<?xml version="1.0" standalone="yes"?>
<Paper uid="C00-1062">
  <Title>LFG Generation Produces Context-free Languages</Title>
  <Section position="6" start_page="427" end_page="430" type="concl">
    <SectionTitle>
4 Consequences and Observations
</SectionTitle>
    <Paragraph position="0"> Our main result oflb.rs a new way to con(:et)tualize the problenl of generation lbr HPG and other lfigherorder context-free-based grainmatical tbr, nalisms.</Paragraph>
    <Paragraph position="1"> The proof of the theorem is constructive: it indicates precisely how to lmild 1;111.' grmnmar GI; whose language is the desired set; of strings. Thus, the 1)rol~lem of LFG generation is divided into two phases, constructing the context-Dee grammar G/,,, an(t then using a standard context-free generation algorithm to produce strings fl'om it.</Paragraph>
    <Paragraph position="2"> \Ve can regard the first t)hase of LFG generation as specializing the original LFG gl'allilllal to Oi11~ that only produces the given input fstructure. This specialization refines the context-fiee backbone of 1;11(; original grannnar, but our theoreln indica.tes that the inl)ut t'-si;ru(:ture l)rovides enough infornmtion so tlmt, in effect, tlm metaw~riables in the functional annotations can all be replaced by variables contained in a tixed tinite set. Thus, in the LFG generation case the st)e(:ialized grammar turns out to be in a less l)owerful tbrmal class than the original.</Paragraph>
    <Paragraph position="3"> \Ve (:an mlderstand different aspects of generation as I)ertaining either to the way the grammar is construtted or to well-known properties of (;Oll(;exl;-free grammars and (~olltoxl;-\]'l'ee generation.</Paragraph>
    <Paragraph position="4"> It follows as an immediate corollary, tbr exampie, that it is (lecidalfle whether the set GcnG,(F) is emt)ty , contains a tinite mmfl)er of strings, or contains all infinite number of strings. This C}lll lie deternfined by inspecting GF with standard context-free tools, once it has l)een constructed. If the language is infinite, we (:an make use of tim context-Dee pumping lemma to identify a tlnite number of short strings Dora which all other strings ('an be produced 1)y rel)el,ition of sul)(lcrivations. Wedekin(1 (19{)5) tirs( estal)lished the de(:idability of I,FG generation and t)roved a lmmping lemma ti)1 the generated string set; our tlwx)r(nn l)rovides alternative ;ul(l very direct 1)root's of the.st previously known results.</Paragraph>
    <Paragraph position="5"> \Y=e also \]lave gtll exl)lanation for another observation of Wedekind (1995). Kaplan and Bre.snan (1982) showed that the Nonbranclfing I)ominance Condition (sometinms called ()flline Parsability) is a sufficient (:on(liti(m to guarantee (le(:idal)ility of lhe meml)ership l)rol)lenL Wedekind noted, how(~ver, (;bat (;hi~ condition is not nex:essary to delermine \v\]mlht~r a given tkstrlletlll'e corresponds 1;o any strings. We now see more clearly why this is the case: if there is a (:olltext-Dee derivation for a given string that involves a nonl)ranching dominance cy(:le, we know (fronl the pumi)ing hmnna) that there is another derivation for tlmt saint string that has no such cycle. Thus, the generated language is the same whether or not derivations with nonbranching dominance (:y(:h;s are allowed.</Paragraph>
    <Paragraph position="6"> There is a practical consequence to the two phases of LFG generation. Tim gralllllHtl' GI,' eaIt t)e provided to a client as a finite representation of the set of 1)crhal)s infinitely many strings that corresl)ond to the given fstrueture, and the client can then ('o11trol the process of enumerating individual strings.</Paragraph>
    <Paragraph position="7"> The client ntay choose simply to produce the shortest ones jllst 1) 3, avoiding recursive category expansions. O1 the client may apply the technology of stochastic context-free grammars to choose the most probable, senI;ence, f1'o111 the set of possibilities. The client may also be ilW;erested in strings that meet further conditions that the shortest or most probable strings fail to satist~y; in this case the client may</Paragraph>
    <Paragraph position="9"> apply the pumt)ing lemma to systematically produce longer strings for exmnination.</Paragraph>
    <Paragraph position="10"> Our recipe tbr constructing GF may produce many categories and expansion rules that ca.ili, ot play a role in any derivation, either because they are inaccessible from the root symbol, they do not lead to a terminal string, or because they involve individual descriptions that F does not sat, is\[y. Having constructed the grammar, we ea.n again api)ly standard context-free methods, this time to trot the grammar in a more ot)timal forln by reinoving useless categories and productions. We can view several difl!erent generation algorithms as strategies tbr avoiding the creation of useless categories in the first place.</Paragraph>
    <Paragraph position="11"> The most obvious optimization, of course, is to incretnentally evaluate all the instantiated descriptions and remove froin consideration categories and rules involving descriptions for which F is not a model.</Paragraph>
    <Paragraph position="12"> A second strategy is to construct the grammar in bottom-up fashion. We begin by comparing the terminal rules of the LFG grannnar with the features of the input f-structure, and construct only the corresponding categories and rules that meet the criteria in (iii) above. We then construct rules that can derive the mother categories of those rules, and so oil. With this strategy we insure that every category we construct can derive a terminal string, but we have no guarantee that every bottom-up sequence will reach the root symbol.</Paragraph>
    <Paragraph position="13"> It is also at)pealing to construct the grmnmar by means of a top-down process. If we start with an agenda containiug the root symbol, create rules only to expand categories on the agenda, and place categories on the agenda whenever they appear for the first time oi1 the right side of a new rule, we get the effect of a top-dowu exploration of the gratnmar. We will only create categories and rules that are accessible fronl the root symbol, but we may still 1)roduce categories that derive no terminal string.</Paragraph>
    <Paragraph position="14"> The toi)-down strategy may not provide ett'ective gui(tance, however, if the set D(F) contains many alternative descriptions of F. But suppose we can associate with every instantiated description D a unique canonical description that has the stone f-structure as its minimal model, and suppose that we then reformulate tlm grammar construction in terms of such canonical descriptions. This can shari)ly reduce the size of the grammar we produce according to any enumeration strategy, since it avoids rules and categories that express only uuinforlnative variation. It can particularly benefit a top-down era&gt; meration because the set D(F) will have at most one canonical member. Presumably any practical generation scheme will define and operate on canonical descriptions of some sort, but our context-Dee result does not depend on whether or how such descriptions inight be specified and maifipulated.</Paragraph>
    <Paragraph position="15"> Just as for context-free parsing, there are a number of mixed strategies that take top-down and bottom-up inibrmation into account at the stone time. We can use a precomputed reachability table to guide the process of top-down exploration, for iilstance. Or we can simulate a left-corner enumeration of tile sem'ch space, considering categories that are reachable froin a current goal category and  nmtch the left; corner of a possible rule. In general, ahnost any of the traditional algorithms tbr process\[llg (;()iltext-frec gt'atillllars call be reforllllllatctl as a strategy tbr tn,oiding the creation of useless categories and rules. Other enmneration strategies focus on the characteristics of the input f-structure. A head-driven strategy (e.g. van Noord 1993) identities the lexical heads first, finds the rules that exl)and them, and then uses information associated with those heads, such as their grmmnatical flmetion assigmnents, to pick other categories to exlmnd.</Paragraph>
    <Paragraph position="16"> Our proof depends on the assmnl~tion that the input \],' is flllly specified so that the set of i)ossible instantiations ix finite, l)ymetman (1991), van Noord (1993), and Wedekind (1999) have shown that it ix ill generM undecidable whether or not there are any strings associated with an f-structure that has units ill addition to those in the input. Indeed, our proof of context-freeness does not go through if we allow new units to be hypothesized arbitrarily, l/eyond the ones that appear in F; if this ix permitted, we cannot establish a finite. 1)ound on the munbcr of l/ossil)le categories. This is unfortmmte, since there may be interesting practical situations ill which it is convenient to leave UnSlmCified tile value of a liartitular feature. However, if there can be, only a iinil, e nlunb(',i' of possible wflues for an underspecitied feature, the (:ontext-free resull: can still be esi;al)lished. We create from F a set of alternative structures F~..F, by filling ill all possible values of the UllSl)eeified features, a.ml we l)roduce the context-Dee grammar corresponding to o, ach of thcln. Since a finite ration of eontext-flee languages is context-Dee, the set of strings generated fl'om any of t, hese structures renmins ill that class.</Paragraph>
    <Paragraph position="17"> A tinal COilllllellt a\])ollt t;he generation l/rolflem for other high-order granmmtical t'ornmlisnis. ()llr proof dcl)ends on se, veral tb, aturcs of LFG: the (:Oll\[:exl;-ti'(?e 1)ase, the pieeewise correspondence of 1)hrase structure, and f-structure units, and the ideml)Otency of the flumtional description language. PATR shares these properties, although the correspondence is ilnplicit in the mechanisnl and not reified as a linglfistically significant concept. So, our proof can be used to establish the context-free result for PATR. On the other hand, it is not clear whether the string set corresponding to an underlying I{PSG structure is context-flee. HPSG (Pollard and Sag 1994) does trot Iltake direct use of a context-free skeleton, and olmrations other than concatenation may be used to assenfl)le a collection of substrings into an entire Selltetlce. \~e canllot extend ore&amp;quot; proof to ttPSG m&gt; less the etli~ct of these mechanisms can be reduced to an equivalent characterization with a context-free base. However, grammars written for the ALE system's logic of typed feature structures (Carl)enter and Penn 1.994) do have a context-free COlll\])Ollelll; and therefore' are, ainell~fl)\]e to the, treatnlent we have outlined.</Paragraph>
  </Section>
class="xml-element"></Paper>