Tests for the LR-, LL-, and LC-Regular Conditions

Most of the linear time parsing strategies (e.g., LL(k) and U(k) type parsers) for context-free grammars operate by looking ahead on the input tape for a fixed number of symbols. The fixed length look-ahead strings partition the set of input strings into classes of strings which are equivalent with respect to parsing decisions. A moment’s thought shows that these look-ahead classes are regular sets. This observation lends itself to a generalization introduced by Culik and Cohen [2]. The idea is to allow arbitr&y sets as look-ahead classes as long as they form a finite partition of the set of input strings. Culik and Cohen applied this idea to LR parsing and obtained the class of LR-regular grammars. Later, the idea was applied to strong LL parsing [7] and to LL parsing [8]. In all cases tests and parsing algorithms for these grammar classes together with various other properties were obtained. This paper had its origin in an example grammar where the criterion of Culik and Cohen fails. This example raises two questions. What is the class of grammars characterized by Culik and Cohen’s criterion? How can we test for the LR-regular condition? Both questions will be answered in the sequel. The first problem will be solved by a simple trick. The answer to the second question will be obtained by generalizing the item method which is well known from M(k) theory. We shall make use of ideas of [4] and shall obtain tests and various other properties for the classes of LL-regular and LC-regular grammars as a by-product. We establish our notation. “Grammar” always means context-free grammar. All grammars are supposed to be reduced, i.e., all nonterminals are reachable and produce terminal strings. The grammar G = (V, 2, P, S) is arbitrary but fixed. We shall make liberal use of the definitions, notations, and facts from [3]. Our conventions for the use of variables are shown in Table I, where ZZ denotes partitions of Z*. These conventions are an essential part of propositions, e.g., L(G) = {xl S %x) 1 0022~0000183 53.00