Simple precedence grammar

A simple precedence grammar is a context-free formal grammar that can be parsed with a simple precedence parser.^[1] The concept was first created in 1964 by Claude Pair,^[2] and was later rediscovered, from ideas due to Robert Floyd, by Niklaus Wirth and Helmut Weber who published a paper, entitled EULER: a generalization of ALGOL, and its formal definition, published in 1966 in the Communications of the ACM.^[3]

Formal definition

G = (N, Σ, P, S) is a simple precedence grammar if all the production rules in P comply with the following constraints:

• There are no erasing rules (ε-productions)
• There are no useless rules (unreachable symbols or unproductive rules)
• For each pair of symbols X, Y (X, Y ${\displaystyle \in }$ (N ∪ Σ)) there is only one Wirth–Weber precedence relation.
• G is uniquely inversible

Examples

${\displaystyle S\to aSSb|c}$

precedence table

${\displaystyle {\begin{array}{c|ccccc}&S&a&b&c&\$\\\hline S&{\dot {=}}&\lessdot &{\dot {=}}&\lessdot &\\a&{\dot {=}}&\lessdot &&\lessdot &\\b&&\gtrdot &&\gtrdot &\gtrdot \\c&&\gtrdot &\gtrdot &\gtrdot &\gtrdot \\\$&&\lessdot &&\lessdot &\end{array}}}$