Deterministic Finite Automaton. A deterministic finite automaton (DFA) is a five-tuple A = (Q, , , q 0, F) where Q is a finite set of states is a finite set of input symbols is a function : Q Q called transition function q 0 Q is called the start state

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Deterministic Finite Automaton

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A deterministic finite automaton (DFA) is a five-tuple A = (Q, , , q0, F) where

Q is a finite set of states

is a finite set of input symbols

is a function : Q Q called transition function

q0 Q is called the start state

F Q is called the set of accepting states

Deterministic Finite Automaton

Transition diagram for a DFA is graph defined as follows:

For each state there is a node labeled by the state

If (q, a) = p, then there is an arc from q to p labeled by a

There is an arrow into the start state q0 labeled start, The arrow does not originate at any node

Nodes corresponding to accepting states marked by double circles

Deterministic Finite Automaton

A transition table is a tabular representation of the transition function. Rows of the table correspond to states and columns correspond to input symbols. The entry if the cell indexed by state q and input a is (q, a). The start state is marked with an arrow, the final states are marked with *.

Deterministic Finite Automaton

0,1

0

1

1

0

q2

q0

q1

Deterministic Finite Automaton

Transition function on strings:

Let w=xa

1)

2)

Deterministic Finite Automaton

A DFA accepts a string w if is a final state

The set of all strings accepted by a DFA, A, is the language of the DFA denoted by L(A). The DFA A is said to accept/define a language L if L(A) = L

1

q1

q0

1

0

0

0

0

1

q2

q3

1

Example L= {w | w has even 0s and even 1s}

Nondeterministic Finite Automaton

A nondeterministic finite automaton (DFA) is a five-tuple A = (Q, , , q0, F) where

Q is a finite set of states

is a finite set of input symbols

is a function : Q 2Q called transition function

q0 Q is called the start state

F Q is called the set of accepting states

Nondeterministic Finite Automaton

Extended transition function:

The language of an NFA A = (Q, , , q0, F):

0,1

start

q0

q1

q2

0

1

Example – L = {w | w ends with 01}

Equivalence of DFA and NFA

Subset construction:

Let N = (QN,, N, q0, FN) be an NFA, construct a DFA

D = (QD, , D, {q0 }, FD ) as follows:

QD is the set of all subsets of QN

FD is the set of subsets S of QN whose intersection with FN is not empty

For each set S QN and each input symbol a

Equivalence of DFA and NFA

Theorem: If D = (QD, , D, {q0 }, FD ) is the DFA constructed from the NFA N = (QN,, N, q0, FN) by the subset construction, then L(D) = L(N)

Theorem: A language L is accepted by some DFA if and only if L is accepted by some NFA

0,1

0

1

q0

q1

q2

q3

Finite Automata with -transitions

An -NFA N = (Q, , , q0, F) where the definitions of Q, , q0, and F are same as for an NFA. is defined as follows:

: Q X {} 2Q

Function ECLOSE

Epsilon-closure (ECLOSE):

/* associates a set of states with a given state */

basis: state q is in ECLOSE(q)induction: if p is in ECLOSE(q) and (p, ) contains r, then r also is in ECLOSE(q)

Eliminating -transitions

Let E = (QE,, E, q0, FE), then the equivalent DFA