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<Paper uid="C94-2148">
  <Title>UNIVERSAL GUIDES AND FINITENESS AND SYMMETRY OF GRAMMAR PROCESSING ALGORITHMS</Title>
  <Section position="3" start_page="0" end_page="0" type="metho">
    <SectionTitle>
1. INTRODUCTION AND MOTIVATION
</SectionTitle>
    <Paragraph position="0"> This research interacted with a Japanese-English machine translation project at New York University. The results reported herein are part of an attempt to establish an evaluation system for grammar processing algorithms (parsing and generation algorithms). The need for evaluation of various competing approaches presently available lbr parser and generator design has been felt strongly in both Theoretical and Computational 1,inguistics. Both fields have been thus fat predominantly empMcal, so that the measuring of actual progress has become very difficult, l lere, we introduce the notion of universal guides in order to discuss two of the most relevant criteria tot the comparison of different parsing and generation algorithms: finiteness aml symmetry.</Paragraph>
    <Paragraph position="1"> Other criteria such as completeness, soundness, efficiency, etc., although equally significant and interesting, are outside of the scope of this paper and are addressed in \[M92\].</Paragraph>
    <Paragraph position="2"> There is a natural appeal to the problem of characterizing parsing and generation within the same fl'amework and in a symmetrical way. The reversibility is by its nature symmetrical: parsing is retrieving a semantic content fiom a phonological one, and generation, a phonological from a semantic content. Several papers (\[$88\], \[N89\], \[SNMP89\], \[DI88\], \[DI90\], and \[DIP90\]) have recognized parsing and generation as instances of a single paradigm and have pointed out the correspondence between certain parsing problems and techniques (leftrecursion, linking, Early deduction) and their correlates in generation.</Paragraph>
    <Paragraph position="3"> It has also been long noticed that adopting a certain technique for a derNation process can lead to termination problems (sometimes referred to as infinite derivations). Perhaps the best known example of this is using the TDLR derivation lor left-recursive rules. Consequently, to specify conditions on grammars, whose liflfillment is necessary and stffficient to guarantee finite derivations under a given evaluation strategy, posed another problem, one that has been given serious attention recently (\[D90\], \[I)lP90\]). These conditions are usually referred to as the finiteness criteria and are often given in the form of sufficient though not necessary conditions (&amp;quot;worst case&amp;quot; analysis of the finiteness of an algorithm).</Paragraph>
    <Paragraph position="4"> What we propose here is to abstract the notion of string index in parsing to the notion of a universal guide. A similar proposal was made in \[DIPg0\] for guides (here called proper guides to distinguish them fiom univetwal guides). Using the new concept, both parsing and generation call be seen as two instances of the same generic process: universal guide consmnption. Universal guides prove to be more general than proper guides because they can be used under any evaluation strategy and not only under TDLR technique as must proper guides. They achieve symmetry in treating parsing and generation but need not be instantiated difl'erently in either case, unlike proper guides. Universal guides can be consnmed anywhere during a derivation, as opposed to proper guides which may be consumed only by the application of lexical rules, therefore restricting the class of grammars for which they can be used. Moreover, we show here that proper guides can be viewed as a special case of our universal guides concept. The introduction of universal guides into a grammar also prevents the infinite repetition of certain grammar rules (i.e. those that do not instantiate any grammar variable and cause infinite loops).</Paragraph>
  </Section>
  <Section position="4" start_page="0" end_page="916" type="metho">
    <SectionTitle>
2. PRIOR RELATED WORK
</SectionTitle>
    <Paragraph position="0"> Some of the most significant findings with respect to characterizing finiteness and symmetry of gratnmar processing algorithms have been published by \[)ymetman, lsabelle and Perrault in \[D1P90\]. The authors pointed out the symmetrical nature of parsing and generation by introducing the notion of (proper) guides. A guide structure is a partially ordered set G that respects the descending chain condition, i.e. the condition that in (3 all  strictly decreasing chaius (1~ &gt;l\[&gt;...&gt;li&gt;.,. ) are finite. Guides were introduced lilt() each of a logic gralllnlar's llon-tcrlllinals in the fOlill ell: IleW, SO called guide variables. These variables added SOlllC redundal/cy which could be exploited for tighter control of the computational process. Aller the guide wuiablcs were added and left recursion was eliminated (by perlbrming the usual trallslbrmation as indicated in \[I)IP901), the creation of a IICW gl'allllllaf, equivalent lo the original one, was completed. Then, a set of conditions was specified that, if the now gIanllnal' satisfied it, guaranteed a finite deriwition lor any given goal. The conditions are: tile guide coilstullplion condition (GCC) alid the no-clmin condition (Nee). The guide coi\]stnnption condition states that the wthies for guide wu'iablcs must initially he finite and must also be consumed (decreased) each time a lexical predicate is expanded. The no-chain condition prohihits tile exclusive appearance el:predicates like 7' 4/ on tile right-hand side of a rule. It was shown that if both (iCC altd NCC held, all derivations ill the gl-allllllal' would lie finite. The notion of guides is applicable Io both parsing and generation btit it is instantiated difl{Jrently in each case (for parsing, the guide wiriablc rcprcscuts file list of words awaitiug to he allalyzed, aild fbr generation, the list of seniantics of subcalegorizcd constituents lonmining to be generated). Tile authors of I I)lPg01 also denlonstrated all application of their main result to the chiss of lexical gl'all/lllars.</Paragraph>
    <Paragraph position="1"> The following inqJrovcments look desirable with respect to the maiil result from 11)11&gt;901:  (i) The guides should bc specified bolero the details o1 Ihe algoritlun (parsin!.,&gt; or gelleration) alld lhc underlying glallllllal&amp;quot; (lexical or other kind) arc awfilablc. They should llot he dependent on these details; (ii) The lnaiu result (concerning linitc derivations) should be stated with respect to ally gfanlniar evahlatioll strategy and not only with respect to tile top-down, left-to-right algorithm; (iii) The constunption of guides Stlould be allowed at ally level, llot only lexical; alld (iv) The very introduction of guides (with no  addilional gralnlllar lrailsforniatioiis) should prevent a certain kind of iulinite derivalions fron/ happeuing (i.e. those due to left recursive rules).</Paragraph>
    <Paragraph position="2"> Wc show here that tile guides' approach hy l)ymctman ct al. can be viewed as a special case of tile universal guides approach lhat we introduce in this paper. We also demonstrate thai universal guides realize the desired iniprovcmcnls.</Paragraph>
  </Section>
  <Section position="5" start_page="916" end_page="977" type="metho">
    <SectionTitle>
3. UNIVERSAl, (;UII)ES
</SectionTitle>
    <Paragraph position="0"> We nlotivatc our introduction of tmivcrsal guides around the idea that a deriwition can be perceived as a process of discovering the set of all wiriabfcs that participate iu it. In other words, a deriwltion is 13nding all logic grammar symbols that arc Uldnstantiatcd at the moment when they appear in the derivation for the first time, and keeping track of changing of their binding status. The logic glammar symbols enter a derivation hy applying a production rule in which they participate either as bound, or partially or totally unbotuld. 'f'hc non-hound ones may or may not get instantiated during the dcriwltion and their number can be finite (in the case era finite derivation) o, inlhiitc (as with an inl\]nitc deriwltion). The term complete derivation we use for the derivation &amp;quot;in which the set of uninstantiated variables eventually gets reduced to all empty set, The set (finite or infinite) of all these variables has properties sinlilar to tile guides as defined by llylnetman, et al. in \[I)IP90\] (only the descending chain condition is not guaranteed) and will as such be a major compoHent of .our notion of universal guides. The comparison relalioil Ibr this partially ordered structure consisting of sets will be based on tile rclatiolJ &amp;quot;being a subset&amp;quot; c-. We forinalizc the previous discussion by the Ibllow ing dclhfitions.</Paragraph>
    <Paragraph position="1">  I)I:,FINITION 3. I. (A \[;SEI:UI, I~ARTIAI,I,Y O1(1)1H~, El) P, 1,;1 ,ATION) Lot S and S' bc two sets and N and N' lwo n()nnegative integers. We say that ordered pair (S,N) is greater than or equal lo ordered pair (S',N') and write (S,N)~&gt;(S',N') iff(S S'&amp; N. N')or(S.~S' or(S :S'&amp; N&gt;N')~  is obviously a reflexive, anti.-symmctrical and transitive relation and therefore a relation e/l partial order.</Paragraph>
    <Paragraph position="2"> I)I&lt;;I:INITI()N 3.2. (UNIVI~,RSA1, GUII)IiS) l,ct ~ he a collection of all subsets of a set ~ (~ '.1~(~)) and I deg set of all non-negative integers. A universal guide structure is a partially ordered structure (P,~), where P: { (S,N)/ S~'~ &amp; N~I deg }, and &gt; is fiom tile definition 3.1..</Paragraph>
    <Paragraph position="3"> The presence of a special kind of universal guides in a grammar is always only implicit, but for tile sake of being able to prove Ihcts about them formally, it can be made explicit. The universal guide structure in a logic gramnmr is hased oil the set of wuiables still uninstantiated at a given moment of a derivation process, l&amp;quot;,xpansion of a production rule may or may not instantiate (consume) some of them. l:or instance, by adding two special extra argtuncnts to each symbol in the following rules:  (1) noun phrase ( Num, NP Str, NP Rest ) --&gt; det(Num,NP Str,I) Rcst),noun(Nun\],l) Rest,NP Rest). (2) del ( sing, laiD Rest\] I) Rest ).</Paragraph>
    <Paragraph position="4"> (3) noun ( sing, I(log~kN Rest\]~ N Rest ).</Paragraph>
    <Paragraph position="5"> tile convergence or diver/_,ence of a derivation using tile rules becomes explicit. The uew arguments are: a set of currently uninstantiated variables (tile so called guide's set component) and a non-negative integer (the guide's  numeric component). The following simple derivation of the phrase a dog: noun~hrase (Num, NP_Str, NP_Rest) -&gt; det (Num, NP. Str, DRest), noun (Nmn, DRest, NP_Rest) --&gt; (1) det (sing, laiD Rest\], D Rest), noun (sing, D Rest, NP Rest) --&gt; (2) det (sing, \[a, doglNP Rest\], \[doglNP - Rest\]), noun (sing, \[dog\[NP_Rest\], NP_Rest). (3) (assuming there are p rules in the grammar) becomes: noun phrase (Num, NP_Str, NP Rest,  {NP_Rest}, 0). (3).</Paragraph>
    <Paragraph position="6"> The new arguments are given in bold case. When a rule to be expanded (partially or completely) instantiates some variables (rules (2) and (3) in the previous example), the guide's set component is reduced by those variables and the guide's numeric component is reset to p (the number of rules in the grammar). On the other hand, when a rule does not instantiate any variable, the set component stays unchanged and the numeric component is decreased by one (rule (1)).</Paragraph>
    <Paragraph position="7"> The numeric component actually counts (down) the number of consecutive occurrences of rules that do not instantiate may variable. If that number is larger than the total number of rules in the grammar (p), then a grammar rule must have been repeated in its unchanged form, and a potentially infinite derivation is caught. Such a rule is always failed. For example, np --&gt; np pp rule will in a TDLR derivation cause an infinite loop.</Paragraph>
    <Paragraph position="8"> However, adding universal guides to it will always, when this rule is applied, decrease the guide's numeric component by one (as no variable gets instantiated). Since the numeric component is always initialized and reset (after a rule that instantiates some variable(s)) to 9 (number of rules in the grammar), it will eventually go down to 0, which in turn will fail this rule, as in the following sequence:</Paragraph>
    <Paragraph position="10"> --&gt; fail.</Paragraph>
    <Paragraph position="11"> All details (additional arguments for original predicates, additional predicates, transformation of the original rules into equivalent ones containing guides) of the procedure tbr introducing universal guides into a grammar and their handling can be lound in \[M92\].</Paragraph>
    <Paragraph position="12"> Thus, the universal guide structure is represented in the new grammar by the pairs (Unln,Num) which stand for the set of all currently uninstantiated variables and the numeric guide's component, respectively. The new grammar is equivalent to the original one. By the introduction of an additional grammar predicate at the end of each rule (called decrease in \[M92\]) the guide consumption condition is demonstrated to hold for any finite and complete derivation. If a variable gets bound then the guide gets stricUy smaller because the set of still uninstantiated variables participating in the derivation has lost one member. If no variabie gets instantiated, the decrease of the number component is there to ensure that the guide itself strictly decreases (by the definition 3.2.). By failing, this new (decrease) predicate will stop any derivation that contains a sequence of more than p consecutive applications of rules that do not instantiate any variable (because the further decrease of the guide's numeric component would make it negative). Otherwise, a production rule would be repeated in the same manner without instantiating any of the present variables which would in turn cause an infinite derivation to take place. Thus, because of the way the universal guides are introduced for any derivation in the new grammar, guide consumption condition holds. The difference between finite and infinite derivations is isolated and solely characterized by the set component of the universal guides being either finite or infinite initially.</Paragraph>
    <Paragraph position="13"> The following theorem establishes a correlation between proper and universal guides.</Paragraph>
    <Paragraph position="14"> Theorem 3.1.: lfthere is a proper guide structure for a class of logic grammars (in its form flom \[DIP90\]) satisfying guide consumption and no-chain conditions (GCC and NCC) under the TDLR grammar evaluation algorithm, then the universal guide structure (under the same algorithm) is a proper guide structure satisfying both GCC and NCC.</Paragraph>
    <Paragraph position="15"> Proof The existence of the proper guides satisfying GCC and NCC nnder the TDLR algorithm guarantees that all derivations will be finite for the given class of grammars (main theorem from \[DIP90\]).</Paragraph>
    <Paragraph position="16"> Since the derivations are finite, the universal guides will initially assume a finite value (its set component will be set to all variables taking part in the derivation and its number component will be assigned value p (number of different production rules in the grammar).</Paragraph>
    <Paragraph position="17"> The universal guides are defined to always satisfy the guide consumption condition, and since NCC is assnmed as well, it only remains to show that the descending  chain condition is respected.</Paragraph>
    <Paragraph position="18"> As every strictly descending chain of universal guides with a finite initial value must be finite (moreover, we know lhat its length is always less than or equal to p*u (u being number of variables taking part in the derivation), the universal guide structure has all properties of a proper guide structure,, Thus, whenever proper guides can be used to establish the finiteness of an algorithm, the universal guides approach may likewise be used.</Paragraph>
    <Paragraph position="19"> Also, tile notion of universal guides proves to be more general than the notion of guides in the sense that it does not asstane any partictflar algorithm under which a grammar will be processed. It is applicable to any algorithm, and to apply it would mean to specify conditions on gtammars thal would (for a given algorithm) guarantee finiteness of tile forclnentioncd sets (set components of the universal guides). Of conrse, the character of tile conditions will ctepcnd oll tile nature of tile gtamtnar processing algorithm. Proper guicles as proposed by I)ymetman, ct al. guarantee finiteness el:one specific (TI)I,R) algorithm if the grammar satisfies GCC and NCC.</Paragraph>
    <Paragraph position="20"> The following example describing a wh-question (here used for tile generation of tile sentence who wrote this \[~l'Olll the given semantics wrote(who, this')) could be helpftfl to illustrate the applicability of tile tmiversal guides approach where tim proper guides would not  When the semantic-head-driven generation algorithm (see \[SNMP89\]) is used, lhe order in which different rules are applied carl best be described by tile analysis trec from tile Fig. 3.2.. The mnnbering for edges indicates the order in which the grammar rules were used. Thus, rule (4) was used first, then rule (1), lollowcd by rules (2) and (5) respectively. The variables that were introduced as uninstantiated are WhQues Sen (by invoking the topmost predicate), Sut?/, Old/, and WhPredRest (by the application of rule (4)). The application of rule (I) (expanded second) unified number components of u,hsub\] and whpred (already instantiatcd by rule (4)), semantic components ofwhsuhj and whol~/ with variables Sul?j and Ot?j fi'om wkf)red ,  respectively, WhSubjRest component with WhPred Sen component of whpred (already partially instantiated by rule (4)), as well as WhPredRest with WhObj Sen. It also unified WhSut~j Sen with WhQues Sen and \[\] with WhObjRest and therefore did not introduce any new uninstantiated variables. The application of rules (2) and (4) did not introduce new variables neither. Thus, the set of all uninstantiated variables participating in this derivation is { WhQues_Sen, Subj, Obj, WhPredRest }.</Paragraph>
    <Paragraph position="21"> Subj gets instantiated by the application of rule (2), Ol~/ and WhPredRest by rule (5), and the instantiation of the variable WhQues Sen is partially done by rules (4), (2), and eventually completed by rule (5). Thus, set component of the universal guide variable { WhQues_Sen, Subj, Obj, WbPredRest } is first consumed by the application of rule (4) (part of WhQues &amp;m), then by rule (2) (another part of WhQuesSen, plus Subj), and eventually reduced to an empty set (represented here as \[\]) by rule (5) (final ingredienf of WhQues Sen, plus Ol~/ and WhPredRest).</Paragraph>
    <Paragraph position="22"> This semantic-head-driven derivation with the universal guides included (and under' assumption that the grammar has p different production rules) can be described by the following steps:  -- whobj (this, \[this\], \[\], {}, p) rule (5) Unlike universal guides proper guides require a specific (TDLR) grammar evaluation strategy and therefore this approach is not applicable at all for the semantic-head-driven generation algorithm since this algorithm assumes a grammar evaluation strategy different fi'om TDI,R.</Paragraph>
    <Paragraph position="23"> Generally and tbr any evaluation strategy, the formal link between the universal guide consumption and termination can be expressed by the following claim: Theorem 3.2.: I,et G be a logic grammar and G' its equivalent aRer the universal guides were introduced into G. If the guide consumption condition is fulfilled lbr a derivation in G' and initial value of the guide structure is finite, then the derivation in question will be finite too. The proof of this theorem as well as a detailed specification of how to introduce tmiversal guides into a grammar can be found in \[M92\].</Paragraph>
    <Paragraph position="24"> Moreover, the common essence of tim parsing and generation process as merely different instances of the stone generic process of consuming the universal guides becomes obvious after making the appearance of universal guides explicit. Universal guide variables do not necessarily have different meaning for&amp;quot; parsing and generation as do proper guides. Even under an evaluation strategy (TDLR) assumed in advance proper guides (as in the case of lexical grammar&gt; floln \[DIP90\]) represent difterenl entities for parsing and generation. For a parsing algorithm guides are difference lists of words remaining to be analyzed, and for' a generation they are lists of subcategorized semantics to be generated next. Unlike proper guides, universal guides exposed the common substance of the two processes. They are always (for parsing as well as for generation) instantiated as sets of all currently uninstantiated variables.</Paragraph>
    <Paragraph position="25"> Another feature of the univcrsal guides that gives them an advantage over proper guides is that they do not impose the restriction that the guides can be consnmed only at the level of lexical predicates. Thus, the class of grammars lbr which this approach can be used is broader than for that of proper guides.</Paragraph>
    <Paragraph position="26"> Also, we in effect presented here a class of grammars the recursivity of which can be proven by induction.</Paragraph>
  </Section>
  <Section position="6" start_page="977" end_page="977" type="metho">
    <SectionTitle>
4. CONCLUSION
</SectionTitle>
    <Paragraph position="0"> This paper' addressed finiteness and symmctry of parsing and generation algorithms using a novel univers'al guides approach. We pointed out some deficiencies of proper guides' approach as advocated in some earlier research. These included the applicability of proper guides only when the evaluation strategy is TDLR, and when it is also known whether a parsing or a generation algorithm is in question. Also, the consmnption of proper guides was allowed only at the lexical level. By the introduction of univers'al guide.s' all of these deficiencies are eliminated and a true symmetry is achieved in treating the parsing and generation problem. Unlike proper guides, univers'al guides' do not need to be constructed and instantiated differently tbr parsing and for generation, and no additional grammar transformation (i.e. left recursion elimination) is needed for them to be applicable.</Paragraph>
  </Section>
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