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Equivalence of Pushdown Automata with Context-Free Grammar - Theory
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Page 1
Equivalence of Pushdown
Automata with Context-Free
Grammar
Page 2
Equivalence of Pushdown
Automata with Context-Free
Grammar
Motivation
CFG and PDA are equivalent in power: a
CFG generates a context-free language and
a PDA recognizes a context-free language.
We show here how to convert a CFG into a
PDA that recognizes the language speci?ed
by the CFG and vice versa
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.
45
Page 3
Equivalence of Pushdown
Automata with Context-Free
Grammar
Motivation
CFG and PDA are equivalent in power: a
CFG generates a context-free language and
a PDA recognizes a context-free language.
We show here how to convert a CFG into a
PDA that recognizes the language speci?ed
by the CFG and vice versa
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.
45
Theorem 2.20
A language is context-free iff some pushdown
automaton recognizes it.
Note: This means that:
1. if a language
is context-free then there is a PDA
that
recognizes it
2. if a language
is recognized by a PDA
then there is a CFG
that generates
.
p.3/45
Page 4
Equivalence of Pushdown
Automata with Context-Free
Grammar
Motivation
CFG and PDA are equivalent in power: a
CFG generates a context-free language and
a PDA recognizes a context-free language.
We show here how to convert a CFG into a
PDA that recognizes the language speci?ed
by the CFG and vice versa
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.
45
Theorem 2.20
A language is context-free iff some pushdown
automaton recognizes it.
Note: This means that:
1. if a language
is context-free then there is a PDA
that
recognizes it
2. if a language
is recognized by a PDA
then there is a CFG
that generates
.
p.3/45
Lemma 2.21
If a language is context-free then some
pushdown automaton recognizes it
Proof idea:
1. Let
be a CFL. From the de?nition we know that
has a CFG
,
that generates it
2. We will show how to convert
into a PDA
that accepts strings
if
generates
3.
will work by determining a derivation of
.
r – p.4/45
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May 16, 2026 Last updated
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