COLING 82, J. Horeck~ (eel) 
North-HoOand Publlshi~ Company 
O A~deml~ 1982 
Lexical-Functional Grammar and Order-Free Semantic Composition 
Per-Kristian Halvorsen 
Norwegian Research Council's Computing Center for the Humanities 
and 
Center for Cognitive Science, MIT 
This paper summarizes the extension of the theory 
of lexical-functional grammar to include a formal, 
model-theoretic, semantics. The algorithmic 
specification of the semantic interpretation 
procedures is order-free which distinguishes the 
system from other theories providing 
model-theoretic interpretation for natural 
language. Attention is focused on the 
computational advantages of a semantic 
interpretation system that takes as its input 
functional structures as opposed to syntactic 
surface-structures. 
A pressing problem for computational linguistics is the development 
of linguistic theories which are supported by strong independent 
linguistic argumentation, and which can, simultaneously, serve 
as a basis for efficient implementations in language processing 
systems. Linguistic theories with these properties make it 
possible for computational implementations to build directly on the 
work of linguists both in the area of grammar-writing, and in the 
area of theory development (cf. universal conditions on anaphoric 
binding, filler-gap dependencies etc.). 
Lexical-functional grammar (LFG) is a linguistic theory which 
has been developed with equal attention being paid to theoretical 
linguistic and computational processing considerations (Kaplan & 
Bresnan 1981). The linguistic theory has ample and broad 
motivation (vide the papers in Bresnan 1982), and it is 
transparently implementable as a syntactic parsing system (Kaplan 
& Halvorsen forthcoming). LFG takes grammatical relations to be of 
primary importance (as opposed to the transformational theory where 
grammatical functions play a subsidiary role). Sentences are 
derived by means of a set of context-free phrase-structure rules 
annotated with functional-schemata. There are no transformational 
rules. The pivotal elements in the theory are the phrase-structure 
rules, and in particular, the lexical rules. A typical 
LFG-analysis is the treatment of the passive-construction implied 
by the rules and lexical entries in (la-c). 
115 
116 P.-K. HALVORSEN 
(1)a.Phrase-structure rules (annotated with schemata determining 
the assignment of grammatical relations): 
S --> NP VP 
(¢ SUBJ)=I ~=~ 
VP --> V NP NP 
=~ (~ OBJ) =~ (~ OBJ2) = 
pp* 
(~' OBLo~ = ¢ 
(~PCASE)= OBLB 
VP 
(¢ VCOMP)=$ PP --> P NP 
(¢=~) (¢:~) 
b.Lexical Rule of Passive: 
OBJ --> SUBJ 
SUBJ --> OBLBy 
Optionally: OBLBy--> 
c.Lexical entries (derived by the lexical rules): 
(i) buy V ( PRED)='buy<(# SUBJ) (~ OBJ)>' 
(ii) buy V ( PRED)='buy<~, ( @SUBJ)>' 
(iii) buy V ( PRED)='buy<(# OBLBy)(@ SUBJ)>' 
(2)a.John bought Cottage, Inc. 
b. 
NPf2 VPf 
i f4 NPf5 Jol n bought 
C. Functional equations 
(fl SUBJ)=f2 
(f2 PRED)= 'John' 
(f2 NUM)=SG 
fl=f3 
f3=f4 
(f4 PRED)='buy<(#SUBJ)'(%OBJ)>' 
(f30BJ)=f5 
Cottage,Inc. (f5 PRED)='C°ttage' Inc.' 
(f5 NUM)=SG 
f4 f3/flFsUBJITENSE f2pREDLNUMpAsT • 'JohnqSG \]. \]I 
I PRED 'buy<( SUBJ), ( OBJ)>' f 5 rPRED OBJ ' Cottage, Inc. 
LNUM SG 
d.Functional Structure (acyclic-graph) 
ORDER-FREE SEMANTIC COMPOSITION 117 
(3)a.Cottage, Inc. was bought by John b. 
/Sfl~ 
\[ vPf3 Pf2 / 
VCOMP= 
f6 /PPf7~ 
Cottage,Incwas bou ht by P \[ 
f8 ? f9 
John 
c.Functional Equations 
(fl SUBJ)=f2 
fl=f3 
fj=f4 
(f4 PRED)='be<~COMP)>' 
(f3 VCOMP)=f5 
(f5 SUBJ)=f2 
f6=f5 
(f6 PRED)='buy<(% OBLBy),(T SUBJ)>' 
d.Functional Structure 
(f50BLBY)=f7 
f7=f8 
f7=f9 
(f7 PCASE)=OBLBy 
(f9 PRED)='John' 
(f9 NUM)=SG 
fI-SUBJ f2\[PRED 'Cottage, Inc.'~ 
f3 ~UM SG 
f4 TENSE PAST 
PRED be<(~ VCOMP)> 
VCOMP f5 SUBJ \[ \] 
f61PRED 'buy<(t OSLBy) (T SUBJ)> 
SJBy f7\[PCASE 
~PRED 'John' LNUM SG \] 
The phrase-structure rules in (i a) generates the 
phrase-structure tree in (2 b). Each phrase-structure tree is indexed. The indices instantiate the up- and down-arrows in the 
functional schemata which are found in the phrase structure rules. An up-arrow refers to the node dominating the node the 
schemata is attached to. A down-arrow refers to the node which carries the functional schemata. The result of the instantiation 
process is the set of functional equations in (2 c). These equations describe the functional structure (f-structure) in (2 
d). 
The functional structures provide, in canonical form, a representation of the meaningful grammatical relations in the 
sentence. The functional equations, mediate between the constituent-structure and the functional-structure. Each 
functional equation determines an aspect of the functional-structure. When the functional equations are solved 
118 P.-K. HALVORSEN 
they uniquely determine the functional-structure. Moreover, the 
order in which the equations are processed is immaterial for the 
final result (see Kaplan and Bresnan 1981). The derivation of 
functional-structures is order-free. The semantic theory which 
accompanies the theory of functional structures provides, in its 
turn, an order-free derivation of semantic representations from 
functional structures. 
The theoretical problem of natural language interpretation can 
be decomposed into three tasks: (i) Unraveling all possible 
thematic relations holding in a sentence; (2) Composition of the 
meaning of the constituents of the sentence into a well-deflned 
and coherent representation of the meaning of the entire 
sentence; (3) Specifying all possible scope and 
control-relations holding in a sentence, 
The unraveling of the thematic relations in a sentence is 
already accomplished in f-structure; no special moves are 
therefore needed in the semantics to establish the propositional 
equivalence of active-passive pairs or pairs with and without 
~, Equi etc.. Notice that both the f-structure for the 
active sentence, (2d), and the f-structure for the passive 
sentence, (3d), contain a f-structure with the predicate 
'buy< .... , .... >', and the arguments John and Cotta@e, Inc. are 
tied to the same argument positions in this predicate in the 
active sentence as well as in the passive sentence, thus 
expressing the truth-conditional equivalence of the two 
utterances. 
The task of semantic composition and the determination of 
scope relations require that semantic representations be derived 
from the functional structures. 
Rather than translating functional structures into formulas of 
standard predicate calculus, f-structures are mapped into acyclic 
graphs called semantic structures (cf. 4a). Since semantic 
structures are acyclic graphs, just like functional structures, 
symmetric constraining equations (cf. 2b and 3b) can be used to 
define an order-free derivation of semantic structures from 
functional structures just as functional equations yield an 
order-free derivation of f-structures from annotated 
phrase-structure trees. Each constraining equation in the 
Mapping from f-structure to semantic structure adds information 
about the semantic structure of a sentence. Specifically if an 
f-structure, f has a PRED whose value is the semantic form~ , 
then the equa\[ion (Mf PREDICATE)=~' is introduced, which tells 
us that the semantic structure corresponding to f, Mr, has a 
PREDICATE attribute whose value is the translation in 
intensional logic of 5. Each constraining equation adds a bit of 
information about the semantic structure, and just as in a 
jigsaw-puzzle what piece is found at what time is of no 
consequence for the final outcome. ° 
It is the functional structure which drives the semantic 
composition, not the application of specific syntactic rules. 
This makes it possible to construct a highly constrained, and 
universal, theory of semantic composition: there are only a small 
number of structurally distinct configurations in f-structure 
(the predicate-, argument-, quantifier-, control-, and 
adjunct-relations). Explicit semantic composition rules for each 
of these configurations have been constructed. This enables the 
interpretation of any well-formed f-structure. Once the semantic 
composition rules for functional structures have been correctly 
ORDER-FREE SEMANTIC COMPOSITION 119 
stated they will extend to cover any sentence, which is assigned an 
f-structure by the grammar. The composition rules do not have to 
be revised as the coverage of the grammar is extended, or as new 
languages are described. The semantics for the LFG-theory is 
clearly more easily transportable than the semantics of systems 
where each syntactic rule requires a special semantic rule (Bach 
1976). 
The semantic translation rules, working off of the 
f-structures in (2d) and (3d) give rise to constraining equations 
that determine the semantic structure in (4a) as the semantic 
representation for both John bought Cottage, Inc___t and Cottage, 
IncL was bought by John,. The semantic structure 
corresponds to the formula of intensional logic in (4b). 
(4) a. 
;REDICATE buy' \] b. b u~.(~,~) 
RGI ~P~ (i) 
RG2 ~'(~) 
In the semantic structure the translations of the basic meaningful 
expressions of the sentence are present. There is no reflex of 
syntactic expressions without independent meaning (expletives, it, 
there, governed prepositions, by, etc.). The values of the 
attribute PREDICATE are semantic functions, and the functional 
arguments are identified by the the attribute ARGi Semantic 
composition consists of the application of the functions to the 
intensions of the meanings of the functional arguments. 
The scope of NP's with quantifier phrases is explicitly 
indicated in semantic structure, as are all occurrences of semantic 
arguments which the quantifiers bind. The semantic reflex of 
functional control is also explicated in semantic structure. 
The construction of meaning representations from f-structures 
proceeds by successive approximation from any point in the sentence 
by way of the symmetrical constraining equations. The algorithmic 
specification of the interpretation process does, therefore, not 
impose any arbitrary constraints on tne order in which the semantic 
composition of the words should proceed. Instead, one is free to 
impose such constraints on the order of steps in the interpretation 
procedure as proves to be psychologically and/or computationally 
motivated. The use of symmetric constraining equations and the 
resulting monotonic character of the mapping between f-structures 
and meaning representations is also useful in clearing the way for 
interpretation of sentence fragments. As it stands, the procedure 
can, if desired, proceed from left to right in a string of words, 
as they are being presented. The composition of sentence meanings 
within the Montague Semantics framework, in contrast, typically 
proceed from the most deeply embedded constituent and outward (see 
Thomason 1976, Dowty et al 1981). Interleaving of syntactic and 
semantic processing is. also facilitated when there are no 
unmotivated constraints on the order of steps in the semantic 
processing of the sentence. Within this approach to semantic 
interpretation it is also possible to let the order of steps in the 
parsing of a sentence be determined by efficiency considerations, 
which may vary~from sentence to sentence and even from phrase to 
phrase within a given sentence. An imposition of order in the 
abstract specification of the composition process would limit 
implementation choices prematurely. 
120 P.-K. HALVORSEN 
In addition to the processing advantages, the semantic theory 
for lexical-functional grammars also offers significant 
simplifications in the analysis of a number of controversial and/or 
recalcitrant constructions such as Raising-, There-insertion, 
Passive, and constructions with dismantled idioms. This is an 
illustration of the efforts to search for evidence for the theory 
bothinlinguistic arguments and through computational efficiency 
considerations. 

References 

I. Bach, E., An extension to classlcal transformational 
grammar, mlmeo, University of Massachusetts, Amherst, 
Mass°° 

2. Bresnan, J. W.,(ed.), The mental representation of 
grammatical relations (MIT Press, Cambridge, 1982). 

3. Dowry, D., Wall, R. E., and Peters, S., Montague 
~emantics (Reldel, Dordrecht, 1981). 

4. Kaplan, R., and Bresnan, J., Lexical-functional 
grammar: a formal system for grammatical representation, 
in: Bresnan, J. W., (ed.),The Rental Representation of 
Grammatical Relations, (MIT Press, Cambridge, Mass). 

5. Kaplan, R., and Halvorsen, P.Kr., Parsing 
Lexical-functional grammars, (forthcoming). 

6. Thomason, R., ed., Formal Philosophy: Selected 
papers of Richard ~ontague (Yale 0niversity Press, New 
Haven, 1974). 
