PUSHDOWN AUTOMATA (PDA)

ALAK ROY.Assistant ProfessorDept. of CSENIT Agartala

CSE-501 FORMAL LANGUAGE & AUTOMATA

THEORYAug-Dec,2010

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PUSHDOWN AUTOMATA (PDA) Informally:

Think of an ε-NFA with the additional power that it can manipulate a stack.

Transitions are modified to accommodate stack operations.

Questions: What is a stack? How does a stack help?

Why PDA? A DFA can “remember” only a finite amount of

information, whereas a PDA can “remember” an infinite amount of (certain types of) information.

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Example:

{0n1n | 0=<n} Is not regular

{0n1n | 0nk, for some fixed k} Is regular, for any fixed k.

For k=3:

L = {ε, 01, 0011, 000111}

0/1

q0

q7

0q1

11

q2

1q5

0q3

11

q4

0

1

00

0/1 q6

0

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In a DFA, each state remembers a finite amount of information.

To get {0n1n | 0n} with a DFA would require an infinite number of states using the preceding technique.

An infinite stack solves the problem for {0n1n | 0n} as follows: Read all 0’s and place them on a stack Read all 1’s and match with the corresponding 0’s on the

stack

Only need two states to do this in a PDA

Similarly for {0n1m0n+m | n,m0}

PDA MODEL

CHARACTERISTIC OF PDA

PDS (Pushdown Stack)

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PDA Being nondeterministic, the PDA can

have a choice of next moves.

Its moves are determined by:1. The current state (of its “NFA”),2. The current input symbol (or ε), and 3. The current symbol on top of its stack.

In each choice, the PDA can:1. Change state, and also2. Replace the top symbol on the stack by

a sequence of zero or more symbols. Zero symbols = “pop.” Many symbols = sequence of “pushes.”

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FORMAL DEFINITION OF A PDA

A pushdown automaton (PDA) is a seven-tuple:

M = (Q, Σ, Г, δ, q0, z0, F)

Q A finite set of states

Σ A finite input alphabet

Г A finite stack alphabet

q0 The initial/starting state, q0 is in Q

z0 A starting stack symbol, is in Г

F A set of final/accepting states, which is a subset of Q

δ A transition function, where

δ: Q x (Σ U {ε}) x Г –> finite subsets of Q x Г*

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Consider the various parts of δ:

Q x (Σ U {ε}) x Г –> finite subsets of Q x Г*

Q on the LHS means that at each step in a computation, a PDA must consider its’ current state.

Г on the LHS means that at each step in a computation, a PDA must consider the symbol on top of its’ stack.

Σ U {ε} on the LHS means that at each step in a computation, a PDA may or may not consider the current input symbol, i.e., it may have epsilon transitions.

“Finite subsets” on the RHS means that at each step in a computation, a PDA will have several options.

Q on the RHS means that each option specifies a new state. Г* on the RHS means that each option specifies zero or

more stack symbols that will replace the top stack symbol.

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CONVENTIONS

a, b, … are input symbols. But sometimes we allow ε as a possible value.

…, X, Y, Z are stack symbols. …, w, x, y, z are strings of input symbols. , ,… are strings of stack symbols.

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THE TRANSITION FUNCTION

Takes three arguments:1. A state, in Q.2. An input, which is either a symbol in Σ or ε.3. A stack symbol in Γ.

δ(q, a, Z) is a set of zero or more actions of the form (p, ).

p is a state; is a string of stack symbols.

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ACTIONS OF THE PDA

If δ(q, a, Z) contains (p, ) among its actions, then one thing the PDA can do in state q, with a at the front of the input, and Z on top of the stack is:

1. Change the state to p.2. Remove a from the front of the input (but a

may be ε).3. Replace Z on the top of the stack by . (or Pop

Z, Push )

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Two types of PDA transitions:

δ(q, a, z) = {(p1, 1), (p2, 2),…, (pm, m)}

Current state is q Current input symbol is a Symbol currently on top of the stack z Move to state pi from q

Replace z with i on the stack (leftmost symbol on top) Move the input head to the next input symbol

:

q

p1

p2

pm

a/z/ 1

a/z/ 2

a/z/ m

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Two types of PDA transitions:

δ(q, ε, z) = {(p1, 1), (p2, 2),…, (pm, m)}

Current state is q Current input symbol is not considered Symbol currently on top of the stack z Move to state pi from q

Replace z with i on the stack (leftmost symbol on top) No input symbol is read

:

q

p1

p2

pm

ε/z/ 1

ε/z/ 2

ε/z/ m

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EXAMPLE: PDA

Design a PDA to accept {0n1n | n > 1}. The states:

q = start state. We are in state q if we have seen only 0’s so far.

p = we’ve seen at least one 1 and may now proceed only if the inputs are 1’s.

f = final state; accept.

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EXAMPLE: PDA – (2)

The stack symbols: Z0 = start symbol. Also marks the bottom of the

stack, so we know when we have counted the same number of 1’s as 0’s.

X = marker, used to count the number of 0’s seen on the input.

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EXAMPLE: PDA – (3)

The transitions: δ(q, 0, Z0) = {(q, XZ0)}. δ(q, 0, X) = {(q, XX)}. These two rules cause one

X to be pushed onto the stack for each 0 read from the input.

δ(q, 1, X) = {(p, ε)}. When we see a 1, go to state p and pop one X.

δ(p, 1, X) = {(p, ε)}. Pop one X per 1. δ(p, ε, Z0) = {(f, Z0)}. Accept at bottom.

ACTIONS OF THE EXAMPLE PDA

q

0 0 0 1 1 1

Z0

ACTIONS OF THE EXAMPLE PDA

q

0 0 1 1 1

XZ0

ACTIONS OF THE EXAMPLE PDA

q

0 1 1 1

XXZ0

ACTIONS OF THE EXAMPLE PDA

q

1 1 1

XXXZ0

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ACTIONS OF THE EXAMPLE PDA

p

1 1

XXZ0

ACTIONS OF THE EXAMPLE PDA

p

1

XZ0

ACTIONS OF THE EXAMPLE PDA

p

Z0

ACTIONS OF THE EXAMPLE PDA

f

Z0

THANK YOU