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<Paper uid="P01-1054">
  <Title>Tractability and Structural Closures in Attribute Logic Type Signatures</Title>
  <Section position="4" start_page="2" end_page="2" type="metho">
    <SectionTitle>
CLAUSALITY HEADEDNESS
</SectionTitle>
    <Paragraph position="0"> &amp;quot;dimensions&amp;quot; of classification.</Paragraph>
    <Paragraph position="1"> mum of approximately 500,000 completion types, whereas only 893 are necessary, 648 of which are inferred without reference to previously added completion types.</Paragraph>
    <Paragraph position="2"> Whereas incremental compilation methods rely on the assumption that the joins of most pairs of types will never be computed in a corpus before the signature changes, this method's efficiency relies on the assumption that most pairs of types are join-incompatible no matter how the signature changes. In LinGO, this is indeed the case: of the 11,655,396 possible pairs, 11,624,866 are join-incompatible, and there are only 3,306 that are consistent (with or without joins) and do not stand in a subtyping or identity relationship. In fact, the cost of completion is often dominated by the cost of transitive closure, which, using a sparse matrix representation, can be completed for LinGO in about 9 seconds on a 450 MHz Pentium II with 1GB memory (Penn, 2000a).</Paragraph>
    <Paragraph position="3"> While the continued efficiency of compile-time completion of signatures as they further increase in size can only be verified empirically, what can be said at this stage is that the only reason that signatures like LinGO can be tractably compiled at all is sparseness of consistent types. In other geometric respects, it bears a close enough resemblance to the theoretical worst case to cause concern about scalability. Compilation, if efficient, is to be preferred from the standpoint of static error detection, which incremental methods may elect to skip. In addition, running a new signature plus grammar over a test corpus is a frequent task in large-scale grammar development, and incremental methods, even ones that memoise previous computations, may pay back the savings in compile-time on a large test corpus. It should also be noted that another plausible method is compilation into logical terms or bit vectors, in which some amount of compilation (ranging from linear-time to exponential) is performed with the remaining cost amortised evenly across all run-time unifications, which often results in a savings during grammar development.</Paragraph>
  </Section>
  <Section position="5" start_page="2" end_page="2" type="metho">
    <SectionTitle>
3 Unique Feature Introduction
</SectionTitle>
    <Paragraph position="0"> LTFS and ALE also assume that appropriateness guarantees the existence of a unique introducer for every feature: Definition 3 Given a type hierarchy, a4a35a148a78a7a40a9a149a11 , and a finite set of features, Feat, an appropriateness specification is a partial function, a150a8a151a24a151a153a152a155a154a123a151 a156  appropriate for certain empirical domains either, although Pollard and Sag (1994) do otherwise observe it. The debate, however, has focussed on whether to modify some other aspect of type infer- null encing in order to compensate for the lack of feature introduction, presumably under the assumption that feature introduction was difficult or impossible to restore automatically to grammar signatures that did not have it.</Paragraph>
    <Paragraph position="1">  Just as with the condition of meet-semilatticehood, however, it is possible to take a would-be signature without feature introduction and restore this condition through the addition of extra unique introducing types for certain appropriate features. The algorithm in Figure 5 achieves this. In practice, the same signature completion type, a88 , can be used for different features, provided that their minimal introducers are the same set, a188 . This clearly produces a partially ordered set with a unique introducing type for every feature. It may disturb meetsemi-latticehood, however, which means that this completion must precede the meet semi-lattice completion of Section 2. If generalisation has already been computed, the signature completion algorithm runs in a189a142a34a83a190a166a58a153a36 , where a190 is the number of features, and a58 is the number of types.</Paragraph>
  </Section>
  <Section position="6" start_page="2" end_page="2" type="metho">
    <SectionTitle>
4 Subtype Covering
</SectionTitle>
    <Paragraph position="0"> In HPSG, it is generally assumed that non-maximally-specific types are simply a convenient shorthand for talking about sets of maximally specific types, sometimes called species, over which the principles of a grammar are stated. In a view where feature structures represent discretely ordered objects in an empirical model, every feature structure must bear one of these species.</Paragraph>
    <Paragraph position="1"> In particular, each non-maximally-specific type in a description is equivalent to the disjunction of the maximally specific subtypes that it subsumes.</Paragraph>
    <Paragraph position="2"> There are some good reasons not to build this assumption, called &amp;quot;subtype covering,&amp;quot; into LTFS or its implementations. Firstly, it is not an appropriate assumption to make for some empirical domains. Even in HPSG, the denotations of 1. Given candidate signature, a191 , find a feature, F, for which there is no unique introducing type. Let a192 be the set of minimal types to which F is appropriate, where a193a192a50a193 a180a59a181 . If there is no such feature, then stop.  parametrically-typed lists are more naturally interpreted without it. Secondly, not to make the assumption is more general: where it is appropriate, extra type-antecedent constraints can be added to the grammar signature of the form:</Paragraph>
    <Paragraph position="4"> straints become crucial in certain cases where the possible permutations of appropriate feature values at a type are not covered by the permutations of those features on its maximally specific subtypes. This is the case for the type, verb, in the signature in Figure 6 (given in ALE syntax, where sub/2 defines the partial order of types, and intro/2 defines appropriateness on unique introducers of features). The combination, AUXa156a209a77a141a210 INVa156a212a211 , is not attested by any of verb's subtypes. While there are arguably better ways to represent this information, the extra type-antecedent constraint: null verba1 aux verba204 main verb is necessary in order to decide satisfiability correctly under the assumption of subtype covering.</Paragraph>
    <Paragraph position="5"> We will call types such as verb deranged types.</Paragraph>
    <Paragraph position="6"> Types that are not deranged are called normal types.</Paragraph>
    <Paragraph position="7"> bot sub [verb,bool].</Paragraph>
    <Paragraph position="8"> bool sub [+,-].</Paragraph>
    <Paragraph position="9"> verb sub [aux_verb,main_verb] intro [aux:bool,inv:bool].</Paragraph>
    <Paragraph position="10"> aux_verb sub [aux:+,inv:bool].</Paragraph>
    <Paragraph position="11"> main_verb sub [aux:-,inv:-].</Paragraph>
    <Section position="1" start_page="2" end_page="2" type="sub_section">
      <SectionTitle>
4.1 Non-Disjunctive Type Inference under
Subtype Covering is NP-Complete
</SectionTitle>
      <Paragraph position="0"> Third, although subtype covering is, in the author's experience, not a source of inefficiency in practical LTFS grammars, when subtype covering is implicitly assumed, determining whether a non-disjunctive description is satisfiable under appropriateness conditions is an NP-complete problem, whereas this is known to be polynomial time without it (and without type-antecedent constraints, of course). This was originally proven by Carpenter and King (1995). The proof, with corrections, is summarised here because it was never published. Consider the translation of a 3SAT formula into a description relative to the signature given in Figure 7. The resulting description is always non-disjunctive, since logical disjunction is encoded in subtyping. Asking whether a formula is satisfiable then reduces to asking whether this description conjoined with trueform is satisfiable. Every type is normal except fortruedisj, for which the combination, DISJ1a156falseforma210 DISJ2a156falseform, is not attested in either of its subtypes. Enforcing subtype covering on this one deranged type is the sole source of intractability for this problem.</Paragraph>
    </Section>
    <Section position="2" start_page="2" end_page="2" type="sub_section">
      <SectionTitle>
4.2 Practical Enforcement of Subtype
Covering
</SectionTitle>
      <Paragraph position="0"> Instead of enforcing subtype covering along with type inferencing, an alternative is to suspend constraints on feature structures that encode subtype covering restrictions, and conduct type inferencing in their absence. This restores tractability at the cost of rendering type inferencing sound but not complete. This can be implemented very transparently in systems like ALE that are built on top of another logic programming language with support for constraint logic programming such as SICStus Prolog. In the worst case, an answer to a query to the grammar signature may contain varibot sub [bool,formula].</Paragraph>
      <Paragraph position="1"> bool sub [true,false].</Paragraph>
      <Paragraph position="2"> formula sub [propsymbol,conj,disj,neg, trueform,falseform].</Paragraph>
      <Paragraph position="3"> propsymbol sub [truepropsym, falsepropsym].</Paragraph>
      <Paragraph position="4">  disjunctive type inferencing.</Paragraph>
      <Paragraph position="5"> ables with constraints attached to them that must be exhaustively searched over in order to determine their satisfiability, and this is still intractable in the worst case. The advantage of suspending subtype covering constraints is that other principles of grammar and proof procedures such as SLD resolution, parsing or generation can add deterministic information that may result in an early failure or a deterministic set of constraints that can then be applied immediately and efficiently. The variables that correspond to feature structures of a deranged type are precisely those that require these suspended constraints.</Paragraph>
      <Paragraph position="6"> Given a diagnosis of which types in a signature are deranged (discussed in the next section), suspended subtype covering constraints can be implemented for the SICStus Prolog implementation of ALE by adding relational attachments to ALE's type-antecedent universal constraints that will suspend a goal on candidate feature structures with deranged types such as verb or truedisj. The suspended goal unblocks whenever the deranged type or the type of one of its appropriate features' values is updated to a more specific subtype, and checks the types of the appropriate features' values. Of particular use is the SICStus Constraint Handling Rules (CHR, Fr&amp;quot;uhwirth and Abdennadher (1997)) library, which has the ability not only to suspend, but to suspend until a particular variable is instantiated or even bound to another variable. This is the powerful kind of mechanism required to check these constraints efficiently, i.e., only when necessary. Re-entrancies in a Prolog term encoding of feature structures, such as the one ALE uses (Penn, 1999), may only show up as the binding of two uninstantiated variables, and re-entrancies are often an important case where these constraints need to be checked. The details of this reduction to constraint handling rules are given in Penn (2000b). The relevant complexity-theoretic issue is the detection of deranged types.</Paragraph>
    </Section>
    <Section position="3" start_page="2" end_page="2" type="sub_section">
      <SectionTitle>
4.3 Detecting Deranged Types
</SectionTitle>
      <Paragraph position="0"> The detection of deranged types themselves is also a potential problem. This is something that needs to be detected at compile-time when sub-type covering constraints are generated, and as small changes in a partial order of types can have drastic effects on other parts of the signature because of appropriateness, incremental compilation of the grammar signature itself can be extremely difficult. This means that the detection of deranged types must be something that can be performed very quickly, as it will normally be performed repeatedly during development.</Paragraph>
      <Paragraph position="1"> A naive algorithm would be, for every type, to expand the product of its features' appropriate value types into the set, a45 , of all possible maximally specific products, then to do the same for the products on each of the type's a208 maximally specific subtypes, forming sets a91a105a138 , and then to remove the products in the a91a105a138 from a45 . The type is deranged iff any maximally specific products remain in a45a78a213a100a34a35a132a140a138a35a91a12a138a6a36 . If the maximum number of features appropriate to any type is a29 , and there are a119 types in the signature, then the cost of this is dominated by the cost of expanding the products, a119a155a214 , since in the worst case all features could have a26 as their appropriate value.</Paragraph>
      <Paragraph position="2"> A less naive algorithm would treat normal (nonderanged) subtypes as if they were maximally specific when doing the expansion. This works because the products of appropriate feature values of normal types are, by definition, covered by those of their own maximally specific subtypes. Maximally specific types, furthermore, are always normal and do not need to be checked. Atomic types (types with no appropriate features) are also trivially normal.</Paragraph>
      <Paragraph position="3"> It is also possible to avoid doing a great deal of the remaining expansion, simply by counting the number of maximally specific products of types rather than by enumerating them. For example, in Figure 6, main verb has one such product, AUXa156a215a77a92a210 INVa156a215a77 , and aux verb has two, AUXa156a212a211a92a210 INVa156a212a211 , and AUXa156a212a211a216a210 INVa156a135a77 . verb, on the other hand, has all four possible combinations, so it is deranged. The resulting algorithm is thus given in Figure 8. Using the smallest normal For each type, a171 , in topological order (from maximally specific down to a217 ): a184 if t is maximal or atomic then a171 is normal. Tabulate normalsa146a112a171a112a147a54a199a59a218a16a171a109a219 , a minimal normal subtype cover of  subtype cover that we have for the product of a119 's feature values, we iteratively expand the feature value products for this cover until they partition their maximal feature products, and then count the maximal products using multiplication. A similar trick can be used to calculate maximal efficiently.</Paragraph>
      <Paragraph position="4"> The complexity of this approach, in practice, is much better: a2a22a34a109a119a4a3a6a5 a214 a36 , where a3 is the weighted mean subtype branching factor of a subtype of a value restriction of a non-maximal non-atomic type's feature, and a28 is the weighted mean length of the longest path from a maximal type to a sub-type of a value restriction of a non-maximal non-atomic type's feature. In the Dedekind-MacNeille completion of LinGO's signature, a3 is 1.9, a28 is 2.2, and the sum of a3 a5 a214 over all non-maximal types with arity a29 is approximately a72a8a7a10a9 . The sum of maximala214 a34a109a119a155a36 over every non-maximal type, a119 , on the other hand, is approximately a72a8a7 a107a12a11 . Practical performance is again much better because this algorithm can exploit the empirical observation that most types in a realistic signature are normal and that most feature value restrictions on subtypes do not vary widely. Using branching factor to move the total number of types to a lower degree term is crucial for large signatures.</Paragraph>
    </Section>
  </Section>
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