English and the Class of Context-Free Languages I 
Paul M. Postal 
IBM Thomas J. Watson Research Center 
Post Office Box 218 
Yorktown Heights, NY 10598 
D. Terence Langendoen 
Brooklyn College and the Graduate Center 
City University of New York 
33 West 42 Street 
New York, NY 10036 
0. Background 
Let L range over all natural languages (NLs). For any L, 
one can consider two collections of strings of symbols, 
one consisting of all strings over the terminal vocabulary 
of L, call it W*(L), the other consisting of that always 
very proper subcollection of W*(L) consisting of all and 
only those members of W*(L) that are well-formed, that 
is, that correspond to sentences of L. Call the latter 
collection WF(L). 
During the early development of generative grammar, 
a number of attempts were made to show, for various 
choices of L, that WF(L) was not a context-free (CF) 
string collection. These attempts all had, naturally, a 
common logical structure. First, it was claimed that there 
was some mathematical property P which, if possessed by 
some collection of strings, C, rendered C non-CF. 
Second, it was claimed that WF(C) had P, so the conclu- 
sion followed. Two sorts of criticisms can be, and have 
• been, directed at such attempted demonstrations. One 
attacks the mathematical foundations and argues, for 
particular choices of P, that a collection manifesting P is 
not necessarily not CF. The other type of criticism 
admits that if a collection manifests a particular property 
P, it is thereby not CF, but contends that the WF(L)s 
claimed to manifest P in fact don't. 
A survey of the various attempts, from roughly 1960 
to 1982, to prove for various L that WF(L) is not CF is 
provided in Pullum and Gazdar (1982). These authors 
conclude, justifiably we believe, that for one or the other 
of the reasons mentioned above, none of these attempts, 
including those by the present authors, stand up. Despite 
widespread belief to the contrary, as of 1982 there had 
been no demonstration that there is some NL L for which 
WF(L) is not CF. 2 
However, Langendoen and Postal (1984) have 
obtained a result infinitely stronger than the claim that 
for some L, WF(L) is not CF. This work shows that for 
any L, WF(L) is a proper class, hence not a set, much less 
a recursively enumerable set. There is thus no question 
of WF(L) being CF. Moreover, WF(L) can then have no 
constructive characterization (generative grammar), 
although there is no reason to doubt that it can be given 
a nonconstructive characterization. But the demon- 
stration of Langendoen and Postal (1984) is based on 
principles that determine WF(L) includes nonfinite strings 
corresponding to nonfinite (transfinite) sentences. It is 
the existence of such sentences that places complete NLs 
beyond generative (constructive) characterization. 
Nevertheless, as noted in Langendoen and Postal (1984: 
103), this novel result still leaves entirely open the ques- 
tion of whether that subpart of WF(L) consisting of all 
and only the well-formed finite strings in W*(L) is CF. 
Let F(inite)WF(L) be that subcollection of WF(L) 
consisting of all and only the finite strings corresponding 
to the finite sentences of L. What follows shows that 
there are dialects of English, E1 and E2, such that: 
l We thank J. Higginbotham for helpful comments on an earlier version 
of this paper. 
2 Recently, Higginbotham (1984) presents another argument that 
English is not CF. The formal part of the demonstration seems impec- 
cable, but the factual premises are questionable; see Pullum (p. 182). 
Copyright1985 by the Association for Computational Linguistics. Permission to copy without fee all or part of this material is granted provided that 
the copies are not made for direct commercial advantage and the CL reference and this copyright notice are included on the first page. To copy 
otherwise, or to republish, requires a fee and/or specific permission. 
0362-613X/84/030177-05503.00 
Computational Linguistics, Volume 10, Numbers 3-4, July-December 1984 177 
Paul M. Postal and D. Terence Langendoen 
1. Neither FWF(E1) nor FWF(E2) is CF. 
The demonstration of (1) makes use of the following 
corollary of a theorem of Arbib (1969) about language 
intersections: 
2. The Intersection Theorem 
Let L be a stringset ar~l let R be a regular stringset. 
If L n R is not a CF stringset, then neither is L. 
One can then show that, for example, FWF(E1) and 
FWF(E2) are not CF by finding some regular set R such 
that R n FWF(EI) and R n FWF(E2) are not CF (Daly 
1974). 
1. The Sluicing Construction 
The present demonstration that FWF(E), E ranging over 
various forms of English, is not CF is based on the sluic- 
ing construction, first discussed by Ross (1969) and more 
recently by van Riemsdijk (1978) and Levin (1982). 
Standard examples of this construction include: 
3a. He stole something but it's not known what. 
b. Someone stole the jewels and I can tell you who. 
c. The police found him in some bar but the paper 
didn't say which one. 
The sluicing construction has the following properties: 
4a. It consists of n (n > 2) clauses, often, but not 
necessarily coordinate clauses joined by but 
b. Each of the second to nth clauses counting from the 
left contains a wh-phrase, WH, which corresponds 
to a potential indefinite phrase found in the first 
clause. 
c. WH has an interpretation equivalent to an entire 
wh-clause, WH+Z, and the 'missing parts' are 
understood as identical to the first clause minus the 
potential indefinite phrase. 
The reason for the strange usage 'potential indefinite 
phrase' in (4b) is the existence of sluicing cases like (5b) 
and (6b) alongside (5a) and (6a): 
5a. Martha was abducted by someone but we don't 
know by who(m). 
b. Martha was abducted but we don't know by 
who(m). 
6a. The doctors screamed for some reason but we don't 
know why. 
b. The doctors screamed but we don't know why. 
Depending on frameworks, one might analyze cases like 
(5b) and (6b) as involving invisible versions of indefinite 
phrases like those in the corresponding (5a) and (6a). 
But this matter need not concern us. We can concentrate 
on cases where alternatives like (5b) and (6b) are not 
possible, as in (7): 
7a. Max visited someone but we don't know who. 
b. *Max discussed Tom but we don't know who. 
c. *Max discussed but we don't know who. 
English and the Class of Context-Free Languages 
That is, we pick cases where WH can only be anaphori- 
cally connected to a visible element in the first clause. 
From this point on, references to the sluicing 
construction only denote cases of this restricted sort. 
The nature of the formal argument to be presented is 
such that this limitation in no way impugns the validity of 
the present result, since one can still construct a model of 
the intersection situation described at the end of the 
previous section. 
In (3a), WH is what, the indefinite phrase correspond- 
ing to it is something and WH is understood as equivalent 
to the wh-clause what he stole, since he stole is the whole 
first clause minus the indefinite phrase. A similar analysis 
holds for (3b). 
In (3), as in most of the examples in the literature 
previously illustrating the sluicing construction, the indef- 
inite phrase in the first clause is an indefinite pronoun, 
and WH is a wh-pronoun. However, both phrases can 
consist of multi-word sequences: 
8a. Sarah considered some proposals but it's unknown 
how many. 
b. If any books are still left on the sale table, find out 
which ones. 
c. The warehouse will ship us several typewriters but 
we have no idea how many machines. 
d. A few physicians still use this drug and Sam can tell 
you how many doctors. 
e. Joe discussed certain formulas but which formulas 
is uncertain. 
Let us from this point on, for simplicity, limit attention to 
sluicing constructions consisting of only two clauses. 
Then it is possible to represent all the relevant cases 
schematically as follows: 
9. V QI XI W \[Y Q2 X2 Z\], where V, W, Y and Z are 
strings; QI is an indefinite quantifier or pronoun; Xl 
is the rest of the nominal quantified by Q1; Q2 is a 
wh-quantifier or pronoun anaphorically related to 
Q1; and X2 is the rest of the nominal quantified by 
Q2. Moreover, Q2 X2 (= WH) is understood as a 
wh-clause that contains material from V or W. 
Table 1 presents the values of Q1, X1, Q2 and x2 in the 
examples in (8a-e). Henceforth, we further restrict the 
class of sluicing constructions under consideration, limit- 
ing attention only to examples like (8c-e), in which X1 
and X2 are neither empty nor pronouns. 
In (8c-e), the main stress on the wh-phrase can fall on 
either Q2 or on X2. If it falls on X2, then the wh-phrase 
can not be anaphorieally related to the corresponding 
indefinite phrase in the first clause. While we cannot, 
and need not, give a theoretical account of 
"anaphorically related", informally it means that the 
potential reference of the wh-phrase is determined by 
that of its antecedent. Hence the lack of anaphoric 
connection in cases of stressed X2 means that in partic- 
ular, in (8c), machines does not denote the same things 
that typewriters does in that sentence; in (8d), physicians 
178 Computational Linguistics, Volume 10, Numbers 3-4, July-December 1984 
Paul M. Postal and D. Terence Langendoen English and the Class of Context-Free Languages 
Table 1. 
EXAMPLE Q1 X1 Q2 X2 
(8a) some proposals how many 4' 
(8b) any books which ones 
(8c) several typewriters how many machines 
(8d) a few physicians how many doctors 
(8e) certain formulas which formulas 
does not denote the same people that doctors does (the 
latter perhaps referring to nonmedical doctors); and in 
(8e) the two occurrences of formulas then denote differ- 
ent things (say, mathematical formulas in the first 
instance and baby milk formulas in the second). On the 
other hand, if phrasal stress falls on Q2, then the 
wh-phrase is anaphorically related to the corresponding 
indefinite phrase in the first clause. In (8c), machines is 
then taken to denote the same things that typewriters 
does; in (8d), physicians is then taken to denote the same 
people that doctors does and in (8e) the two occurrences 
of formulas denote the same objects, whether mathemat- 
ical or nutritional. 
Henceforth, we-limit attention only to examples in 
which phrasal stress falls on Q2 (indicated by small caps). 
These are therefore structures where the wh-phrase in the 
second clause is anaphorically related to the correspond- 
ing indefinite phrase in the first clause. 
It turns out that variants of English differ with respect 
to the class of wh-phrases that can be used 
anaphorically. 3 So, for many speakers, but not all, (8c-e) 
are fully acceptable with phrasal stress on Q2. For 
others, only (8e) is fully acceptable; in fact, (8c,d) are 
judged to be ungrammatical. For the latter dialect, 
henceforth referred to as El, the subpart of the sluicing 
construction on which we have focused is subject to the 
constraint informally stated as in (10), henceforth 
referred to as the strong matching condition (SMC). 4 
10. If WH is the anaphoric wh-phrase in the second 
clause, then, if X2 is neither null nor a pronoun, the 
sequence of linguistic elements up to and including 
the head noun of X2 must be identical to the material 
up to and including the head noun of X1. 
According to SMC, only the posthead modifiers of X1 
and X2 can differ in El, as in: 
1 1. Joe discussed several attempts to grow corn on Mars 
but WHICH attempts is unknown. 
If prehead modifiers differ, then the resulting structure is 
ungrammatical in El: 
12. El*Joe discussed lots of curious proposals but 
everyone has forgotten WHICH proposals. 
Now consider another dialect, call it E2, in which 
(8c-e) are fully grammatical when phrasal stress falls on 
Q2. For E2 speakers, the relation between X1 and X2 is 
governed by condition (13), which we refer to as the 
weak matching condition (WMC). 
13. X2 is a possible anaphor of X1. 
According to WMC, X2 can either be a complete repe- 
tition of Xt, as in (8e); a synonym of X1, as in (8d); or a 
term whose denotation includes that of X1, as in (8c), 
(11) and (12), the last two examples being well-formed 
in E2. However, if X2 is not a possible anaphor of X1 
and phrasal stress falls on Q2, then the resulting structure 
is ungrammatical even in E2. Since the following exam- 
ples are ungrammatical in both E1 and E2, they are 
marked with double asterisks. 
14a. **The warehouse will ship several machines to our 
office but we have no idea how MANY typewriters. 
b. **A few physicians still use this drug and Sam can 
tell you how MANY nurses. 
c. **Joe discussed certain formulas but WHICH 
equations is uncertain. 
In (14a), typewriters cannot be used as an anaphor for 
machines, presumably because the reference of the 
former fails to subsume that of the latter. This judge- 
ment is rendered by E2 speakers even for contexts in 
which the words typewriters and machines are otherwise 
used interchangeably, such as an office with limited word 
processing equipment. This shows that one is dealing 
here with a grammatical restriction, not a pragmatic 
property that varies with context. Similarly, in (14b), 
nurses cannot be used as an anaphor for physicians, since 
some physicians are not nurses and some nurses are not 
physicians; again this is true even in a context in which 
all the nurses under discussion happen to be physicians 
and vice versa. Finally, equations cannot be used as an 
anaphor for formulas in (14c), since again an equation is 
only a certain kind of formula (statements of inequality 
are also formulas). One who judges that (14c) is in fact 
grammatical might well do so under the mistaken belief 
that all mathematical formulas are equations. Alterna- 
tively, one could assume that such a judge has a different 
dialect of English, with different anaphoric conditions. 
Now consider examples of the sluicing construction in 
which a compoun~i noun occurs as X1. If X2 exactly 
matches X1, then the result is grammatical in both E1 and 
E2. If only the head of the compound occurs as X2, then 
the results are always ungrammatical in El, and either 
3We have not investigated whether this correlates with more general 
differences in anaphoric usages for these distinct forms of the language. 
Moreover, this is not relevant to the present demonstration. 
4El is the dialect of the first author. 
Computational Linguistics, Volume 10, Numbers 3-4, July-December 1984 179 
Paul M. Postal and D. Terence Langendoen English and the Class of Context-Free Languages 
grammatical or ungrammatical in E2, depending on the 
relation between X1 and X2. As before, double asterisks 
mark examples that are ungrammatical in both E1 and 
E2. 
15a. Joe discussed some candy store but it's not known 
WHICH candy store. 
b. El*Joe discussed some candy store but it's not 
known WHICH store. 
c. **Joe discussed some fire escape but it's not known 
WHICH escape. 
d. **Joe discussed some bourbon hater but it's not 
known WHICH hater. 
e. **Joe discussed some bourbon lover but it's not 
known WHICH lover. 
The whole compound can be used as an anaphor for 
itself in both E1 and E2, as in (15a). But store cannot be 
used as an anaphor for candy store in (15b) in El, since 
SMC is not satisfied. On the other hand, store can be 
used as an anaphor for candy store in (15b) in E2, 
presumably since a candy store is a certain kind of store; 
that is, candy store is an endocentric compound. Howev- 
er, escape cannot be used as an anaphor for fire escape, 
even in E2, since a fire escape, which is a certain kind of 
physical object, is not an escape, which is a certain kind 
of event; that is, fire escape is an exocentric compound. 
Finally, hater and lover cannot be used as anaphors for 
bourbon hater or bourbon lover in (15d,e), since the agen- 
tive noun hater is used only in compounds and lover; by 
itself has a limited (sexual) meaning which makes it 
unsuitable as an anaphor for compounds such as bourbon 
lover 
An important consequence for the present discussion 
is that if one limits the vocabulary over which sluicing 
constructions are formed in the right way, the conditions 
of linkage in E1 and E2, though intensionally distinct, 
become extensionally identical. That is, for fixed cases, 
SMC and WMC have the same consequences. For any 
such situation, one can thus equate them and refer simply 
to the matching condition (MC). 
It is possible to embed English compounds within 
compounds; in particular, there are well-formed 
compounds such as bourbon hater lover 'one who loves 
bourbon haters', bourbon lover hater 'one who hates 
bourbon lovers', bourbon hater lover hater 'one who hates 
those who love bourbon haters', etc. Now, if any such 
compound occurs as X1 in a sluicing construction, then 
for speakers of both E1 and E2 the only possible anaphor 
drawn exclusively from the vocabulary used to construct the 
compound that can occur as X2 is the whole compound 
itself. 
16a. Joe discussed some bourbon hater lover but it's not 
known WHICH bourbon hater lover. 
b. **Joe discussed some bourbon hater lover but it's 
not known WHICH hater lover. 
c. **Joe discussed some bourbon hater lover but it's 
not known WHICH lover. 
It follows that if attention is limited to instances of the 
sluicing construction like (16), in which the only possible 
anaphonc wh-expressions are whole compounds, then 
SMC and WMC are equivalent, permitting one to speak 
simply of MC with no loss of accuracy. Given this fact, 
we are now in a position to demonstrate simultaneously 
that neither FWF(E1) nor FWF(E2) is CF. 
2. The Proof 
We first define the following notion: 
17. A copying language is any language of the form: 
L {cxdxe \] x E (a,b)* and a,b,c,d,e are fixed strings} 
Given that English contains a sluicing construction char- 
acterized as in section 1, one can prove that FWF(E), E 
ranging over E1 and E2, is not CF by means of the Inter- 
section Theorem of (2) and the fact that copying 
languages are not CF (Langendoen 1977). To prove that 
FWF(E) is not CF, one must find a regular language R 
whose intersection I with FWF(E) is not CF. Such an R is 
given in (18): 
18. R = {Joe discussed some bourbon x but WHICH 
bourbon y is unknown t x,y E (hater, lover)*} 
Since R is a concatenation of regular languages, it is itself 
regular. 
Now consider the intersection of R with FWF(E). The 
matching condition on sluicing constructions guarantees 
that this intersection is the copying language I in (19): 
19. I = {Joe discussed some bourbon x but WHICH 
bourbon x is unknown \] x,y ~ (hater, lover)*} 
Since I is not CF, by the Intersection Theorem, neither is 
FWF(E). 
As has been stressed, limitations on the vocabulary 
render the distinct sluicing conditions of E1 and E2 
equivalent over certain subcollections of sluicing cases. 
It follows that the demonstration just presented holds not 
only for E1 and E2 but more generally for any variant of 
English whose matching condition for sluicing, even if 
different from that of both E1 and E2 for the full 
collection of English sentences, has the same extension 
for the language R of (18). We see no current reason to 
doubt that this will include every variant of English. 
3. Conclusion 
Gazdar (1983: 86), summarizing inter alia the conclu- 
sions of Pullum and Gazdar (1982), makes the following 
claims: 
20a. "There is no reason, at the present time, to think 
that NLs are not CFLs." 
b. "There are good reasons for thinking that the 
notations we need to capture significant syntactic 
generalisations will characterise CF-PSGs, or some 
minor generalisations of them, such as Indexed 
Grammars." 
180 Computational Linguistics, Volume 10, Numbers 3-4, July-December 1984 
Paul M. Postal and D. Terence Langendoen English and the Class of Context-Free Languages 
But as Langendoen and Postal (1984) shows, NLs are 
proper classes not sets, so the question of WF(L)s as 
wholes being CF no longer arises. Restricting attention 
to FWF(L)s, the result of Section 2 shows that for any 
dialect E of English for which MC holds, FWF(E) is not 
CF. 
Since, however, neither FWF(E) nor any other FWF(L) 
has been shown to lie outside the domain of indexed 
languages (ILs) in the sense of Aho (1968), it would 
appear that one can conclude that while the collection of 
all sentences in an NL K is a proper class, FWF(K) is an 
IL. Consequently, a correct account of NL grammars 
must be such that a proper grammar for K specifies K as 
an appropriate proper class 5 and entails that FWF(K) is 
an IL. A grammatical theory with just these properties 
remains to be constructed. 
5By 'appropriate', we mean one which satisfies inter alia the axiom 
called Closure Under Coordinate Compounding of Sentences in 
Langendoen and Postal (1984: 53). This is necessary for the proof that 
NLs are proper classes. 

References 
Aho, A.V. 1968 Indexed Grammars-An Extension of Context-Free 
Grammars, Journal of the Association for Computing Machinery 15: 
647-671. 
Arbib, M. 1969 Theories of Abstract Automata. Prentice-Hall, Engle- 
wood Cliffs, New Jersey. 
Daly, R.T. 1974 Applications of the Mathematical Theory of Linguistics. 
Mouton and Company, The Hague. 
Gazdar, G. 1983 NLs, CFLs and CF-PSGs. In: Sparck Jones, K. and 
Wilks, Y., Eds., Automatic Natural Language Parsing. Ellis Horwood 
Ltd., West Sussex, England. 
Higginbotham, J. 1984 English is Not a Context-Free Language. 
Linguistic Inquiry 15:119-126. 
Langendoen, D.T. 1977 On the Inadequacy of Type-2 and Type-3 
Grammars for'Human Languages. In: Hopper, P.J., Ed., Studies in 
Descriptive and Historical Lingistics. John Benjamins, Amsterdam, 
Holland. 
Langendoen, D.T. and Postal, P.M. 1984 The Vastness of Natural 
Languages. Basil Blackwell, Oxford, England. 
Levin, L. 1982 Sluicing: A Lexical Interpretation Procedure. In: Bres- 
nan, J., Ed., The Mental Representation of Grammatical Relations. 
The M1T Press, Cambridge, Massachusetts. 
Pullum, G.K. and Gazdar, G. 1982 Natural Languages and Context- 
Free Languages: Linguistics and Philosophy 4:471-504. 
Ross, J. R. 1969 Guess Who. In: Binnick, R. et al., Eds., Papers from 
the Fifth Regional Meeting Chicago Linguistic Society. University of 
Chicago, Chicago, Illinois. 
van Riemsdyk, H. 1978 ,4 Case Study in Syntactic Markedness: The 
Binding Nature of Prepositional Phrases. Foris Publications, 
Dordrecht, Holland. 
