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<Paper uid="P98-2197">
  <Title>Transforming Lattices into Non-deterministic Automata with Optional Null Arcs</Title>
  <Section position="2" start_page="0" end_page="0" type="abstr">
    <SectionTitle>
Abstract
</SectionTitle>
    <Paragraph position="0"> The problem of transforming a lattice into a non-deterministic finite state automaton is non-trivial. We present a transformation algorithm which tracks, for each node of an automaton under construction, the larcs which it reflects and the lattice nodes at their origins and extremities. An extension of the algorithm permits the inclusion of null, or epsilon, arcs in the output automaton. The algorithm has been successfully applied to lattices derived from dictionaries, i.e. very large corpora of strings.</Paragraph>
    <Paragraph position="1"> Introduction Linguistic data -- grammars, speech recognition results, etc. -- are sometimes represented as lattices, and sometimes as equivalent finite state automata. While the transformation of automata into lattices is straightforward, we know of no algorithm in the current literature for transforming a lattice into a non-deterministic finite state automaton. (See e.g. Hopcroft et al (1979), Aho et al (1982).) We describe such an algorithm here. Its main feature is the maintenance of complete records of the relationships between objects in the input lattice and their images on an automaton as these are added during transformation. An extension of the algorithm permits the inclusion of null, or epsilon, arcs in the output automaton.</Paragraph>
    <Paragraph position="2"> The method we present is somewhat complex, but we have thus far been unable to discover a simpler one. One suggestion illustrates the difficulties: this proposal was simply to slide lattice node labels leftward onto their incoming arcs, and then, starting with the final lattice node, to merge nodes with identical outgoing arc sets.</Paragraph>
    <Paragraph position="3"> This strategy does successfully transform many lattices, but fails on lattices like this one: Figure 1 For this lattice, the sliding strategy fails to produce either of the following acceptable solutions. To produce the epsilon arc of 2a or the bifurcation of Figure 2b, more elaborate measures seem to be needed.</Paragraph>
    <Paragraph position="4"> a.</Paragraph>
    <Paragraph position="5"> a b. ~ Figure 2 a We present our datastructures in Section 1; our basic algorithm in Section 2; and the modifications which enable inclusion of epsilon automaton arcs in Section 3. Before concluding, we provide an extended example of the algorithm in operation in Section 4. Complete pseudocode and source code (in Common Lisp) are available from the authors.</Paragraph>
  </Section>
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