meljun cortes -- automata theory lecture - 10

7/31/2019 MELJUN CORTES -- AUTOMATA THEORY LECTURE - 10

1/23

CSC 3130: Automata theory and formal languages

Andrej Bogdanov

http://www.cse.cuhk.edu.hk/~andrejb/csc3130

Pushdown automata

Fall 2008


2/23

Motivation

We had two ways to describe regular languages

How about context-free languages?

regular

expressionDFANFA

syntactic computational

CFG pushdown automaton

syntactic computational


3/23

Pushdown automata versus NFA

Since context-free is more powerful than regular,pushdown automata must generalize NFAs

state control0 1 0 0

input

NFA


4/23

Pushdown automata

A pushdown automaton has access to a stack,which is a potentially infinite supply of memory


input

pushdown automaton (PDA)

stack


5/23

Pushdown automata

As the PDA is reading the input, it can push / popsymbols in / out of the stack


input

pushdown automaton (PDA)

Z0 0 1

stack

1


6/23

Rules for pushdown automata

The transitions are nondeterministic Stack is always accessed from the top

Each transition can pop a symbol from the stack

and / orpush another symbol onto the stack Transitions depend on input symbol and on last

symbol popped from stack

Automaton accepts if after reading whole input, it

can reach an accepting state


7/23

Example

L= {0n#1n: n 0}state control

0 0 0 #

input

Z0 a a

stack

a

1 1 1

read 0push a

read #

read 1pop a

pop Z0


8/23

Shorthand notation

read 0push a

read #

read 1pop a

pop Z0

0, e / a

#, e / e

e, Z0 / e

1, a / e

read, pop / push


9/23

Formal definition

A pushdown automaton is (Q, ,, , q0, Z0, F): Q is a finite set ofstates;

is the input alphabet;

is the stack alphabet, including a special symbol Z0;

q0in Q is the initial state;

Z0in is the start symbol;

F Q is a set of final states;

is the transition function

: Q ( {e}) ( {e}) subsets of Q ( {e})

state input symbol pop symbol state push symbol


10/23

Notes on definition

We use slightly different definition than textbook Example

0, e / a

#, e / e e, Z0 / e

1, a / e

: Q ( {e}) ( {e}) subsets of Q ( {e})

q0 q1 q2

(q0, 0, e) = {(q0, a)}(q0, 1, e) =

(q0, #, e) = {(q1, e)}

(q0, 0, a) = ...


11/23

A convention

Sometimes we denote transitions by:

This will mean:

Intuitively, pop b, then push c1, c2, and c3

a, b/ c1c2c3q0 q1

e, e / c2

q0 q1a, b/ c1 e, e / c3

intermediatestates


12/23

Examples

Describe PDAs for the following languages: L = {w#wR: w *}, = {0, 1, #}

L = {wwR: w *}, = {0, 1}

L = {w: whas same number of 0s and 1s}, = {0, 1}

L = {0i1j: ij 2i}, = {0, 1}


13/23

Main theorem

A language Lis context-free if and only if it

is accepted by some pushdown automaton.

context-free grammar pushdown automaton


14/23

From CFGs to PDAs

Idea: Use PDA to simulate (rightmost) derivationsA 0A1

A B

B #

A 0A1 00A11 00B11 00#11

PDA control:

CFG:

write start variable

stack:

Z0A

replace production in reverse Z01A0

pop terminals and match Z01A

e, e / A

0, 0 / e

e, A / 1A0

input:

00#11

00#11

0#11

replace production in reverse Z011A0e, A / 1A0 0#11

pop terminals and match Z011A0, 0 / e #11

replace production in reverse Z011Be, A / B #11


15/23

From CFGs to PDAs

If, after reading whole input, PDA ends up with anempty stack, derivation must be valid

Conversely, if there is no valid derivation, PDAwill get stuck somewhere

Either unable to match next input symbol,

Or match whole input but stack non empty


16/23

Description of PDA for CFGs

Repeat the following steps: If the top of the stack is a variableA:

Choose a ruleA a and substituteA with a

If the top of the stack is a terminal a:

Read next input symbol and compare to aIf they dont match, reject (die)

If top of stack is Z0, go to accept state


17/23

Description of PDA for CFGs

q0 q1 q2e, e / S

a, a / efor every terminal a

e, A / ak...a1for every productionA a1...ak

e, Z0 / e


18/23

From PDAs to CFGs

First, we simplify the PDA:

It has a single accept stateqf Z0 is always popped exactly before accepting

Each transition is either a push, or a pop, but not both

context-free grammar pushdown automaton


19/23

From PDAs to CFGs

We look at the stack in an acceptingcomputation:

a

Z0 Z0

a

Z0

a

Z0

a

Z0

a

Z0

a

Z0

a

Z0

a

Z0

a

Z0 Z0

b a c c c

a

portions that preserve the stack

q0 q1 q3 q1 q7 q0 q1 q2 q1 q3 q7

A03 = {x: xleads from q0 to q3 and preserves stack}

e 1 1 e 0 1 e e 0 00input

state

stack

qf


20/23

From PDAs to CFGs

a

Z0 Z0

a

Z0

a

Z0

a

Z0

a

Z0

a

Z0

a

Z0

a

Z0

a

Z0 Z0

b a c c c

a

q0 q1 q3 q1 q7 q1 q2 q1 q7e 1 1 e 0 1 e e 0 00input

state

stack

A11

A03

0 eA11 0A03e

q0 q3 qf


21/23

From PDAs to CFGs

a

Z0 Z0

a

Z0

a

Z0

a

Z0

a

Z0

a

Z0

a

Z0

a

Z0

a

Z0 Z0

b a c c c

a

q0 q1 q3 q1 q7 q1 q2 q1 q7e 1 1 e 0 1 e e 0 00input

state

stack

A03

A13

A13 A10A03

q0 q3

A10

qf


22/23

From PDAs to CFGs

qiqj

qi

qj Aij aAijb

qi qj qk Aik AijAjk

qi Aii e

variables:Aij

qfa, Z0 /eqi A0f A0ia

start variable:A0f


23/23

Example

0, e / a

#, e / e e, Z0 / e

1, a / e

q0 q1 q2

start variable: A02productions:

A00 A00A00A00 A01A10

A00 A03A30A01 A01A11A01 A02A21

A00 e

...

A11 eA

22

e

A01 0A011A01 #A33

A33 e

0, e / a

#, e / $ e, Z0 / e

1, a / e

q0 q1 q2q3e, $ / e

A02 A01

meljun cortes -- automata theory lecture - 10

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