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Pushdown AutomataPushdown Automata
Section 2.2
CSC 4170
Theory of Computation
Components of a pushdown automaton (PDA) 2.2.a
(Q,,,,s,F)
xyxz
.
.
.
a a b a c …Stack Input
Q is the set of states is the input alphabet is the stack alphabet is the transition functions is the start stateFQ is the set of accept states
a,xyq1 q2
If the input symbol is a andthe top stack symbol is x,go from q1 to q2, pop x and push y
If a=, the read head is not advanced If x=, nothing is poppedIf y=, nothing is pushed
Push: write a symbol on the top of the stackPop: delete a symbol from the top of the stack
How a PDA works 2.2.b1
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 0 0 1 1 1
Stack Input
How a PDA works 2.2.b2
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 0 0 1 1 1
Stack Input
$
How a PDA works 2.2.b3
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 0 0 1 1 1
Stack Input
0$
How a PDA works 2.2.b4
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 0 0 1 1 1
Stack Input
00$
How a PDA works 2.2.b5
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 0 0 1 1 1
Stack Input
000$
How a PDA works 2.2.b6
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 0 0 1 1 1
Stack Input
00$
How a PDA works 2.2.b7
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 0 0 1 1 1
Stack Input
0$
How a PDA works 2.2.b8
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 0 0 1 1 1
Stack Input
$
How a PDA works 2.2.b9
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 0 0 1 1 1
Stack Input
Accept
How a PDA works 2.2.b10
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
What language does this automaton recognize?
How a PDA works 2.2.b11
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 0 1
Stack Input
How a PDA works 2.2.b12
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 0 1
Stack Input
$
How a PDA works 2.2.b13
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 0 1
Stack Input
0$
How a PDA works 2.2.b14
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 0 1
Stack Input
00$
How a PDA works 2.2.b15
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 0 1
Stack Input
0$
Reject
How a PDA works 2.2.b16
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 1 1
Stack Input
How a PDA works 2.2.b17
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 1 1
Stack Input
$
How a PDA works 2.2.b18
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 1 1
Stack Input
0$
How a PDA works 2.2.b19
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 1 1
Stack Input
$
How a PDA works 2.2.b20
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 1 1
Stack Input
Reject
How a PDA works 2.2.b21
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 1 0
Stack Input
How a PDA works 2.2.b22
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 1 0
Stack Input
$
How a PDA works 2.2.b23
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 1 0
Stack Input
0$
How a PDA works 2.2.b24
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 1 0
Stack Input
$
How a PDA works 2.2.b25
, $q1 q2
q4 q3,$
1,0
0, 0
1,0
0 1 0
Stack Input
Reject
Designing pushdown automata 2.2.c
s
0
Design a pushdown automaton that recognizes the language
{w | w has an equal number of 0s and 1s}
1
=
Converting NFA into PDA2.2.d
3 2
1
a
a b
ab
Every NFA can be understood as a PDA that never pushes or pops.
Just replace every label a of the NFA by a,
3 2
1
a,
a, b,
a,b,
,
Main theorems 2.2.e
Theorem 2.20: A language is context-free iff some pushdown automaton recognizes it.
Theorem: Not every nondeterministic PDA has anequivalent deterministic PDA.
Example 2.18: There is a nondeterministic PDA recognizing {wwR | w{0,1}* }(wR means w reversed),but no deterministic PDA can recognize this language.
Proofs omitted.
