On the Universal Generation Problem for Unification Grammars

The universal generation problem for unification grammars is the problem of determining for an arbitrary grammar G and an arbitrary feature structure F whether there exists at least one sentence that G derives with F. If F is acyclic, Wedekind and Kaplan (2012) have shown that the problem is decidable for LFG (Kaplan and Bresnan 1982) and PATR (Shieber et al. 1983) grammars. They prove that the set of strings that a grammar relates to an acyclic feature structure can be described by a context-free grammar. Decidability of the problem then follows because the emptiness problem is decidable for context-free languages. For cyclic feature structures they demonstrated by example that the set of strings that a grammar relates to an input might not be context-free, but they did not further investigate the formal properties of the languages that are in general related to cyclic structures.

In this brief note, we show the undecidability of the universal generation problem by reduction from the undecidable emptiness problem for the intersection of two context-free languages. We provide a proof for LFG- or PATR-style grammars that associate feature structures with trees derived in accordance with a context-free grammar. Our result also applies to other systems such as HPSG (Pollard and Sag 1994) whose formal devices are powerful enough to simulate, albeit indirectly, the effect of context-free derivation.

To state the universal generation problem more formally, recall that a unification grammar G defines a binary derivation relation Δ_G between terminal strings and feature structures, as given in (1).

• (1) Δ_G(s,F) iff G derives terminal string s with feature structure F

For the reduction of the emptiness problem for the intersection of two context-free languages, we can, without loss of generality, assume that the context-free languages are ε-free. These languages can be described by grammars in Chomsky normal form, that is, by context-free grammars G = (N,T,S,P) with nonterminal vocabulary N, terminal vocabulary T, and start symbol S where every rule in P is of the form A →BC with B,C ∈ N, or A →a with a ∈ T.

For the proof we first define for each context-free grammar G in Chomsky normal form two LFG grammars that both derive L(G) and that associate with each derivable terminal string feature structures (f-structures) that provide slightly different encodings of the derivable string.

Figure 1 illustrates a c-structure and the f-structures associated with it by type 1 and type 2 string grammar derivations.¹ The attributes l, r, b, and e are mnemonic for ‘left’, ‘right’, ‘begin’, and ‘end’, respectively. For later reference, we also depicted the constant root that we uniformly use to instantiate the ↑ of a derivation that refers to the c-structure root; root then labels the f-structure element to which it refers in the minimal model of the f-description. (In Kaplan and Bresnan’s 1982 terminology, root corresponds to the f-structure variable associated with the c-structure root, usually notated by f₁.)

Both types of string grammars have in common that they have G as their context-free skeleton and that for every string in L(G), the f-structure for each string grammar encodes both the string itself and also the branching structure of a derivation in G that leads to that terminal string. The f-structures derived by the two types of grammars vary only slightly in the labels that they use to encode those properties. An f-structure of a type 2 grammar derivation for a given string shares the ‘begin’ attribute (b) with the f-structure of a corresponding type 1 grammar derivation, but it has distinct ‘left’, ‘right’, and ‘end’ attributes (l', r', e').

Because the derived f-descriptions can never be unsatisfiable (the string grammars do not contain atomic values), the f-structure constraints of the string grammars do not actually filter the language of the context-free grammar. Thus G and its string grammars must have the same language L(G) = L(String₁(G)) = L(String₂(G)). By induction on the depth of the derivation trees it is also easy to see that the minimal solution of the f-description of a derivation of a terminal string s is acyclic and single-rooted, and satisfies (rootbs′) = (roote) and (rootbs′) = (roote'), respectively, if and only if s′ = s. That is, these grammars both encode their terminal strings in their respective (rootb) to (roote)/(roote') paths.

Before going into the details of the undecidability proof, we first give an outline of the proof idea. For the reduction, we have to construct for two arbitrary ε-free context-free languages L₁ and L₂ an LFG grammar G and an input structure F such that the set of terminal strings that G derives with F is empty if and only if the intersection of L₁ and L₂ is empty. Because every ε-free context-free language is derivable by a context-free grammar in Chomsky normal form, we can make the LFG grammar G by combining the productions of String₁(G₁) and String₂(G₂), for two arbitrary context-free grammars G₁ = (N₁,T₁, S₁,P₁) and G₂ = (N₂,T₂, S₂,P₂) in Chomsky normal form. To avoid undesired interactions between the rules of the two string grammars, we assume that the sets of nonterminals of G₁ and G₂ are disjoint (this is without loss of generality because nonterminals can always be renamed).

We observed already that the string grammars String₁(G₁) and String₂(G₂) associate with any c-structure derivation of a terminal string s₁ in G₁ and any c-structure derivation of a terminal string s₂ in G₂ f-structures that encode s₁ and s₂ as their respective (rootb) values. By construction of the string grammars, the only paths that the two f-structures share are the paths (rootbs′) where s′ is a common prefix of s₁ and s₂. Thus, if we define G to consist of the rules of String₁(G₁) and String₂(G₂), and a start rule that expands S to S₁ S₂ and forces the f-structures for s₁ and s₂ to unify, their (roote) and (roote') paths become reentrant ((roote) = (roote')) if and only if s₁ and s₂ are identical. G then assigns to a terminal string an f-structure with reentrant (roote) and (roote') paths if and only if it has the form s′s′ and s′ is in L(G₁) ∩ L(G₂).

The folding into F is accomplished by annotations of G’s start rule whose contribution to a derivation in G is depicted in Figure 2. As earlier, we include root for later use. Obviously, if (roote) = (roote') holds in the unified f-structures of the string grammars, then the unification of the string grammar f-structures and the structure in Figure 2 yields F. Otherwise, their unification results in a structure that only properly subsumes F. This is because neither (roote) nor (roote'e) exists in the unified f-structures of the two string grammars, and therefore their values in the structure in Figure 2 are not merged when the structures are unified. Thus G derives with F exactly the set of strings s′s′ with s′ in L(G₁) ∩ L(G₂). Hence, this set is empty if and only if L(G₁) ∩ L(G₂) is empty.

We now give a rigorous statement and proof of our undecidability theorem.

Theorem

For an arbitrary LFG grammar G and an arbitrary f-structure F it is undecidable whether {s|Δ_G(s, F)} = ∅.

Proof

If s₁ = s₂, then FD ⊢ (roote) = (roote'), since (rootbs₁) = (roote) and (rootbs₂) = (roote') follow from FD. From (roote) = (roote') and the instantiated annotations of the S rule, we get (root x = root), for all x ∈ {b, l, r, l', r'} ∪ (T₁ ∩ T₂). With these equations we can then derive from (rootbs₁) = (roote) and (rootbs₂) = (roote') equations root = (root x), for x ∈ {e, e'}. Thus F must be the f-structure that we obtain from a minimal model of FD.

Now suppose s₁ ≠ s₂. Let FD₁ and FD₂ be the f-descriptions of the string grammars, , , with n₁ and n₂ instantiating ↓ in the annotations at S₁ and S₂, and . By construction of G, the only terms shared by the deductive closures of and are the common subterms of (rootbs₁) and (rootbs₂). Thus , because otherwise and would imply s′ = s₁ and s′ = s₂, as we saw earlier. Because obviously (rootee) and (rootee) do not occur in any equation derivable from FD′, (roote) = (roote′) cannot follow from FD either, and F cannot be the f-structure for FD.

Thus {s|Δ_G(s,F)} = {s′s′|s′ ∈ L(G₁) ∩ L(G₂)} and hence {s| Δ_G(s,F)} = ∅ if and only if L(G₁) ∩ L(G₂) = ∅. Since the emptiness problem for the intersection of context-free languages is in general undecidable, the generation problem must be undecidable too.

As a consequence of this theorem we know that there does not exist a general generation algorithm, at least if cyclic input structures are considered as legitimate inputs.

We note that the grammars constructed in this proof are off-line parsable (cf., e.g., Kaplan and Bresnan 1982; Johnson 1988; Jaeger et al. 2005). Off-line parsability is sufficient to guarantee the decidability of the recognition/parsing problem even for cyclic f-structures. But Wedekind and Kaplan (2012) have shown that off-line parsability is not necessary to guarantee that generation from acyclic structures is decidable, and the grammars in this proof demonstrate that it is not sufficient for cyclic structures.

Off-line parsability typically bounds the size of the c-structures of a string by a function of the length of that string. This works for parsing because the size of the f-structure is bounded by the size of the c-structure, but it is insufficient for generation because it does not constrain the structural correspondence between the c- and f-structure (see also Dymetman 1991). A single constraint that guarantees decidability for both parsing and generation must not only bound the size of the f-structures for a terminal string by the length of the string, but it must also ensure, as we have learned from the proof herein, that the determination of the terminal strings for an f-structure can be achieved with finite control.

The author wishes to thank Ron Kaplan for his insightful comments and helpful suggestions during the preparation of this paper, and the four anonymous reviewers for their valuable feedback on an earlier draft.