Features and Agreement 
Sam Bayer and Mark Johnson* 
Cognitive and Linguistic Sciences, Box 1978 
Brown University 
{ b ayer,mj } @cog.brown.edu 
Abstract 
This paper compares the consislency- 
based account of agreement phenomena 
in 'unification-based' grammars with an 
implication-based account based on a sim- 
ple feature extension to Lambek Catego- 
rim Grammar (LCG). We show that the 
LCG treatment accounts for constructions 
that have been recognized as problematic 
for 'unification-based' treatments. 
1 Introduction 
This paper contrasts the treatment of agreement 
phenomena in standard complex feature structure 
or 'unification-based' grammars such as HPSG (Pol- 
lard and Sag, 1994) with that of perhaps the sim- 
plest possible feature extension to Lambek Catego- 
rial Grammar (LCG) (Lambek, 1958). We iden- 
tify a number of situations where the two accounts 
make different predictions, and find that gener- 
ally the LCG account is superior. In the pro- 
cess we provide analyses for a number of construc- 
tions that have been recognized as problematic for 
'unification-based' accounts of agreements (Zaenen 
and Karttunen, 1984; Pullum and Zwicky, 1986; In- 
gria, 1990). Our account builds on the analysis of 
coordination in applicative categorial grammar in 
Bayer (1994) and the treatment of Boolean connec- 
tives in LCG provided by Morrill (1992). Our anal- 
ysis is similiar to that proposed by Mineur (1993), 
but differs both in its application and details. 
The rest of the paper is structured as follows. The 
next section describes the version of LCG we use in 
this paper; for reasons of space we assume familiar- 
ity with the treatment of agreement in 'unification- 
based' grammars, see Shieber (1986) and Pollard 
and Sag (1994) for details. Then each of the follow- 
*We would like to thank Bob Carpenter, Pauline Ja- 
cobson, John Maxwell, Glynn Morrill and audiences at 
Brown University, the University of Pennsylvania and 
the Universit£t Stuttgart for helpful comments on this 
work. Naturally all errors remain our own. 
ing sections up to the conclusion discusses an impor- 
tant difference between the two approaches. 
2 Features in Lambek Categorial 
Grammar 
In LCG semantic interpretation and long distance 
dependencies are handled independently of the fea- 
ture system, so agreement phenomena seem to be 
the major application of a feature system for LCG. 
Since only a finite number of feature distinctions 
need to be made in all the cases of agreement we 
know of, we posit only a very simple feature system 
here. Roughly speaking, features will be treated as 
atomic propositions (we have no need to separate 
them into attributes and values), and a simple cat- 
egory will be a Boolean combination of such atomic 
'features' (since we have no reason to posit a re- 
cursive feature structures either). In fact we are 
agnostic as to whether more complex feature sys- 
tems for LCG are linguistically justified; in any event 
Dorre et. al. (1994) show how a full attribute-value 
feature structure system having the properties de- 
scribed here can be incorporated into LCG. 
Following the standard formulation of LCG, we 
regard the standard LCG connectives '/' and 'V as 
directed implications, so we construct our system so 
that a//~ fl~ can combine to form a if fl' is logically 
stronger than/~. 
Formally, we adopt Morrill's treatment (Morrill, 
1992) of the (semantically impotent) Boolean con- 
nectives '^' and 'v' (Morrill named these 'lq' and '11' 
respectively). Given a set of atomic features 5, we 
define the set of feature terms 7- and categories g 
as follows, where '/' and 'V are the standard LCG 
forward and backward implication operators. 
7- ::= Y= + 7-^7- + 7-v7- 
C ::= 7- + C/C + C\¢ 
In general, atomic categories in a standard catego- 
rim grammar will be replaced in our analyses with 
formulae drawn from 7-. For example, the NP Kim 
might be assigned by the lexicon to the category 
np^sg^3, the verb sleeps to the category s\npnsg^3, 
70 
and the verb slept (which does not impose person 
or number features on its subject) to the category 
s\np. 
To simplify the presentation of the proofs, we for- 
mulate our system in natural deduction terms, and 
specify the properties of the Boolean connectives us- 
ing the single inference rule P, rather than providing 
separate rules for each connective. 
~P where I- in the calculus. 1 ¢ ¢ propositional 
The rule P allows us to replace any formula in T 
with a logically weaker one. For example, since Kim 
is assigned to the category np^sgA3, then by rule P 
it will belong to np as well. 
Finally, we assume the standard LCG introduc- 
tion and elimination rules for the directed implica- 
tion operators. 
A/B B B A\B 
A /~ A 
\[B\]" \[B\] n 
A A 
A/B ~in A\B \i~ 
For example, the following proof of the well- 
formedness of the sentence Kim slept can be derived 
using the rules just given and the lexical assignments 
described above. 
Kim 
np^sg^3 slept 
P up s\np 
8 
This example brings out one of the fundamental dif- 
ferences between the standard treatment of agree- 
ment in 'unification-based' grammar and this treat- 
ment of agreement in LCG. In the 'unification-based' 
accounts agreement is generally a symmetric rela- 
tionship between the agreeing constituents: both 
agreeing constituents impose constraints on a shared 
agreement value, and the construction is well-formed 
iff these constraints are consistent. 
However, in the LCG treatment of agreement pro- 
posed here agreement is inherently asymmetric, in 
1Because conjunction and disjunction are the only 
connectives we permit, it does not matter whether we 
use the classical or intuitionistic propositional calcu- 
lus here. In fact, if categories such as np and ap are 
'decomposed' into the conjunctions of atomic features 
+nounA--verb and q-noun^+verb respectively as in the 
Sag et. at. (1985) analysis discussed below, disjunction 
is not required in any of the LCG analyses below. How- 
ever, Bayer (1994) argues that such a decomposition is 
not always plausible. 
that an argument must logically imply, or be sub- 
sumed by, the antecedent of the predicate it com- 
bines with. Thus in the example above, the rule 
P could be used to 'weaken' the argument from 
npAsgA3 to rip, but it would not allow np (with- 
out agreement features) to be 'strengthened' to, say, 
npA SgA 3. 
Abstracting from the details of the feature sys- 
tems, we can characterize the 'unification-based' ap- 
proach as one in which agreement is possible be- 
tween two constituents with feature specifications ¢ 
and ¢ iff ¢ and ¢ are consistent, whereas the LCG 
approach requires that the argument ¢ implies the 
corresponding antecedent ¢ of the predicate (i.e., 
Interestingly, in cases where features are fully 
specified, these subsumption and consistency re- 
quirements are equivalent. More precisely, say that 
a formula ¢ from a feature constraint language fixes 
an atomic feature constraint X iff ¢ ~ X or ¢ 
-~X- For example, in single-valued feature systems 
(person) = 1 and (person) = 3 both fix (person) = 1, 
(person) = 2, (person) = 3, etc., and in general all 
fully-specified agreement constraints fix the same set 
of formulae. 
Now let ¢ and ¢ be two satisfiable formulae that 
fix the same set of atomic feature constraints. Then 
A ¢ is consistent iff ¢ ~ ¢. To see this, note 
that because ¢ and ¢ fix the same set of formulae, 
each condition holds iff ¢ and ¢ are elementarily 
equivalent (i.e., for each feature constraint X, ¢ ~ X 
iff ¢ ~ X)- 
However, the role of partial agreement feature 
specifications in the two systems is very different. 
The following sections explore the empirical conse- 
quences of these two approaches. We focus on co- 
ordination phenomena because this is the one area 
of the grammar where underspecified agreement fea- 
tures seem to play a crucial linguistic role, and can- 
not be regarded merely as an abbreviatory device for 
a disjunction of fully-specified agreement values. 
3 Coordination and agreement 
asymmetries 
Interestingly, the analysis of coordination is the one 
place where most 'unification-based' accounts aban- 
don the symmetric consistency-based treatment of 
agreement and adopt an asymmetric subsumption- 
based account. Working in the GPSG framework 
Sag et. al. (1985) proposed that the features on 
a conjunction must be the most specific category 
which subsumes each conjunct (called the general- 
ization by Shieber (1992)). Shieber (1986) proposed 
a weaker condition, namely that the features on the 
conjunction must subsume the features on each con- 
junct, as expressed in the annotated phrase struc- 
71 
VP 
bec~rae wealthy and a Republican 
wealthy a Republican 
and np ap P 
became npvap eonj npvap 
vp/npvap npvap 
vp 
Figure 2: The LCG analysis of (2b). 
,p 
GO 
Figure 1: The feature structure subsumption analy- 
sis of (2b). 
ture rule below (Shieber, 1992).2 In all of the exam- 
pies we discuss below, the features associated with 
a conjunction is the generalization of the features 
associated with each of its conjuncts, so our conclu- 
sions are equally valid for both the generalization 
and subsumption accounts of coordination. 
(1) Xo , Xl conj X2 
where X0 E X1 and X0 E X2 
Consider the sentences in (2). Decomposing the cat- 
egories N(oun) and A(djective) into the Boolean- 
valued features {(noun) = +,(verb) = -} and 
{(noun) = +, (verb) = +} respectively, the fact that 
became can select for either an NP or an AP comple- 
ment (2a) can be captured by analysing it as subcat- 
egorizing for a complement whose category is under- 
specified; i.e., its complement satisfies (noun) = +, 
and no constraint is imposed on the verb feature. 
(2) a. Kim \[v became \] \[hv wealthy \] / \[NP a Re- 
publican \] 
b. Kim \[vP \[v became \] lAP wealthy \] and \[NP 
a Republican \] \] 
Now consider the coordination in (2b). Assum- 
ing that became selects the underspecified category 
(noun) = +, the features associated with the coor- 
dination subsume the features associated with each 
coordinate, as required by rule (1), so (2b) has the 
well-formed structure shown in Figure 1. 
On the other hand, a verb such as grew which 
selects solely AP complements (3a) requires that 
its complement satisfies (noun) = +, (verb) = +. 
Thus the features on the coordinate structure in (3b) 
must include (verb) = + and so do not subsume the 
(verb) = - feature on the NP complement, correctly 
predicting the ungrammatieality of (3b). 
(3) a. Kim grew lAP wealthy\]/*\[Np a Republican\] 
2Note that the LFG account of coordination provided 
by Kaplan and Maxwell (1988) differs significantly from 
both the generalization and the subsumption accounts of 
coordination just mentioned, and does not generate the 
incorrect predictions described below. 
wealthy a Republican 
ap and np .p p 
grew npvap conj npvap 
'CO vp/ap npvap 
Figure 3: A blocked LCG analysis of the ungram- 
matical (3b) 
b. *Kim \[vP \[v grew \] \[hP wealthy \] and \[r~P a 
Republican \] \] 
Our LCG account analyses these constructions in 
a similar way. Because the LCG account of agree- 
ment has subsumption 'built in', the coordination 
rule merely requires identity of the conjunction and 
each of the conjuncts. 
A conj A 
CO A 
Condition: No undischarged assumptions 
in any conjunct. 3 
We provide an LCG derivation of (2b) in Fig- 
ure 2. Roughly speaking, rule P allows both the 
AP wealthy and the NP a Republican to 'weaken' to 
npvap, so the conjunction satisfies the antecedent of 
the predicate became. (This weakening also takes 
place in non-coordination examples such as Kim be- 
came wealthy). On the other hand, (3b) is correctly 
predicted to be ill-formed because the strongest pos- 
sible category for the coordination is npvap, but this 
does not imply the 'stronger' ap antecedent of grew, 
so the derivation in Figure 3 cannot proceed to form 
a vp. 
Thus on these examples, the feature-based sub- 
sumption account and the LCG of complement co- 
ordination constructions impose similiar feature con- 
straints; they both require that the predicate's fea- 
ture specification of the complement subsumes the 
features of each of the arguments. In the feature- 
based account, this is because the features associ- 
ated with a conjunction must subsume the features 
3This condition in effect makes conjunctions into is- 
lands. Morrill (1992) shows how such island constraints 
can be expressed using modal extensions to LCG. 
72 
associated with each conjunct, while in the LCG ac- 
count the features associated with the complement 
specification in a predicate must subsume those as- 
sociated with the complement itself. 
Now consider the related construction in (4) in- 
volving conjoined predicates as well conjoined argu- 
ments. Similar constructions, and their relevance 
to the GPSG treatment of coordination, were first 
discussed by Jacobson (1987). In such cases, the 
feature-based subsumption account requires that the 
features associated with the predicate conjunction 
subsume those associated with each predicate con- 
junct. This is possible, as shown in Figure 4. Thus 
the feature structure subsumption account incor- 
rectly predicts the well-formedness of (4). 
(4) *Kim \[ grew and remained \] \[ wealthy and a 
Republican \]. 
Because the subsumption constraint in the LCG 
analysis is associated with the predicate-argument 
relationship (rather than the coordination construc- 
tion, as in the feature-based subsumption account), 
an LCG analysis paralleling the one given in Figure 4 
does not exist. By introducing and withdrawing a 
hypothetical ap constituent as shown in Figure 5 it 
is possible to conjoin grew and remained, but the re- 
sulting conjunction belongs to the category vp/ap, 
and cannot combine with the wealthy and a Repub- 
lican, which belongs to the category npvap. 
Informally, while rule P allows the features associ- 
ated with an argument to be weakened, together with 
the introduction and elimination rules it permits the 
argument specifications of predicates to be strength- 
ened (e.f. the subproof showing that remained be- 
longs to category vp/ap in Figure 5). As we re- 
marked earlier, in LCG predicates are analysed as 
(directed) implicational formulae, and the argument 
features required by a predicate appear in the an- 
tecedent of such formulae. Since strengthening the 
antecedent of an implication weakens the implica- 
tion as a whole, the combined effect of rule P and 
the introduction and elimination rules is to permit 
the overall weakening of a category. 
4 Consistency and agreement 
Complex feature structure analyses of agreement 
require that certain combinations of feature con- 
straints are inconsistent in order to correctly reflect 
agreement failure. For example, the agreement fail- 
ure in him runs is reflected in the inconsistency of the 
constraints (case) = acc and (case) = nora. In the 
LCG account presented above, the agreement fail- 
ure in him runs is reflected by the failure of acc to 
imply nora, not by the inconsistency of the features 
acc and nora. Thus in LCG there is no principled 
reason not to assign a category an apparently con- 
tradictory feature specification such as np^nom^acc 
(this might be a reasonable lexical category assign- 
ment for an NP such as Kim). 
COMP = 
V V 
finder und hilft 
VP 
NP ~OBJ = + \] 
Frauen 
Figure 6: The feature structure subsumption analy- 
sis of (5c). 
Consider the German examples in (5), cited by 
Pullum and Zwicky (1986) and Ingria (1990). These 
examples show that while the conjunction finder und 
hilft cannot take either a purely accusative (5a) or 
dative complement (5b), it can combine with the NP 
Frauen (5c), which can appear in both accusative 
and dative contexts. 
(5) a. * Er findet und hilft Miinner 
he find-ACC and help-DAT men-ACC 
b. * Er findet und hilft Kindern 
he find-ACC and help-DAT children-DAT 
c. Er findet und hilft 
he find-ACC and help-DAT 
Frauen 
women-ACC+DAT 
Contrary to the claim by Ingria (1990), these exam- 
ples can be accounted for straight-forwardly using 
the standard feature subsumption-based account of 
coordination. Now, this account presupposes the ex- 
istence of appropriate underspecified categories (e.g., 
in the English example above it was crucial that ma- 
jor category labels were decomposed into the fea- 
tures noun and verb). Similarly, we decompose the 
four nominal cases in German into the 'subcase' fea- 
tures obj (abbreviating 'objective') and dir (for 'di- 
rect') as follows. 
Nominative 
Accusative 
Dative 
Genetive 
{(air) = +, (obj) = -} 
= +, (obj) = +} 
{(air) = -, (obj) = +} 
{(d,r) = -, (obj) = -} 
By assigning the NPs Mh'nner and Kindern the fully 
specified case features shown above, and Frauen the 
underspecified case feature (obj) = +, both the fea- 
ture structure generalization and subsumption ac- 
counts of coordination fail to generate the ungram- 
matical (5a) and (hb), and correctly accept (5c), as 
shown in Figure 6. 
73 
VP 
COMP -- \[V~ 
, coN, v -71 
I I-VERB = +7 FVE  = - 1 L NOUN=+IJ L 
I - j I L I- 'j I I  ouN-+ NooN-+ 
grew and remained wealthy and a Republican 
Figure 4: The feature structure subsumption analysis of the ungrammatical (4). 
remained \[ap\] 1 .p 
vp/npvap npvap/e wealthy a Republican 
grew and vp ap and np 
vp/ap conj vp/ap /il npvap P conj npvap "P 
vp/ap eo npvap eo 
Figure 5: A blocked LCG analysis of the ungrammatical (4). 
As in the previous example, the LCG approach 
does not require the case feature to be decom- 
posed. However, as shown in Figure 7 it does as- 
sign the conjunction finder und hilfl to the cat- 
egory vp/np^ace^dat; hence the analysis requires 
that Frauen be assigned to the 'inconsistent' cat- 
egory np^accAdat. Such overspecified or 'inconsis- 
tent' features may seem ad hoc and unmotivated, 
but they arise naturally in the formal framework of 
Morrill's extended LCG. 
In fact, they seem to be necessary to obtain a 
linguistically correct description of coordination in 
German. Consider the ungrammatical 'double coor- 
dination' example in (6). Both the feature structure 
generalization and subsumption accounts incorrectly 
predict it to be well-formed, as shown in Figure 8. 
(6) * Er findet und hilft M~nner und 
he find-ACC and help-DAT men-ACC and 
Kindern 
children-DAT 
However, the LCG analysis systematically distin- 
guishes between Frauen, which is assigned to the cat- 
egory npAaccAdat, and Mdnner und Kindern, which 
is assigned to the weaker category np^(accvdat). 
Thus the LCG analysis correctly predicts (6) to 
be ungrammatical, as shown in Figure 9. The 
distinction between the categories npAacc^dat and 
np^(accvdat), and hence the existence of the appar- 
ently inconsistent categories, seems to be crucial to 
the ability to distinguish between the grammatical 
(5c) and the ungrammatical (6). 
5 Conclusion 
This paper has examined some of the differences 
between a standard complex feature-structure ac- 
count of agreement, which is fundamentally orga- 
nized around a notion of consistency, and an ac- 
count in an extended version of LCG, in which agree- 
ment is fundamentally an asymmetric relationship. 
We have attempted to show that the LCG account 
of agreement correctly treats a number of cases of 
coordination which are problematic for the stan- 
dard feature-based account. Although we have not 
shown this here, the LCG account extends straight- 
forwardly to the cases of coordination and morpho- 
logical neutralization discussed by Zaenen and Kar- 
tunen (1984), Pullum and Zwicky (1986) and In- 
gria (1990). 
The nature of an appropriate feature system for 
LCG is still an open question. It is perhaps surpris- 
ing that the simple feature system proposed here 
can handle such complex linguistic phenomena, but 
additional mechanisms might be required to treat 
other linguistic constructions. The standard account 
of adverbial modification in standard LCG, for in- 
stance, treat.~ adverbs as functors. Because the verb 
74 
findet \[npAaccAdat\] I hilft \[npAaccAdat\] ~ P P 
vp/npAacc npAacc /~ vp/npAdat npAdat /e 
vp und vp 
vp/npAaccAdat /il conj vp/npAaccAdat ~iS Frauen 
vp/npAaccAdat ~o npaaccAdat 
vp 
Figure 7: The LCG analysis of (5c) 
VP 
OMP = 
v v \[ ro~##+ll c\]~J \[ ro~+l l ~ N~ 
COMP = COMP = FOBJ = + F OBJ = + l CONJ LDIR =_ LDm=+ JJ LDm=-JJ LD,~=+J \] 
I I I I findet und hilft Manner und Kindern 
Figure 8: The feature structure subsumption analysis of the ungrammatical (6). 
findet \[npAaccAdat\] 1 hilft \[npAaccAdat\] 2 P P 
vp/npAacc npAacc vp/npAdat npAdat Miinner 
vp und vp npAacc und 
vp/npAaccAdat /il conj vp/npAaccAdat /i2 npA(accvdat)P conj 
vp/npAaccAdat 
Kindern 
npAdat 
npA(accvdat) P 
npA(accvdat) 
Figure 9: The blocked LCG analysis of the ungrammatical (6) 
75 
heading an adverbial modified VP agrees in number 
with its subject, the same number features will have 
to appear in both the antecedent and consequent of 
the adverb. Using the LCG account described above 
it is necessary to treat adverbs as ambiguous, assign- 
ing them to the categories (s\np^sg)\(s\np^sg) and 
( s\ np^pl) \ ( s\ np^pl). 
There are several approaches which may eliminate 
the need for such systematic ambiguity. First, if the 
language of (category) types is extended to permit 
universally quantified types as suggested by Mor- 
rill (Morrill, 1992), then adverbs could be assigned 
to the single type 
VX.((s\np^X)\(s\np^X)). 
Second, it might be possible to reanalyse adjunction 
in such a way that avoids the problem altogether. 
For example, Bouma and van Noord (1994) show 
that assuming that heads subcategorize for adjuncts 
(rather than the other way around, as is standard) 
permits a particularly elegant account of the double 
infinitive construction in Dutch. If adjuncts in gen- 
eral are treated as arguments of the head, then the 
'problem' of 'passing features' through adjunction 
disappears. 
The comparative computational complexity of 
both the unification-based approach and the LCG 
accounts is also of interest. Despite their simplic- 
ity, the computational complexity of the kinds of 
feature-structure and LCG grammars discussed here 
is largely unknown. Dorre et. al. (1992) showed 
that the satisfiability problem for systems of feature- 
structure subsumption and equality constraints is 
undecidable, but it is not clear if such problems 
can arise in the kinds of feature-structure gram- 
mars discussed above. Conversely, while terminat- 
ing (Gentzen) proof procedures are available for ex- 
tended LCG systems of the kind we presented here, 
none of these handle the coordination schema, and 
as far as we are aware the computational proper- 
ties of systems which include this schema are largely 
unexplored. 

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