n $\backslash$epsi $-$free PDA is a PDA with two modifications:

$$ that does not have any $\backslash$epsi transitions $($having $\backslash$epsi for stack actions is OK$),$ and

$$ it can push multiple symbols in one transition $($however it can pop at most one symbol per transition$).$

That is$,($Q$,\backslash$Sigma $,\backslash$Gamma $,\backslash$delta $,$ q$0$ F$)$ is an $\backslash$epsi $-$free PDA iff it is a PDA with the following modified transition function:

$\backslash$delta : Q $\backslash$times $\backslash$Sigma $\backslash$times $\backslash$Gamma $\backslash$epsi $->$ P $($Q $\backslash$times $\backslash$Gamma $).$

Here, P $($A$)$ is the powerset of A that contains only finite sets. Note that there are $2$ differences to $\backslash$delta :

$1.$ The second argument of $\backslash$delta cannot be $\backslash$epsi $.$

$2.$q in Q$,\backslash$sigma in $\backslash$Sigma $,\backslash$gamma in $\backslash$Gamma $\backslash$epsi $.\backslash$delta $($q$,\backslash$sigma $,\backslash$gamma $)$ returns a finite set of pairs made up of a state, and a sequence of symbols

from the stack alphabet $($as opposed to a state and a single stack symbol or $\backslash$epsi from the definition of a normal

PDA$).$

The accepting criterion for $\backslash$epsi $-$free PDA follow the same form presented for PDA in the lectures.

Prove or disprove: the class of languages $\backslash$epsi $-$free PDA recognize is the class of context$-$free languages.