<?xml version="1.0" standalone="yes"?>
<Paper uid="P01-1011">
  <Title>Underspecified Beta Reduction</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2 Examples
</SectionTitle>
    <Paragraph position="0"> In this section, we show what underspecified a0 -reduction should do, and why the task is nontrivial. Consider first the ambiguous sentence Every student didn't pay attention. In first-order logic, the two readings can be represented as</Paragraph>
    <Paragraph position="2"> A classical compositional semantics construction first derives these two readings in the form of two HOL-formulas:</Paragraph>
    <Paragraph position="4"> An underspecified description of both readings is shown in Figure 2. For now, notice that the graph has all the symbols of the two HOL formulas as node labels, that variable binding is indicated by dashed arrows, and that there are dotted lines indicating an &amp;quot;outscopes&amp;quot; relation; we will fill in the details in Section 3.</Paragraph>
    <Paragraph position="5"> Now we want to reduce the description in Figure 2 as far as possible. The first a0 -reduction step, with the redex at a51a52a26 is straightforward. Even though the description is underspecified, the reducing part is a completely known a1 -term. The result is shown on the left-hand side of Figure 1.</Paragraph>
    <Paragraph position="6"> Here we have just one redex, starting at a25 a26 , which binds a single variable. The next reduction step is less obvious: The a21 operator could either belong to the context (the part between a22a64a23 and a25a37a26 )</Paragraph>
    <Paragraph position="8"> or to the argument (below a25a35a34 ). Still, it is not difficult to give a correct description of the result: it is shown in the middle of Fig. 1. For the final step, which takes us to the rightmost description, the redex starts at a41a44a42 . Note that now the a21 might be part of the body or part of the context of this redex. The end result is precisely a description of the two readings as first-order formulas.</Paragraph>
    <Paragraph position="9"> So far, the problem does not look too difficult.</Paragraph>
    <Paragraph position="10"> Twice, we did not know what exactly the parts of the redex were, but it was still easy to derive correct descriptions of the reducts. But this is not always the case. Consider Figure 3, an abstract but simple example. In the left description, there are two possible positions for the a21 : above a51 or below a25 . Proceeding na&amp;quot;ively as above, we arrive at the right-hand description in Fig. 3. But this description is also satisfied by the term a69 a54 a21 a54a74a70a16a54 a71 a56a11a56a11a56 , which cannot be obtained by reducing any of the terms described on the left-hand side. More generally, the na&amp;quot;ive &amp;quot;graph rewriting&amp;quot; approach is unsound; the resulting descriptions can have too many readings. Similar problems arise in (more complicated) examples from semantics, such as the coordination in Fig. 8.</Paragraph>
    <Paragraph position="11"> The underspecified a0 -reduction operation we propose here does not rewrite descriptions. Instead, we describe the result of the step using a &amp;quot;a0 -reduction constraint&amp;quot; that ensures that the reduced terms are captured correctly. Then we use a saturation calculus to make the description more explicit.</Paragraph>
  </Section>
class="xml-element"></Paper>