By Seppo Sippu, Eljas Soisalon-Soininen

This paintings is quantity II of a two-volume monograph at the idea of deterministic parsing of context-free grammars. quantity I, "Languages and Parsing" (Chapters 1 to 5), was once an creation to the fundamental ideas of formal language idea and context-free parsing. quantity II (Chapters 6 to ten) includes a thorough deal with ment of the speculation of the 2 most crucial deterministic parsing equipment: LR(k) and LL(k) parsing. quantity II is a continuation of quantity I; jointly those volumes shape an built-in paintings, with chapters, theorems, lemmas, and so on. numbered consecutively. quantity II starts off with bankruptcy 6 within which the classical con structions bearing on LR(k) parsing are offered. those contain the canonical LR(k) parser, and its decreased variations reminiscent of the LALR(k) parser and the SLR(k) parser. The grammarclasses for which those parsers are deterministic are referred to as LR(k) grammars, LALR(k) grammars and SLR(k) grammars; houses of those grammars also are investigated in bankruptcy 6. loads of consciousness is paid to the rigorous improvement of the speculation: distinctive mathematical proofs are supplied for many of the consequences presented.

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43 Let G = (V, T, P, S) be a grammar and M its canonical LR(k) parser, where k ~ 1. Then M detects an error in any input string in T*\L(G). Proof Let w be a string in T*\L(G), We consider two cases: (1) k:w"# k:w' for all W'EL(G); (2) k:w = k:w' for some W'EL(G). 42. Then the initial configuration for w is an error configuration. In case (2) T* contains strings x, y, and y' such that the following statements are true. (a) w = xy. (b) k:y = k:y' and xY'EL(G). (c) For all y" in T*, xY"EL(G) implies k + l:y"# k + l:y", Now y "# y' because xy~L(G) and xy' EL(G).

4 LR(k) Grammars 45 where y' = k:xly$. This means that (11) Combining (9), (11), and (4) we then have: (12) in M , where n' = n~ r' n'l' Here r(n') = r(n~)r(r')r(n'd = rn l = n, and (13) In'l = In~1 = Inl as claimed. 33 If (M, r) is the canonical LR(k) parser for grammar G, then L(G) £; L(M), andfor any right parse n of sentence w in G, r(n') = nfor some parse n' ofw in M. Moreover, TIMEM(W) ~ TIMEG(w) + Iwl. Proof. 32. 0 --t (x. 34 The canonical LR(k) parser M for grammar G is a right parser for G.

For k = 1 the relation can be computed via a simple relational expression (see the exercises). In the proofs that follow we often use induction on the initial derivation segment S =* c5Az in the definition of LR(k)-validity. Therefore it is convenient rm 22 6. LR(k) Parsing to define explicitly, for each n ~ 0, a set denoted by VALIDk,n(Y) consisting of those items [A -+ (t. f3, y] for which S= rm n(;Az =rm (;(tf3z = yf3z and k:z = y for some (;E V* and ZE T*. 16 For all k ~ 0 and yE V*, U VALIDk,n(Y) .