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<Paper uid="P99-1013">
  <Title>Compositional Semantics for Linguistic Formalisms</Title>
  <Section position="7" start_page="101" end_page="102" type="concl">
    <SectionTitle>
5 Conclusions
</SectionTitle>
    <Paragraph position="0"> This paper discusses alternative definitions for the semantics of unification-based linguistic formalisms, culminating in one that is both compositional and fully-abstract (with respect to grammar union, a simple syntactic combination operations on grammars). This is mostly an adaptation of well-known results from h)gic programming to the ti'amework of unification-based linguistic tbrmalisms, and it is encouraging to see that the same choice of semantics which is compositional and fiflly-abstra(:t for Prolog turned out to have the same desirable properties in our domain.</Paragraph>
    <Paragraph position="1"> The functional semantics '~.\].f,' defined here assigns to a grammar a fimction which reflects the (possibly infinite) successive application of grammar rules, viewing the lexicon as input to the parsing process. We, believe that this is a key to modularity in grammar design. A grammar module has to define a set of items that it &amp;quot;exports&amp;quot;, and a set of items that can be &amp;quot;imported&amp;quot;, in a similar way to the declaration of interfaces in programming languages. We are currently working out the details of such a definition. An immediate application will facilitate the implementation of grammar development systems that support modularity in a clear, mathematically sound way.</Paragraph>
    <Paragraph position="2"> The results reported here can be extended in various directions. First, we are only concerned in this work with one composition operator, grammar union. But alternative operators are possible, too. In particular, it would be interesting to define an operator which combines the information encoded in two grammar rules, for example by unifying the rules. Such an operator would facilitate a separate development of grammars along a different axis: one module can define the syntactic component of a grammar while another module would account for the semantics. The composition operator will unify each rule of one module with an associated rule in the other. It remains to be seen whether the grammar semantics we define here is compositional and fully abstract with respect to such an operator.</Paragraph>
    <Paragraph position="3"> A different extension of these results should provide for a distribution of the type hierarchy among several grammar modules. While we assume in this work that all grammars are defined  over a given signature, it is more realistic to assume separate, interacting signatures. We hope to be able to explore these directions in the future. null</Paragraph>
  </Section>
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