\[
\newcommand{\tr}{\Rightarrow}
\newcommand{\trs}{\tr^{\!\ast}}
\newcommand{\rlnm}[1]{\mathsf{(#1)}}
\newcommand{\rred}[1]{\xrightarrow{#1}}
\newcommand{\rreds}[1]{\mathrel{\xrightarrow{#1}\!\!^*}}
\newcommand{\cl}{\mathsf{Cl}}
\newcommand{\pow}{\mathcal{P}}
\newcommand{\matches}{\mathrel{\mathsf{matches}}}
\newcommand{\kw}[1]{\mathsf{#1}}
\newcommand{\andop}{\mathrel{\&\!\&}}
\newcommand{\orop}{\parallel}
\newcommand{\ff}{\mathsf{false}}
\newcommand{\tt}{\mathsf{true}}
\newcommand{\abra}[1]{\langle #1 \rangle}
\newcommand{\bnfnt}[1]{\abra{\small \textsf{#1}}}
\newcommand{\llbracket}{[\![}
\newcommand{\rrbracket}{]\!]}
\newcommand{\first}{\mathsf{First}}
\newcommand{\nullable}{\mathsf{Nullable}}
\newcommand{\follow}{\mathsf{Follow}}
\newcommand{\tm}[1]{\mathsf{#1}}
\newcommand{\nt}[1]{\mathit{#1}}
\newcommand{\Coloneqq}{::=}
\newcommand{\abs}[1]{|#1|}
\]
A Context Free Grammar (CFG) consists of four components:
- An alphabet of terminal symbols.
- A finite, non-empty set of non-terminal symbols, disjoint from the terminals.
- A finite set of production rules, each of shape $X \Coloneqq \alpha$.
- A designated non-terminal called the start symbol, which we will usually write as $S$.
A sentential form is just a sequence of terminals and nonterminals.