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, q₀, F), consisting of

• Q − a finite set of states

• ∑ − a finite set of input symbols

• δ − a transition function δ : Q × (∑ ∪ {ε}) → 2^Q

• q₀ − an initial state q₀ ∈ 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 q₁ and q₂, so let q₁ is X and q_f is 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 q₁ without changing the edges from q_f and get the following FA −

Step 3 −

Here q₁ 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 q₁ also a final state.

So the FA becomes −