the constrained rewriting properties of CFG's, HG's, and TAG's.</Paragraph>
    <Paragraph position="3"> In order to observe the similarity between these constrained systems, it is crucial to abstract away from the details of the strucUwes and operations used by the system. The similarities become apparent when they are studied at the level of derivation structures: derivation tree sets of CFG's, HG's, TAG's, and MCTAG's are all local sets. Independence of paths at this level reflects context freeness of rewriting and suggests why they can be recognized efficiently. As suggested in Section 4.3.2, a derivation with independent paths can be divided into subcomputatious with limited sharing of information.</Paragraph>
    <Paragraph position="4"> We outlined the definition of a family of constrained grammatical formalisms, called Linear Context-Free Rewriting Systems. This family represents an attempt to generalize the properties shared by CFG's, HG's, TAG's, and MCTAG's. Like HG's, TAG's, and MCTAG's, members of LCFRS can manipulate structures mere complex than terminal strings and use composition operations that are more complex that concatenation.</Paragraph>
    <Paragraph position="5"> We place certain restrictions on the composition operations of LCFRS's, restrictions that are shared by the composition operations of the constrained grammatical systems that we have considered. The operations must be linear and nonerasing, i.e., they can not duplicate or erase structure from their arguments. Notice that even though IG's and LFG's involve CFG-like productions,  they are (linguistically) fundamentally different from CFG's because the composition operations need not be linear. By sharing stacks (in IG's) or by using nonlinear equations over f-structares (in FUG's and LFG's), structures with unbounded dependencies between paths can be generat_~i_. LCFRS's share several properties possessed by the class of m//d/y context-sensitive formalisms discussed by Joshi (1983/85). The results described in this paper suggest a characterization of mild context-sensitivity in terms of generalized context-freeness.</Paragraph>
    <Paragraph position="6"> Having defined LCFRS's, in Section 4.2 we established the sem/1/nearity (and hence constant growth property) of the languages generated. In considering the recognition of these languages, we were forced to be more specific regarding the relationship between the structures derived by these formalisms and the substrings they span. We insisted that each slzucture dominates a bounded number of (not necessarily adjacent) substrings. The composition operations are mapped onto operations that use concatenation to define the substrings spanned by the resulting strucntres. We showed that any system defined in this way can be recocniTed in polynomial time. Members of LCFRS whose operations have this property can be translated into the ILFP notation (Rounds, 1985). However, in order to capture the properties of various grammatical systems under consideration, our notation is more restrictive that ILFP, which was designed as a general logical notation to characterize the complete class of languages that are recognizable in polynomial time. It is known that CFG's, HG's, and TAG's can be recognized in polynomial time since polynomial time algorithms exist in for each of these formalisms. A corollary of the result of Section 4.3 is that polynomial time recognition of MCTAG's is possible.</Paragraph>
    <Paragraph position="7"> As discussed in Section 3, independent paths in tree sets, rather than the path complexity, may be crucial in characterizing semilinearity and polynomial time recognition. We would like to relax somewhat the constraint on the path complexity of formalisms in LCFRS. Formalisms such as the restricted indexed grammars (Gazdar, 1985) and members of the hierarchy of grammatical systems given by Weir (1987) have independent paths, but more complex path sets. Since these path sets are semillnear, the property of independent paths in their tree sets is sufficient to cause semilinearity of the languages generated by them. In addition, the restricted version of CG's (discussed in Section 6) generates Use sets with independent paths and we hope that it can be included in a more general definition of LCFRS's containing formalisms whose tree sets have path sets that are themselves LCFRL's (as in the case of the restricted indexed grammars, and the hierarchy defined by Weir).</Paragraph>
    <Paragraph position="8"> LCFRS's have only been loosely defined in this paper; we have yet to provide a complete set of formal properties associated with members of this class. In thi s paper, our goal has been to use the notion of LCFRS's to classify grammatical systems on the basis of their strong generative capacity. In considering this aspect of a formalism, we hope to better understand the relationship between the structural descriptions generated by the grammars of a formalism, and the properties of semilinearity and polynomial recognizability.</Paragraph>
  </Section>
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