A note on left-recursive rules and the partitioning of a recognition matrix for syntax-directed translation

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Given a context-free grammarG with a cycle of productionsA 1 →A 2 α 1,A 2 →A 3 α 2, ...,A m →A 1 α m, it is shown that there is a grammarG 1 in which the only left-recursive rules are of the formA 1 →A 1 α, andL(G 1)=L(G). Using this normalization it is then shown that a Boolean matrix used in the analysis phase of some syntax-directed translation schemes may be partitioned into rectangular and upper-triangular submatrices.