File Information
File: 05-lr/acl_arc_1_sum/cleansed_text/xml_by_section/metho/94/c94-1065_metho.xml
Size: 23,208 bytes
Last Modified: 2025-10-06 14:13:35
<?xml version="1.0" standalone="yes"?> <Paper uid="C94-1065"> <Title>LIl~(k)- I'msing of C( ut 1 A-Cont(xt,-\]~ (c Grammars :) -) 0 &quot; ~ ~ ~ '~* ~ *</Title> <Section position="2" start_page="0" end_page="401" type="metho"> <SectionTitle> LR(k) Coupled--Context.-\]:ree (Iranlmars. \] Introduction </SectionTitle> <Paragraph position="0"> ill order to process ll&ttlra\[ I;t/l~ll;l~,eS 3 ;Vfl first }lave to model the syntax formally. Many investigations ;is~ e.g., \[lligS,l \] show th~tt this cannot be done by context-free graliilllars ((Jl!'(ls). I&quot;or context-sensitive ~.,,l'allllllars which are powerful enough, it is kn(iwlt that the anal5'sis is PSl)ACl:,--coml~lete. q'hus, there is a trade-off' he tween tile power of the. formalism and it;; analysis colnplexlty. To solve this dilenlma, much work has \]men done tO char;t(:terize l;tllgl|al~e classes ill Detweett coltt(!xt-free and context-sensitlve languages he{rig powerful enough to model the syntax o\[' natural \]allgllages hut endowed with at Imlynomial time analysis. (2mll)\[{:(l-(;onl.ext-I&quot;ree Gram mars represent such a. forma\]isnl yeneralizhlg CI&quot;Cs. Their suitability to model syntactical phenmnen:t fo\]h)ws \[rom the fact that they include tit{.' \]itllgllaJ!,eS ~.;CllCl'il(Cd by (he Tree Adjoining (h'ammars (TACs) (i\]' \[.hls87\] as (me subclass. Anlung other properties, both f()rmallsms are ;d)le to model tim linguistic phenomenon of cross s(:ri;d depen.</Paragraph> <Paragraph position="1"> dencies, which is not context.free hut \]'rcquently appears in natural languages (of. \[Shi86\]).</Paragraph> <Paragraph position="2"> The formalism of (fOul)led-Context-Free (',ramnlars \]tits been introdu<:ed in \[CIIR92\] and \[Cua92\]. It h,'.hmt, s to ~l~(: family (ff v'egulate<l striwg rewriting syslems investigated in \[1)P89\]. The inc:ruased generative capa<:ity is o\])laincd by allowing to rewrite sintultanemlsly a {retain iHllllbCl' of elements. Other regulated string rewriting systems its, e.g., the Scatte.red (Jontext (ll'allllllars of \[(',llft.q\] generalize CFCs by allowing simultaneous rewriting of arbitrary combinations o\] elements. \[n \[I)P89\], it is shown that this results in languages which are not sentillnear. But semilin= earity is important since it formalizes the &quot;conslant-growth lm)pe.rty&quot; of natural lanyuages (cf. \[3os85\]). In co,,t,'ast to these, all languages defined by our I'urmallsnl are semi\[{near *'\['his research has been supported hy a (h'aduh:rtenkol\]cgfellowship of the I)cutsche l&quot;orschtmgsgenleinschaf(. because of two restrlctions. First, only those elementsean he rewritteu simultaneously whh:h were produced by the same re.writing. Second, the Coupled-Context-Free (\]ramllial'S coltsi(ler e\]ellleltts Iewr\[tten SillllllLalleo|tsly ;ts COil|portents of ;t parenthesis. Those can only l)e substituted if they form a parenthesis and t\]ley can only he suhstitutetl \[)y seqlteltc(!s of pa.r(.*lll,heses cov'rectly nested.</Paragraph> <Paragraph position="3"> When characterizhq4 Cuup\]ed-Context-I'~ree (~\]r&ll~l\[Htrs by lhe maximal number o1&quot; elements rewritten simultan(> ously - width we call the rm& o\[&quot; the grammar we get an infinite hierarchy. The generative capacity grows with the rank. The smallest element of the hierarc\] W - the one of rank I * are (~l;'(;s. The next element, namely Couple.d-Context-Free (Iramnlars of rank 2, generates t}le same class of Tanguag.es as the Tree Adioini,g (~raIlllnaFs of \[Jl/I'75\] and \[Jos87\]. llence, all noth)ns and algorithms designled for Couph~d-Conl.ext-Free (il'ilIiIllt;tl's Of riLItk 2 can easily be translated onto 't'A(ls (eL \[Cua92\]).</Paragraph> <Paragraph position="4"> l~ec:allse of the enJarl,e(I geIterat, lve capacity, it is not sutprising that the coutllh:xity of analysing I:tl<gIHtges gener~tled by (Jouph:d-Context-l&quot;ree (h'anm,ars is larger than it is in the context free case. \[t even increa.ses with growi,m rattk (c:L \[11P9,1\]). Therefore, we aim to characterize suhclasses of the set of all lall,~ll~t.ff, es geller~tl.ed hy (,'Oul)led(.~oiHexl,-.Vree (~'4t'atllllltaFs which are powerful ellOltgh to model the important phenomena of natural liu,guages, I)ut which are of at lower c(mq)lexity.</Paragraph> <Paragraph position="5"> The deternliuistic (:onlext-free parsing with Llt(k)~,~&quot; ............ leads t .... li ...... r tinl ......... lysis (of. \[Knu65\]), the best lmssible. Therefbre, its generalization is very attractive. A fh'st altempt in this direction was done in \[.'qVg()\]. I~u(. there, only TA(Is are ilwestiga|.ed. \]\]ere, we invesilgatc~ the whole hi(Hallchy Of Couph:d Context-l&quot;ree (lramma~s. Allhou,;h their enlarged generative (Stl)a{:ity seems to I)e c(mlradictory Io a. Ihm;u' time cOnll)\]exlty of q\]le pals{rig algmithm, we can present an Ll~(k)-notion t'{, (:oup\[(:d-Co,ttext-l&quot;vee Crammars dcs{:ril,ing a class of laugual'j:s, which can actnal\]y bc anMysed in linear time.</Paragraph> <Paragraph position="6"> This increase ill power a.s to the linear time ~tlHtlysis is ilaid hy an expensive ln'eprocessil~g. It in taking into account (he complex rclntlons \])etwe{'n parewthese.s that involves ILL(: ill(:rease ill conll)lexity, l\[owever, these costs are to he paid only once for each grammar. The suhchuss described hy our I,l~(k)-notion for a lixed k i.rows with the rank.</Paragraph> <Paragraph position="7"> &quot;\['he al~orithm of \[.q\:90\] fnr l,lC/(k)/l'AGs does not fullill theimp(,tant Valid Prefix I)roperty. This means that for any prefix o1&quot; the iUl)Ul: aheady accepted, there exists a suffix such that the whoh', word is in the language analysed.</Paragraph> <Paragraph position="8"> It allows to detect illegal inputs as soon its possible, which is necessary for efllcient parsing. Our algorithm fulfills this property. Addition;lily, the algorithm its well as the notion defined here ,'eprescnt genera.lizations o\] their context-free C<)lllt(,et'pltrts which are ll~ttllra\] ill tit<: sense that they strictly contain tit{: context-free sitnation as the special case of (;ouple(\[-(Jonl, exl.-Free (\]ranLulars o\] rank 1. An example of an important LR(k)-Coupled-Context-Free Grammar is the one generating the language {w$w I w ~ {a, b}*} which reflects the syntactical construction of cross-serial dependencies.</Paragraph> <Paragraph position="9"> The paper starts by defining the Coupled-Context-Free Grammars. Thcn, wc shortly recall the context-free Lib parsing procedure. Subsequently, the deterministic finite automaton used there to guide the analysis is modified such that it can handle Couplcd-Cm~text-Frec Grammars.</Paragraph> <Paragraph position="10"> Based on it, the parsing algorithm for LR(O) Coupled-Context-Free Grammars is derived. Tiffs results in the generalized definition of the LR(O)-notlon. As for CFCs, the LR(k)-Coupled-Context-Free Grammars result from the LR(0)-ones by resolving decision conflicts using a lookahead of at most k symbols.</Paragraph> </Section> <Section position="3" start_page="401" end_page="402" type="metho"> <SectionTitle> 2 Coupled-Context-Free Grammars </SectionTitle> <Paragraph position="0"> Coupled-Context-Free Grammars are defined over extended semi-Dyck sets which are a generalization of scml-Dyck sets. Elements of these sets can be regarded as sequences of parentheses that are correctly nested. Senti-Dyek sets play an important role in the theory of formal languages. To extend the family of context-free languages by using them wc consider parentheses of arbitrary finite order define(\[ as follows: Definition I (Parentheses Set) A finite sctK := ((ki,~ ..... ki,,,,,)li, mi ~ N} is a Parentheses Set iff it satisfies ki,j # kt,,,, Jot&quot; i C/ I or j # m. The elements eric are erdled Parentheses. All parentheses of a fixed length r are summarized as</Paragraph> <Paragraph position="2"> Straightforward frmn this, we get Definition 2 (Extended Semi-Dyck Set) Let ~ be a parentheses set and 7' an arbiteary set where 7&quot; VI K = T m comp( K) = O. E D( K, T), the extended semi-Dyck set over E and T, is indnetively defined by</Paragraph> <Paragraph position="4"> Now, we define how to generate new elements in ED(K, T) starting from given ones.</Paragraph> <Paragraph position="5"> Definition 3 (Parenthesis Rewriting System) A Parenthesis Rewriting System over ED(/C, T) is a fiaite, nonempty set P of productions of the form {(k, ..... k~) ~ (.~ ..... o,.) I (~, ..... ~,) C- ,~,., ..... ,,. e ED(~, &quot;r)}. The left and the right side of p := (X~,...,X~) -* (o:1 ..... o6,) G P is denoted by * S(p):= (x, ..... x~), thc sonrce oh,, a,,d</Paragraph> <Paragraph position="7"> Now, we can deline our grammars. The term &quot;coupled&quot; expresses that a certain number of (:ontext-free rewritings is executed in parallel and controlled by K\[.&quot; Definition 4 (Couph;d-Context-l, Yeo Grammar) A Coupled-Context-Free Grammar over ED(K,T) is au ordered ~-luple (IC, T, l 7 ,S) whcr'e l' is a Parentheses Rewritin 9 Syste,n over ED(K~,7') and S 6 KIll. Therefore, IC can be regaeded as a set of couph:d nonterminctls. The set of (all these granmmrs is denoted by CG' I,'G.</Paragraph> <Paragraph position="8"> - ~(~ At last, we give the definition of derivation in CCI ,.</Paragraph> <Paragraph position="9"> Let (;' = (K,T, P,S) C- ,(.1 G and V := cornp(lPS)UT.</Paragraph> <Paragraph position="10"> We define the relation ::~zc; as a subset of V* x V' consisting of all derivation steps of rank r for G' with &quot;r > l. p =>a '0 holds for 99,~/, G V* if and only if there ex\[st (k, .... ,k,) -~ (-* ..... -,) e P, ,*,,,*,,,, e V', and</Paragraph> <Paragraph position="12"> -'~6' denotes the reflexive, transitive CIOSII|'(: Of ~'(~', ()bviously, 'al .u,+\] (5 I')D(/C,'/') follows from S =*>c; (p for ~2 and ~b siuce the result of the substitution is a sequence of parentheses correctly nestcd if and only if the original word was. The language generated by C is defined as</Paragraph> <Paragraph position="14"> A sequence ~1,..., 90, with 9~i -~c; (~'~i-{-1 for all I < i < n and ~1 -- g, (P. = C/ is called a derivation o\]'(, from g in C. A deriwttion is righhaost if and only if in each derivation step, the parenthesis ending at the rlghtnlost point is substituted. In analogy to C\]&quot;Cls, it is obvions that for any derivation in CC1;'C there exists exactly one ri~htmost (h'.riwtt k,i.</Paragraph> <Paragraph position="15"> Ex,,,np\[,, ~ c: _- ({s, (x, x)), {a, (',S,'t}, P, s) is i,, cc/,'6'(2) ,.here 1' := {S -. XSYf, (X, X) -~ (aXb, e~,t) I (.~, ea)}. c ~e,,e,'(,.,s the lan:tt,.g. {.&quot;~*'e&quot;d&quot; I '* -> ~}, e.g. S :~'6' X$.~ =>(~ aXb$c~;d :>c; aaXbb$ccX-dd =>c1 aaabbbeccddd In order to be able to describe the generative cal)acity of Coupled-Context-Free Crammars of different ranks exltctly, we need the following notions: Definition 5 (Rank, CCFG(l)) Foe any (5' = (K,7', P,S) G C(21,'C,', let the rank of (l be d,'finc,l ,,s ,'ank(G) := ,,,ax {&quot; I (< ..... k,.) C ,~}. The.at, we define for all l > l: CCl,'C(l) := {c, E ccrc;' l ,.a,,k(c') < l} The following theorem prowm in \[Gua92\] shows that CCI,'(I In,\[his up an infinite hierarchy of languages and, at the same time, represents a prel)er extension of CI&quot;Cs not exceeding the lmwer of context-sensitive granlolars: Theorem 1 (llierarehy) Let CI&quot;L be the family of all eontext-free, CSL the family of all context-sensitive languages, TAL the family of all languages generated by 7'A (.Is and CCI;'L(I) Ihe one gener- null ated by CCFG(l). It hohls: (1) c't,u, = cc/,'t,(1), &quot;eAL = c'cl.'t,(2).</Paragraph> <Paragraph position="16"> (2) cct.'t,(l) ~ C'C'FL(I + ~) /o,. .. l > i.</Paragraph> <Paragraph position="17"> (a) 6'CVL(I) c cs/, /or ,n l > i.</Paragraph> <Paragraph position="18"> Sometimes, i* is useful to &quot;neglect&quot; the relations be- g tween the components of a parenthesis for a short time. Then, wc investlg~tte C/ :== (eomp()C),7', P',S) instead of G = (K, T, P, S) C~ .CI'G for</Paragraph> </Section> <Section position="4" start_page="402" end_page="402" type="metho"> <SectionTitle> 3 Context-Free L\]?,-Parsing </SectionTitle> <Paragraph position="0"> Now, we shortly recall the deterministic context-free LR(k)-parsing strategy of I(nuth (or. \[Kmt65\]). 1,'or slmplicity, we restrict ourselves on the case k = O. The strategy essentially renlains unchanged if lookahead is necessary. It uses a deterministic finite antoniaton (dfa) to drive a pushdown stack while scilnliiiig the inpnt from left to right. 'l.'}uis, it COllstrllcts a riglitnlost derivation bottom-up. The states of the d.fa for a given LR(0)-CI&quot;G consist of subsets of the set of all context-free items for (; (N,T, P,S'), i.e. of the set {\[X --~ ~,.fit\] I x .... i~ e v).</Paragraph> <Paragraph position="1"> '\['hey result from dcternilning the deternlinistic w~rslon of the following nondeternfinistic automaton for (:l: \]!\]aeh context-free item is a state.</Paragraph> <Paragraph position="2"> There are. three kinds of state transitions: \[x ...... yf~\] &quot;~Z \[x .... Y.fq, \[X .~,r . * -~ ..,,i7} , b': -' <,,./~\], a,,d \[Y > &quot;~.X~\]-<, \[X --, .<,.1.</Paragraph> <Paragraph position="3"> In the (leternllnlstic version, all those context-free \tents are grouped in one state which can he reaclled from tile initial st~tte by the sail)e Setltle.itce of symbols) with ally possible number of e-transltions in-between.</Paragraph> <Paragraph position="4"> The stack symbols are the states of the dfa. At first, the state containing the item \[S' -, .S\] is llushed. (The addithmal l)rodnction .S '~ --, ,S' serves to define exactly the start and the end of the analysis.) Then, we iterate the following actions delmndhlg on the toImlost state q: (Shift) lfq contains \[X-,~*.a/f\] and a is the next input synt\])ol to be read, we push the state reached froln r 1 vl ..... (it coiitaiits at least {X ~ <,a.fl\].) (Reduce) if q contai,,s \[X -~ c<.}, we I,Ol, tim I-I tov,il,ist states. Let q' be the state now OH top of the stack.</Paragraph> <Paragraph position="5"> Then, we push the state reached via X from q'. (q' eoutains at least one it.e,n \[Y--. T.XS\] and \[X -~ .~\] while the new topniost state contains \[Y -~ 7X.@) 'l?he pushdown is driven determlnlsth:ally hy the Ilia if this d fa contains no state wliere th('~e are two different Reduce-items (Ih:duce-lteduce conllict) or as well it Shift- as a lt.ednee-itent (Shift-Reduce conflict). A (~'1&quot;(1 is LR(O) ill&quot; the states of its dfa show no Shift-i\[e.dnce and 11o Reduce-l/.educe conflict. For LR(k)-grammars, conflicts in the LR(O)-dfa are solved by it h)okahead of k syml)ols.</Paragraph> </Section> <Section position="5" start_page="402" end_page="403" type="metho"> <SectionTitle> 4 The Finite Automaton </SectionTitle> <Paragraph position="0"> One possibility to generalize dfa is to construct the usnal dJa for 6'F(G), (I E CCI,'G. In l)rinciph!, this idea is used in \[SVg0\]. &quot;|'lie fl)llowing example shows that this produces unnecessary contlicts: I,et (; --.-</Paragraph> <Paragraph position="2"> hi Figure l. (~ is not LI~(O) in this way since this dSa ohvlously has a Sllift-l\[educe conflict (in the box doubly lined). This conflict cannot he solved by loekahead since at this point, the lookahead is always d x. Therefore, (l is not LR(k) for ~tlly k ~ 0. But this conflict is liOt necessary. I&quot;,g., when analysiug bedd bottoni-up, we first haw.. to reduce X --+ b. This inlplies that lie|ere coming to the conflict state, we have to choose X -, e in order to get a correct derlwttion. This is the case \])ecause X an(I X resnlthlg from applyh~g the productiou 5; '-~ XX D$ ar(:</Paragraph> <Paragraph position="4"> <:mq)h'.d and therefm'e haw~ to be substitut~M by coupled I)roducLiolls.</Paragraph> <Paragraph position="5"> To awfid these conflicts, we extend the dfa. If&quot; we use the context-free LR-I)arsing strate.gy, we know which product\o,, we have to choose for any Xi G eoinp(K.) \ compl (K.) because we first encounter alld reduce the corresponding X1 E co,lpj (hT). Suppose that we can store the infornlatlm, ahout X ...... X,., (Xl ..... X,.) E K.\[,'\], when X~ is reduced, let us say as the &quot;future&quot;. (\[low to do this is shown in Section 5.) Can we use this to awfid the conflict? Now, our automaton needs additim,fl transitions under s,,ch ~,~ c cv(P) where S(7,~) C/ ,,o,,,V~(JC) h,)las. &quot;rh,,s, we split ways inside the dfa which lead to conflict si.ates.</Paragraph> <Paragraph position="6"> &quot;Fo formalize our atltOlt)aiA}ll) We need the following Definlt.hni 6 (1-Closure) /,'o,. <,..\'c= ,.o,,,p4sc ), t,.~ ,'e.<,h.bte(X):= (v c ...... ~,,(E.) I -Ix -. Y. e or(v)}.</Paragraph> <Paragraph position="7"> reachabb:+(X) denotes its reflexive transitive closure. For .,y q E f,({\[X --, a./7\] \[ X --+ c~fl G CI&quot;(I')}), wc define the I-Closure(q) as q untied to Ihe set</Paragraph> <Paragraph position="9"> i- Closure formalizes the construction of the deterministic version of a nondeterministic finite autonhM, On as it is done for the dJ'a of (Jl&quot;Cls. Its special feature is that it uses o,,ly those X ~ ,* E CI&quot;(P) f,,lfilling X C- e,,,,,p,(K). If 6<dq, ~):: 1-Cto.~,,i.4{\[xs->-s~.~J\] I \[A'j .... J.~fls\] c q)), 5c,,(q, fi) := i-CIosurc({\[S(fi) --~ &quot;P(I')\] I ~\[x: -, .~.s(fd/~s\] c q}), (2(;' is the set of the states given by {'l I =~&quot; C= (eo,.p(IC) U &quot;rU OF(P))&quot; : ~;(S., u) = 'd, ,,,,~ l,'~, := (,~ c O. I\[X ~ -.\] c q,x -. <,< ~ cs,'(v)) is the set of the final states.</Paragraph> <Paragraph position="10"> sl /x ~ ~_ I d s,-~ .s_ ~s ~ xX.D, I s~'xxD*l Iv_.,, I D ~ 'D' I I x -~'~ I ~_12_~ .... ID ~.,l The first difference to the usual context-free automaton is that we allow transitions nnder fl C CI;'(P), if we have S(fl) C/~ comps(It). The second point is that we use 1-Closure instead of the usual closure. DFA(G') for the example grammar is shown in Figure 2. The conflict is removcd because we can now distinguish two cases by looking at the information additionally stored. Ill \[SV90\], only the first i(te~t was realized ohviously leading to a weaker antolnaton.</Paragraph> </Section> <Section position="6" start_page="403" end_page="403" type="metho"> <SectionTitle> 5 The Analysis </SectionTitle> <Paragraph position="0"> &quot;Po use DFA(G), the usual pnshdown is extended by a data-structure consisting in a list of partial derivation trees. This list future collects all information determined by Reduce's relative to first nonterminal components and is used to drive the transitions under p G CF(P) in DFA(G) as soon as we have to investigate nontermlnal components Xi ~ compl(K. ). The change between the two different kinds of control leads to a new characterization of conflicts.</Paragraph> <Paragraph position="1"> For better explanation, we use a list past paralM to future where all Reduce operatimls performed so far ale stored. An example for tile new data-structnres is shown in Figure 3. We use it to explain how they arc built up during the analysis. The first operations on this past were</Paragraph> <Paragraph position="3"> know that any Reduce takes place at the end of the sententlal {'orm generated so far. This remains true. Thus, we Call a.rgue completely analogolls as f~kl' its l)(18l iS COtlC(!l'lICd. But we investigate coupled productions as, e.g., (Z,,Z~) --, (w~A,U,V~), A, (Z~,Z~), (Vl,U~) C- *C. We know that coupled nonterminal components are located at the same depth of the derivation tree and that they are substitnted by components of tile same coupled production. Therefore, when insertiilg any p, S(p) G eomlh(K. ), in past, e.g., Zl -~ ,viA, we additionally insert the coupled productions, e.g., Z2 --+ UtU2, in future. In general, Yt -~w4 &quot;-'-tll &quot;,~-past fld.&quot;~ B2 -&quot; B:l -~ V2 2-, i&quot;-. I//%, Z, N ws wo a D Z~ Ql (22 5,\ A</Paragraph> <Paragraph position="5"> there are two cases to distinguish depending oil l)l inserted in pnst. If Dips) cont,(ins only sYmbols in /C\[1\] O 'F (i.e.</Paragraph> <Paragraph position="6"> only uncoupled ones), the conpled p2,...,p,, are inserted as the first up to the (r - 1)th element in future. (E.g. for (Z1,Z~) ---+ (wlA, U1U'~).) Otherwise, we behave as it is do,le for (Y,,~) ~ (Z~N,Z./&Q2)in V@,re a. ~ .... the sllbtr~es i,, Iut,,re for those sy,nbols i. &quot;(V~) ..... &quot;D(V,,) coupled to first comlmnents in 7)(pt) become the sons of these elements. T|ms, wc maintain the property that the symbols at each fixed depth ill past and future together form an element of 13D(~,T).</Paragraph> <Paragraph position="7"> Thereby, in addition to Shift's which are handled as usual, we know what to do during a se(ltlelice Of Reduce operatk)ns relative to elements of co,np~(K). Now, let us be in the situation that we have to use the information in future, e.g. a transition under B2 --+ a =: pi frmn the topnlost state. Then, we create a pointer prise(tee walking on future. We ptlsh ~Sc;(qtov, pl) ;rod make presence point onto the first son {&quot; o1' &quot;D(pi). Let q be the new topmost state. We have to distinguish three cases: GT': If ~&quot; is tile next input symbol, we push ~a(q,~).</Paragraph> <Paragraph position="8"> Otherwise, the whole, input is rejected, preseuce now points on the brother of ~'.</Paragraph> <Paragraph position="9"> e G eomp(~) \ comlh(J~): fulurealready stores theexp;tnsion ~ --+ ft. We push 6c~(q,( -- fl). presence now points on the first symbol in ft.</Paragraph> <Paragraph position="10"> (7_ eompl(K): \],du,'e does not store information +tl)oltt (, but ~ and its coupled components rel)resent ;t complete independent analysis probh:n~ which has to bc solved rccurslvely. E.g., this is the casc for D, (I:~, U~) ...... I (q~, Q~). The recursive call of the procedure starts with tlu'. topmost state since, it contains all items \[~&quot; -~ .~,\]. Each recursion needs separate data-structures. I)etails are described in \[Pitg3\].</Paragraph> <Paragraph position="11"> If l)Feserlce encollllters no brot\]ler~ We }13.ve to redllce. Let Y -~ 3' be the production at whose last synlbol presence points. We pop 171 -I- 1 states. The additional Imp conll)ared to the context.-fi'ee case results from the transition under }: ~ 7. peesenee walks to tim brother of Y in fut,,re and we push 6c;(q', Y) if q' is the IleW toi)nlost state. 1\[ Y is the root of the tirst tree in f,l,,'e, its complete subtree is nlovcd \['rom flttm'e to past and presence is deleted.</Paragraph> <Paragraph position="12"> ~Ve Ollllllt 19 ~ 1) whell redueiug its last conll)onent.</Paragraph> <Paragraph position="13"> &quot;\['hus, ollr result ix a uiphtmost dcrlval.ion in inverse or(h:r.</Paragraph> </Section> class="xml-element"></Paper>