# Construction of an FA from an RE

We can use Thompson's Construction to find out a Finite Automaton from a Regular Expression. We will reduce the regular expression into smallest regular expressions and converting these to NFA and finally to DFA.

Some basic RA expressions are the following −

** Case 1** − For a regular expression ‘a’, we can construct the following FA −

** Case 2** − For a regular expression ‘ab’, we can construct the following FA −

** Case 3** − For a regular expression (a+b), we can construct the following FA −

** Case 4** − For a regular expression (a+b)*, we can construct the following FA −

### Method

**Step 1** Construct an NFA with Null moves from the given regular expression.

**Step 2** Remove Null transition from the NFA and convert it into its equivalent DFA.

**Problem**

Convert the following RA into its equivalent DFA − 1 (0 + 1)* 0

*Solution*

We will concatenate three expressions "1", "(0 + 1)*" and "0"

Now we will remove the **ε** transitions. After we remove the **ε** transitions from the NDFA, we get the following −

It is an NDFA corresponding to the RE − 1 (0 + 1)* 0. If you want to convert it into a DFA, simply apply the method of converting NDFA to DFA discussed in Chapter 1.

## Finite Automata with Null Moves (NFA-ε)

A Finite Automaton with null moves (FA-ε) does transit not only after giving input from the alphabet set but also without any input symbol. This transition without input is called a **null move**.

An NFA-ε is represented formally by a 5-tuple (Q, ∑, δ, q_{0}, F), consisting of

**Q**− a finite set of states**∑**− a finite set of input symbols**δ**− a transition function δ : Q × (∑ ∪ {ε}) → 2^{Q}**q**− an initial state q_{0}_{0}∈ Q**F**− a set of final state/states of Q (F⊆Q).

The above **(FA-ε)** accepts a string set − {0, 1, 01}

## Removal of Null Moves from Finite Automata

If in an NDFA, there is ϵ-move between vertex X to vertex Y, we can remove it using the following steps −

- Find all the outgoing edges from Y.
- Copy all these edges starting from X without changing the edge labels.
- If X is an initial state, make Y also an initial state.
- If Y is a final state, make X also a final state.

**Problem**

Convert the following NFA-ε to NFA without Null move.

*Solution*

**Step 1** −

Here the ε transition is between **q _{1}** and

**q**, so let

_{2}**q**is

_{1}**X**and

**q**is

_{f}**Y**.

Here the outgoing edges from q_{f} is to q_{f} for inputs 0 and 1.

**Step 2** −

Now we will Copy all these edges from q_{1} without changing the edges from q_{f} and get the following FA −

**Step 3** −

Here q_{1} is an initial state, so we make q_{f} also an initial state.

So the FA becomes −

**Step 4** −

Here q_{f} is a final state, so we make q_{1} also a final state.

So the FA becomes −