A Specification Language for 
Lexical Functional Grammars 
Patrick Blackburn and Claire Gardent 
Computerlinguistik 
Universit£t des Saarlandes 
Postfach 1150, D-66041 Saarbrficken 
Germany 
{patrick, claire)©coli, uni-sb, de 
Abstract 
This paper defines a language Z~ for spe- 
cifying LFG grammars. This enables 
constraints on LFG's composite onto- 
logy (c-structures synchronised with f- 
structures) to be stated directly; no ap- 
peal to the LFG construction algorithm 
is needed. We use f to specify schemata 
annotated rules and the LFG uniquen- 
ess, completeness and coherence princip- 
les. Broader issues raised by this work 
are noted and discussed. 
1 Introduction 
Unlike most linguistic theories, LFG (see Kaplan 
and Bresnan (1982)) treats grammatical relations 
as first class citizens. Accordingly, it casts its lin- 
guistic analyses in terms of a composite ontology: 
two independent domains -- a domain of consti- 
tuency information (c-structure), and a domain of 
grammatical function information (f-structure) -- 
linked together in a mutually constraining man- 
ner. As has been amply demonstrated over the 
last fifteen years, this view permits perspicuous 
analyses of a wide variety of linguistic data. 
However standard formalisations of LFG do not 
capture its strikingly simple underlying intuitions. 
Instead, they make a detour via the LFG con- 
struction algorithm, which explains how equatio- 
nal constraints linking subtrees and feature str.uc- 
tures are to be resolved. The main point of the 
present paper is to show that such detours are 
unnecessary. We define a specification language 
£ in which (most of) the interactions between c- 
and f-structure typical of LFG grammars can be 
stated directly. 
The key idea underlying our approach is to 
think about LFG model theoretically. That is, 
our first task will be to give a precise -- and trans- 
parent -- mathematical picture of the LFG onto- 
logy. As has already been noted, the basic enti- 
ties underlying the LFG analyses are composite 
structures consisting of a finite tree, a finite fea- 
ture structure, and a function that links the two. 
Such structures can straightforwardly be thought 
of as models, in the usual sense of first order model 
theory (see Hodges (1993)). Viewing the LFG on- 
tology in such terms does no violence to intuition: 
indeed, as we shall see, a more direct mathemati- 
cal embodiment of the LFG universe can hardly 
be imagined. 
Once the ontological issues have been settled we 
turn to our ultimate goal: providing a specifica- 
tion language for LFG grammars. Actually, with 
the ontological issues settled it is a relatively sim- 
ple task to devise suitable specification languages: 
we simply consider how LFG linguists talk about 
such structures when they write grammars. That 
is, we ask ourselves what kind of constraints the 
linguist wishes to impose, and then devise a lan- 
guage in which they can be stated. 
Thus we shall proceed as follows. After a brief 
introduction to LFG, 1 we isolate a class of models 
which obviously mirrors the composite nature of 
the LFG ontology, and then turn to the task of de- 
vising a language for talking about them. We opt 
for a particularly simple specification language: a 
propositional language enriched with operators for 
talking about c- and f-structures, together with a 
path equality construct for enforcing synchronisa- 
tion between the two domains. We illustrate its 
use by showing how to capture the effect of sche- 
mata annotated rules, and the LFG uniqueness, 
completeness and coherence principles. 
Before proceeding, a word of motivation is in 
order. Firstly, we believe that there are practical 
reasons for interest in grammatical specification 
languages: formal specification seems important 
(perhaps essential) if robust large scale grammars 
are to be defined and maintained. Moreover, the 
essentially model theoretic slant on specification 
we propose here seems particularly well suited to 
this aim. Models do not in any sense "code" the 
LFG ontology: they take it pretty much at face va- 
lue. In our view this is crucial. Formal approaches 
1This paper is based upon the originM formula- 
tion of LFG, that of Kaplan and Bresnan (1982), and 
will not discuss such later innovations as functional 
uncertainty. 
39 
to grammatical theorising should reflect linguistic 
intuitions as directly as possible, otherwise they 
run the risk of being an obstacle, not an aid, to 
grammar development. 
The approach also raises theoretical issues. The 
model theoretic approach to specification langua- 
ges forces one to think about linguistic ontologies 
in a systematic way, and to locate them in a well 
understood mathematical space. This has at least 
two advantages. Firstly, it offers the prospect of 
meaningful comparison of linguistic frameworks. 
Secondly, it can highlight anomalous aspects of 
a given system. For example, as we shall later 
see, there seems to be no reasonable way to deal 
with LFG's --c definitions using the simple models 
of the present paper. There is a plausible model 
theoretic strategy strategy for extending our ac- 
count to cover =c; but the nature of the required 
extension clearly shows that =c is of a quite diffe- 
rent character to the bulk of LFG. We discuss the 
matter in the paper's concluding section. 
2 Lexical Functional Grammar 
A lexical functional grammar consists of three 
main components: a set of context free rules anno- 
tated with schemata, a set of well formedness con- 
ditions on feature structures, and a lexicon. The 
role of these components is to assign two interrela- 
ted structures to any linguistic entity licensed by 
the grammar: a tree (the c-structure) and a fea- 
ture structure (the f-structure). Briefly, the con- 
text free skeleton of the grammar rules describes 
the c-structure, the well-formedness conditions re- 
strict f-structure admissibility, and the schemata 
synchronise the information contained in the c- 
and f-structures. 
(1) S , NP VP 
(T soB J)=l T= 
(2) NP , Det N 
(3) VP , V 
a 
girl 
walks 
Det, (T SPEC) ---- a, (T NUM) ---- sing 
N, (~ PRED) ~- girl, (1" NUM) "-" sing 
V, (1" PRED) = walk( (subj) ), 
(~ TENSE) ---- pst 
Figure 1: An LFG grammar fragment 
To see how this works, let's run through a sim- 
ple example. Consider the grammar given in Fi- 
gure 1. Briefly, the up- and down-arrows in the 
schemata can be read as follows: T Feature denotes 
the value of Feature in the f-structure associated 
with the tree node immediately dominating the 
current tree node, whereas ~ Feature denotes the 
value of Feature in the f-structure associated with 
the current tree node. For instance, in rule (1) the 
NP schema indicates that the f-structure associa- 
ted with the NP node is the value of the SUBJ 
feature in the f-structure associated with the mo- 
ther node. As for the VP schema, it requires that 
the f-structure associated with the mother node is 
identical with the f-structure associated with the 
VP node. 
Given the above lexical entries, it is possible 
to assign a correctly interrelated c-structure and 
f-structure to the sentence A girl walks. Moreo- 
ver, the resulting f-structure respects the LFG 
well formedness conditions, namely the uniquen- 
ess, completeness and coherence principles discus- 
sed in section 5. Thus A girl walks is accepted by 
this grammar. 
3 Modeling the LFG ontology 
The ontology underlying LFG is a composite one, 
consisting of trees, feature structures and links 
between the two. Our first task is to mathemati- 
cally model this ontology, and to do so as trans- 
parently as possible. That is, the mathematical 
entities we introduce should clearly reflect the in- 
tuitions important to LFG theorising -- "No co- 
ding!", should be our slogan. In this section, we 
introduce such a representation of LFG ontology. 
In the following section, we shall present a formal 
language L: for talking about this representation; 
that is, a language for specifying LFG grammars. 
We work with the following objects. A mo- 
del M is a tripartite structure (7.,zoomin,J:), 
where 7- is our mathematical picture of c- struc- 
ture, 9 r our picture of f-structure, and zoomin 
our picture of the link between the two. We 
now define each of these components. Our de- 
finitions are given with respect to a signature of 
the form (Cat, Atom, Feat), where Cat, Atom and 
Feat are non-empty, finite or denumerably infinite 
sets. The intuition is that these sets denote the 
syntactic categories, the atomic values, and the 
features that the linguist has chosen for some lan- 
guage. For instance, Cat could be {S, NP, VP, V}, 
Atom might be {sg,pl, third, fem, masc} and 
Feat might be { subj, obj, pred, nb, case, gd}. 
Firstly we define 7". As this is our mathema- 
tical embodiment of c-structure (that is, a cate- 
gory labeled tree) we take it to be a pair (T, Vt), 
where T is a finite ordered tree and Vt is a function 
from the set of tree nodes to Cat. We will freely 
use the usual tree terminology such as mother-of, 
daughter-of, dominates, and so on. 
Second, we take jr to be a tuple of the form 
(W, {fa}c, EFeat, initial, Final, VI) , where W is aft- 
nite, non-empty set of nodes; f~ is a partial func- 
tion from W to W, for all a E Feat; initial is a 
unique node in W such that any other node w' 
of W can be reached by applying a finite number 
40 
of fa to initial; Final is a non-empty set of no- 
des such that for all fa and all w E Final, f~(w) 
is undefined; and V! is a function from Final to 
Atom. This is a standard way of viewing feature 
structures, and is appropriate for LFG. 
Finally, we take zoomin, the link between c- 
structure and f-structure information, to be a par- 
tial function from T to W. This completes our 
mathematical picture of LFG ontology. It is cer- 
tainly a precise picture (all three components, and 
how they are related are well defined), but, just 
as importantly, it is also a faithful picture; models 
capture the LFG ontology perspicuously. 
4 A Specification Language 
Although models pin down the essence of the LFG 
universe, our work has only just begun. For a 
start, not all models are created equal. Which 
of them correspond to grammatical utterances of 
English? Of Dutch? Moreover, there is a practical 
issue to be addressed: how should we go about 
saying which models we deem 'good'? To put in 
another way, in what medium should we specify 
grammars? 
Now, it is certainly possible to talk about mo- 
dels using natural language (as readers of this pa- 
per will already be aware) and for many purposes 
(such as discussion with other linguists) natural 
language is undoubtedly the best medium. Ho- 
wever, if our goal is to specify large scale gram- 
mars in a clear, unambiguous manner, and to do 
so in such a way that our grammatical analyses 
are machine verifiable, then the use of formal spe- 
cification languages has obvious advantages. But 
which formal specification language? There is no 
single best answer: it depends on one's goals. Ho- 
wever there are some important rules of thumb: 
one should carefully consider the expressive capa- 
bilities required; and a judicious commitment to 
simplicity and elegance will probably pay off in 
the long run. Bearing this advice in mind, let us 
consider the nature of LFG grammars. 
Firstly, LFG grammars impose constraints on 
7". Context free rules are typically used for this 
purpose -- which means, in effect, that constraints 
are being imposed on the 'daughter of' and 'sister 
of' relations of the tree. Secondly, LFG grammars 
impose general constraints on various features in 
2-. Such constraints (for example the completen- 
ess constraint) are usually expressed in English 
and make reference to specific features (notably 
pred). Thirdly, LFG grammars impose constraints 
on zoomin. As we have already seen, this is done 
by annotating the context free rules with equati- 
ons. These constraints regulate the interaction of 
the 'mother of' relation on 7", zoomin, and specific 
features in 2-. 
Thus a specification language adequate for LFG 
must be capable of talking about the usual tree re- 
lations, the various features, and zoomin; it must 
also be powerful enough to permit the statement 
of generalisations; and it must have some mecha- 
nism for regulating the interaction between 7" and 
2-. These desiderata can be met by making use 
of a propositional language augmented with (1) 
modal operators for talking about trees (2) modal 
operators for talking about feature structures, and 
(3) a modal operator for talking about zoomin, 
together with a path equality construct for syn- 
chronising the information flow between the two 
domains. Let us build such a language. 
Our language is called Z: and its primi- 
tive symbols (with respect to a given signature 
(Cat, Atom, Feat)) consists of (1) all items in Cat 
and Atom (2) two constants, c-struct and f-struct, 
(3) the Boolean connectives (true, false, -~, A, ~, 
etc.), (4) three tree modalities (up), (down) and 
,,, (5) a modality (a), for each feature a E Feat, 
(6) a synchronisation modality (zoomin), (7) a 
path equality constructor ~, together with (8) the 
brackets ) and (. 
The basic well formed formulas (basic wits) of £ 
are: {true, false, c-struct, f-struct}UCatUAtomU 
Patheq, where Patheq is defined as follows. Let t, 
t I be finite (possibly null) sequences of the moda- 
lities (up) and (down), and let f, f' be finite (pos- 
sibly null) sequences of feature modalities. Then 
t(zoomin)f ~ t'(zoomin)f' is in Patheq, and no- 
thing else is. 
Tim wffs of/:: are defined as follows: (1) all basic 
wffs are wffs, (2) all Boolean combinations of wffs 
are wffs, (3) if ¢ is a wff then so is Me, where 
M E {(a) : a E Feat} U {(up}, (down), (zoomin)} 
and (4) if n > 0, and ¢1,...,¢n are wffs, then so 
is *(¢1,...,¢n)- Nothing else is a wff. 
Now for the satisfaction definition. We induc- 
tively define a three place relation ~ which holds 
between models M, nodes n and wffs ¢. Intui- 
tively, M, n ~ ¢ means that the constraint ¢ holds 
at (is true at, is satisfied at) the node n in model 
M. The required inductive definition is as follows: 
M, n ~ true always 
M, n ~ false never 
M, n ~ c-struct iff 
n is a tree node 
M, n ~ f-struct iff 
n is a feature structure node 
M,n ~ c iff 
Vt(n) = c, (for all e E Cat) 
M,n ~ a iff 
Vf(n) = a, (for alia ~ Atom) 
M, n ~ -,¢ iff 
not M,n~¢ 
M,n~fA¢ if 
M,n~ff and M,n~f 
M, n ~ (a)¢ iff 
f~(n) exists and M, f~(n) ~ ¢ 
(for all a E Feat) 
41 
M, n ~ (down)¢ iff 
n is a tree node with 
at least one daughter n' such that 
M,n'~¢ 
M, n ~ (up)¢ ig 
n is a tree node with 
a mother node m and M,m~¢ 
M, n ~ (zoomin)¢ iff 
n is a tree node, 
zoomin(n) is defined, and 
M, zoomin(n) ~ ¢ 
M,n ~ *(¢x,...,¢k) ig 
n is a tree node with 
exactly k daughters nl ... nk and 
M, n I ~ ~1,... ,M, nk ~ ¢k 
M, n ~ t(zoomin)f ~, t'(zoomin)f' iff 
n is a tree node, and there is a 
feature structure node w such that n(St; 
zoomin; Sl )w and 
n( S,, ; zoomin; S 1, )w 
For the most part the import of these clauses 
should be clear. The constants true and false play 
their usual role, c-struct and f-struct give us 'la- 
bels' for our two domains, while the elements of 
Cat and Atom enable us to talk about syntactic 
categories and atomic f-structure information re- 
spectively. The clauses for --, and A are the usual 
definitions of classical logic, thus we have all pro- 
positional calculus at our disposal; as we shall 
see, this gives us the flexibility required to for- 
mulate non-trivial general constraints. More in- 
teresting are the clauses for the modalities. The 
unary modalities (a), (up), (down), and (zoomin) 
and the variable arity modality * give us access 
to the binary relations important in formulating 
LFG grammars. Incidentally, • is essentially a 
piece of syntactic sugar; it could be replaced by a 
collection of unary modalities (see Blackburn and 
Meyer-Viol (1994)). However, as the * operator 
is quite a convenient piece of syntax for captu- 
ring the effect of phrase structure rules, we have 
included it as a primitive in/3. 
In fact, the only clause in the satisfaction "de- 
finition which is at all complex is that for ~. 
It can be glossed as follows. Let St and St, be 
the path sequences through the tree correspon- 
ding to t and t ~ respectively, and let S I and Sf, 
he the path sequences through the feature struc- 
ture corresponding to f and f' respectively. Then 
t(zoomin)f ~ t'(zoomin)f' is satisfied at a tree 
node tiff there is a feature structure node w that 
can be reached from t by making both the tran- 
sition sequence St;zoornin; S! and the transition 
sequence S,,;zoomin; S!,. Clearly, this construct 
is closely related to the Kasper Rounds path equa- 
lity (see gasper and Rounds (1990)); the princi- 
ple difference is that whereas the Kasper Rounds 
enforces path equalities within the domain of fea- 
ture structures, the LFG path equality enforces 
equalities between the tree domain and the fea- 
ture structure domain. 
If M, n ~ ¢ then we say that ¢ is satisfied in M 
at n. If M, n ~ ¢ for all nodes n in M then we say 
that ¢ is valid in M and write M ~ ¢. Intuitively, 
to say that ¢ is valid in M is to say that the 
constraint ¢ holds universally; it is a completely 
general fact about M. As we shall see in the next 
section, the notion of validity has an important 
role to play in grammar specification. 
5 Specifying Grammars 
We will now illustrate how/3 can be used to spe- 
cify grammars. The basic idea is as follows. We 
write down a wff ¢ a which expresses all our desi- 
red grammatical constraints. That is, we state in 
/3 which trees and feature structures are admissi- 
ble, and how tree and feature based information is 
to be synchronised; examples will be given shortly. 
Now, a model is simply a mathematical embodi- 
ment of LFG sentence structure, thus those mo- 
dels M in which ¢ a is valid are precisely the sent- 
ence structures which embody all our grammatical. 
principles. 
Now for some examples. Let's first consider how 
to write specifications which capture the effect of 
schemata annotated grammar rules. Suppose we 
want to capture the meaning of rule (1) of Figure 
1, repeated here for convenience: 
S , NP VP (IsuBJ) 
=l T=~ 
Recall that this annotated rule licenses structures 
consisting of a binary tree whose mother node m 
is labeled S and whose daughter nodes nl and n2 
are labeled NP and VP respectively; and where, 
furthermore, the S and VP nodes (that is, m and 
n2) are related to the same f-structure node w; 
while the NP node (that is, nl) is related to the 
node w ~ in the f-structure that is reached by ma- 
king a SUBJ transition from w. 
This is precisely the kind of structural cons- 
traint that /3 is designed to specify. We do so 
as follows: 
S --* *(NP A (up)(zoomin)(subj) ~ (zoomin), 
VP A (up)(zoomin) ,~ (zoomin)) 
This formula is satisfied in a model M at any node 
m iff m is labeled with the category S, has exactly 
two daughters nx and n2 labeled with category 
NP and VP respectively. Moreover, nl must be 
associated with an f-structure node w ~ which can 
also be reached by making a (sub j) transition from 
the f-structure node w associated with the mother 
node of m. (In other words, that part of the f- 
structure that is associated with the NP node is 
re-entrant with the value of the subj feature in 
42 
the f-structure associated with the S node.) And 
finally, n2 must be associated with that f-structure 
node w which m. (In other words, the part of the 
f-structure that is associated with the VP node is 
re-entrant with that part of the f-structure which 
is associated with the S node.) 
In short, we have captured the effect of an an- 
notated rule purely declaratively. There is no ap- 
peal to any construction algorithm; we have sim- 
ply stated how we want the different pieces to fit 
together. Note that . specifies local tree admissi- 
bility (thus obviating the need for rewrite rules), 
and (zoomin), (up) and ~ work together to cap- 
ture the effect of ~ and T- 
In any realistic LFG grammar there will be se- 
veral -- often many -- such annotated rules, and 
acceptable c-structures are those in which each 
non-terminal node is licensed by one of them. We 
specify this as follows. For each such rule we form 
the analogous £ wff Cr (just as we did in the pre- 
vious example) and then we form the disjunction 
V Cr of all such wffs. Now, any non-terminal node 
in the c-structure should satisfy one of these dis- 
junctions (that is, each sub-tree of c-struct must 
be licensed by one of these conditions); moreover 
the disjunction is irrelevant to the terminal nodes 
of c-struct and all the nodes in f-struct. Thus we 
demand that the following conditional statement 
be valid: 
(e-struct A (down)true) --~ V ¢~" 
This says that if we are at a c-struct node which 
has at least one daughter (that is, a non-terminal 
node) then one of the subtree licensing disjuncts 
(or 'rules') must be satisfied there. This picks pre- 
cisely those models in which all the tree nodes are 
appropriately licensed. Note that the statement 
is indeed valid in such models: it is true at all the 
non-terminal nodes, and is vacuously satisfied at 
terminal tree nodes and nodes of f-struct. 
We now turn to the second main component 
of LFG, the well formedness conditions on f- 
structures. 
Consider first the uniqueness principle. In es- 
sence, this principle states that in a given f- 
structure, a particular attribute may have at most 
one value. In £ this restriction is 'built in': it fol- 
lows from the choices made concerning the ma- 
thematical objects composing models. Essenti- 
ally, the uniqueness principle is enforced by two 
choices. First, V! associates atoms only with fi- 
nal nodes of f-structures; and as V/ is a func- 
tion, the atom so associated is unique. In ef- 
fect, this hard-wires prohibitions against constant- 
compound and constant-constant clashes into the 
semantics of £. Second, we have modeled featu- 
res as partial functions on the f-structure nodes 
- this ensures that any complex valued attribute 
is either undefined, or is associated with a uni- 
que sub-part of the current f-structure. In short, 
as required, any attribute will have at most one 
value. 
We turn to the completeness principle. In LFG, 
this applies to a (small) finite number of attributes 
(that is, transitions in the feature structure). This 
collection includes the grammatical functions (e.g. 
subj, obj, iobj) together with some longer transiti- 
ons such as obl; obj and to; obj. Let GF be a meta- 
variable over the modalities corresponding to the 
elements of this set, thus GF contains such items 
as (subj), (obj), (iobj), (obl)(obj) and (to)(obj). 
Now, the completeness principle requires that any 
of these features appearing as an attribute in the 
value of the PRED attribute must also appear as 
an attribute of the f-structure immediately con- 
taining this PRED attribute, and this recursively. 
The following wff is valid on precisely those mo- 
dels satisfying the completeness principle: 
(wed) GF true --* GF true. 
Finally, consider the counterpart of the com- 
pleteness principle, the coherence principle. This 
applies to the same attributes as the completen- 
ess principle and requires that whenever they oc- 
cur in an f-structure they must also occur in the 
f-structure associated with its PRED attribute. 
This is tantamount to demanding the validity of 
the following formula: 
( GF true A (pred)true) ~ (pred) GF true 
6 Conclusion 
The discussion so far should have given the reader 
some idea of how to specify LFG grammars using 
£. To conclude we would like to discuss =c defi- 
nitions. This topic bears on an important general 
issue: how are the 'dynamic' (or 'generative', or 
'procedural') aspects of grammar to be reconciled 
with the 'static', (or 'declarative') model theoretic 
world view. 
The point is this. Although the LFG equations 
discussed so far were defining equations, LFG also 
allows so-called constraining equations (written 
=e). Kaplan and Bresnan explain the difference as 
follows. Defining equations allow a feature-value 
pair to be inserted into an f-structure providing 
no conflicting information is present. That is, 
they add a feature value pair to any consistent f- 
structure. In contrast, constraining equations are 
intended to constrain the value of an already exi- 
sting feature-value pair. The essential difference 
is that constraining equations require that the fea- 
ture under consideration already has a value, whe- 
reas defining equations apply independently of the 
feature value instantiation level. 
In short, constraining equations are essentially 
a global check on completed structures which re- 
quire the presence of certain feature values. They 
have an eminently procedural character, and there 
43 
is no obvious way to handle this idea in the pre- 
sent set up. The bulk of LFG involves stating 
constraints about a single model, and /: is well 
equipped for this task, but constraining equations 
involve looking at the structure of other possible 
parse trees. (In this respect they are reminiscent 
of the feature specification defaults of GPSG.) The 
approach of the present paper has been driven by 
the view that (a) models capture the essence of 
LFG ontology, and, (b) the task of the linguist is 
to explain, in terms of the relations that exist wi- 
thin a single model, what grammatical structure 
is. Most of the discussion in Kaplan and Bres- 
nan (1982) is conducted in such terms. However 
constraining equations broaden the scope of the 
permitted discourse; basically, they allow implicit 
appeal to possible derivational structure. In short, 
in. common with most of the grammatical forma- 
lisms with which we are familiar, LFG seems to 
have a dynamic residue that resists a purely de- 
clarative analysis. What should be done? 
We see three possible responses. Firstly, we 
note that the model theoretic approach can al- 
most certainly be extended to cover constraining 
equations. The move involved is analogous to the 
way first order logic (a so-called 'extensional' lo- 
gic) can be extended to cope with intensional no- 
tions such as belief and necessity. The basic idea 
-- it's the key idea underlying first order Kripke 
semantics -- is to move from dealing with a sin- 
gle model to dealing with a collection of models 
linked by an accessibility relation. Just as quan- 
tification over possible states of affairs yields ana- 
lyses of intensional phenomena, so quantification 
over related models could provide a 'denotational 
semantics' for =~. Preliminary work suggests that 
the required structures have formal similarities to 
the structures used in preferential semantics for 
default and non-monotonic reasoning. This first 
response seems to be a very promising line of work: 
the requisite tools are there, and the approach 
would tackle a full blooded version of LFG head 
on. The drawback is the complexity it introduces 
into an (up till now) quite simple story. Is such 
additional complexity reMly needed? 
A second response is to admit that there is a 
dynamic residue, but to deal with it in overtly 
computational terms. In particular, it may be 
possible to augment our approach with an ex- 
plicit operational semantics, perhaps the evolving 
algebra approach adopted by Moss and Johnson 
(1994). Their approach is attractive, because it 
permits a computational treatment of dynamism 
that abstracts from low level algorithmic details. 
In short, the second strategy is a 'divide and con- 
quer' strategy: treat structural issues using model 
theoretic tools, and procedural issues with (reve- 
aling) computational tools. It's worth remarking 
that this second response is not incompatible with 
the first; it is common to provide programming 
languages with both a denotational and an opera- 
tional semantics. 
The third strategy is both simpler and more 
speculative. While it certainly seems to be the 
case that LFG (and other 'declarative' forma- 
lisms) have procedural residues, it is far from clear 
that these residues are necessary. One of the most 
striking features of LFG (and indeed, GPSG) is 
the way that purely structural (that is, model 
theoretic) argumentation dominates. Perhaps the 
procedural aspects are there more or less by ac- 
cident? After all, both LFG and GPSG drew on 
(and developed) a heterogeneous collection of tra- 
ditional grammar specification tools, such as con- 
text free rules, equations, and features. It could 
be the case such procedural residues as --¢ are 
simply an artifact of using the wrong tools for tal- 
king about models. If this is the case, it might be 
highly misguided to attempt to capture =¢ using 
a logical specification language. Better, perhaps, 
would be to draw on what is good in LFG and 
to explore the logical options that arise naturally 
when the model theoretic view is taken as pri- 
mary. Needless to say, the most important task 
that faces this third response is to get on with the 
business of writing grammars; that, and nothing 
else, is the acid test. 
It is perhaps worth adding that at present the 
authors simply do not know what the best re- 
sponse is. If nothing else, the present work has 
made very clear to us that the interplay of sta- 
tic and dynamic ideas in generative grammar is 
a delicate and complex matter which only further 
work can resolve. 
References 
Patrick Blackburn and Wilfried Meyer-Viol. 1994. 
Linguistics, Logic and Finite Trees. Bulletin 
of the 1GPL, 2, pp. 3-29. Available by an- 
onymous ftp from theory.doc.ic.ac.uk, directory 
theory/forum/igpl/Bulletin. 
Wilfrid Hodges. 1993. Model Theory. Cambridge 
University Press. 
Ron Kaplan and Joan Bresnan. 1982. Lexical- 
Functional Grammar: A formal system for 
grammatical representation. In The Mental Re- 
presentation of Grammatical Relations, pp. 173 
- 280, MIT Press. 
R. Kasper and W. Rounds. 1990. The Logic of 
Unification in Grammar. Linguistics and Phi- 
losophy, 13, pp. 33-58. 
Lawrence Moss and David Johnson. 1994. Dyna- 
mic Interpretations of Constraint-Based Gram- 
mar Formalisms. To appear in Journal of Logic, 
Language and Information. 
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