The document discusses the equivalence between context-free grammars (CFGs) and pushdown automata (PDAs). It states that for any CFG, an equivalent PDA can be constructed to accept the language generated by the grammar, and vice versa. This allows a programming language to be specified by a CFG and implemented with a PDA in a compiler. The document also provides procedures for converting between CFGs and PDAs, including an example of constructing a PDA from a given CFG.

1. Equivalence of
Pushdown
automata and
CFL
-Sampath Kumar S,
AP/CSE, SECE

2. Equivalence of PDA and CFL
 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)
21 November 2017
Sampath Kumar S, AP/CSE, SECE
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3. Application:
 This equivalence allows a CFG to be used to
specify a programming language and the
equivalent PDA to be used to implement its
compiler.
21 November 2017
Sampath Kumar S, AP/CSE, SECE
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4. Theorem
Theorem: A language is context-free iff some
pushdown automaton recognizes it.
Note:
This means that:
1. If a language L is context-free then there is a
PDA ML that recognizes it
2. If a language L is recognized by a PDA ML then
there is a CFG GL that generates L.
21 November 2017
Sampath Kumar S, AP/CSE, SECE
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5. CFG to PDA Conversion:
 PDA accepts CFL
 CFL is generated by CFG
 If CFG is given to us then we can construct a PDA
which will accept the language generated by this
grammar.
 CFG G can be defied as G= (N, T, P, S) and M is
a PDA
 M accepts the language generated by G by empty
stack
21 November 2017
Sampath Kumar S, AP/CSE, SECE
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6. CFG to PDA Conversion procedure:
 Input −A CFG, G= (V, T, P, S)
 Output −Equivalent PDA, M= (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 M.
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 A → aX where a is terminal
and A, X are combination of non-terminals, make a
transition δ (q, a, A) = (q, X)
21 November 2017
Sampath Kumar S, AP/CSE, SECE
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7. Problem to discuss:
125. Construct a PDA from the following CFG.
G = ({S, X}, {a, b}, P, S) where the production P are −
S → aA2|aA3
A2 → aA2A3|aA3A3
A3 → b
Solution
Let the equivalent PDA,
P = ({q}, {a, b}, {a, b, S, A2, A3 }, δ, q,
S)
where δ −
δ(q, a, S) = {(q, A2), (q, A3)}
δ(q, a, A2) = {(q, A2A3),(q, A3 A3)}
δ(q, b, A3) = {(q, ε )}
21 November 2017
Sampath Kumar S, AP/CSE, SECE
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8. 21 November 2017
Sampath Kumar S, AP/CSE, SECE
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9. நன்றி
21 November 2017
Sampath Kumar S, AP/CSE, SECE
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