In Ullman's Introduction to Automata, Languages and Computation (1979):
A context-free grammar (CFG or just grammar) is denoted $G = (V, T, P, S)$, where $V$ and $T$ are finite sets of variables and terminals, respectively. We assume that $V$ and $T$ are disjoint. $P$ is a finite set of productions; each production is of the form $A\to \alpha$, where $A$ is a variable and $\alpha$ is a string of symbols from $(V \cup T)^*$.
Lemma 43 Define an $A$-production to be a production with variable $A$ on the left. Let $G = (V, T, P, S)$ be a CFG....
Is an $A$-production defined as a production $A\to \alpha$, where $A$ is a variable and $\alpha$ is a string of symbols from $(V \cup T)^*$?
Is every production in a CFG an $A$-production?
Thanks.
No. The $A$-productions defined in Lemma $\mathbf{4.3}$ are specifically those productions that have the variable $A$ on the lefthand side. In that terminology a production $B\to\beta$ would be a $B$-production. If you look at the second paragraph of the proof of Theorem $\mathbf{4.6}$, you’ll see this usage: the authors talk about ‘each $A_j$-production’ in a context in which they are clearly referring to productions with $A_j$ on the lefthand side.