The book I am reading have proof for the statement
Every context-free language there exist a pushdown automata $M$ s.t. $L=L_{e}(M)$
For the case $\epsilon\not\in L$. The proof uses greibach normal form (hence the reason for the condition $\epsilon\not\in L$)
How can I prove this statement (preferably not having to re-prove everything again) for the general case ?
I understand that we can add the single rule $S\to\epsilon$ to the grammar after it's in greibach normal form, but how can I make the pushdown automata also accept $\epsilon$ ?
I’m assuming that $L_e$ means that you’re accepting by empty stack. In that case you should be able simply to modify $\delta(q_0,\epsilon,Z)$, where $Z$ is the initial stack symbol, by adding $(q_0,\epsilon)$. That is, if $\delta(q_0,\epsilon,Z)=A$ in the original PDA, let $\delta(q_0,\epsilon,Z)=A\cup\{(q_0,\epsilon)\}$ in the modified PDA.