\[
\newcommand{\tr}{\Rightarrow}
\newcommand{\trs}{\tr^{\!\ast}}
\newcommand{\rlnm}[1]{\mathsf{(#1)}}
\newcommand{\rred}[1]{\xrightarrow{#1}}
\newcommand{\rreds}[1]{\mathrel{\xrightarrow{#1}\!\!^*}}
\newcommand{\cl}{\mathsf{Cl}}
\newcommand{\pow}{\mathcal{P}}
\newcommand{\matches}{\mathrel{\mathsf{matches}}}
\newcommand{\kw}[1]{\mathsf{#1}}
\]
Big Step Semantics
The relation “\(R \matches w\)” is defined inductively using the following rules:
\[\begin{prooftree}
\AxiomC{}
\LeftLabel{$\rlnm{MEmpty}$}
\UnaryInfC{$\epsilon \matches \epsilon$}
\end{prooftree}
\qquad\qquad
\begin{prooftree}
\AxiomC{}
\LeftLabel{$\rlnm{MChar}$}
\UnaryInfC{$a \matches a$}
\end{prooftree}\]
\[\begin{prooftree}
\AxiomC{$R \matches v$}
\AxiomC{$S \matches w$}
\LeftLabel{$\rlnm{Concat}$}
\BinaryInfC{$R \cdot S \matches vw$}
\end{prooftree}\]
\[\begin{prooftree}
\AxiomC{$R \matches w$}
\LeftLabel{$\rlnm{ChoiceL}$}
\UnaryInfC{$R + S \matches w$}
\end{prooftree}
\qquad\qquad
\begin{prooftree}
\AxiomC{$S \matches w$}
\LeftLabel{$\rlnm{ChoiceR}$}
\UnaryInfC{$R + S \matches w$}
\end{prooftree}\]
\[\begin{prooftree}
\AxiomC{}
\LeftLabel{$\rlnm{StarB}$}
\UnaryInfC{$R^* \matches \epsilon$}
\end{prooftree}
\qquad\qquad
\begin{prooftree}
\AxiomC{$R \matches v$}
\AxiomC{$R^* \matches w$}
\LeftLabel{$\rlnm{StarS}$}
\BinaryInfC{$R^* \matches vw$}
\end{prooftree}\]