In the literature various proofs of the inclusion of the class of LL(k) grammars into the class of LR(k) grammars can be found. Some of these proofs are not correct, others are informal, semi-formal or contain flaws. Some of them are correct but the proof is less straightforward than demonstrated here

Nijholt, A.

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Nijholt, Anton

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On the relationship between the LL(k) and LR(k) grammars

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Volume 15, Number 3 INFORMATION PROCESSING LETTERS 11 October 1982 ON THE RELATIONSHIP BETWEEN THE LL(k) AND LR(k) GRAMMARS * Anton NIJHOLT Faculty of Science, Department of Infwmatics, Nijmegen Universi?v, 6525 ED Nijmegen, The Netherlands Received 14 March 1981; revised ver:sion received 33 July 1982 In the literature various proofs of the inclusion of the class of LL(k) grammars ir.to the class of LR(k) grammars can be found. Some of these proofs are not correct, others are informal, semi-formal or contain flaws. Some of them are correct but the proof is less straightforward than demonstrated here. Keywords: Formal languages, LR(k) g?ammars, LL(k) grammars, parsing 1. Introduction In [ 191 the LR(k) grammars were introduced. In [22] and [20] definitions of LL(k) grammars ap- pear. The three main theoretical problems which are associated with these grammar classes are the following. Problem A. The decidability of the equivalence problems for LL(k) and LR(k) grammars. Problem B. The proof that each deterministic lan- guage can be generated by an LR( 1) grammar. Problem C. The proof that each LL(k) grammar is an LR(k) grammar. This paper is concerned with Problem C. We give an alternative proof that each LL(k) grammar is an LR(k) grammar. There arc g& reasons to do so. Firstly, we note that some of the published proofs are poor from a mathematical point of view. The arguments which are used are sometimes intuitive and sometimes contain flaws. In [20] it is * The preparation of this paper was partially supported by a Natural Sciences and Engineering Research Council of Canada Grant No. A-7700 during the author’s stay at Mcc Master University, Hamilton, Ontario. remarked that a careful and clear proof of this inclusion result would make a suitable master’s thesis. We give a short survey of the relevant literature on this inclusion problem. From the transduction results for LL(k) and LR(k) grammars in [22] (obtained by observations on deterministic pushdown transducers and syn- tax-directed translation schemes) it becomes clear that every LL(k) grammar is also an LIP(k) gram- mar. A similar approach can be found in [ 121 and, in a much more formal setting, in (51. Yet another approach results if, instead of the general device of a deterministic pushdown trans- ducer made suitable for LR(k) analysis, we use the practical parsing algorithm which is based on the construction of LR(k) state sets (cf. [8]). Then it is possible to show that the state sets which can be constructed for an LL(k) grammar satisfy the properties of the state sets of LR(k) grammars. Examples of this kind of argument can be found in [ 13,14,17,21,4]. Moreover such an argument has also been used to show the inclusion of other classes of grammars in the class of LR(k) gram- mars (cf., e.g., (18,251). In the third approach which we want to men- tion the inclusion can be shown by observations on transition diagrams which are used as parsers for a grammar (cf. [6,23,9]). It should be men- 0020-0190/82/0000-OUOO/$O2.75 @ 1982 North-Holland 97 Vohme IS, Number 3 INFORMATION PROCESSING LETTERS 1 I October 1982 tioned that in the latter papers the inclusion prob- lem is not explicitly considered. Instead of establishing the inclusion result by looking at properties of the parsing methods it is possible to study properties of the grammatical trees of LIL(k) and LR(k) grammars. Examples of this argument can be found in [ 161 (e.g.9 the inclu- sion of the strict deterministic grammars in the class of LR(0) grammars) and in [3]. Finally we come to the approach which we prefer. ILL(k) grammars are usually defined by introducing conditions on the leftmost derivations of a grammar. Similarly, LR(k) grammars are de fined with the help of rightmost derivations. It is then natural to prove the inclusion result by slkow- ing that if the leftmost derivations of a grammar satisfy the LL(k) conditions, then the rightmost derivations satisfy the LR(k) conditions. Such proofs can be found in [29] (with a different LR-definition), [l J and [2]. Part of Aho and Ull- man’s proof [l] has been further formalized by Beatty [3] by studying properties of grammatical trees. Other authors who introduce classes of grammars between the LL(k) and LR(k) grammars obtain the inclusion result by showing the inclu- sion of the LL(k) grammars in this newly intro- duced grammar class and then by showing the inclusion of the new class of grammars in the class’ of LR(k) grammars (cf. Soi4on-Soininen and Uk- konen [31] and Pitt1 [28]). Iln [24] related inclusion results can be found. In [30] an alternative for the proof in [ 1] is given. That is, a straightforward and formal proof is given which only uses the defini- tions of L-L(k) and LR(k) grammar s and some properties of derivations. This paper gives another short and formal proof of the inclusion result. It should be mentioned that the proof method is already available in the litera- ture. It is an adaptation of a method which has been used in [ 1 l,28,26]. However, we think it is useful to give a straightforward textbook-like proof of the inclusion result using this proof method. 1. I. Preliminaries we assume that the reader is f&miliar with [ 151. Our notation follows this book. Let G = (V, Z, P. S) be a context-free grammar, let cx E V* and let k be 98 a non-negative integer. Define FIRST&u) = (khw~a SW . 1 \ J For convenience we repeat the definitions of LL(k) and LR(k) grammars. ilefinition IJ. Let k a 0 and G L= (V, 2, P, S) be a reduced context-free grammar such that S&S is impossible in G. G is LR(k) if, for each w, w’, x in C*; a, ar9 /3, ,/3’, y in V*; A, A’ in N, if (i) S GaAw =+a/3w- yw, R R (ii) S Sa’A’x *a'/#'~ = yw', R R . . . () 111 00 W=(wW' 9 then (A --* /3, t!g( a/3) = (A’ --* fi’, tg(a'j3')). Definition 1.2. Let k a 0 and let G = (V, Z, P, S) be a context-free grammar. G is LL(k) if, for each w, x, y in Z*; a, j3, w in V*; A in N, if (i) SswAw-waohx, L L (ii) S~wAw=,wh+vy, L L . . . () ill OOX =OOy 9 then a = j3. Reduced LL(k) grammars are non-left-recur- sive. That is, for any A in N and a in V* a derivation A ;f Aa is impossible. If an LL(k) gram- mar is not reduced, then it is_ not necessarily non-left-recursive. For example, for any k a 0, the left-recursive grammar G with the productions S-A, S-,a and A+A is an LL(k) grammar. Moreover, for any k a 0 this grammar is not LR(k). Proposition 1.3. Let G = (N, =E, P, S) be a reduced LL(k) grammar. For any n * 0, A in N, w in P, X, Y in V and a, /3 in V*, if (i) A:wXaandA:wY@, (ii) FIRSTk(Xa) C3 FIRSTk(Yp) +I& then Xa = Yfl. Volume 15, Numb= 3 INFORMATION PROCESSING LETTERS 11 October 1982 Prwf. Straightforwrd induction on the length n of the derivations. 2. LL@) grammars are LR(kJ grammars We now show the inclusion of the class of LL(k) grammars in the class of LR(k) grammars. Theorem 2.1. Let k a 0. Any reduced LL(k) grum- mar is an I.&(k) gram,mar. Roof. Let G = (V, Z, P, SJ be a reduced LL(k) grammar. Since reduced LL(ki grammars are non- left-recursive a derivation S&S is impossible. Now consider the derivations (i) and (ii) of Defini- tion 1.1. We may assume that. there exist n, m 2 0 and AiEN, ai,yiEV*, lai<n, A; EN, a;, y;EV*, iGjG:m, with A,, =A, A’,= A’, a -=a,(~~ ---a,, a’= a; l l l al,, A, =A6 = S, and production!; Ai-1 +aiAiyi, 1 <iin, AS-1 + oljl Alj yj’, 1 GjGrn, suchthatyn-y,~wandy~-&x. Claim 2.2. n = m, p= @‘, ai = al, Ai = A: and yi = $, 1 s i 6 n. Proof of Claim 2.2. First we prove that ai = ai for all i such that 1 <i SG mix+, m). Assume for the sake of contradiction that this is not the case. Let j be the smallest integer such that aj # ai. THUS one of the words aj, a; must be a prefix of the other. First assume aj ‘=ajXa for some XEV and aE V*. We can write By rewriting a, . l l aj these derivations can be brought into a form which makes it possible to use Proposition 1.3. Notice that yw’ = t$w’ = a’p’x. Hence, al ‘*‘ajcYj+~ l - l Qw’= =a[ 1 . . . ajXtUYi+ I . l l akp’x and cYj+l ’ l l a,pw’ = Xaa;, , l l l a;p’x. Since tk) w = (k) w’ 9 FIRSTk(aj+l l l l a,pw) = = FIRSTk(Xaaj+I l l l a:p’x), (1) and since w E L( y,, l l l y, ), x E L( yk l l l y;) F!RSTk(aj+, l l 0 a,Pv, - - l y,) n nFIRST,(XCUY;+, l l l tYl,p’& l l l y;) #8: . From Proposition 1.3 and derivations (a) and (b) we now may conclude that Aj = X. Hence, X is a nonterminal symbol and from (1) we conclude that (1) aj+l l l l a,P = X = A,. .it follows from deriva- tion (a) that Aj is left-recursive. However, this is not possible for a reduced LwL(k) grammar. Simi- larly the case aj = a:Xa leads to a contradiction. Therefore ai =a: for 1 ~i~min{n,m}. Next we show that n = m. First assume n -C m. Thus ai= ai f9r 1GiGn. Instead of (a) and (b) we now have derivations By rewriting q l l l a, these derivations can be brought into a form which makes it possible to use Proposition 1.3. We know that yw’ = @w’ = a’p’x. Therefore, 99 Volume 15, Number 3 INFORMATION PROCESSING LETTERS 11 October 1982 SW’ = u’,,, . l l al,#Vx. Since dk)w ztk)W’ , FIRSTk(BW) = FIRsIj&,+, l l l ah/?“‘) and (2) nFIRsT&Y:,+, e=‘cUl,~‘~&* l -I’;)#@- From Proposition 1.3 and derivations (a’) and (b’) we may conclude that A = A’, and since G is LL(k) it follows that fl= a’,+,A’,+,y~+I. Thus, instead of (2) we may write (~:,+~A’,,+,y;+y~‘=a:,+, l -+,$‘x and we may conclude that A’,, + , =(‘)a~ +2 . II . akfl’x. That is, A’,,, =(‘)a’,+2 l l l al,j3’. Therefore A’, + f is left-recursive. However, this is not possible for a reduced LL(k) grammar. Simi- larly the case n > m leads to a contradiction and we conclude that n = m and ai = ai, 1 G i G n. From the U-definition and from Proposition 1.3 we now may conclude that Ai = Ai and yi = y[ for 1 <iin. It remains to show that /3 = p’. Notice that the derivations (a’) and (b’) reduce to the derivations S&Ay*a@y and S 9 aAy *~~fl’y with FIRST~( by) n FIRSTk( fl’y) # 8: Since G is LL(k) it follows that /? = fl’. Proof of Theorem 2.1 I(cOnfinued). It follows from Claim 2.2 that (A --, /% fg( a/3)) = (A’ + B’, tg( up’)). That is, G is an LR(k) grammar. Acknowledgemat I am grateful to the referee for some helpful suggestions and comments on an earlier version of this paper. 100 References [ 1] A.V. Aho and J.D. 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In the literature various proofs of the inclusion of the class of LL(k) grammars ir.to the class of LR(k) grammars can be found. Some of these proofs are not correct, others are informal, semi-formal or contain flaws. Some of them are correct but the proof is less straightforward than demonstrated here. Keywords: Formal languages, LR(k) g?ammars, LL(k) grammars, parsing 1

On the relationship between the LL(k) and LR(k) grammars

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