complexity and computability theory i lecture #4 rina zviel-girshin leah epstein winter 2002-2003

Complexity and Computability Theory I

Lecture #4

Rina Zviel-Girshin

Leah Epstein

Winter 2002-2003

Rina Zviel-Girshin @ASC 2

Overview

NondeterminismExamplesEquivalence of the nondeterministic model

to the deterministic model


Nondeterminism (briefly)

• Each state can be accepting or not.• The automaton reads one input word symbol at

each time unit.• An NFA also can change it state without reading

an input - transition• If a state in which it stops reading an input

(reading it till the end) is an accepting state- the automaton accepts (recognizes) the input.


Nondeterminism (briefly)• On each symbol:

– it moves to a new state (defined by the transition function) as a function of:

• the current state

• the symbol read or an transition

qj qkqj

qk

qi

• or halts if no valid move definedqj ?


Formal Definition

A finite automaton is a 5-tuple (Q, , , q0, F), where:

• Q - a finite set of states - alphabet - transition function - Q()2|Q| or P(Q)(given a

state and an input symbol - what can be the next state)

• q0 - q0Q is the start state

• F - FQ the set of accept states


DFA: ExampleConsider the following language: L = (ab aba)*

The deterministic automaton :

q0 a q1

q4

a

a,b

q0

b

b q1q2a

q1q3

b

b

a


NFA: ExampleThe nondeterministic automaton :

q0a

q1

q2

b

q0

a

b

This simplifies the deterministic automaton.

The basic idea of the construction:

•An aba path

•An ab path


How does an NFA compute?

• A nondeterministic computation is a tree of possibilities.

• The root of the tree is a start state of the automaton.

• Each branching point is a computation at which automaton has multiple choices.

• An NFA accepts an input string if there exists at least one computation that leads to an accept state.


NFA vs. DFA

• The difference between a deterministic finite automaton DFA and a non-deterministic one NFA is the number of paths in a computation tree.

• In DFA’s we have only one path – one chain.

• In NFA’s we have a tree.


Deterministiccomputation

Nondeterministiccomputation

accept orreject

accept

reject

.

.

....

.

.

.


The language of an NFA • Informally:

The words that an NFA A accepts (recognizes), i.e. when reading them it stops in an accepting state.

• Formally:

L(A)={w | w*, ’( q0,w)F}

means:

There exist a path from q0 to qiF for

the input string w.


Example

Construct a nondeterministic finite automaton for the following language:

L = { w1 | w is over *={0,1}*}

A:

q0

0,1

q1


What does A do on the input w=110?

q01

q1q1

1

q01

q1q1

1

q0

0

q0

no movefor 1

no movefor 0


What does A do on the input w=101?

q01

q1q1

1

q0

0

q0

1

no movefor 0

q1q1


Additional examples

Construct a nondeterministic finite automaton recognizing the following language:

L={w | w(={0,1,2})*,

w starts with 01 and ends with 1}

q0 0 q1 q4q2

0,2

11

q4

0,2

1


Additional examples

Construct a nondeterministic finite automaton recognizing the following language:

L={0i1j | i mod 3 = 1, j mod 2 = 1}

q0 0 q1 0 q2

0

0i

i mod 3 = 1


Additional examples

q0 0 q1

q4q3

1

1

q4

0 q2

0

1

L={0i1j | i mod 3 = 1, j mod 2 = 1}

0i1j i mod 3 = 1j mod 2 = 1


Equivalence of FA

Definition:

Two automata are equivalent if they

recognize the same language.


Example

q0

0,1

q2

0,1

q4q1

0,1

0,1

q0

0,1

0,1

q4q1

A:

B:

L(A)=L(B)


Equivalency

Theorem

Every NFA has an equivalent DFA.

Proof idea:

• We will give a constructive proof - by giving an algorithm to build a DFA equivalent to a given NFA.


How can we do it?• Consider an example in which we can get from some

state q on an input letter to more than one state.

q

qj

qkqi

• The next state - (q,) - is one of {qi, qj, qk}.


How can we do it?

• That means that the next state in an NFA is one state among several states.

• But in a DFA only one state can be the next state.

Conclusion:

One state in a DFA = A set of states in an NFA

q

qj

qkqi


How can we do it?

• Consider an example. What states can we reach starting from some state q on two letters input 1.

q

qj

qkqi

qn

qmql

qp


How can we do it?

Conclusion:

’(q, 1)= ({qi,qj,qk},1) =

(qi,1) (qj,1) (qk,1)

Informally: The next state in a DFA = A set

of all the states in an NFA to which you can get from the set of all the current states in an NFA.

q

qj

qkqi

qn

qmql

qp


The - closure

Formally:The - closure of a state is a set of all the states

reached from the current state using -transitions only.

E(q)= { pQ | (q,i) = p, i>=0 }Informally: Use all the transitions you can. Add the states

you reach to -closure. Try to reach as many states as you can.


The - closure

q0 q1 q2 q3 q4

E(q0) = {q0, q1, q2}

E(q1) = {q1, q2}

E(q2) = {q2}

E(q3) = {q3, q4}

E(q4) = {q4}


A DFA construction algorithm Let A be an NFA where A = (QA, , A, q0A, FA) .We construct a DFA M equivalent to A,

where M = ( Q, , , q0, F).

• Q = P(QA) = • For each R in Q and in , (R,) is

)....,())),((( qqEERqRq

•q0 = E(q0A)

•F = { RQ | there exists rR such that rFA}


Explanation• Create all the subsets of set of states of A. These

subsets will become the states of M.• The alphabet remains the same.• Transition function : for each state m in M and a letter find what are

the states qi..qj in A included in m are:

qi

qk

qlqi,qk,qlm:


Explanation

for each qim find a set R of states which you can reach in NFA A using -closure, the letter and the -closure

qiqt

qp

R

i j

k l


Explanation

the next state of m on in M is S.

qt

qpRi

qi,qk,qlm:qs

qfRk

qt

qpRl

S

.. ..

unite all the sets R you reach into one set : S


Explanation

• The initial state of M is the set which includes only E(q0A)

• The final states of M are all the sets in which at least one state is a final (accepting) state of A.

• Eliminate all the unreachable states in M - states to which the is no path from the initial state of M.

• The automaton you have is a deterministic automaton equivalent to A.


Example

Convert a given nondeterministic finite automaton into a deterministic finite automaton.

q0 q1

q4q3

a

a,b

a,

b

q2q0a


Example

Construction of the DFA:({q0},a)={q2,q3}({q0},b)={q2}({q1},a)={q2,q3}({q1},b)={q2} Therefore:({q0,q1},a)={q2,q3}({q0,q1},b)={q2}

q2

q2,q3

q0q0

b

aq0,q1


Example

({q2},a)={q1}

({q2},b)={q2}

q2

q2,q3

q0q0

b

a

b

q1

a

q0,q1


Example

({q1},a)={q2,q3}

({q1},b)={q2}

q2

q2,q3

q0q0

b

a

b

q1

b

a

a

q0,q1


Example

({q2,q3},a)={q1,q2}

({q2,q3},b)={q2}

q2

q2,q3

q0q0

b

a

b

b

aq1,q2

q1

b

a

a

q0,q1


Example

({q1,q2},a)={q1,q2,q3}

({q1,q2},b)={q2}

q2

q2,q3

q1,q2,q3

q0q0

b

a

b

b

aq1,q2

aq1

b

a

b

a

q0,q1


({q1,q2,q3},a)={q1,q2,q3}

({q1,q2,q3},b)={q2}

q2

q2,q3

q1,q2,q3

q0q0

b

a

b

b

aq1,q2

aq1

b

a

ab

b

a

q0,q1


Any Questions?

complexity and computability theory i lecture #4 rina zviel-girshin leah epstein winter 2002-2003

Documents

state q

defined slide

asc10 slide

deterministic model

ab path slide

nondeterministic automaton

accepting state

start state f f q