<?xml version="1.0" standalone="yes"?>
<Paper uid="P01-1054">
  <Title>Tractability and Structural Closures in Attribute Logic Type Signatures</Title>
  <Section position="3" start_page="0" end_page="2" type="intro">
    <SectionTitle>
2 Meet-semi-latticehood
</SectionTitle>
    <Paragraph position="0"> In LTFS and ALE, partial orders of types are assumed to be meet semi-lattices: Definition 1 A partial order, a4a6a5a8a7a10a9a12a11 , is a meet semi-lattice iff for any a13a14a7a16a15a18a17a19a5 , a13a21a20a22a15a24a23 . a20 is the binary greatest lower bound, or meet operation, and is the dual of the join operation, a25 , which corresponds to unification, or least upper bounds (in the orientation where a26 corresponds to the most general type). Figure 1 is not a meet semi-lattice because a27 and a28 do not have a meet, nor do a29 and a30 , for example.</Paragraph>
    <Paragraph position="1"> In the finite case, the assumption that every pair of types has a meet is equivalent to the assumption that every consistent set of types, i.e., types with an upper bound, has a join. It is theoretically convenient when discussing the unification of feature structures to assume that the unification of  a meet semi-lattice.</Paragraph>
    <Paragraph position="2"> two consistent types always exists. It can also be more efficient to make this assumption as, in some representations of types and feature structures, it avoids a source of non-determinism (selection among minimal but not least upper bounds) during search.</Paragraph>
    <Paragraph position="3"> Just because it would be convenient for unification to be well-defined, however, does not mean it would be convenient to think of any empirical domain's concepts as a meet semi-lattice, nor that it would be convenient to add all of the types necessary to a would-be type hierarchy to ensure meet-semi-latticehood. The question then naturally arises as to whether it would be possible, given any finite partial order, to add some extra elements (types, in this case) to make it a meet semi-lattice, and if so, how many extra elements it would take, which also provides a lower bound on the time complexity of the completion.</Paragraph>
    <Paragraph position="4"> It is, in fact, possible to embed any finite partial order into a smallest lattice that preserves existing meets and joins by adding extra elements. The resulting construction is the finite restriction of the Dedekind-MacNeille completion (Davey and Priestley, 1990, p. 41).</Paragraph>
    <Paragraph position="5"> Definition 2 Given a partially ordered set,</Paragraph>
    <Paragraph position="7"> This route has been considered before in the context of taxonomical knowledge representation (A&amp;quot;it-Ka'ci et al., 1989; Fall, 1996). While meet semi-lattice completions are a practical step towards providing a semantics for arbitrary partial orders, they are generally viewed as an impractical preliminary step to performing computations over a partial order. Work on more efficient encoding schemes began with A&amp;quot;it-Ka'ci et al. (1989), and this seminal paper has  MacNeille completion at a58a59a41a61a60 .</Paragraph>
    <Paragraph position="8"> in turn given rise to several interesting studies of incremental computations of the Dedekind-MacNeille completion in which LUBs are added as they are needed (Habib and Nourine, 1994; Bertet et al., 1997). This was also the choice made in the LKB parsing system for HPSG (Malouf et al., 2000).</Paragraph>
    <Paragraph position="9"> There are partial ordersa5 of unbounded size for which a49a62a31a22a32a2a34a35a5a37a36a40a49a37a41a64a63a50a34a35a65a67a66a68a42a66a51a36 . As one family of worst-case examples, parametrised by a58 , consider a set a69a70a41a71a43a73a72a24a7a40a74a10a74a40a74a75a7a16a58a76a56 , and a partial order a5 defined as all of the size a58a78a77a59a72 subsets of a69 and all of the size a72 subsets of a69 , ordered by inclusion. Figure 2 shows the case where a58a79a41a80a60 . Although the maximum subtype and supertype branching factors in this family increase linearly with size, the partial orders can grow in depth instead in order to contain this.</Paragraph>
    <Paragraph position="10"> That yields something roughly of the form shown in Figure 3, which is an example of a recent trend in using type-intensive encodings of linguistic information into typed feature logic in HPSG, beginning with Sag (1997). These explicitly isolate several dimensions1 of analysis as a means of classifying complex linguistic objects. In Figure 3, specific clausal types are selected from among the possible combinations of CLAUSAL-ITY and HEADEDNESS subtypes. In this setting, the parameter a58 corresponds roughly to the number of dimensions used, although an exponential explosion is obviously not dependent on reading the type hierarchy according to this convention. null There is a simple algorithm for performing this completion, which assumes the prior existence of a most general element (a26 ), given in Figure 4.</Paragraph>
    <Paragraph position="11"> 1It should be noted that while the common parlance for these sections of the type hierarchy is dimension, borrowed from earlier work by Erbach (1994) on multi-dimensional inheritance, these are not dimensions in the sense of Erbach (1994) because not every a81 -tuple of subtypes from an a81 -dimensional classification is join-compatible. Most instantiations of the heuristic, &amp;quot;where there is no meet, add one&amp;quot; (Fall, 1996), do not yield the Dedekind-MacNeille completion (Bertet et al., 1997), and other authors have proposed incremental methods that trade greater efficiency in computing the entire completion at once for their incrementality. null Proposition 1 The MSL completion algorithm is correct on finite partially ordered sets, a5 , i.e., upon termination, it has produced a31a33a32a82a34a83a5a37a36 . Proof: Let a84a59a34a35a5a37a36 be the partially ordered set produced by the algorithm. Clearly, a5a85a39a86a84a59a34a35a5a37a36 . It suffices to show that (1) a84a59a34a35a5a37a36 is a complete lattice (with a87 added), and (2) for all a88a89a17a44a84a22a34a35a5a37a36 , there  Suppose there are a88a100a7a102a101a103a17a61a84a59a34a35a5a37a36 such that a88a18a20 a101a105a104 . There is a least element, so a88 and a101 have more than one maximal lower bound, a106a83a107a38a7a16a106a109a108 and others. But then a43a110a106 a107 a7a16a106 a108 a56 is upper-bounded and a106a83a107a8a25a111a106a112a108a113a104 , so the algorithm should not have terminated. Suppose instead that a88a33a25a114a101a105a104 . Again, the algorithm should not have terminated. So a84a59a34a35a5a37a36 with a87 added is a complete lattice.</Paragraph>
    <Paragraph position="12"> Given a88a19a17a79a84a22a34a83a5a37a36 , if a88a19a17a79a5 , then choose a45a105a115a93a41 a91a105a115a116a41a117a43a110a88a118a56 . Otherwise, the algorithm added a88 because of a bounded set a43a40a119 a107 a7a120a119 a108 a56 , with minimal upper bounds, a121a122a107a123a7a10a74a40a74a40a74a124a121a100a125 , which did not have a least upper bound, i.e., a126a128a127 a72 . In this case, choose</Paragraph>
    <Paragraph position="14"> Termination is guaranteed by considering, after every iteration, the number of sets of meetirreducible elements with no meet, since all completion types added are meet-reducible by definition. null In LinGO (Flickinger et al., 1999), the largest publicly-available LTFS-based grammar, and one which uses such type-intensive encodings, there are 3414 types, the largest supertype branching factor is 19, and although dimensionality is not distinguished in the source code from other types, the largest subtype branching factor is 103. Using supertype branching factor for the most conserva- null density, respectively, of a144 in a145a8a146a112a144a42a147 (Davey and Priestley, 1990, p. 42).</Paragraph>
    <Paragraph position="15"> fin-wh-fill-rel-cl inf-wh-fill-recl-cl red-rel-cl simp-inf-rel-cl wh-subj-rel-cl bare-rel-cl fin-hd-fill-ph inf-hd-fill-ph fin-hd-subj-ph wh-rel-cl non-wh-rel-cl hd-fill-ph hd-comp-ph hd-subj-ph hd-spr-ph imp-cl decl-cl inter-cl rel-cl hd-adj-ph hd-nexus-ph clause non-clause hd-ph non-hd-ph</Paragraph>
  </Section>
class="xml-element"></Paper>