NPDA Transitions

Given transitions are:

$($q$0,$$,$Z$)=\{($q$1,$AZ$)\}$

$($q$2,$$,$Z$)=\{($q$3,$$)\}$

$($q$2,$a$,$Z$)=\{($q$1,$AZ$)\}$

$($q$1,$b$,$A$)=\{($q$2,$$)\}$

Explanation:

Step $1$: Define Grammar Variables

For each pair of states $($qi$,$qj$),$ we define a grammar variable Aij$.$ This variable represents the language that can take the NPDA from state qi$$ to state qj$$ with an empty stack.

Step $3$

The variables we need are:

A$01,$A$02,$A$03$

A$10,$A$11,$A$12,$A$13$

A$20,$A$21,$A$22,$A$23$

A$30,$A$31,$A$32,$A$33$

However, since not all of these transitions will be relevant, we$$ll focus only on the variables directly involved in the given transitions.

Step $2$: Write Initial Production Rules

Using the transitions of the NPDA, we write the initial production rules. We will define productions based on each transition by incorporating the stack symbols and input symbols.

Production Rules Based on Transitions

Transition $($q$0,$$,$Z$)=($q$1,$AZ$)$

This transition means that in state q$0$ with Z on the stack, the automaton can move to state q$1,$ replacing Z with AZ$.$ Therefore, we get the following production:

A$01$AZ

Transition $($q$2,$$,$Z$)=($q$3,$$)$

This transition means that in state q$2$ with Z on the stack, the automaton can go to state q$3$ and remove Z from the stack $($thus making the stack empty$).$ This gives us:

A$23$

Transition $($q$2,$a$,$Z$)=($q$1,$AZ$)$

This transition tells us that from state q$2$ on input a with Z on the stack, the automaton moves to q$1$ and replaces Z with AZ$.$ This results in the production:

A$21$aA$11$Z

Transition $($q$1,$b$,$A$)=($q$2,$$)$

In state q$1$ on input b with A on the stack, the automaton moves to q$2$ and removes A from the stack. This gives us:

A$12$b

Step $3$: Simplify the Grammar

Now that we have the initial set of production rules, we simplify by eliminating unreachable variables and productions that do not contribute to generating strings from the language of the NPDA.

Final Production Rules After Simplification

A$01$AZ

A$23$

A$21$aA$11$Z

A$12$b

After simplification, these production rules represent the CFG equivalent to the given NPDA. The grammar generates the same language recognized by the NPDA.

Answer

The equivalent context$-$free grammar $($CFG$)$ for the given NPDA is as follows

Final Productior Rules

These production rules represent the CFG the generates the same langugaes as the given NPDA

$1.$A$01$AZ

$2.$A$23$

$3.$A$21$aA$11$Z

$4.$A$12$b

The equivalent context free grammar for the given NPDA is as follows