SIMULTANEOUS-DISTRIBUTIVE COORDINATION 
AND CONTEXT-FREENESS 
Michael B. Kac 
Department of Linguistics 
University of Minnesota 
Minneapolis, Minnesota 55455 
Alexis Manaster-Ramer 
Department of Computer Science 
Wayne State University 
Detroit, MI 48202 
William C. Rounds 
Department of Electrical Engineering and Computer Science 
University of Michigan 
Ann Arbor, MI 48109 
English is shown to be trans-context-free on the basis of coordinations of the respectively type that involve 
strictly syntactic cross-serial agreement. The agreement in question involves number in nouns and reflex- 
ive pronouns and is syntactic rather than semantic in nature because grammatical number in English, like 
grammatical gender in languages such as French, is partly arbitrary. The formal proof, which makes crucial 
use of the Interchange Lemma of Ogden et ai., is so constructed as to be valid even if English is presumed 
to contain grammatical sentences in which respectively operates across a pair of coordinate phrases one of 
whose members has fewer conjunets than the other; it thus goes through whatever the facts may be 
regarding constructions with unequal numbers of conjuncts in the scope of respectively, whereas other argu- 
ments have foundered on this problem. 
respective(ly). Delight in these words 
is a widespread but depraved taste. 
Fowler (1937: 500) 
INTRODUCTION 
Pullum and Gazdar (1982) systematically review and 
critique a large number of arguments for trans-context- 
freeness of natural languages, t finding each one defective 
conceptually, empirically, or mathematically. Among 
these are various ones (e.g., that of Bar-Hillel and Shamir 
(1960) 2 ) appealing to the existence in English of 
sentences like 
(1) John and Bill dated Mary and Alice respectively. 
Pullum (1984) cites a number of more recent arguments, 
involving languages other than English, which do appear 
to establish their trans-context-freeness and remarks (p. 
117), in connection with a suggestion regarding Swedish 
gender agreement, that if this type of agreement can be 
shown to be a purely syntactic matter, then sentences 
analogous to English instances of the schema The N, N, 
... and N are respectively A, A ... and A r, might provide 
the basis of an argument for trans-context-freeness of 
the language (or of any other with similar facts). In this 
paper, we shall produce a rigorous argument along 
comparable lines to show that English is trans-context- 
free (trans-CF), though we shall rely on facts regarding 
grammatical number rather than gender. 
The relevance of a strictly negative result such as the 
one we have obtained is not restricted to the narrow 
question of where natural languages do (or don't) place 
in the Chomsky hierarchy. Given that proving the trans- 
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Computational Linguistics, Volume 13, Numbers 1-2, January-June 1987 25 
Michael B. Kac, Alexis Manaster-Ramer, William C. Rounds Simultaneous-Distributive Coordination and Context-Freeness 
context-freeness of particular natural languages has 
turned out to be considerably more difficult than anyone 
had expected it to be, and that solutions to difficult prob- 
lems are likely to bear fruit outside the parochial confines 
of the original problem area and to call attention to hith- 
erto unnoticed facts, an exercise such as this goes well 
beyond lily gilding or dead horse beating. To develop 
our argument, for example, we shall turn the spotlight on 
the linguistics phenomenon of arbitrary number, some- 
thing which is rarely mentioned in standard treatment of 
grammatical phenomena (in vivid contrast to arbitrary 
gender), but which turns out to be more than a mere 
curiosity. 
The mathematical approach that we employ is also 
noteworthy in making crucial use of the Interchange 
Lemma for CFLs (Ogden et al. 1985) and of a "sepa- 
ration" technique that allows trans-context-freeness to 
be demonstrated by showing that certain strings are 
included in a language while others are excluded. 
Neither of these has, to our knowledge, been used in a 
natural language context before. The interest of the 
separation method in particular lies in the way in which it 
simplifies the following problem. Since natural languages 
are large, complex, and (most important) lacking in ante- 
cedent definitions, the only practical way to argue about 
their mathematical properties is to examine sublanguages. 
If one is not careful, however, one runs the risk of 
committing the "trickle-up" fallacy, which consists in 
showing that a certain set S has a property P and then 
attributing P to some proper superset of S. The usual 
way of circumventing this difficulty is to capitalize on 
closure properties of languages under intersection with a 
regular set; the separation technique provides an alterna- 
tive in cases where appeal to such closure properties is 
not sufficient (or at least not obviously so). 
Partly in the interest of a terminology free of English 
bias, we call constructions like (1) simultaneous-distribu- 
tive (SD) coordinations. This label reflects the fact that a 
sentence such as (1) can be "unpacked" to yield, salva 
veritate, a coordination of noncoordinate sentences by 
means of the following procedure: 
• first, put a copy of the verb directly before each NP in 
the second coordinate phrase that is not already imme- 
diately preceded by a verb, thus creating a coordination 
of VPs; 
• then "distribute" the NPs in the first coordinate phrase 
among the VPs by simultaneously associating a copy of 
each of the former with exactly one of the latter, name- 
ly the one in the corresponding positions; 
• finally, suppress the first coordinate phrase, the first 
and and respectively. 
1 MATHEMATICAL TECHNIQUES FOR ESTABLISHING 
TRANS-CONTEXT-FREENESS 
We shall rely here on a number of established mathemat- 
ical results which, taken together, give us a way of estab- 
lishing trans-context-freeness for a language with certain 
syntactic properties. 
Theoreml (Bar-Hillel et al. 1961) 
The set of context-free languages is closed under 
homomorphism. 
Theorem 2 (Interchange Lemma, Ogden et al. 
1985) 
Let L be a CFL, and let L n be the set of length n 
strings in L. Then there is a constant C L such that 
for any n, any nonempty subset Qn of L n, and any 
integer m such that n > m >_ 2, the following holds: 
Let k = rllQnll/(CLnZ)l, where rxl denotes x 
rounded up to the nearest integer, and II Qn II is the 
cardinality of Qn" Then there are k distinct strings 
z 1, ..., z k in Qn such that z i can be written w i xy i 
forl <i<k, and: 
(i) \[wi\[ = Iwj\[ for alli, j<_k; 
(ii) lYil = lYjl foralli, j_<k; 
(iii) m _> I xi\[ > m/2; 
(iv) \[xi\[ = Ixjl for alli, j<_k;and 
(v) w i xj Yi e L for alli, j, <_ k. 
Since this result is likely to be unfamiliar to some readers, 
we shall provide some commentary that should prove 
helpful in following the remainder of the presentation. 
Less forbiddingly stated, the Interchange Lemma says 
(in part) that in a CFL it is possible, for any n, to find at 
least two strings of length n with internal parts of the 
same length that can be exchanged for each other to 
produce strings that are also in L, providing that there 
are at least two distinct strings of length n. This makes it 
possible to prove the trans-context-freeness of a certain 
kind of language (what kind will be stated in a moment) 
by showing that once n become sufficiently large, the 
possibility of interchange no longer exists. For this strat- 
egy to work, it is required that the cardinality of L n grow 
very rapidly as a function of n, which we can illustrate 
with the case of the copying language over the vocabu- 
lary {a, b} (call this language CP): For every n < 2, 
II cPn II -- 2n2, thus growing exponentially, while Ccpn 2 
grows only polynomially; the necessary conditions for 
use of the Interchange Lemma to prove trans-context- 
freeness are thus satisfied by this case. 
Making use of the Interchange Lemma, we have 
proved the following further result: 
Theorem 3. 
LetH = {xy I x E {a, b}*, y E {c, d}*, y = h(x), 
where h(a) = c, h(b) = d} and G = {xy I x E 
{a,b}*,y c {c, d}*, Ixl # lyl} Then any set Iis 
trans-CF if I is a superset of H and a subset of G LI 
H. 
The proof is presented as an appendix to the paper. As 
an immediate corrollary, we obtain the following result: 
26 Computational Linguistics, Volume 13, Numbers 1-2, January-June 1987 
Michael B. Kac, Alexis Manaster-Ramer, William C. Rounds Simultaneous-Distributive Coordination and Conlext-Freeness 
Theorem 4. 
Let G, H, and I be as defined in Theorem 3, and let 
J = {xy I x • {a, b}*,y • {c, d}*, Ixl = lyl} 
and K = J - H. Then any set L containing H and 
disjoint from K is trans-CF. 
Proof: Intersect L with the regular set H = {xy I x 
• {a, b}*,y • {c,d}*3 and letN = Lt3M. Hisa 
subset of N, K is disjoint from N, and the only 
other strings that might be in N are those in G. 
Therefore, N contains H and some subset (possibly 
empty) of G, and is trans-CF by Theorem 3. Since 
the intersection of any CFL with any regular set is 
CF, L is trans-CF.n 
Theorem 4 says that an arbitrary language L is 
trans-CF if it meets the following conditions: 
• It includes H. 
• It excludes every string not in H that is nonetheless 
divisible into two equal parts, the first over {a, b} 
and the second_over {c, d}. (This is the set K.) 
In order to apply these results, we sill actually 
need to consider not quite the sets G through N as 
defined above but the corresponding sets G r through 
N r, where the latter differ from the former in includ- 
ing only strings whose x and y parts are of at least 
length 2. The subtraction of a finite subset obviously 
changes nothing essential, and Theorems 3 and 4 will 
hold, mutatis mutandis, of sets G t through N r. 
Hence: 
Theorem 5. 
Let H p and K r be as defined above. Then any set 
L' containing H ~ and disjoint from K p is trans-CF. 
Our strategy in applying these results to English will be 
to show that there is a subset F of English that can be 
homomorphically mapped to some L p, and that F is the 
intersection of English with a regular language. This 
suffices to show that English itself is trans-CF. 
2 GRAMMATICAL NUMBER AGREEMENT IN ENGLISH 
Our empirical argument rests on the claim that number 
agreement between reflexive pronouns and their antece- 
dents is a syntactic phenomenon in English. For exam- 
ple, the string 
(2) *The girl likes themselves. 
must be considered ungrammatical rather than merely 
semantically ill-formed by virtue of the impossibility 
(because of number incompatibility) of supplying an 
intraelausal antecedent for the reflexive pronoun. The 
reason that this is so has to do with a fact about gram- 
matical number in English that has not been generally 
recognized; namely, that it is, like grammatical gender in 
languages such as French and German, partly arbitrary. 
This can be shown by a number of different kinds of 
examples, among them the following. First, there are 
synonym pairs in English, each consisting of a grammat- 
¢ 
Computational Linguistics, Volume 13, Numbers 1-2, 
ically singular member and a grammatically plural one. A 
partial list is given in the table below) 
SINGULAR PLURAL 
apparel clothes 
forest woods 
underwear underpants 
car wheels 
kibble crunchies 
location whereabouts 
merchandise goods 
Pamir Pamirs* 
Hellespont Dardanelles 
corpse remains 
pant** pants 
hosiery stockings 
military armed forces 
issue offspring 
* a mountain range in Central Asia 
** as used in the garment trade 
Further, these examples can be elaborated in various 
ways. For example, names of some mountain ranges are 
strictly singular (Caucasus, Hindu Kush), while those of 
others are strictly plural (Alps, Rockies); items from simi- 
lar semantic fields may vary as to their grammatical 
number properties (compare odds-probability, wheat-oats, 
yoghurt-curds, pasta-noodles, mush-grits (in some 
dialects), Granola-Rice Krispies). Note further that there 
is dialect variation regarding the grammatical number of 
certain collective nouns (such as government and 
company), which are strictly singular in American 
English, but which can be used as plurals in British 
English. 
A further phenomenon on which we shall capitalize is 
the existence in English of an idiomatic way of express- 
ing the ease with which an activity can be performed 
involving the use of reflexive constructions, as illustrated 
by, for example, This land will rent itself, or These woods 
will sell themselves. With this in mind, compare now 
(3) This land and these woods can be expected to rent 
itself and sell themselves respectively. 
(4) *This land and these woods can be expected to rent 
themselves and sell itself respectively. 
It is clear that strings like (3), in which each reflexive 
pronoun agrees with the corresponding noun, are gram- 
matical, while those in (4), in which each pronoun disa- 
grees with the corresponding noun, are not. This fact 
will be the basis of our demonstration that English is 
trans-CF. 
3 ARGUMENT REGAINED 
Let A = {{this land, those woods} and {this land, these 
woods} + can be expected to {{rent, sell} {itself, 
themselves}} + and {{rent, sell} {itself, themselves}} 
respectively}, and note that A is regular. Now let B be the 
subset of A that satisfies the following condition: 
January-June 1987 27 
Michael B. Kac, Alexis Manaster-Ramer, William C. Rounds Simultaneous-Distributive Coordination and Context-Freeness 
In case the number of occurrences of members of 
{this land, these woods} is equal to the number of 
occurrences of members of {itself, themselves}, then 
for all i _> l, if the ith noun in the string is land, 
then the ith pronoun is itself and if the ith noun is 
woods, then the ith pronoun is themselves (i.e., 
number agreement obtains between the nouns and 
the reflexive pronouns). 
Now let C = A - B. Every string in C contains exactly 
as many pronouns as it does nouns, but for some i > 1, 
the ith pronoun fails to agree in number with the ith 
noun. Finally, let D be the subset of B consisting of just 
those strings that contain exactly as many nouns as 
pronouns. 
It is clear that D is part of English, and that C is 
disjoint from English, inasmuch as D exhibits the 
required number agreement and C does not. If D were 
the intersection of English with the regular set A, then 
the result that English is trans-CF would follow 
immediately. 4 However, things are not that simple, and it 
is conceivable that the intersection of English with A is 
some proper superset F of D that is a subset of B (possi- 
bly B itself). The point of uncertainty here is the status 
of strings of A with unequal numbers of occurrences of 
/ nouns and pronouns, such as the following: 
(5) This land, these woods, and this land can be 
expected to rent itself and sell themselves respec- 
tively. 
(6) This land and these woods can be expected to rent 
itself, sell themselves, and rent itself respectively. 
While it might seem that such strings are ungrammatical, 
this assumption is called into question by Pullum and 
Gazdar's (1982) observation that there is no syntactic 
constraint in English governing the number of conjuncts 
in SD-coordinations. Thus, contrary to conventional 
wisdom, there are perfectly well-formed SD-co-ordina- 
tions with mismatched numbers of conjuncts; for exam- 
ple, The last two people in this picture live in Columbus 
and Chicago respectively. This undermines a number of 
older arguments that English is trans-CF that crucially 
assume that grammatical SD-coordinations must have 
equal numbers of conjuncts. In light of this, it may be 
that strings like (5-6) are to be considered syntactically 
well-formed, albeit lacking sensible interpretations, and 
so an argument presupposing the contrary cannot be used 
to show that English is trans-CF. 
We now show that Theorems 3, 4, and 5 allow us to 
get around this obstacle by, in effect, ignoring the strings 
with mismatched numbers of conjuncts in constructing 
our argument. If strings like (5-6) are grammatical, then 
the intersection of English with the regular language A is 
not the trans-CF language D but some proper superset F 
thereof. Theorem 5 tells us in effect that, no matter what 
its exact identity, if there is a sublanguage of English 
homomorphic to H r but none homomorphic to K p, then 
English is trans-CF. This "separation" strategy yields the 
conclusion that so long as English contains D and 
excludes C, it is trans-CF. 
Recall now that, according to our definition, B 
includes strings like (5-6), along with sentences like (3) 
but, crucially, excludes (4) and all strings like it. To be 
precise, B includes all strings of A which, like (3) have 
exactly as many nouns as pronouns and cross-serial 
number agreement, but also all strings in A that, like (5) 
and (6), have more pronouns than nouns or vice versa. 
In virtue of Theorem 5, we will be able to show that 
English is trans-CF no matter what position we take on 
the grammaticality of strings like (5-6), so long as there 
are no English sentences in the subset C of A consisting 
of strings in which there are as many nouns as pronouns 
but at least one of the pronouns fails to agree with the 
corresponding noun in the first part of the string. Thus, 
the intersection of English with A is some subset F of B 
that is disjoint from C; it is of no consequence whether F 
is equal to all of B, or only to D, or to some proper 
subset of B that is a proper superset of D. 
We now define the homomorphism h such that 
h (this land) = a 
h (these woods) = b 
h (itself) = c 
h (themse-lves) = d 
For all z • {can be expected to, sell, rent, and, 
respectively}, h(z) ~ 
This homomorphism maps F to L t. Since the CFLs are 
closed under homomorphism, and L r is trans-CF, F is 
trans-CF. And since the CFLs are also closed under 
intersection with regular sets, and the intersection of 
English with the regular set A has turned out to be 
trans-CF, it follows that English is also trans-CF. 
It should be immediately apparent that a similar strate- 
gy can be applied in instances such as the one mentioned 
in Section 1, where grammatical gender agreement is 
involved, providing that the language in question has 
instances of arbitrary gender. So, for example, we can 
construct for French a sublanguage parallel to D consist- 
ing of sentences like 
(7) Cette nation et ce pays sont respectivement une 
alli6e et un associ6 des Etats-Unis. 
'This nation and this country are respectively an 
ally and a partner of the United States.' 
In this example, we capitalize on the fact that the inani- 
mate nouns nation 'nation' and pays 'country' belong to 
different gender classes, reflected in the predicate nomi- 
nals une alli~e 'an ally' and un associd 'a partner'. 5 Inver- 
sion of the predicate nominals yields an ungrammatical 
string, but corresponding inversion of the subjects 
restores grammaticality: 
(8) *Cette nation et ce pays sont respectivement un 
associ6 et un alli6e des Etats-Unis. 
(9) Ce pays et cette nation sont respectivement un 
assoei6 et une alli6e des Etats-Unis. 
28 Computational Linguistics, Volume 13, Numbers 1-2, January-June 1987 
Michael B. Kac, Alexis Manasler-Ramer, William C. Rounds Simultaneous-Distributive Coordination and Conlexl-Freeness 
Comparable examples can be constructed in other 
languages, a case in point being the Polish sentence 
(10) Francja i Kongo s~ przeciwniczka wzgl~dnie zwol- 
ennikiem traktatu. 
which translates literally as 'France and (the) Congo are 
opponent respectively supporter (of the) treaty'; here, as 
in the French example, the two nouns in the subject 
phrase are of different grammatical genders, and are 
matched with corresponding gender-compatible nouns in 
the predicate phrase. Example (11) is ungrammatical 
(notice the suffixes of the predicate nouns) while (12) is 
grammatical again. 
(1 1) *Francja i Kongo s~ zwolennikiem wzgl~dnie 
przeciwniczk~ traktatu. 
(12) Kongo i Francja s~ zwolennikiem wzgl~dnie przec- 
iwniczk~ traktatu. 
4 CONCLUSION 
We would like to close by pointing out the importance 
and interest of,the following question: Given that it is 
logically possible for a language to have an operator just 
like English respectively except that the conjuncts to be 
linked with each other are paired in center-embedding 
rather than cross-serial fashion, and given that the prop- 
erties of such an operator can be characterized by formal 
apparatus apparently more elementary than what is 
required to characterize respectively, why do operators of 
this seemingly more elementary type appear not to exist 
in any natural language? The use of apparently is impor- 
tant here: from the standpoint of the Chomsky hierarchy, 
nesting is less complex than mutual intercalation, in the 
sense that the type of grammar required to handle the 
former is more restricted than the type required to handle 
the latter, but the possibility is always open that this type 
of complexity is not germane to human psychological 
capacity. We strongly suspect that this is the case (see 
Manaster-Ramer and Kac 1985), though that is a topic 
for another time. 
APPENDIX: PROOF OF THEOREM 3 
Assume that I c G U H is context-free and apply the 
Interchange Lemma to 
Q2n = {xh(x) \] x E {a, b} n} 
Since H _c I, Qzn _c Izn, the set of length 2n substrings in 
I. Choose n suitably large (for the exact choice, see the 
proof of Claim 1 below), and let m = n. Let k = 
r llQznl\[/(Cl(2n)2)'l be the number defined in the 
lemma. We get k distinct z i in Q2n satisfying the conclu- 
sion of the lemma. Our result will follow from the next 
two claims: 
Claim 1. Let x i, ..., x k be the middle parts of z i, ..., 
z k respectively. Then there are i and j such that x i 
xj, provided that n is suitably chosen. 
Claim 2. If x i ¢ Xj, then wixjy i is not in G O H and, 
afortiori, not in I. 
Proof of Claim 1. Suppose that all the x i were equal. By 
(iii) of the Interchange Lemma, I xil > n/2. Therefore, 
at least n/4 characters from the x i are in the {a, b} half 
or in the {c, d} half of z i. We may assume the former 
possibility since the argument works exactly the same 
way in the latter case. Each z i is determined by the n 
characters of its {a, b} half and, by supposition, at least 
n/4 of these characters are fixed. So there can be at most 
2 n-n~4 = 2 3n/4 strings z t ..... z k. But JJ Q2n IJ = 2 n, and if n 
is chosen so that 23n/4 < 2n/(Cl n2) = Jl Qzn JJ/(el n2), we 
contradict the Interchange Lemma, which says that all 
the z's are distinct. Note that n can always be chosen this 
way, no matter what C I is, by elementary inequalities 
from college algebra and calculus. 
Proof of Claim 2. Observe first that interchanging x i with 
xj does not produce a string in G. The substrings x i and 
xj disagree in some position, which is also a position in 
one half or the other of the strings z i and zj. Thus the 
matching position in z i and zj (in the other half of the 
word) does not occur in x i or xj because \] Xi\] -~ \]Xjl _< 
n, by (iii) of the Interchange Lemma, so interchanging x i 
and xj produces a string not in H. 
Since Claim 1 and Claim 2 violate clause (v) of the Inter- 
change Lemma, I cannot be context-free.II 
NOTES 
1. We use the term trans-contexl-free in preference to non-context- 
free, to distinguish the class of languages outside the weak genera- 
tive capacity of type 2 grammars. 
Computational Linguistics, Volume |3, Numbers 1-2, January-June 1987 29 
Michael B. Kac, Alexis Manaster-Ramer, William C. Rounds Simultaneous-Distributive Coordination and Context-Freeness 
2. This argument is reiterated in Postal (1964); however the relevance 
of SD-coordination to the question of eontext-freeness was evident- 
ly first noticed in 1959 by Ray Solomonoff in personal communi- 
cation to Chomsky (see Chomsky 1963). 
3. Some of these examples were suggested to us by Rosemarie Whit- 
ney. 
4. See, for example, the argument, due to H. Brandt Corstius, cited in 
Levelt (1974: 31-32). 
5. These nouns were selected because they have counterparts of the 
opposite gender: thus, corresponding to allide we have alli~, and 
corresponding to associd we have assoCi~e. The masculine and femi- 
nine forms of 'partner' and 'ally' are phonologically indistinguish- 
able in isolation, but can be distinguished in the context of 
preceding indefinite articles, as our examples show.  

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