Briefly on Bottom-up

These short notes provide descriptions and references relative to the construction of parsing tables for SLR(1), for LR(1), and for LALR(1) grammars. 1 Notation and basic definitions Basic definitions and notational conventions are summerized below. A context-free grammar is a tuple G = (V, T, S,P), where the elements of the tuple represent, respectively, the vocabulary of terminal and nonterminal symbols, the set of terminal symbols, the start symbol, and the set of productions. Productions have the shape A → β where A ∈ V \ T is called the driver, and β ∈ V ∗ is called the body. The one-step rightmost derivation relation is denoted by “⇒”, and “⇒” stands for its reflexive and transitive closure. A grammar is said reduced if it does not contain any useless production, namely any production that is never involved in the derivation of strings of terminals from S. In what follows we assume grammars be reduced. The following notational conventions are adopted. The empty string is denoted by ǫ. Lowercase letters early in the Greek alphabet stand for strings of grammar symbols (α, β, . . . ∈ V ), lowercase letters early in the alphabet stand for terminals (a, b, . . . ∈ T ), uppercase letters early in the alphabet stand for nonterminals (A,B, . . . ∈ (V \ T )), uppercase letters late in the alphabet stand for either terminals or nonterminals (X, Y, . . . ∈ V ), and strings of terminals, i.e. elements of T , are ranged over by w,w0, . . .. For every α, first(α) denotes the set of terminals that begin strings w such that α ⇒ w. Moreover, if α ⇒ ǫ then ǫ ∈ first(α). For every A, follow(A) denotes the set of terminals that can follow A in a derivation, and is defined in the usual way. Given any context-free grammar G, parsing is applied to strings followed by the symbol $ / ∈ V used as endmarker. Also, the parsing table is produced for an enriched version of G, denoted by G ′ = (V , T, S ,P ). The enriched grammar G ′ is obtained from G by augmenting V with a fresh nonterminal symbol S , and by adding the production S ′ → S to P. An LR(0)-item of G ′ is a production of G ′ with the distinguished marker “·” at some position of its body, like, e.g., A → α · β. The single LR(0)-item for a production of the shape A → ǫ takes the form A → ·. The LR(0)-items S ′ → ·S and S ′ → S· are called, respectively, initial item and accepting item. The LR(0)-item A → α · β is called