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<Paper uid="C00-1062">
  <Title>LFG Generation Produces Context-free Languages</Title>
  <Section position="3" start_page="425" end_page="426" type="metho">
    <SectionTitle>
2 A Context-free Grammar
</SectionTitle>
    <Paragraph position="0"> for Gena(F) An inl)ut structure F for generation is t)resented as a hierarchical attribute-value matrix such as the oue in Figul&amp;quot;e 1, repeated here in (5).</Paragraph>
  </Section>
  <Section position="4" start_page="426" end_page="427" type="metho">
    <SectionTitle>
TENSE PAST
</SectionTitle>
    <Paragraph position="0"> An fstructure is an attrilmte-valut sl;ructure where the values a.re either subsidiary atl;rilxlte-vahm rimtrices, symliols, semantic forms, or sei;s of subsidiary structures (not shown in this example).</Paragraph>
    <Paragraph position="1"> (6) A structure 9 is contained in a structm'e J' if and only if: .q= f, f is a set and g is eonl;aintd in an dement of f, or f is an f-structm'e and 9 is contained in (fa) for some attribute a.</Paragraph>
    <Paragraph position="2"> in tssence, 9 is conl;ained in f if 9 can 11o located ill f by ignoring sonm enclosing SUl)erstructure. For any f-structure f, the sel; of all units contained in f is then defined as in (7).</Paragraph>
    <Paragraph position="3"> (7) Units(f) - {glo is contained in f} Note t;hat Units(f) is a tinit;e set for any f, and U'nits(f) is the range of any C/ that A(; associai;es with a parl;icular intmC F.</Paragraph>
    <Paragraph position="4"> The. (:-strucl;m'es and (/) corresliondences tbr F are the unknowns i;hai; nmsI; be discovered in the process of generation so l;hat the 1)rol)er instantiatcd descrip-Lions can \])e constructed and cvahtal;e(l, llowever, since thtre ix only a tinite mlml)er of l)ossible terms thai: can be used i;o designate the ltnil;s of t ?, we can produce a (Iinite) SUl)Cxsct of the, 1)r(/t)er instantiaW, d descriptions without knowing in advance the details of either the (;-sl;rucl;ure or ;4 1)articular (/). Let l;' be an f-structure tlmt has m (m &gt; 0) set elements. We introduce m + 1 distinct variables v0,..,v,~, which denote biuniquely the root refit of F (v0) and each net element of F (vi, i &gt; 0). 1 We consider the set of all designators of the tbrm (vi c,) which art defined in F, where a is a (possibly empty) sequence of attributes. The set of designators for a particular unit corresponds, of course, to the set of all possible fstrueture paths fl'om one of the vi roots to that unit. Thus, the set of designa.t;ors for all units of F in finitt, since the number of units of F is tinite and there art no cycles in F.</Paragraph>
    <Paragraph position="5"> The set of variables that we will use to construct the instantiated descriptions is the set 1/- consisl;ing of all vt where t in a designator of the set just defined. If l is the maximal arity of the rules in G, we will conskltr for the instantiation the set Z consisting of all sequences &lt;vto, vt,,.., vt; ) of variables of V of length 1., ..,n + 1, not containing any set tit;1 Multi-rooted sl, ructures would require ~ whole set of reel wu'iables, similm&amp;quot; I,o set elements.</Paragraph>
    <Paragraph position="6"> merit w~rial)le v,,~ (i = O, .., m) more dlan once. On the basis of this (finite) set of sequences, we define a (partial) fmwtion 1D which assigns to eat:h rule 7' E 1{ and each sequence I E 27 that is apl)ropriate for r an instantiated description.</Paragraph>
    <Paragraph position="7"> Let r be an n-ary LFG rule</Paragraph>
    <Paragraph position="9"> with annotated flmctional schemata SI...S,z. A sequence of variables I G 27 is appr'opr'iatc for r if I = @t0,vt,,.-,vl.) is of length n + 1 and ('Oi O-t(7) if ~,(1 ~--- (V i (Y') all(l (j&amp;quot; (7) =,Le Sj tj = a set element varial)lt vj if SE (j&amp;quot; o-) E Sj for all j = 1, .., n ((7' and C/ are (possibly tlnpty) sequences of attributes). (Note that (? (7) =$ reduces to J'=$ if a is empty.) If I is al)prot)riatt for r, then ID(r, I), the instantiated description for r and l, is defined as follows: N (s) m(,., n = U l ,&lt;sj, V,o, %), j=l where l:nsl.(,gj, vto, vtj) in the instantiated description produced by substituting vt0 for all occurrences of 1&amp;quot; in ,5'j and substituting vtq for all occurences of $ in Sj.</Paragraph>
    <Paragraph position="10"> If r is a lexical rule with a context-free skeleton of the fl)rm X ~ a every sequence I = (v,0&gt; of length \] is ~@propriate for r mMID is detined by:</Paragraph>
    <Paragraph position="12"> The instantiation using a.pprotn'iate sequences of variables, all;hough tinite, permits an elfectivt discrinfinal;ion of l;he fst, ructure variables, since it prorides diflbXeld; varial)les for the $% associated with diti'erent daughters i;hat have different flmction assigmnents (i.e., mmotations of the form (1&amp;quot; c,) =$ and (t (7') =$ with (7 C/ J), but identifies variables where fstructure variables are identified explicitly (j'=$) or where the identity tbllows by ratification, as in cases where the annotations of two diflbrent (laughters contain the same function-assigning equal;loll (J&amp;quot; (7) =$. Hence, we in fact have enough variables to make all the distinctions that could arise from any c-si;rueturt and C/ correspondence for the given f-structurt.</Paragraph>
    <Paragraph position="13"> The set of all possible instantiated descriptions is large but finite, since R. and Y are finite. Thus, the set IP(F) of all possible instantiated propositions for G and F is also large but finite.</Paragraph>
    <Paragraph position="15"> For the construction of the eonttxt-fi'ee grmmnar we have to consider those subsets of IP(F) which have F as their minimal model. This is the set D(F), again finite.</Paragraph>
    <Paragraph position="16">  (11) D(F) is tim set of all D C_ IP(F) such that F is a minimal model for D.</Paragraph>
    <Paragraph position="17"> We are now prepared to establish the main result of tiffs paper: (12) Lct G be an LFG grammar conforming to thc restrictions we have described. Then for&amp;quot; any f-structure F, the set GenG(F) is a context-free languaf\]e.</Paragraph>
    <Paragraph position="19"> empty, then Gena(F) is again the empty context-free language. If D(F) is not empty, we construct a context-free grammar Gr = (ARE, Tr, SF, RE} its the following way.</Paragraph>
    <Paragraph position="20"> The collection of nonterlninals ~rj,, is the (finite) set {SF} U N x V x I)ow(IP(F)), wtsere SF is a new root; category. Categories in NI; other than SF are written X:v:D, where X is a category in N, v is contained in 17, and D is an instantiated description in Pow(IP(F)). 2),, is the set T x {(/)} x {0}. The rules RF are constructed from the annotated rules R of G. We include all rules of the form:  (i) S,,, ~ S:v~o:D, for every D d D(F) (ii) X0:vto:D0--+ Xl:Vtl:Dl..Xn:vt:Dn s.t.</Paragraph>
    <Paragraph position="21"> (a) there is an r E R expanding X0 to X1..X,~, (b) Do = m(,., ..,v,o))u UD,, i=1 (c) if vv~ 6 (vtj c~) belongs to Dj then v,,, C/ vt,, (k = 1, .., v,) and (1,, C/ j) s.t. v,,, c (v,,,, o-') c (iii) X:vl:D -~ a:(/):~ s.t.</Paragraph>
    <Paragraph position="22"> (a) there is an r E R expanding X to a, (b) D = ZD(r, (vt)).</Paragraph>
    <Paragraph position="23">  We define the projection Cat(a::?\]:z)= a: for every category in NF U Tl,, and extend this function in the natural way to strings of categories and sets of strings of categories. Note that the set</Paragraph>
    <Paragraph position="25"> is context-free, since the set of context-free languages is closed under homolnorphisms such as Cat. We show that the language Cat(L(GF)) = Gena(F).</Paragraph>
    <Paragraph position="26"> We prove first that Gcna(F) C Cat(L(aA). Let c be an annotated c-structure of a string s with f-structure F in G. On the basis of c and F we construct a derivation tree of a string s' in G j,, with Cat(s') = s in two steps. In the first step we relabel each terminal node with label a by a:(~, the rook by S:vv0, each node introducing a set element with label X biuniquely by X:v~, and each other node ~This condition captures LFG's special interpretation of membership statements. The proper treatment of LFG's semantic forms requires a similar condition.</Paragraph>
    <Paragraph position="27"> labelled X by X:vt where * is a designator that is constructal)le from the function-assigning equations of the mmotations along the path from the unique root or set element to that node. On the basis of this relabelled c-structure we construct a derivation tree of s' in Gt,' bottom-up. We relabel each terurinal node with label a:(/) by a:(/):~) and each preterminal node with label X:vt by X:vt:D where D is defined as in (iiib) with r expanding X in c to a. Suppose we have constructed the subtrees dominated by X1 :Vtl:D1..X,z:vt.:D,, the corresponding subtrees in c are derived with r expanding X0 to X1..X m and the nlother node is relabelled by X0:vt0. We then relabel this mother node by Xo:vto:Do where Do is determined according to (iib). By induction on the depth of the subtrees it is then easy to verify that the instantiated description D of a subtree donfinated by X:vt:D is equivalent to the f-description of the corresponding annotated subtree its c. Thus, F must be a minimal model of the instantiated description of the root label S:v~0:D~, ~, Sl,. derives S:v~o:DF in GI, ~ and Cat(J) = s.</Paragraph>
    <Paragraph position="28"> We now show that Cat(L(GI~)) C Geno(F). Let c&amp;quot; be a derivation tree of s' in Gr with Uat(s') = s and supl)ose that the root (with label SF) expands to S:vv0:DF. We construct a new derivation tree c' that results from c&amp;quot; by eliminating the root. We then define a fimction C/' such that for each nonterminal node /t of c': C/'(IL) = vt if # is labelled by X:vt:D in c'. According to our rule construction it can easily be seen by induction on the depth of the subtrees that the, re nmst be an annotated c-structure c of G with the same underlying tree structure as c' such that for each node tt labelled by z:~/:D in c': (i) t* is labelled by a: in c, (ii) D is identical with the description that results from Dr, , the f-description of the sub-c-structure dominated by tt in c, by replacing each occurrence of an f-structure variable 'qS0/)' (usually abbreviated by f,) in D,~ by 4/(,,). Since (/'(It) = qS(,,) follows for two f-structure designators if (b'(#) = 4/(u), tim f description of the whole c-structure must be equivalent to DE mid thus Ac,,(s, c, C/, F) where ~ = ~b' o Ov and Cv is the unique flmction ttmt maps each ut to the unit of F that is denoted by t. QEI)</Paragraph>
  </Section>
  <Section position="5" start_page="427" end_page="427" type="metho">
    <SectionTitle>
3 An Example
</SectionTitle>
    <Paragraph position="0"> As a simple illustration, we produce the context-fl'ee gramnmr GF for the input (5) and the grmnmar in (2,3) above. The only designator variables that will yield useful rules are v~ 0 mid v(~ o sui33), in tim tbllowing abbreviated by v aim Vs. Consider first the context-fl'ee rules that correspond to the rules that generate NP's. If we choose the sequence I = (vs), the instantiated description for the determiner rule in (33)is (13).</Paragraph>
    <Paragraph position="2"> For the NP rule and the sequence {vs,vs,vs}, both daughter annotatiolls instantiate to the trivial description vs = vs, and this can combine, with many daughter descriptions. Two of these are the basis for the rules (16) and (17). The (laughter categories of rule (1.6) match the mother categories of rules (14) a,nd (15), all(1 the tlll&amp;quot;ee rllles together can derive the stting a:(~:0 student:(/}:{~. Rule (17), Oll the other hand, is a legi(;iinate rule but does not combine with any others to l)roduce a terminal string. 1\]; is a useless, albeit harmless, production; if desired, it tan be removed froln the set of productions 1) 3, standard algorithnts tbr COll(;exl;-\['l.ee gramnmrs.</Paragraph>
    <Paragraph position="3"> llf we contimm along in this rammer, we find that the rules in (18,1.9,20) are the only other useful rules that belong to G1,'.</Paragraph>
    <Paragraph position="4"> The grammar GI~' also includes (;he following sl;arl;ing rule:</Paragraph>
    <Paragraph position="6"> This grammar provides one derivation for a single string, a:(/):(/) student:(/):(/) Dll:(/):{/}. Applying Cat to this string gives 'a stlldent Dll', tim only sentence that this grammar associates with the inlmt f'd;ructure.</Paragraph>
  </Section>
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