# NDFA to DFA Conversion

## Problem Statement

Let X = (Qx, ∑, δx, q0, Fx) be an NDFA which accepts the language L(X). We have to design an equivalent DFA Y = (Qy, ∑, δy, q0, Fy) such that L(Y) = L(X). The following procedure converts the NDFA to its equivalent DFA −

## Algorithm

Input − An NDFA

Output − An equivalent DFA

Step 1 − Create state table from the given NDFA.

Step 2 − Create a blank state table under possible input alphabets for the equivalent DFA.

Step 3 − Mark the start state of the DFA by q0 (Same as the NDFA).

Step 4 − Find out the combination of States {Q0, Q1,... , Qn} for each possible input alphabet.

Step 5 − Each time we generate a new DFA state under the input alphabet columns, we have to apply step 4 again, otherwise go to step 6.

Step 6 − The states which contain any of the final states of the NDFA are the final states of the equivalent DFA.

## Example

Let us consider the NDFA shown in the figure below. q δ(q,0) δ(q,1)
a {a,b,c,d,e} {d,e}
b {c} {e}
c {b}
d {e}
e

Using the above algorithm, we find its equivalent DFA. The state table of the DFA is shown in below.

q δ(q,0) δ(q,1)
[a] [a,b,c,d,e] [d,e]
[a,b,c,d,e] [a,b,c,d,e] [b,d,e]
[d,e] [e]
[b,d,e] [c,e] [e]
[e]
[c, e] [b]
[b] [c] [e]
[c] [b]

The state diagram of the DFA is as follows − 