How do I convert this PDA to CFG?

Question

I am currently stuck with this, any help would be appreciated, thank you in advance!

As you may have seen already, the language of this PDA is $L = {a^n b^m \mid n \geq m}$ (assuming the PDA accepts a string if the stack is empty). If you are not interested in the algorithm, you might want to construct the grammar from this description which should be fairly easy. — ttnick, Apr 20 '20 at 10:25

score 1 · Answer 1 · answered Apr 20 '20 at 10:28

From the PDA diagram we get the transition function as follows:

$\delta(q,a,Z_0)=(q,ZZ_0)$
$\delta(q,a,Z)=(q,ZZ)$
$\delta(q,\epsilon,Z)=(r,Z)$
$\delta(r,b,Z)=(r,\epsilon)$
$\delta(r,\epsilon,Z)=(p,\epsilon)$
$\delta(p,\epsilon,Z)=(p,\epsilon)$
$\delta(p,\epsilon,Z_0)=(p,\epsilon)$

Now let $G$ be the context free grammar and V be the set of variables. Now we represent a variable depending on the state $p$ which the PDA was in before having the ultimate effect of popping an element $X$ from the stack and ultimately landing in the state $q$, as $[pXq]$ where $p$,$q$ are any arbitrary states in the PDA and $X$ is any arbitrary stack symbol.

So our variables will be of the form $[\{p,q,r\} \times \{Z,Z_0\} \times \{p,q,r\}]$

So corresponding to the transition 1 we have the following productions:

1.$[qZ_0q] \rightarrow a[qZq][qZ_0q]$

2.$[qZ_0q] \rightarrow a[qZr][rZ_0q]$

3.$[qZ_0q] \rightarrow a[qZp][pZ_0q]$

4.$[qZ_0p] \rightarrow a[qZq][qZ_0p]$

5.$[qZ_0p] \rightarrow a[qZp][pZ_0p]$

6.$[qZ_0p] \rightarrow a[qZr][rZ_0p]$

7.$[qZ_0r] \rightarrow a[qZq][qZ_0r]$

8.$[qZ_0r] \rightarrow a[qZp][pZ_0r]$

9.$[qZ_0r] \rightarrow a[qZr][rZ_0r]$

From transition 2 we have:

10.$[qZq] \rightarrow a[qZq][qZq]$

11.$[qZq] \rightarrow a[qZp][pZq]$

12.$[qZq] \rightarrow a[qZr][rZq]$

13.$[qZp] \rightarrow a[qZq][qZp]$

14.$[qZp] \rightarrow a[qZp][pZp]$

15.$[qZp] \rightarrow a[qZr][rZp]$

16.$[qZr] \rightarrow a[qZq][qZr]$

17.$[qZr] \rightarrow a[qZp][pZr]$

18.$[qZr] \rightarrow a[qZr][rZr]$

From transition $3$ we have:

19.$[qZq] \rightarrow [rZq]$

20.$[qZp] \rightarrow [rZp]$

21.$[qZr] \rightarrow [rZr]$

from transition $4$ we have:

22.$[rZr] \rightarrow b$

from transition $5$ we have:

23.$[rZp] \rightarrow \epsilon$

From transition $6$ we have:

24.$[pZp] \rightarrow \epsilon$

From transition $7$ we have:

25.$[pZ_0p] \rightarrow \epsilon$

Now let us rename our variables which occur on the $LHS$ of the productions using simpler names as:

$[qZ_0q] = A$, $[qZ_0p] = B$,$[qZ_0r] = C$,$[qZq] = D$,$[qZp] = E$,$[qZr] = F$, $[rZr] = G$,$[rZp] = H$,$[pZp] = I$,$[pZ_0p] = J$

Now we rewrite the productions with the new variable names substituted but we omit those productions whose body has a variable of the form $[pXq]$ which cannot be expanded so, for this reason productions $2,3,6,8,9,11,12,17,19$ are discarded.

So we have:

$A \rightarrow aDA$

$B \rightarrow aDB | aEJ$

$C \rightarrow aDC$

$D \rightarrow aDD$

$E \rightarrow aDE | aEI | aFH | H$

$F \rightarrow aDF | aFG | G$

$G \rightarrow b$

$H \rightarrow \epsilon$

$I \rightarrow \epsilon$

$J \rightarrow \epsilon$

Now let $S$ be the start symbol and we add productions:

$S \rightarrow A | B | C $

So the resulting CFG $G'$ is

$G' = (\{S,A,B,C,D,E,F,G,H,I,J\},\{a,b\},P,S)$ where $P$ contained the productions described above.

How do I convert this PDA to CFG?

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