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<Paper uid="E95-1007">
  <Title>Some Remarks on the Decidability of the Generation Problem in LFGand PATR-Style Unification Grammars</Title>
  <Section position="3" start_page="0" end_page="47" type="intro">
    <SectionTitle>
2 Preliminaries
</SectionTitle>
    <Paragraph position="0"> The unification grammars we want to consider consist of rules with a context-free skeleton and a set of annotations associated with the constituents mentioned in the rules. Typical examples taken from</Paragraph>
    <Paragraph position="2"> Figure 1 Examples of rules in LFG (left) and PATR-II format (right).</Paragraph>
    <Paragraph position="3"> mal definition of those grammars we reconstruct the annotations as formulas of a quantifier-free sublanguage of a classical first-order language with equality whose (nonlogical) symbols are given by a finite set of unary partial function symbols and a finite set of constants. For the translation of LFG and PATR-II annotations we regard the attributes (in figure 1: SUB J, PRED, AGR, NUM, PER) 58 unary partial function symbols and the atomic values (in figure 1: JOHN,  SG, 3RD) as individual constants. Furthermore, we assume for a context-free rule of the form A ---&gt; w (w e (VN U VT)*) that the variable x0 is associated with A and that for each occurrence wi in w there is a variable xi which is associated with wi. For the formal reconstruction of LFG's we assume that each occurrence of $ in the annotation of w~ corresponds to an occurrence of xi and that each occurrence of 1&amp;quot; corresponds to an occurrence of x0. For grammars in PATR-II format we suppose that occurrences of categories in the annotations correspond to the associated variables.</Paragraph>
    <Paragraph position="4"> Before we give the definition of the grammars we want to investigate, we introduce the following notation. In the following we use S\[xl, .., x~\] to indicate that the variables occurring in the set of formulas S are included in {xl, .., Xn} and S(Xl, .., xn) if the set of variables occurring in S is exactly {Xl,.., xn}.</Paragraph>
    <Paragraph position="5">  1. DEFINITION. A unification grammar is a tuple  (VN , VT , S, F1, V, V, R&gt;, consisting of a finite nonterminal vocabulary VN, a finite terminal vocabulary VT, a start symbol S E VN and a feature-description language L determined by a finite set of unary partial function symbols F~, a finite set of atomic values V and a denumerable set of variables 1 V= {x~ I a e N*} with x~ #x,, for a # a'.</Paragraph>
    <Paragraph position="6"> All vocabularies are pairwise disjoint. R is a finite set of rules of the form r = ((A,w),S~\[xo,..,xl~l\] } (zi E 1;), with (A, w) e VN x (VN U VT)* (a context-free phrase structure rule) and S~\[x0, .., xl~l\] a finite set of (quantifier-free) literals of L. 2 According to our definition the LFG rules in figure 1 are now expressed as depicted in (la) and the PATR-II rules as given in (lb). Note that the structure of the terms is now &amp;quot;mirror imaged&amp;quot;, since we assume the attributes to be unary partial function symbols.  (1) (a) (S -+ NP VB, {SUBJ Xo ..~ xl,xo ~ x2}) (NP --~ John, {PRED XO ~ JOHN}) (b) (S --&gt; NP VP, {AOa z2 ~ AGR xl}) SNUM AGR XO ~ SO,1 \ (NP --+ Uther, \].PEg AOR Xo ~ 3RD~/ For the definition of the sentences derivable by a unification grammar we have to specify first what derivations are.</Paragraph>
    <Paragraph position="7"> 2. DEFINITION. A sequence of pairs ~r0...~rn with</Paragraph>
    <Paragraph position="9"> n iff for each 7ri = (B \[..A~..\]~, S) (0 &lt; i &lt; n) there is a rule r (A -+ w~ k = ..win, S~) such that = .., ,~\]~..\]~,S~).</Paragraph>
    <Paragraph position="10"> In the definition we assume that the order of the arcs of a tree is encoded by numbering the arcs and that each node is identified with the sequence of integers numbering the arcs along the path from the  root (O) to that node. In our bracket notation we add to a constituent its root node as the right and its root node label as the left index. In order to be able to refer to the c-structure derivation and to the sequence of feature descriptions and to have access to the nodes which are substituted in each step of a derivation, we define for a derivation 7r three other sequences.</Paragraph>
    <Paragraph position="11"> 3. DEFINITION. Let ~r be a derivation of length n. We then define two sequences w and &amp;quot;)' for each i=O,..,n with lh=(Tc, S) by wi=Tc and</Paragraph>
    <Paragraph position="13"> Let S be a set of literals and 0 a unary partial mapping over the set of terms. Then the expression S\[0\] denotes the set of expressions obtained from S by simultaneously replacing each occurrence of a term ~- in each formula in S by 0(T). The feature description derived by zr is then defined by means of the following operation.</Paragraph>
    <Paragraph position="14"> 4. DEFINITION. If ~r is a derivation of length n then the feature description derived by 7r from h to k</Paragraph>
    <Paragraph position="16"> and apply the S-rule in (la) and the following VP-</Paragraph>
    <Paragraph position="18"> For the steps depicted above the sequence w is given by wl = 0 and w2 = 2 and the feature description derived by 7r from 0 to 2 (S~_~2) is</Paragraph>
    <Paragraph position="20"> Sentences are then defined as follows.</Paragraph>
    <Paragraph position="21">  5. DEFINITION. A terminal string w (w E V~) is  a sentence iff there is a derivation (So, 0) = r0..Trn with Wn = S\[w\]0 and 3x~1 ..x,~ A S~-&amp;quot;~n(X~tl' &amp;quot;&amp;quot; Z~tm) satisfiable. 3 In the following we write S&amp;quot; for S~_+n if the interval covers the whole derivation, i.e. if ~r is of length n.</Paragraph>
    <Paragraph position="22"> Since a specific reduction algorithm and a few model-theoretic facts required in the proofs later on can be introduced by showing how satisfiability of such existential prenex formulas can be decided, we will continue with a short excursion on satisfiability.</Paragraph>
    <Section position="1" start_page="46" end_page="47" type="sub_section">
      <SectionTitle>
2.1 Satisfiability
</SectionTitle>
      <Paragraph position="0"> In order to test whether for a given finite set of literals S of a feature-description language (2) (2) 3z~..zt A S(x~,.., zl) is satisfiable, we can exploit by skolemization well-known test procedures available for quantifier- and variable-free sets of such literals. Let C be a set of Skolem-constants (\[{xl, ..,xz}\[ = ICl) and 0 be a bijective function from {Xl, .., xt} to C, then (2) can be tested by testing the set of literals (3) over L(C) 4 (3) S\[0\], since (2) and (3) are equi-satisfiable. In the following we complete the procedure by introducing a reduction algorithm that reduces a set of literals (3) according to a measure in a sequence of measure decreasing rewrite steps to a deductively equivalent set (4) (in reduced form) (4) (S\[e\])p, which is satisfiable iff the terms 7- of all inequalities T ~ 7- of (4) do not occur as subterms in equations of (4).5 For the proof we first introduce a few definitions and some notation. Let 7- be the set of terms of a variable-free feature-description language L(C). Then an injective function m * \[7- ~ ~l*\] is a measure iff it satisfies the following conditions for all T, T' * 7&amp;quot; and a * FI*:</Paragraph>
      <Paragraph position="2"> For literals and sets of literals S we extend a measure m as usual by m((.~)7- ~ 7-')= m(7-)+ m(7-') and re(S) = Era(C).</Paragraph>
      <Paragraph position="3"> Ces In the following we use 7- ~7-' iff m(7-) &gt; m(7-') and 7-~7-' to denote ambiguously 7- ~ 7-' or 7-~ ~ 7-. Let S be a set of literals then E denotes the set of all equations in S, 7-s the set of terms occurring in the formulas of S (7-s = {~-, 7-' \[ (&amp;quot;~)7- ~ 7-' * S}) and SUB(Ts) the set of all subterms of the terms in 7~</Paragraph>
      <Paragraph position="5"> For the construction of a reduced form we need a specific partial choice function p which satisfies p(S) * {7- ~7-' * SIT * SVS(Ts\{r~.,.,})} if the specified set is nonempty and undefined otherwise. null  6. DEFINITION. For a given finite set of literals S and a choice function p we define a sequence of sets</Paragraph>
      <Paragraph position="7"> aThe feature-description language which in addition to L provides a distinct set of Skolem-constants C'. Cf.</Paragraph>
      <Paragraph position="8"> the appendix for more details.</Paragraph>
      <Paragraph position="9"> ~The algorithm is adapted from Statman 1977 and Knuth and Bendix 1970 and first applied to feature-description languages by Beierle and Pletat (1988). Since m(Sm) &gt; m(Sp,+l ) ifp is defined for Sin, the construction terminates with a finite set of literals.</Paragraph>
      <Paragraph position="10"> If we set Sp = Spt ; with t = min{i \[ Sp, = Sin+ ~ } the following lemma can easily be proven by induction on the construction of Sp. 6  7. LEMMA. For Sp it holds that: (i) S ~F S o, (ii) if T~T' C S o then T C/ SUB(Tsp\{r~r,}).  Since Sp is obviously not satisfiable if it contains an inequality T ~ 7 and 7 occurs as a subterm in Ep, the whole proof is completed by showing that we can construct a canonical model satisfying Sp if Sp does not contain such an inequality.</Paragraph>
      <Paragraph position="11"> For the model construction we need the set T~p = {r e SUB(TE,) \[ -~3T'(T ~T' e Ep)} and the function h c E \[SUB(7-Ep) ~ 7-~,\] which is defined for each 7- e SUB(TE,) by f ,T'(7-~T' Ep) if 7- E, h e (T) = * f\[ T'c \[7- otherwise.</Paragraph>
      <Paragraph position="12"> That h e is well-defined results of course from 7(ii). 8. DEFINITION. For a set of literals S o the canonical term model is given by the pair Mp = (Hp, .~p), consisting of the universe</Paragraph>
      <Paragraph position="14"> and the interpretation function ~p, which is defined forc*VUC, f*/'l and 7-*Hpby:</Paragraph>
      <Paragraph position="16"> For Mp which is well-defined the following lemma holds:  9. LEMMA. If 7- is a subterm of Ts, then (i) ~p(7-) = he(7-), if 7- * SUB(TE~), (ii) 7- * SUB(T~), if T * Dom(.~o). PROOF. (By induction on the length of 7-.) The lemma is trivial for constants. By showing (i) before (ii) we get the induction step for a subterm fTof Ts, in both cases according to</Paragraph>
      <Paragraph position="18"/>
      <Paragraph position="20"> according to lemma 7(ii). Now, if (i) fT * SUB(TEp) then ~p(f)(T) is defined and equal to h~(fr) and  (ii) if fr * SUB(Ts,) and .~o(fT) is defined then fr * SUB(TE~). \[\] On the basis of lemma 9 it is now easy to prove: 10. LEMMA. VT ~ T * So(7&amp;quot; C/ SVB(&amp;quot;fEp)) --~PMp S O. PROOF. (If the condition is satisfied ~M, C/ holds for every C/ * So. ) If C/ = ~'~T' * S o with m(T') &lt; m(r), then v' * T~o by 7(ii) and hence hC(T ') = T'. We get then h~(~ -) = T' for m(T') = m(T) by T' = T and for m(~-') &lt; m(~-) by the definition of h ~, since r C/f T~.  Thus ~p(T) = ~p(T') by 9(i) and hence ~Mo C/.</Paragraph>
      <Paragraph position="21"> Assume C/=TCT'. If T~7' were satisfied by Mp, we would get ~p(T)= ~p(T') and by 9(ii) T,T'*SUB(TE,). Since 7(ii) ensures he(r) = h~(~ -') = v = ~-', we would have ~- C/ r * Sp with T * SUB(TEo). \[\]  Finally it should be mentioned that Mp is a unique (up to isomorphism) minimal model for Sp, i.e. if M is a model for So, homomorphic to Mp, then every minimal submodel of M that satisfies S o is isomorphic to Mp.</Paragraph>
    </Section>
  </Section>
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