If a grammar G is context-free, we can build an equivalent nondeterministic PDA which accepts the language that is produced by the context-free grammar G. A parser can be built for the grammar G.
Also, if P is a pushdown automaton, an equivalent context-free grammar G can be constructed where
L(G) = L(P)
In the next two topics, we will discuss how to convert from PDA to CFG and vice versa.
Algorithm to find PDA corresponding to a given CFG
Input − A CFG, G = (V, T, P, S)
Output − Equivalent PDA, P = (Q, ∑, S, δ, q0, I, F)
Step 1 − Convert the productions of the CFG into GNF.
Step 2 − The PDA will have only one state {q}.
Step 3 − The start symbol of CFG will be the start symbol in the PDA.
Step 4 − All non-terminals of the CFG will be the stack symbols of the PDA and all the terminals of the CFG will be the input symbols of the PDA.
Step 5 − For each production in the form A → aX where a is terminal and A, X are combination of terminal and non-terminals, make a transition δ (q, a, A).
Problem
Construct a PDA from the following CFG.
G = ({S, X}, {a, b}, P, S)
where the productions are −
S → XS | ε , A → aXb | Ab | ab
Solution
Let the equivalent PDA,
P = ({q}, {a, b}, {a, b, X, S}, δ, q, S)
where δ −
δ(q, ε , S) = {(q, XS), (q, ε )}
δ(q, ε , X) = {(q, aXb), (q, Xb), (q, ab)}
δ(q, a, a) = {(q, ε )}
δ(q, 1, 1) = {(q, ε )}
Algorithm to find CFG corresponding to a given PDA
Input − A CFG, G = (V, T, P, S)
Output − Equivalent PDA, P = (Q, ∑, S, δ, q0, I, F) such that the non- terminals of the grammar G will be {Xwx | w,x ∈ Q} and the start state will be Aq0,F.
Step 1 − For every w, x, y, z ∈ Q, m ∈ S and a, b ∈ ∑, if δ (w, a, ε) contains (y, m) and (z, b, m) contains (x, ε), add the production rule Xwx → a Xyzb in grammar G.
Step 2 − For every w, x, y, z ∈ Q, add the production rule Xwx → XwyXyx in grammar G.
Step 3 − For w ∈ Q, add the production rule Xww → ε in grammar G.
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