TECHNICAL CORRESPONDENCE 
SUBJECT-VERB AGREEMENT IN RESPECTIVE 
COORDINATIONS AND CONTEXT-FREENESS 
Langendoen (1977) advanced an argument against 
English being a context-free language involving cross- 
serial subject-verb agreement in respectively constructions 
such as (1). 
(1) The man and the women dances and sing, 
respectively. 
As noted by Pullum and Gazdar (1982), however, and 
acknowledged subsequently by Langendoen (personal 
communication), such examples are unacceptable, and 
the argument collapses on empirical grounds. 
However, at least some speakers reject examples like 
(2) as well. 
(2) The man and the women dance and sing, 
respectively. 
This fact leads directly to a demonstration that there is, 
after all, a cross-serial dependency involving the gram- 
matical number of subject NPs and verbs in respectively 
constructions. However, it is not clear at present how 
representative such speakers are, and so instead of 
making claims about English in general, we will confine 
them to just those varieties of the language that stigma- 
tize examples like (2), which will be denoted as English1, 
leaving to one side any varieties of which this may not be 
true (English2). I 
In English1, a verb that formally distinguishes singular 
from plural, i.e., a non-auxiliary present tense verb, 
cannot occur in a respectively construction if the corre- 
sponding subject NP is singular. This cannot be 
accounted for merely by barring marked singular verbs 
from occurring in coordinate predicates of respectively 
constructions. Such a move would correctly exclude 
examples like (1), but it would allow sentences like (2), 
with plural verbs corresponding to singular subjects. 
Nor is it possible to simply bar singular subjects from 
occurring in respectively constructions, since they are 
perfectly possible provided the corresponding verb is 
either a past tense, as in (3a), or an auxiliary, as in (3b). 
(3) a.The man and the women danced and sing, 
respectively. 
b.The women and the man sing and can dance, 
respectively. 
This means that a singular subject can only co-occur 
with a past tense or an auxiliary verb, whereas a plural 
subject can take a non-auxiliary present tense verb as 
well. The difference in the co-occurrence possibilities of 
singular as opposed to plural subject NPs amounts to a 
peculiar kind of number agreement. 2 This fact leads 
quite directly to a demonstration that English 1 is not 
context free. 3 
Consider the regular set (4). 
(4) {the man x and the women danced y and sing 
respectively \[ x e {the man, the women}+; y E 
{danced, sing} + } 
This is the set of all strings 4 (only some of them gram- 
matical in English) consisting of .any number of occur- 
rences in any order of the phrases the man and the 
women, with an and between the last two, followed by 
any number of occurrences in any order of the words 
danced and sing, with an and between the last two, 
followed by the word respectively. 
According to what has been said, the intersection of 
(4) with English 1 must be (5). 
(5) {the man x and the women danced y and sing 
respectively I x • {the man, the women}+; y = 
%(x) • {danced, sing}+; ol(the man) = {danced}; 
% (the women) = {danced, sing}} 
This is the set of all those strings of (4) that meet the 
additional condition that every occurrence of the man 
must be matched by an occurrence of danced and every 
occurrence of the women by an occurrence of either 
danced or sing. This matching is achieved by defining the 
substitution s o~ of the man to be the set {danced} and that 
of the women to be the set {danced, sing} and requiring y 
to be equal to o 1 of x. 
We now define a substitution o 2 such that 
o2(the man) = {a} 
oz(the woman) = {b} 
oz(danced) = {c} 
o2(sing) = {d} 
oz(and ) = {~}6 
o2(respectively) = {~} 
This substitution maps (5) to (6). 
(6) {a x b c y d I x e {a, b}+; 
y = 03 c {c, d}+; o3(a ) = c; o3(b) = {c, d}&rbrc. 
We now intersect (6) with the regular set 
{a + b + a + c + d + c +} 
to obtain (7). 
(7) {a nb ma Ic kdjc i I n < k;m < j; 
l_<i;n+m+l=k+j+i} 
The set in (7) can be shown trans-context-free by 
pumping. Take the string z = akbkakckdkc k (where k is 
the constant of the pumping lemma). On the one hand, 
if we pump only in the first or only in the second half of 
the string, the resulting string will violate the condition 
that n + m + l= k +j+ i. On the other hand, sincek 
64 Computational Linguistics, Volume 13, Numbers 1-2, January-June 1987 
Technical Correspondence A Note on a Study of Cases 
is the constant of the pumping lemma, the pumped 
substring cannot be longer than k, and therefore the only 
other place we might be able to pump would be in the 
middle as and the middle cs, But this would result in 
violating the condition that l may not be greater than i. 
Thus, z cannot be pumped without violating the pumping 
lemma, and hence (7) is not context free. Since 
context-free languages are closed under intersection with 
regular sets, it follows that (6) is not context-free either. 
Since context-free languages are also closed under substi- 
tution, this means that (5) is also not context-free. Final- 
ly, since (5) is the intersection of English, with the 
regular set (4), it follows that Englishj is not context- 
free. Q.E.D. 
Alexis Manaster-Ramer 
Computer Science Department 
Wayne State University 
Detroit, MI 48202 
NOTES 
1. Pullum and Gazdar (1982) state that they "'can tolerate" examples 
like (2), and Langcndoen (personal communication) agrees. 
2. In other terms, we must be able to tell which verb would agree with 
which subject given the chance, and disallow just those combina- 
tions where the result would be a marked singular verb. 
3. Ignoring, for the sake of simplicity, the arguments advanced in 
Manaster-Ramer (1983; in press) about the need to state formal 
results about natural language in other than weak generative capaci- 
ty terms. 
4. Ignoring, for the sake of readability, the punctuation that would 
normally be required in written English and the suprasegmental 
features that would occur in the spoken language. 
5. In the discussions of formal properties of natural languages, substi- 
tutions have not figured at all prominently, whereas homomor- 
phisms, which are just a special case of substitutions, have. it may 
be helpful, therefore, to point out that a substitution is a mapping 
like a homomorphism except that it is usually denoted by o rather 
than h and that it may associate each element in the vocabulary of a 
language with a whole set (possibly infinite) of strings rather than 
with just one string, as in the case of a homomorphism. In the pres- 
ent case, we needed to employ a (slightly more general kind of) 
substitution in order to be able to associate the women with sing as 
well as danced. It should also be noted that, while the man and the 
women are linguistically analyzable, we have for technical conven- 
ience treated them as single elements of the terminal vocabulary in 
dcl'ining Ihc substitutions. 
6. )~ denotes the empty string. 

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Kac, M.B.; Manaster-Ramer, A; and Rounds, W.C. 1987 Simultane- 
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Langendoen, D.T. 1977 On the Inadequacy of Type-3 and Type-2 
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Descriptive and Historical Linguistics: Festschrift for Win fred P. 
Lehman. John Benjamins, Amsterdam: 159-172. 
Manaster-Ramcr, A. 1983 The Soft Formal Underbelly of Theoretical 
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Manaster-Ramer, A. In press. Dutch as a Formal Language. Linguis- 
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