COSC 3340: Introduction to Theory of
Computation
University of Houston
Dr. Verma
Lecture 11
1
Lecture 11
UofH - COSC 3340 - Dr. Verma
Push Down Automaton (PDA)


Language Acceptor Model for CFLs
It is an NFA with a stack.
Finite
State
control
Input
Accept/Reject
Stack
2
Lecture 11
UofH - COSC 3340 - Dr. Verma
PDA (contd.)

In one move the PDA can :
–
–
–
–

3
change state,
consume a symbol from the input tape or ignore it
pop a symbol from the stack or ignore it
push a symbol onto the stack or not
A string is accepted provided the machine
when started in the start state consumes the
string and reaches a final state.
Lecture 11
UofH - COSC 3340 - Dr. Verma
PDA (contd.)

If PDA in state q can consume u, pop x from
stack, change state to p, and push w on
stack we show it as
u, x  w
q0
4
u, x ; w
In JFLAP
Lecture 11
q1
UofH - COSC 3340 - Dr. Verma
Example of a PDA

PDA L = {anbn |n  0}
Push S to the stack in the beginning and
then pop it at the end before accepting.
5
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
6
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
7
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
8
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
9
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
10
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
11
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
12
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
13
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
14
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
15
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
16
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
17
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
18
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
19
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
20
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
21
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
22
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
23
Lecture 11
UofH - COSC 3340 - Dr. Verma
JFLAP Simulation
24
Lecture 11
UofH - COSC 3340 - Dr. Verma
Definition of PDA

Formally, a PDA M = (K, , , , s, F), where
–
–
–
–
–
–
25
K -- finite set of states
 -- is the input alphabet
 -- is the tape alphabet
s  K -- is the start state
F  K -- is the set of final states
  (K X  X ) X (K X )
Lecture 11
UofH - COSC 3340 - Dr. Verma
Definition of L(M)

Define * as:
(1) *(q, , ) = {(q, , )}  {(p, , ) |((q, , ), (p, ))  }
(2) *(q, uv, xy) = U {*(p, v, wy) | ((q, u, x), (p, w))  }
i.e., first compute * for all successor configurations and
then take the union of all those sets



26
M accepts w if (f, , x) in *(s, w, )
Alternative: if (f, ,  ) in *(s, w, ) [we use]
L(M) = {w  * | M accepts w}
Lecture 11
UofH - COSC 3340 - Dr. Verma
Example

What is L(M)?
Push S to the stack in the beginning and
then pop it at the end before accepting.
27
Lecture 11
UofH - COSC 3340 - Dr. Verma