LIBRARY OF THE 
 
 UNIVERSITY OF ILLINOIS 
 
 AT URBANA-CHAMPAIGN 
 
 510.84 
 U<or 
 no. £19- £>26> 
 cop. 2 
 
„ , UIUCDCS-R-74-62^ 
 
 r Uju 
 
 
 ON THE COVERING PROBLEM FOR 
 UNAMBIGUOUS CONTEXT-FREE GRAMMARS 
 
 by 
 M. Dennis Mickunas 
 
 February, 197^ 
 
UIUCDCS-R-7^-62^ 
 
 ON THE COVERING PROBLEM FOR 
 UNAMBIGUOUS CONTEXT-FREE GRAMMARS 
 
 by 
 
 M. Dennis Mickunas 
 
 February, 197 4 
 
 Department of Computer Science 
 University of Illinois at Urb ana-Champaign 
 Urbana, Illinois 6l801 
 
 This work supported by the Department of Computer Science 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/oncoveringproble624mick 
 
ABSTRACT 
 
 It is shown that every unambiguous grammar which does not 
 generate the empty string is covered by a A-free grammar. Every unambiguous 
 grammar which does generate the empty string is covered by a grammar which is 
 partitioned into a A-free portion and a portion which generates only the empty 
 string. Finally, every unambiguous grammar is covered by such a partitioned 
 grammar in operator form. 
 
1. 
 
 1. INTRODUCTION 
 
 The presence of A-rules in a grammar often poses tricky problems 
 for practical translators. Thus, practitioners usually restrict themselves 
 to considering A-free grammars, arguing that the restriction imposes no 
 significant hardships. In this paper, we present supporting evidence for 
 that position by showing that every unambiguous grammar is completely 
 covered by a grammar in which the A-generating portion (if present) is 
 completely isolated from the remaining portion. That remaining portion is 
 then shown to be covered by a A-free grammar. As a followup, it is then 
 shown that such an unambiguous "A-isolated" grammar is covered by an 
 operator grammar. This strengthens a result of Gray and Harrison [3] • 
 
 Definition. A context-free (CF) grammar is a *+-tuple G = (V,Z, P, S) where: 
 
 (a 
 (b 
 (c 
 
 (a 
 
 (e 
 
 V is a finite non-empty set of symbols (vocabulary); 
 Zc V ( terminal vocabulary ) ; 
 N=V-Z (non-terminal vocabulary) ; 
 
 SeN ( goal symbol ), and; 
 
 P is a finite subset of N x V* ( production ) . 
 
 We will denote an element (u, v) of P by u->v, and we will often 
 ascribe indices to productions: it. = u->v. We also employ the usual binary 
 relation => e V* x V*, writing u => v instead of (u,v) e =^ . 
 
 Let X and Y be sets of words. Write XY = {xy |xeX,yeY} where xy is the 
 
 concatenation of x and y. Define X = [A) where A is the null word. For 
 
 each i > 0, define X 1+1 = X X X and X* = U.^-X 1 . Let X + = X*X and let 
 — i>0 
 
 denote the empty set . 
 
2. 
 
 Definition . Let u, v e V . Define u=>vif there exist words x, y, w e V 
 and A € N so that u = xAy, v = xwy, and A ^w is in P. If y e £ , we write 
 u i|> v. Furthermore, we will write the reflexive-transitive closure of =^> 
 as i • If we wish to make clear that the grammar G is being used, we will 
 
 write =^> . 
 G 
 
 Definition . Let x. e V (o < i < r). If x. =^x. , by applying the production 
 
 it. , then we say that x. directly derives x. , via jt. . If x => x, => . . . =» x 
 1' J i x+1 i o 1 r 
 
 where x. directly derives x. via jt. (for all < i < r) then we say that 
 
 r-1 r-1 
 
 x derives x via (^.). " and that (n. )• is a derivation of x„ from x . 
 o r i y i=o l i=o T o 
 
 R r-1 
 
 If x. =^>x. t (for all < i < r) then (jt. ). is called a canonical de- 
 
 rivation of x from x . 
 r o 
 
 Definition . We define L(G;X) = {xeZ |X=»x) for all X e V. L(G;X) is called 
 
 G 
 
 the language generated by G from X , If X = S, the goal symbol of G, then L(G;S) 
 
 is called simply the language generated by G , and is denoted by L(G). 
 
 Definition . A CF grammar G is said to be unambiguous if and only if 
 
 for all x e L(G;S) and for all canonical derivations (n.)- n? («•)• -, which 
 v ' ' v i / i=l' v i'x=l 
 
 derive x from S, n = n' and n. = n.' (for all 1 < i < n). A CF grammar which 
 
 l l v — — ' 
 
 is not unambiguous is said to be ambiguous . 
 
 Definition . A context-free grammer G = (V, Z, P, S) is said to be 
 
 (a) /y-free if P c N x V + ; 
 
 (b) reduced if 
 
 (i) for each A e V, there exist y, y e V so that S => XAy, and 
 
 -x- -■* 
 (li) for each A e N there exists x e Z so that A ==> x; 
 
 (c) in operator form if P e N x (V - V 8 V ) ; 
 
 P 
 
 (d) in canonical two form if P c W x (CaI U V U N ). 
 
3. 
 
 Following Gray and Harrison [3]> we define the notion of a cover. 
 Definition : Let G = (V, Z, P, S) be a grammer and let H c P. Let 
 
 D = (A. _x.) n , 
 be a canonical derivation in G. Then the corresponding H- sparse derivation 
 
 is 
 
 D TT = (A. _»x. |a. _»x. is in H) n , . 
 H 1 l ' i l '1=1 
 
 Definition . Let G = (V, E, P, S) and G ' = (V, Z, P', S') be context-free 
 
 grammars. Let H c P and H' c P'. Let cp be a map from H' into H. For any 
 
 .n * 
 
 canonical derivation D = (A. ->x.). . in G' of some x e Z , define the 
 
 image of D under cp to be cp (D) = (cp(A. _>x. ) |a. _»x. is in H')._ . C p(D) 
 
 is an element of H . (G' 5 H') is said to cover (G, H) under C p iff 
 
 (a) L(G) = L(G'), and 
 
 (b) for each x e L(G), 
 
 (i) if D is an H-sparse derivation of x in G then there is an 
 H'-sparse derivation D' of x in G' so that <pD ' = D, and 
 
 (ii) if D' is an H'-sparse generation of x in G' then cpD ' is 
 an H-sparse generation of x in G. 
 
 G' is said to cover (G, H) if some H' and cp exist such that (G', H') 
 covers (G, H) under cp. If G' covers (G, P) we say G' completely covers G. 
 2. Main Results 
 
 We turn immediately to the development of our claims . 
 Claim 1 . Every CF grammar G = (V, Z, P, S) is completely covered by a 
 grammar G' = (V Z, P', S') for which: 
 
 a) S' ^S g P'J 
 
 b) S' ^S^ e P'J 
 
 c) S' does not appear in the right part of any production in P'; 
 
 d) L(G';S) - L(G;S) v (A), and; 
 
 e) L(G';S A ) = L(G;S) - L(G';S). 
 
k. 
 
 (Thus, L(G';S) consists of all non-null sentences of L(G;S) Furthermore, 
 
 L(G';S ) = {A) if A e L (G;s) and L(G*;S A ) = <? otherwise.) 
 
 Proof. We may assume [3], without loss of generality that G is in canonical 
 
 two form. Define V* = V U {S ' } U {A |a e Nj and P'=P L U P g U P 3 U P^ U P 5 
 
 where 
 
 p 1 = {s- _>s, s- ->s A ) 
 
 P 2 = {A _»B, A A -»B A |A, B e N; A _B e P} 
 P = {A _>BC, A -^B A C, A ->B C^, 
 
 A A ^B A C Ia, B, C e N; A' -»B C e P} 
 
 A A A [ 
 
 P^ = {A A ^A|A € N, A _>A 6 P} 
 
 P r = {A ^a|A e N, a e Z, A _» a e P} . 
 
 5 
 
 Properties a) - e) hold by construction. It is also clear that 
 
 (G', P 2 U P 3 U P^ U P 5 ) covers (G,P). 
 
 Hence G' completely covers G. Moreover, the canonical two form is preserved. D 
 
 A grammar G* = (V, Z, P'. S) which satisfies a) - c) of Claim 1 
 and for which A | L (G*;S») and L (G';S )c {a}, is called a A-split grammar . 
 
 Having obtained such a A- split grammar, we wish to obtain a A- free 
 cover for the portion which generates L(G';S). The following claim estab- 
 lishes that ability. 
 
 Claim 2 . Every unambiguous CF grammar G' = (V 1 , Z, P 1 , S) for which A ^ L (G';S) 
 is completely covered by a A- free grammar. 
 
 Proof . We may assume without loss of generality that G : is in reduced canoni- 
 cal two form and is A-split via the transformation of Claim 1. Since G' is 
 
 unambiguous, it follows that for each B. e N' > the length of the canonical 
 
 *R i \ 
 
 derivation B => A is finite. Let uAB ) denote that length, and let 
 A G , A 
 
 M = {1, 2,... ,max (u(B ))} 
 B eN' A 
 
 Let G" (V", Z, P", S) where V" = V U (N'xN'xM). 
 
5. 
 
 Define P" = P ' U P ' U P' where 
 
 P^ = {A^v | A e N', v e v ,+ , A _>v e P'} 
 
 P' = {A _^(B A ,C,1),(B A ,C,1) _>(B A ,C,2), ... , (B A ,C, |i (B A )-l) _» (B^C,^) ) , 
 (B A ,C,u(B A )) -»C | A, B A , C e N'; A _> B^ e P'}. 
 
 P^ - {A ^(B,C A ,1), (B,C A ,1) _+(B,C A ,2), ... , (B,C a , u (C a )-1) _» (B,C A , U (C A ) ) , 
 
 (B,C A ,n(C A )) _>B | A, B, C A G N'j A _» BC A € P'}. 
 We define cp by cases : 
 
 a) cp (A _>v e P£) = A ->v; 
 
 b) en (A ^(B A ,C,1) 6 P£) = a ->B A C; 
 
 e) cp ((B A C,i) _*(B A ,C,i+l) e Pp = ^ for 1 < i < U ( B ); 
 
 d) cp ((B ,C Jkl (B )) _C e P' ) '= * , , where B. *4 A via (* )^ ( V; 
 
 e) cp (A -»(B,C A ,1) e P3) - A ^BC A ; 
 
 f ) rp ((B,C A ,i) _» (B,C A ,i+l) e P^) = ^ for 1 < i < ^(c ), and; 
 
 g) cp ((B,C A ,n(C A )) ^B e Pp = n' (c } where C A *4 A via (^)^A } . 
 
 ^ A G ' 
 
 It is easily seen that G" completely covers G' under cp. Furthermore, G" is, 
 
 by construction, A-free and in canonical two form. 
 
 □ 
 
 Theorem 2. 1 . Every unambiguous CF grammar G = (V, 2, P, S) is completely 
 covered by a A-split grammar G' = (V, £, P', S') for which 
 
 a) S' ^S € P*s 
 
 b) S' _,S A e P*; 
 
 c) L (G';S) - L (G;S) {A}, and; 
 
 d) the reduced grammar obtainable from (V, I, P', S) is A- free. 
 
 Proof . The proof follows directly from Claims 1 and 2. 
 
 □ 
 
6. 
 
 A A-split grammar G' which satisfies a) - d) of Theorem 2.1 
 is called a A -isolated grammar. If A 4 L (G';S'), then G' is clearly 
 
 A -free as well. As an example of the application of these transforma- 
 tions, we see that the grammar with productions 
 
 S ^AB 
 
 A _»a | A 
 
 B -*b | CD 
 
 C _»A 
 
 D _>A 
 is completely covered by the A-split grammar with productions 
 
 s- _s | s A 
 
 S _AB | A A B | AB A 
 
 A _^>a 
 
 
 
 B _»b 
 
 
 
 S A - A A B A 
 
 (u(S A ) 
 
 = 5) 
 
 A A ^A 
 
 (u(A A ) 
 
 = 1) 
 
 B A -°A D A 
 
 (H(B A ) 
 
 = 3) 
 
 C A ^A 
 
 (u(C A ) 
 
 = 1) 
 
 D A ^A 
 
 ( W (D A ) 
 
 = 1) 
 
7- 
 which is completely covered by the A-isolated grammar with productions 
 
 S' ->S | s A 
 
 S ^AB |fA A ,B,l) | (A,B A ,1) 
 A ^a 
 B ^b 
 
 (A A ,B,1) ^B 
 
 (A,B A ,1) _»(A,B A ,2) 
 
 (A,B A ,2) ^(A,B A ,3) 
 
 (A,B A ,3) ^A 
 
 s a-Va 
 
 A A- A 
 
 B A^ C A D A 
 
 °A- A 
 
 D A- A 
 
 Gray and Harrison have shown [3] that the A - free portion of a 
 
 A - isolated grammar can be completely covered by an operator grammar. 
 
 We will now show that, in fact the entire grammar can be so covered. 
 
 Claim 3 . Every unambiguous grammar G = (V, 0, P, S.) for which L(G;S ) = { A} 
 
 is completely covered by an operator grammar. 
 
 Proof : We may assume without loss of generality, that G is in reduced 
 
 canonical two form. Since G is unambiguous, it follows that the length 
 
 of the canonical derivation S =^> A is finite. Denote that length by ^(S ). 
 
 A Q 
 
 and let M = {1,2, ... , |i(S )}. Let G' = (V, 0f, P', S ) where 
 V" = {S } U (WxM). Define P» by 
 
 P . = {S A ^(S A ,1),(S A ,D ->(S A ,2), ... , (S A ,n(S A )-l) -♦lB A ,n(B A )) 1 
 
 (S A ,u(S A )) -.A). 
 
 Clearly, the derivation in G' isomorphic to the derivation in G. Thus G' 
 
 completely covers G. Furthermore, G' is, by construction, an operator grammar, 
 
 D 
 
8. 
 
 By a construction similar to that used in Claim 2, it is possible 
 to prove the following: 
 
 Theorem 2.2 . Every operator grammar is completely covered by aA-isolated 
 operator grammar. 
 Proof . Omitted. □ 
 
 We may now state our main result. 
 Theorem 2.3 . Every unambiguous CF grammar is completely covered by a A - 
 isolated operator grammar. 
 
 Proof : The proof follows immediately from Theorem 2.1, Gray and Harrison's 
 Theorem 1.2 [3], Claim 3, and Theorem 2.2. 
 
 D 
 
9. 
 3. Summary and Conclusions 
 
 We have shown that it is often possible to remove A-rules from a 
 grammar without significantly altering its parse trees, and hence its at- 
 tached semantics. This may be taken as supporting evidence of the commonly 
 held notion that in practical situations (i.e. in the case of translators 
 for unambiguous grammars which do not generate the empty string), one can, 
 without difficulty, dispense with A-rules. 
 
 More surprisingly, we have been able to strengthen the operator 
 grammar results of Gray and Harrison [3]« There is, of course, no contra- 
 diction between their Theorem 1.3 and our Theorem 2.3. Their Theorem 1.3 
 holds that there is a grammar (namely the one with productions S _» SS f A) 
 which cannot be covered by any operator grammar. Our results show that 
 their Theorem 1.3 is rooted, not in the fact that the grammar contains a 
 A-rule, but rather in the ambiguity of the grammar. 
 
10. 
 
 REFERENCES 
 
 1. Eloyd, R.W. Syntatic Analysis and Operator Precedence. J. ACM 10,3 
 (July, 1963), 316-333. 
 
 2. Ginsburg, S. The Mathematical Theory of Context Free Languages . 
 McGraw-Hill, New York (1966). 
 
 3. Gray, J.N., and Harrison, M.A. On the Covering and Reduction 
 
 Problems for Context-Free Grammars. J. ACM 19, h (October, 1972), 
 675-698. 
 
BIBLIOGRAPHIC DATA 
 SHEET 
 
 1. Report No. 
 
 UIUCDCS-R- 7^-62^ 
 
 3. Recipient's Accession No. 
 
 i. Title .ind Subtitle 
 
 5. Report Date 
 
 ON THE COVERING PROBLEM FOR UNAMBIGUOUS CONTEXT-FREE 
 GRAMMARS 
 
 February , 197^ 
 
 1. Author(s ) 
 
 M. Dennis Mickunas 
 
 8. Performing Organization Rept. 
 No. 
 
 >. Performing Organization Name and Address 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract/Grant No. 
 
 2. Sponsoring Organization Name and Address 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 13. Type of Report & Period 
 Covered 
 
 Research 
 
 14. 
 
 5. Supplementary Notes 
 
 6. Abstracts 
 
 It is shown that every unambiguous grammar which does not generate 
 the empty string is covered by a A- free grammar. Every unambiguous 
 grammar which does generate the empty string is covered by a grammar 
 which is partitioned into a A-f ree portion and a portion which 
 generates only the empty string. Finally, every unambiguous 
 grammar is covered by such a partitioned grammar in operator form. 
 
 7. Key Words and Document Analysis. 17a. Descriptors 
 
 overs, parsing, ambiguity, A-free, 
 >perator grammars, context-free grammars 
 
 b. Identifiers , Open-Ended Terms 
 
 c. COSATI Field/Group 
 
 Availability Statement 
 
 RELEASE UNLIMITED 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 
 12 
 
 22. Price 
 
 IRM NTIS-35 (10-70) 
 
 USCOMM-DC 40329-P7 1 
 
JUL 2^ 
 
 1974 
 
WW 5 
 
 1977 
 
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