Context sensitive grammar and generation of noncontext free language

Book, R. V.

English

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Ronald V .  book 
6-71 
Ccnter for Kesearch in Computing Technology 
Harvard Oniversity 
Cambridge, Plassact-rusetts 0213s 
 his paper was written w h i l e  the author was v i s i t  inp the 
Department of System Science, University of California, 1.0s 
Arlgeles. I t  was supported i n  part by the National Science 
Foundation under Grant No. CJ-803, and by thc National 
Aeronautics and Space Administration under Grant No. NGK 
22-007-176. 
Introduction. 
It is well @awn that the family of cantext-sensitive grammars generate 
languages which a m  not context-free and that  it is undecidable whether a context- 
sensitive gnanmu. generates a mtext - f ree  language. However the mechanism by 
- which the use of context allows a non-context-free language t o  be generated is not 
well understood--in fact, the question i t s e l f  is vague: what does context do for  
you? In  this paper we attempt t o  make this questim mre precise by surveying so= 
of the results which speak t o  two mre specific questions: (i) w h a t  constraints 
can be placed on the form of the rules of cmtext-sensitive g r a ~ ~ ~ n a r s  without re- 
s t r ic t ing the weak generating capacity? ; and ( ii) what (nm.triviaU constraints can 
be placed on the form of the rules of context-sensitive grarro~.lars such tha t  only 
context-free languages w i l l  be gemated? 

Section 1. 
In this section we review sorrre basic definitions and facts about context- 
sensitive granmws and languages. 
A gr- G = (V, C, R, X) is context-sensitive if each rule is of the form 
* 
+ clyB where a, B, y e V , y # e , and Z E V - C. A language is context-sensitive 
if it is generated by s m  context-sensitive pamar. Let us recall a few facts 
about context-sensitive gr- and languages. 
A language is context-sensitive if and only if it is accepted by a nondeter- 
ministic linear bounded automaton (a nondeterministic Turing acceptor which uses 
an amount of tape proportional to the length of the input). Hence every context- 
sensitive language has a primitive recursive characteristic function but there are 
sets with primitive recursive characteristic functions which are not context-sensitive. 
It is not known whether every context-sensitive language is accepted by a deter- 
ministic linear bounded automton . 
The family of context-sensitive languages possesses rrany of the positive closure 
properties of the context-free languages; for example, this family is a principal 
AFL closed under e-free substitution. H~wever, this family is not closed under 
arbitrary hommrphic map~ings; in fact the imge of the family of context-sensitive 
languages under arbitrary homarorphic mappings is the family of recursively enumer- 
able sets. In addition, this fimily is closed under intersection and it is an open 
- 
question whether it is closed under complementation. 
The form of the rules of a context-sensitive g r m  places no restrictions on 
either left or right context. It is clear that no generative capacity is lost if one 
requires that the context be free of terminal symbols. Similarly, no generative 
capacity is gained (or lost) if mre than one symbol is transfomd by a rule, that 
is, if the non-context-fme rules do not transform terminal symbols and obey the 
restriction that length is preserved, i. e. , p + 0 where I 1 < 1 0 1 . In this case 
-
the gramar is called mxlo~onic. 
Perhaps thk best known noMndl form for context-sensitive granmrs is that 
.. introduced by K u m d a .  [lo]. A context-sensitive gramm G = (V, C , R, X) is i n  
Kumda Normdl Form if each rule i n  R is of one of the following forms: 
ZY -t Z'Y 
YZ -t YZ1 Z,Z',Y,Y1 E V - C  , ~ E C .  
z -t Z' 
Z-ta 
Given any context-sensitive gearmar G1, one can effectively construct a cmtext- 
sensitive grammr G2 in K w d a  No& Form such that L(G2) = L(G1). 
The restriction on the form of the miles of a context-sensitive pamap prevents 
the empty word fmmbeing generated. Often the restriction is altered so that a 
context-sensitive (or extended context-sensitive) language is allowed to cant& the 
empty word, usually by adding same ad hoc rule which generates only the empty word. 
In general, if erasing ( i . e . , length-decreasing) miles are present, then a non-context- 
sensitive language m y  be generated. Ginsburg and Greibadh C51 have shown that 
limited use of erasing rules in derivations preserve the property of generating only 
a context-sensitive language--in this case, "limited use" mans applying no mre than 
a number proportional to the length of the terminal string generated. In general, the 
- upper bound for such "limited use" is not known C11. 
Section 2. 
In th i s  section we consider som of the constraints which can be placed on the 
f o m  of the rules of context-sensitive granmrs without forcing the language generated 
t o  be only context-free. This is done in order t o  gain sow insight in to  the question, 
- 
"what does context do for  you?t' 
I f  one is constructing a context-sensitive g r m  t o  generate some given non- 
context-free language, then one often proceeds as i f  context can be used t o  "store 
and transmit information." Thus one builds miles so that  in derivations "messages" 
(or are transmitted dong a string. Sometimes th i s  effect  is achieved by 
building a granmar which imitates the action of some l inear bounded autarraton; hence, 
the action of the read-write head rrrust be imitated as it travels back and forth 
across the tape. 
The "ability t o  send messages" has not been formalized i n  such way as t o  explz5n 
"what context does for  you," although som properties of the structure of derivatj-ona 
nave been studied [l, 6,  71. However tkis notion does provide an intuit ive "hxfid3.e9' 
for  studying som questions and for  gaining perspective on sow results on contexr- 
sensitive grammars and languages. (For a different approach t o  som of these questions, 
see C 3  1 particularly Section 3 . )  
Haines [81 has established an important result on the structure of context- 
sensitive grammars. A l e f t  context-sensitive gramrim is a context-sensitive gramr 
G ( V ,  C ,  R, X) such that  every non-context-free rule is of the form 
.*I 
ctZ + ay where a ,  y E v", Z E V - C and y # e. 
(Thus, every non-context-free rule has only l e f t  context.) Haines has shown that  
each context-sensitive language is generated by a l e f t  context-sensitive pamar; 
i n  addition, the context need not be more ti-~dcl a single symbol, i . e . ,I a 1 = 1. 
Interpreting this resul t  in terms of l'mssage-sending," we see that  it is enou* 
t o  "send messages" only in one direction--in th i s  case, t o  the right--so that  one 
need only "guess" that  the "mssage w a s  received." (Of course one has the sam 
result for right context-sensitive g r a m r ~ ~  since the theory is symmetric. This 
result  m y  be interpreted cis a new normal form theorem, that  is, it is enough t o  
consider just those g r m  with context-free rules and l e f t  context-sensitive 
rules. 
From the results of Kuroda [ l o ]  ( in particular, Kuroda N o m l  Form), it is 
easy t o  see that one loses no generative capacity i f  one conside~s only those g r m  
- 
such that i n  any derivation only context-free rules are applied f i r s t  and then m'\i 
length-preserving context-sensitive d e s  are applied. One might well ask whethey 
the same result is obtained i f  one relaxes the r e q u i r e m t  that  the non-context-fme 
be length-preserving. It is sham i n  El1 that  this is not the case. 
If one considers those context-sensitive gr-s such that  every rule is ei ther 
length-increasing o r  generates only a terminal symbol, then the family ,&' of languzges 
generated has the following properties: 
I (i) every e-free context-free language is i n  X /  ; 
(ii) for every recursively enumrable s e t  L, there exists  a language Lo in .& and 
a hommrphism h such that  h[iOl = L - hence, .d contains languages which are 
not context-free; 
.r' (iii) .+ is  a proper subfamily of the context-sensiti~~e languages--in particular, 
R .% 
wcw cw I w E {a,b]"} is not i n  $ . 
Note that  (ii) implies that a gr- of this form must have some "message- 
sending capacity" but the proof of (ii) given i n  C11 depends on the grammar also 
generating some "padding1' which i s  eventually erased by the hom~~~rphism. O n  the 
other hand, (iii) shms that  the power t o  "send mssages" is quite restricted. 
One effect oq context is the abil i ty to  achieve a permutation of symbols, that  
t': 
is, using context-sensitive rules one can obtain AE --t BA. One might well ask i f  
context-sensitive rules d1m anything more than the composition of permutations 
along with the type of substitution a l lwed by context-free rules. Si l lars  C131 has 
shown that  this is not the case. He did -this by investigating l lpemtat ion g r m , "  
g r m  which have only pennutation rules in addition t o  the cmtext-free miles. 
A mnotanic gr- G = ( V  , C , R, X3 is a permutation gramar i f  each non- 
context-free rule in R is of the form Y1 ... Ym+ Y ... Y where each Y. E V-C 
~(1)  d m )  I 
and where n is some permutation of 1 . , m with no fixed points. L e t  TiR be tlle 
s e t  of llpenrutationll rules of R and l e t  Gr = (V, E ,  R-lIRy XI so that  Gr is a con- 
t e x t - f ~ e  grannnar. It can be shown that L(G) i s  inf ini te  i f  and only i f  L(Gr) is 
infini te,  so that  L(G) i s  inf ini te  i f  and only if it has an inf ini te  context-free 
subset. Thus the pemta t ion  gr- c m o t  generate all the context-sensitive 
n n n  languages; for  example, {a b c ] n > 11 has no inf ini te  context-free subset. On 
- 
the other hand, it is clear that  the pemta t ion  gramars do generate non-context-free 
languages; for  example, it is easy t o  construct a permutation g r m  t o  generate 
54 
" { W E  Ca,b,c) 1 whas anequalnumberof a's, b ' s ,  a n d c t s )  . 
Note that a permutation gramrm has only length-preserving rules in addition to  
i t s  context-free rules. However it is not clear how t o  analyze such grammars in terms 
of l'message-sending. 'I I n  fact ,  these grammrs may be best analyzed by considering them 
t o  be weak extensions of context-free grammm. 
It should be noted that there are other types of gramam which extend the 
context-free languages but generate only a proper s u b f q l y  of the oontext-sensitive 
4 
languages. For example, there are matrix gr-, indexed grammrs, programmed 
g r m ,  t h - v a r i a n t  granmws, and others. In these cases either the type of rule 
is different fmmthe definition of a context-sensitive rule o r  the way that  the miles 
are applied is regulated in som fashion. It is not clear whether the study of these 
g r m  would throw l ight  on the questions considered in this paper. 
Section 3 .  
In this section we consider certain restrictions on the form of the rules of 
a context-sensitive gr- which force the language generated t o  be context-free. 
In each case it is decidible whether the rules of a g r m  satisfy the restriction. 
Recall that  a context-free language may be represented in various ways: by a 
context-free panunar; by a nondeterministic pushdown store acceptor; by a formila 
-1 
hl(h2 (Dl n R) ,  where hl andh2 are hommorphisms, D is aDyck set ,  and R i s  a 
regular set .  However every panmr has a particular context-free language associated 
with it, the language obtained by left-to-right derivations. 
In an arbitrary rewriting system or  gramnnar G = (V, C, R, X I ,  a derivation 
$, --+ ;=+ . . . -+ Qn is a left-to-right derivation i f  for  each i=l ,. . . , n, there 
:'t C 
arre strings a E X , 6 E V , and a rule p + 9 E R such that  $ = a and = a%$ 
- 
--that is, at each step the remithg rule is applied at the l e f t m s t  possible position 
in the string. It has been sham by Evey C41 and by Matthews U l l ]  tha t  fo r  any re- 
writing system or grammar, the s e t  of terminal strings obtained by left-to-right 
derivations from the i n i t i a l  symbol is  a context-free language. I f  one relaxes the 
"leftmst" condition to  allcrw the rewriting rule t o  be applied w i t h i n  so= fixed dis- 
tance of the leftmost possible position in the s t r i i g ,  then once again one obtains a 
context-free language. Clearly one can substitute right-to-left for  left-to-right 
since the general theory of rewriting systems is syrmetric. 
In an arbitrary rewriting system or  gr- G = (V , C , R , X) , a derivation 
@, + Q 1  + * * *  9% is a two-way derivation i f  for  each i-1, ..., n, there are 
.L 
strings a, 6 and a rule p -+ 0 E R such that  Qi = ape, Jli = a96, and ei ther a E 'Z 
- 
.% 
or  B E Ce' --that is, at each step the rewriting rule is applied at ei ther the l e f t -  
most o r  the right most possible position in the string. It has been sham by Matthews 
C121 that  for  any rewriting system or grarranar the set of teminal  strings obtained by 
two-way derivations from the initial symbol is a context-free language. (A short 
and elegant p m f  of this result  w a s  given by Ginsburg and Greibach C51.1 Again 
one can a l l o w  the rewriting rule to  be applied w i t h i n  some fixed distance of the ends 
and still generate only a context-free language. 
These results  provide a useful tosl for  showing that  a given g r m  generates 
a context-free grammr. Thus, if one can shw that  for  some given grammar the s e t  
of left-to-right (or right-to-left o r  two-way) derivations yields the entire language 
generated by that  gramzw, then one can conclude that  the gran-mr generates only a 
context-free language. This m-thod of attack is i l lustrated by the followbg result. 
Let G = ( V ,  C,  R, X) be any context-sensitive grammar such that  each non-cmtext- 
23 I 
free rule is of the form aZB-+ay(3 where a E C , Z E V - C, 6 , y E V , and 
]a\ 2 I B I (i .e. ,  the left context is a terminal string and is at  leas t  as long as the 
right context). It is sham in [21 that  L(G) must be context-free. This is dme by 
showing that  L(G) may be obtained f r o m  derivations such tha t  a t  each step the rewriting 
rule is  applied within some fixed distance of the l e f t m s t  possible position in the 
string--in th i s  case the fixed distance is m(m +1) /2  where 
rn = max /a1 I aZB + ay8 E R 1 .  
In this case the terminal context r e q u i r e m t  allows the strings t o  be generated 
while "sending messages" only a bounded distance, in fact ,  a distance bounded by 
m(m + 1 ) / 2 .  
It should be noted that the requiremnt that  the terminal l e f t  context be at 
least as long as the right context cannot be relaxed C21--that is, i f  it is relaxed, 
then it is possible to  generate non-context-free languages. 
There are several other results whose known proofs do not depend on reduction 
t o  the yield of left-to-right derivations. In some cases the restrictions on the 
form of the rules my  be viewed as creating "barriers" which keep "mssages" fmm 
being "passed." 
r )  
Let G = (V,C,R, X) be a mnotonic g r m .  Suppose < is a par t ia l  order on V 
w i t h  the property that for  each rule Z1.. . Z -# Y1.. .Y in R, there exists Y. in P Q I 
{Y1 , . . . ,Y 1 such that  fo r  i = 1,. . . , p, Z. < Y. . In t h i s  case L(G) is context- 9 = 1 
free C9l. 
Let G = (V,  C,  R, X) be a mnotonic gramnar. Suppose that every non-context- 
1 Jc free rule in R is of the foam p + 9 where p E ( v - E l f i  and 0 E V E V (so that every 
application of a non-context-free rule generates a new terminal symbol which cannot 
be used as part of context). In this case L(G) is context-free C51. This result  
strongly suggests the notion of "barriern: temimal symbols cannot be used as part 
of context and a new terminal symbol is generated whenever context is  used --thus 
"wssagesl' cannot be lltransmitted'l mre than som bounded distance. 
L e t  G = (V, C ,  R, X) be a context-sensitive gr-. Suppose that  each non- 
.% 5': 
context-free rule in R is of the form clZB -t ~6 where a, 6 E I", Z E V- c , y E v , 
In this case L(G) is context-free C21. 
In each case cited above the known proofs depend on the representation of a 
context-free language as h 2 - % ~ )  f7 R) o r  as hl(D n R) where hl and h2 are horn-r- 
i 
phisms, D is a Dyck s e t  and R is a regular set. In each case the proof suggests the 
P idea of a tlbarrier'l but there is no forrralization of this notion. 
A careful analysis of the proofs of the results cited above suggests that  a 
context-sensitive pammr generates a non-context-free language drily i f  an infinite 
number of strings i n  the language are generated by derivations which tnmsforrn syrrilx)ls 
arbitrari ly far f r o m  e i ther  the leftnost o r  rightrnost nonteMnindl symbol, and that  
arbitrari ly many steps of the derivation "interactw in som way. What is needed is 
a fonrdization of the notion of "nessage-s&ding" and of "barrier" in order t o  prove 
I 
the appropriate necessary conditions for a gmmnw to generate a non-context-fme 
language. Such results would provide a reasonable answer t o  the question: 
context do for  you? 
What does 
References 
1. R. Bwk, Tim-bounded g r m  and their  languages, J. Computer and System Sciences 
S(19711, t o  appear. 
2. R. Book, On the effect of terminal cantext on context-sensitive g r m ,  Math. 1 
Linguistics and Autamatic Translation, NSF-18 (1967 1 , Computation L a b o r a t o c  j 
Harvard Wvers i ty  ,' Cabridge, Mass. 
I 
3. N . Chomsky, Formal pmperties of grammars, in Luce , Bush, Galanter (Eds .1 , I 
Handbook of Math. Psychology, Vol. 11, John Wiley , New York, 1963. 
4. R. Evey, "The Theory and Application of Pushdawn Store Machines," Ph.D. Disse*ation 
Haward Univ. , 196 3. Also appears as Math. L@guistics- ad Automatic Translation 
NSF-10 _I_ (19631, Compu-bation Lab., Harvard lhiv., Cambridge, Mass. 
5. $. Ginsburg and S.A. h i b a d h ,  Mappings which preserve context-sensitive languages, 
Info. and Cmtrol 9 (19661, 563-582. 
6. A. G l a d k i i ,  On the complexity of derivations in phrase structures grurmnars, 
Algebri i Logika Sern. 3 (19641, 29-44. 
7. T. Griffiths, Sam remarks on derivations in general rewriting systems, Info, and 
--- Control 1 2  (19681, 27-54. 
8. L. Haines, Representation theorems for  context sensitive laiflflguages, submitted for  
publication. 
9. T. Hibbard, "Scan Limited Automta and Context Limited Grarmars," Ph.D. Dissertation 
Univ. of California, Los Angeles, 1966. 
10.  S.Y. Kuroda, Classes of languages and linear-bounded automata, Info. and Control 
- --. 7 (19641, 207-223. 
U. G.H. Matthews, A note on asymetry in phrase stmchu?e grammm, Info. and Control 
7 (19641, 360-365. 
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13. W . Sil lars  , "Fo;rrral Properties of Essentially Context-Dependent Languages , " Ph . D ,
Dissertation, Penn. State Univ., 1968. 


On the structure of context-sensitive grammars

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