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<Paper uid="P83-1004">
  <Title>Formal Constraints on Metarules*</Title>
  <Section position="3" start_page="22" end_page="24" type="metho">
    <SectionTitle>
3. For:hal Constraints
</SectionTitle>
    <Paragraph position="0"> ~. a, e it appears unlikely that a reinterpretation of MPS grammars can be found that solves their complexity problem, formal constraints on the MPS formalism itself have to be explored if we want to salvage the basic concept of metarules. In the following examination of currently proposed constraints, the two criteria for evaluation are their effects on computational tractability and on the ezplanatory adcquaeltof the formalism.</Paragraph>
    <Paragraph position="1"> As an example of constraints that satisfy the criterion of computational tractability but not that of explanatory adequacy, we examine the issue of essential variables. These are variables in the metarule pattern that can match an arbitrary string of items in a phrase structure rule. Uszkoreit and Peters have shown that, contrary to an initial conjecture by Jcehi (see \[Gazdar, 1982, fn. 28\]), allowing even one such variable per metarule extends the power of the formalism to recursive enumerability. Gazdar has recommended \[1982, p.160\] that the power of metarules be controlled by eliminating essential variables, exchanging them for abbreviatory variables that can stand only for strings in a finite and cztrinsieally determined range. This constraint yields a computationslly tractable system with only context-free power.</Paragraph>
    <Paragraph position="2"> Exchanging essential for abbreviatory variables is not, however, as attractive a prospect as it appears at first blush.</Paragraph>
    <Paragraph position="3"> Uszkoreit and Peters \[1982\[ show that by restricting MFS grammars to using abbreviatory variables only, some significant generalizations are lost. Consider the following metarule that is proposed and motivated in \[Gazdar 1982\] for endowing VSO languages with the category VP. The metarule generates fiat VSO sentence rules from VP rules.</Paragraph>
    <Paragraph position="4"> (2) VP-.V U~ S-.V NPU Since U is an abbreviatory variable, its range needs to be stated explicitly. Let us imagine 'h:,t the VSO language in question has the follo~ ;~ small set of VF rules:</Paragraph>
    <Paragraph position="6"> Therefore, the range of U has to be {e, NP, ~, \]77~, NP V'P}.</Paragraph>
    <Paragraph position="7"> 3As statements about the object ~'~mmar, however, metxrules might play s role in language acquisition or in dia~hronie processes.</Paragraph>
    <Paragraph position="8">  If these VP rules are the only rules that satisfy the left-hand side of (2), then (2) generates exactly the same rules am it would if we declared U to be an essential variable--i.e., let its range be (Vr O VN) deg. But now imagine that the language adopts a new subcategorizatiun frame for verbs, 4 e.g., a verb that takes  an NP and an S am complements. VP rule (4) is added: (4) VP -- I/&amp;quot; NP -S  Metarule (2) predicts that VPs headed by this verb do not have a corresponding fiat V$O sentence rule. We will have to change the metarule by extending the range of U in order to retain the generalization originally intended by the metarule. Obviously, our metarule did not encode the right generalization (a simple intension-extensiun problem).</Paragraph>
    <Paragraph position="9"> This shortcoming nun also surface in cases where the input to a metarule is the output of another metaruh. It might be that metarule (2) not only applies to basic verb rules but also includes the output of, say, a passive rule. The range of the variable \[.r would have to be extended to cover these tames too, and, moreover, might have to be altered if its feeding metarules change.</Paragraph>
    <Paragraph position="10"> Thus, if the restriction to abbreviatury variables is to have no effect on the weak-gensrative capacity of a grammar, the range assigned to each variable must include the range that would have actually instantiated the variable on an expansion of the MPS grammar in which the variable was treated as essential. The assignment of a range to the variable can only be done po,t /actum. This would be a satisfactory result, were it not for the fact that finding the necessary range of a variable in this way is an undecidable problem in general. Thus, to exchange essential for abbreviatory variables is to risk affecting the generative capacity of the grammar~with quite unintultive and unpredictable results. In short, the choice is among three options: to affect the language of the grammar in ways that are linguistically unmoti~at4ed and arbitrary, to solve an undecidable problem, or to discard the notion of exchanging essential for abbreviatory variables--in effect, a Hobsun's choice.</Paragraph>
    <Paragraph position="11"> An example of a constraint that satisfies the second criterion, that of explanatory adequacy, hut not the first, computational tractability, is the leziesl-head constraint of GPSG \[Gazdar and Pullum, 1982\[. This constraint allows metarules to operate only on rules whose stipulated head is a lexical (preterminal) category. Since the Uszkoreit and Peters results are achieved even under this restriction to the formalism, the cow straint does not provide a solution to the problem of expressive power. Of course, this is no criticism of the proposal, since it was never intended as a formal restriction on the class of languages, but rather ~ a restriction un linguistically motivated grammars. Unfortunal,ely, the motivation behind even this use of the lexical-head constraint may be lacking. One of the few analyses that relies on the lexical-head constraint is a recent GPSG analysis of coordination and extraction in English (Gazdar, 1981\]. In this ease--indeed, in general-one could achieve the desired effect simply by specifying that the coefficient of the bar feature be lezical. It remains to be seen whether the constraint must be imposed for enough metarules so as to justify its incorporation as a general principle.</Paragraph>
    <Paragraph position="12"> Even with such motivation one might raise a question about the advisability of the lexical-head constraint on a metatheoretical level. The linguistic intuition behind the constraint is that the role of metarules is to &amp;quot;express generalizations about possibilities of subeategorizatiun&amp;quot; exclusively \[Gaadar, Klein, Pullum, and Sag, 1982, p.391, e.g., to express the p~mive-active relation. This result is said to follow from principles of ~ syntax \[Jackendoff, 1077\], in which just those categories that are sub-categorized for are siblings of a lexieal head. However, in a language with freer word order than English, categories other than those subcategorized for will be siblings of lexieal heads; they would, thus, be affected by metarules even under the lexical-head constraint. This result will certainly follow from the liberation rule approach to free word order \[Pullum, 1982\]. The original linguistic generalization intended by the hxical-head constraint, therefore, will not hold cross-linguistically.</Paragraph>
    <Paragraph position="13"> Finally, there is the current proposal of the GPSG community for constraining the formal powers of metarules by allowing each metaruh to apply only once in a derivation of a rule. Originally dubbed the once.through hgpothe~is, this constraint is now incorporated&amp;quot; into GPSG under the name finite closure \[Gazdar and Pullum, 1982\]. Although linguistic evidence for the constraint has never been provided, the formal motivation is quite strong because, under this constraint, the metarule formalism would have only context-free power.</Paragraph>
    <Paragraph position="14"> Several linguistic constructions present problems with respect to the adequacy of the finite-closure hypothesis. For instance, the liberation rule technique for handling free-word-order languages {Pullum, 1982\] would require ffi noun-phrase liberation rule to be applied twice in a derivation of a rule with sibling noun phrases that permute their subconstituents freely among one another. As a hypothetical example of this phenomenon, let us suppose that English allowed relative clauses to be extraposed in general from noun phrases, instead of allowing just one extraposifion. For instance, in this quasi-English, the sentence (5) Two children are chasing the dog who are small that is here.</Paragraph>
    <Paragraph position="15"> would he a grammatical paraphrase of (0) Two children who are small axe chasing the dog that is here.</Paragraph>
    <Paragraph position="16"> Let us suppose further that the analysis of this phenomenon involved liberation of the NP-S substructure of the noun phrases for incorporation into the main sentence. Then the noun-phrase liberation rule would apply once to liberate the subject noun phrase, once again to liberate the object noun phrase. That these are not idle concerns is demonstrated by the following sentence in the free-word-order Australian aboriginal language Warlpiri. s 4Note that it does not matter whether the grammar writer discovers an additional subcateKorization, or the language develops one diachronically; the same problem obtains. 5This example is t,.ken from \[van Riemsdijk, 1981\].</Paragraph>
    <Paragraph position="18"> Two 8mall children are cha,ing that dog.</Paragraph>
    <Paragraph position="19"> The Warlpiri example is analogous to the quasi-English example in that both sentences have two discontinuous NPs in the same distribution. Furthermore, the liberation rule approach has been proposed as a method of modeling the free word order of Waripiri. Thus, it appears that finite closure is not consistent with the liberation rule approach to free word order.</Paragraph>
    <Paragraph position="20"> Adverb distribution presents another problem for the hypothesis. In German, for example, and to a lesser extent in Engiish, an unbounded number of adverbs can be quite freely interspersed with the complements of a verb. The following German sentence is an extreme example of this phenomenon \[Uszkoreit, 1982\]. The sequence of its major constituents is given under (9).</Paragraph>
    <Paragraph position="21"> (8) Gestern hatte in dec Mittagspause yesterday had during lunch break der Brigadier in dec Werkzeugkammer the foreman (NOM) in the tool shop dam Labeling au~ Boehaftigkeit lancaam the apprentice (DAT) maliciously slowly zehn schmierige Gasseisenscbeiben unbemerkt ten greasy cast iron disks (ACC) unnoticed in die Hosentasche gesteckt in the pocket put )'*aerdav, durin~ lunch break in the tool shop, the foreman, malicioedy and unnoticed, put ten grea,y caJt iron disks tlowist into the apprentice's pocket.</Paragraph>
    <Paragraph position="22"> (9) ADVP VrrN ADVP NPsuuJ ADVP NProaJ ADVP ADVP NPDoa.t ADVP PP VIN e A metarule might therefore be proposed that inserts a single adverb in a verb-phrase rule. Repeated application of this rule (in contradiction to the finite-closure hypothesis) would achieve the desired effect. To maintain the finite-closure hypothesis, we could merely extend the notion of context-free rule to allow regular expressions on the right-hand side of a rule. The verb phrase rule would then be accurately, albeit clumsily, expressed as, say, VP -.* V NP ADVP* or VP -* V NP ADVP* PP ADVP* for ditransitives.</Paragraph>
    <Paragraph position="23"> Similar constructions in free-word-order languages do not permit such naive solutions. As an example, let us consider the Japanese causative. In this construction, the verb sutRx &amp;quot;-sase&amp;quot; signals the causativization of the verb, allowing an extra NP argument. The process is putatively unbounded (ignoring performance limitations). Furthermore, Japanese allows the NPs to order freely relative to one another (subject to considerations of ambiguity and focus), so that a fiat structure with some kind of extrinsic ordering is presumably preferable.</Paragraph>
    <Paragraph position="24"> One means of achieving a fiat structure with extrinsic ordering is by using the ID/LP formalism, a subformalism of GPSG that allows immediate dominance (ID) information to be specified separately from linear precedence (LP) notions. (Cf. context-free phrase structure grammar, which forces a strict one-to-one correlation between the two types of information.) ID information is specified by context-free style rules with unordered right-hand sides, notated, e.g., .4 ~ B, C, D. LP informa,Aon is specified as a partial order over the nonterminals in the ..orr-,m max, notated, e.g., B &lt; C (read B precedes C). These two rules can be viewed as schematizing a set of three context-free rules, namely, A -- B C D, A -- B D C, and A -- D B C.</Paragraph>
    <Paragraph position="25"> Without a causativization metarule that can operate more than once, we might attempt to use the regular expression notation that solved the adverb problem. For example, we might postulate the ID rule VP -, NP*, V, sane* with the LP relation NP &lt; V &lt; sase, but no matching of NPs with sases is achieved. We might attempt to write a liberation rule that pulls NP.saee pairs from a nested structure into a flat one, but this would violate the finite-closure hypothesis (as well as Pullum's requirement precluding liberation through a recursive category). We could attempt to use even more of the power of regular-expression rules with ID/LP, i.e., VP -, {NP, 8a,e} deg, V under the same LP relation. The formalism presupposed by this analysis, however, has greater than context-free power, deg so that this solution may not be desirable. Nevertheless, it should not be ruled out before the parsing properties of such a formalism are understood. T Gunji's analysis of Japanese, which attempts to solve such problems with the multiple application of a tlash introduction metarule \[Gunji, 1980 l, again raises the problem of violating the 6nite-closure hypothesis (as well as being incompatible with the current version of GPSG which disallows multiple slashes). Finally, we could always move ca~ativization into the lexicon as a lexical rule. Such a move, though it does circumvent the difficulty in the syntax, merely serves to move it elsewhere without resolving the basic problem.</Paragraph>
    <Paragraph position="26"> Yet another alternative involves treating the right-hand ~ides of phrase structure rules as sets, rather than multisets as is implicit in the ID/LP format. Since the nonterminal vocabulary is finite, right-hand sides of ID rules must be subsets of a finite set and therefore finite sets themselves. This hypothesis is quite similar in effect to the finite-closure hypothesis, albeit even more limited, and thus inherits the same problems aa were discussed above.</Paragraph>
  </Section>
  <Section position="4" start_page="24" end_page="25" type="metho">
    <SectionTitle>
4. The Ultimate Solution
</SectionTitle>
    <Paragraph position="0"> An obvious way to constrain MPS grammar, is to eliminate metarules entirely and replace them with other mechanisms. In fact, within the GPSG paradigm, several of the functions of metarules have been replaced by other metagrammatical devices.</Paragraph>
    <Paragraph position="1"> Other functions have not, as of the writing of this paper, though  directly that includes a method for utilizing the Kleene star device. It could be extended to even more of the regular expression notation, though the effect of such extenslon-on the time complexity of the algorithm is an open question.</Paragraph>
    <Paragraph position="2">  it i$ instructive ~.o co=ider ~.he c~es covered ~y this cia~s. In the discussion to follow we have isolated thxee of the primary functions of metarules. This is not intended az an exhaustive taxonomy, and certain metarules may manifest more than one of these functions.</Paragraph>
    <Paragraph position="3"> First, we consider generalizations over linear order. If metarules are metagrammatical statements about rules encoding linear order, they may relate rules that differ only in the linear order of categories. With the introduction of ID/LP format, however, the hypothesis i, that this latter metagrammatical device will suffice to account for the linear order among the categories within rules. For instance, the problematic adverb and causative metarnles could be replaced by extended contex.t-free rules with \[D/LP, as was suggested in Section 3 above. Shieber \[forthcoming\[ has shown that a pure ID/LP formalism (without metarules, Kleene star, or the like) is no le~ computationally tractable than context-free grammars themselves. Although we do not yet know what the consequences of incorporating the extended context-free rules would be for computational complexity, ID/LP format can be used to replace certain word-ordervariation metarules.</Paragraph>
    <Paragraph position="4"> A second function of metarnles wa~ to relate sets of rules that differed only in the values of certain specifed features. It has been suggested \[Gat~iar and Pullum 1982\] that such features are distributed according to certain general principles. For instance, the slash-propagation metarule haz been replaced by the distribution of slash features in accord with such a principle. A third function of metarules under the original interpretation has not been relegated to other metagr~nmatical devices. \Ve have no single device to suggest, though we axe exploring alternative ways r,o account for the phenomena. Formally, this third class can be characterized as comprising those metacules that relate sets of rules in which the number of categories on the right- and left-hand sides of rules differ. It is this sort of metarule that is essential for the extension of GPSGs beyond context-free power in the Uszkoreit and Peters proofs {1982\]. Simply requiring that such metarules be disallowed would not resolve the linguistic issues, however, since this constraint would inherit the problems connected with the regular expression and set notations discussed in Section 3 above. This third cl~s further breaks down into two cases: those that have different parent categories on the rightand left-hand sides of the metarule and those that have the same category on both sides. The ~rst c~e includes those liberation rules that figure in analyses of free-word-order phenomena, plus such other rules as the subject-auxiliary-inversion metarule in English. Uszkoreit \[forthcoming\] is exploring a method for isolating liberation rules in a separate metagrammaticul formalism. It also appears that the subject-auxiliary inversion may be analyzed by already existing principles governing the distribution of features. The second case (those in which the categories on the right- and left-hand sides are the same) includes such analyses as the passive in English. This instance, at least, might be replaced by a lexicai-redundancy rule. Thus, no uniform solution has yet been found for this third function of metarules.</Paragraph>
    <Paragraph position="5"> We conclude that it may be possible to replace MPS-style metagrammatical formalisms entirely without losing generalization~. '~Ve ~re consequently pursuing re~eaxcu tu ~u,o o~,,.</Paragraph>
  </Section>
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