# Can we convert a non-deterministic finite automata into a deterministic finite Automata?

Yes, we can convert a NFA into DFA. For every NFA there exists an equivalent DFA. The equivalence is defined in terms of languages acceptance. Since NFA is nothing but a finite automata in which zero, one or more transitions on an input symbols are permitted. It can always construct finite automata which will simulate all moves of DFA on a particular input symbol in parallel, then get a finite automata in which there will be exactly one transition on every input symbol. Here, corresponding to a NFA there exist a DFA. To construct DFA equivalent to NFA, it should be remembered that states of DFA are a collection of states of NFA.

## Algorithm NFA – to – DFA

Input − NFA with set of states N = {n0,n1……nn}, with start state n0.

Output − DFA, with set of states D′={d0,d1,d2……dn}, with start state d0.

d0=ε−closure (n0)

D′={d0}

set d0 unmarked

while there is an unmarked state d in D′. {
set d marked {
For each input symbol 'a' {
Let T be a set of states in NFA to which there is a transition on 'a' from some state ni in d d′=ε−closure (T).
If d′ is not already present in D′ {
D′=D′∪{d′}
set d′ unmarked
}
}
}
}

Example − Design Lexical Analyzer for the following LEX Program.

AUXILIARY DEFINITION S
letter = A|B|C|…….|Z
digit = 0 |1|2|………|9
TRANSLATION RULES
begin    {return 1}
end      {return 2}
If       {return 3}
then     {return 4}
else     {return 5}
letter  (letter+digit)* {value=Install ( );return 6}
digit + {value=Install ( );return 7}
<      {value=1;return 8}
<=     {value=2;return 8}
=      {value=3;return 8}
< >    {value=4;return 8}
>      {value=5;return 8}
>=     {value=6;return 8}

Solution

• The combined NFA for various patterns will be

• Convert NFA to DFA − The corresponding DFA will be.

States {0, 1, 7, 11, 14, 19, 24, 26, 28, 30, 33, 35, 38, 40} are combined and named as q0to make a starting state of DFA.

In combined NFA, transitions from state 1 to 2 and from state 24 to 25 are the same because Input 'b' is a letter.

States 2, 25 are combined. Similarly, other states are combined.

States 29, 31 & 36 are combined because they all reach after getting the input '<'.

Similarly, other states are combined.