Since the f-structures assigned to the sentences are always subsumed by the se­mantic representations they contain, a semantic-driven generator has to compute for a given semantic representation a sentence with an f-structure $> that is subsumed by the input (in the following, notated by <£>' C $). To state the underlying decidability problem more formally, we need the fact that a unification grammar G defines a binary relation Ac between terminal strings w and f-structures <É>, as given in (2) (2) Ag (w, <3>) iff G assigns <£> to w.

The problem of determining for a given semantic representation <£>' whether there is a sentence with an f-structure <É> that is subsumed by the input turns out then to be an instance of the problem of whether we can decide (3) for any given input <&'.

The undecidability of the generation problem in (3) was shown for definite clause grammars by Dymetman (1991), who reduced the problem to Hilbert's Tenth Prob-

* Institute for Natural Language Processing, University of Stuttgart, Azenbergstr. 12, D-70174 Stuttgart, Germany. E-mail: juergen@ims.uni-stuttgart.de

Let G be a context-free grammar in Chomsky normal form whose nonterminal vo­cabulary, terminal vocabulary, start-symbol, and rules are given by (Vn, Vt,S,R). I.e., each rule has the form A —> e, A—» a or A -> BC with A, B, C G V^, a G VT and e de­noting the empty string. A string grammar String(G) for G is a unification grammar (Vn,Vt,S,Rs) whose rule set is determined as follows. In the first step we construct for each context-free rule r = A —> w a set of annotations Sr:

{(x0 in) « (xi in), (xi out) « (x2 in), (x0 out) fa (x2 out)} if w — BC.

Figure 1 illustrates the f-structure encoding of a terminal string generated by a simple string grammar. By induction on the depth of the derivation trees, it can eas­ily be shown that G and String(G) have the same language and that the f-structure assigned to a terminal string w encodes w, as stated more precisely in the following Lemma:

1 We separated this construction out of the main proof, since it might be useful for analyzing other problems.

2 We used PATR-style notation, since it facilitates the construction of string grammars. For LFG grammars where we do not have the possibility to refer from one daughter to her sister (necessary for (x\ out) rs (%2 in)) we need a slightly more complex construction. If w — BC then B has to be annotated by (T Bi) rs j. and (t in) rs (j. in) and C by (t cfc) rs L (T out) rs q out), and

(î Bi out) ps (l in). If w = a we need (î in first) rs a and (t in rest) rs (î out) and for w = e the equation (f in) rs (| out). With this construction we get the same undecidability results for classical LFG grammars. The only difference is that the constructed grammars are tree grammars rather than string grammars.

A sample constituent structure and the associated f-structure provided by a simple string grammar. The metavariables of the rules are instantiated by the variables attached to the nodes of the constituent structure. To each variable x, is assigned the f-structure element

Let String(G) be a string grammar. Then L(G) = L(String(G)) and if there is a deriva­tion of a terminal string w with root SXo and f-structure $ then the substructure of $ which comprises the elements accessible from a0 in $ is a minimal solution of

If we combine two arbitrary string grammars in such a way that the string encod­ings of the derived terminal strings get unified, we can show the undecidability of the emptiness problem by a simple reduction to the emptiness problem of the intersection of arbitrary context-free languages.

It is undecidable for an arbitrary unification grammar G whether L(G) = 0. Proof

Let G = {Vjj, V\, S, R) and G = (V^, V,S,R) be context-free grammars for two arbitrary context-free languages. Without loss of generality, we can assume that Vjj n Vjy = 0 and that each rule in Rl (i = 1,2) is in Chomsky normal form. On the ba­sis of String(G) and String(G) we construct a unification grammar G = (VN, Vj, S,R) such that # is a new atomic value not in VY- If we assume for G constant-consistency (i.e., axioms of the form h a sé b for all atomic values a, i> € Vt U {#} with a^b) then the problem whether L(G) = 0 reduces to the undecidable problem whether L(Ga) n L(G) = 0. In order to get a derivation of a well-formed terminal string wwfrom S with w? derived from S and w from S, w must be identical with xv, since both string encodings get unified by the S-rule and (xo out first) « # ensures that one string is not a proper prefix of the other. Thus, L(G) = {ww \ w e L(Ga) n L(G)} and L(G) = 0 iff L(GX) n L(G) = 0. ■

3 The whole f-structure encodes the complete difference list derivation of wx — x, which is induced by the derivation tree by relabeling each (nonterminal) node dominating substring v of uvz = w by vzx — zx, since the annotations of each rule of the form A —► BC encode the difference list of the mother as the concatenation of the lists of its daughters (X — X2 = X — Xi + Xi — X2).

4 The annotation (xq out first) sa # is not necessary if acyclicity is assumed.

For an arbitrary unification grammar G and an arbitrary f-structure it is undecidable whether there is an f-structure $ and a terminal string w such that $' C $ and Ag(b;,$).

Although it might be argued that we show the undecidability on the basis of a rather special case, namely the smallest f-structure, the undecidability of the empti­ness problem is nevertheless sufficient, since we always get a (superficially) less triv­ial direct proof of the corollary by using any proof of the theorem and adding some (new) nontrivial input information to the S-rule. If we add, for example, the equation (xq Sem) « 1 to the S-rule of our proof then the problem whether we can find for [sem 1] (= $') an f-structure $ and a terminal string w such that [sem 1] ç $ and Ag(w, $) reduces to the undecidable problem whether L(G) = 0 as well.

Our construction shows that an LFG or PATR grammar G can simulate the valid computations of an arbitrary Turing machine M, since they are known to be speci­fiable by the intersection of two context-free languages. Since L(M) = 0 is undecid­able, the emptiness problem of L(G) must be undecidable too. By adding a bit of semantic representation to the S-rule these properties are trivially carried over from L(G) to the set of possible realizations assigned to by G, given by the lan­guage {w I 3i>($' C$A Ac(w, <&))}. Our proof construction works, of course, even if the grammatical formalisms satisfy the off-line parsability restriction. Thus, the decid­ability of the membership problem—similar to context-sensitive grammars—does not imply the decidability of the emptiness (and the semantic-driven generation) problem.

From a cognitive point of view it seems quite unrealistic that our language gen­eration capabilities require mathematical models of Turing machine power. Hence, natural language grammars (of the LFG and PATR formalisms) must satisfy condi­tions that do not allow us to show the undecidability of the problem. We assumed the semantic representations to be structurally unrelated to the f-structures they sub­sume. It seems more plausible that there is a proportion k that bounds the size of an

5 Van Noord (1993) used the equation (xg solution) rs yes in his proof.

6 If the context-free grammars G and G are off-line parsable then the unification grammars G used in the undecidability proofs are off-line parsable as well. Since we can decide e e L(G') for any context-free grammar G' and can reduce G' to an off-line parsable grammar G" with L(G') - {e} = L(G"), ^(G) n L(G) = 0 and hence L(G) = 0 must be undecidable even if the grammars satisfy the off-line parsability restriction.

L(G) = {w I 3$(AG(w, *))} = {w I 3$(_L ç $ A AG(w, *))}.

Ron Kaplan had the idea that the proof construction which we used in Wedekind and Kaplan (1996) might be useful for other purposes. Thanks to him for valuable suggestions.