Epsilon NFA (∈-NFA)

Epsilon NFA (∈-NFA) is similar to the NFA but with a little difference in that, it has an epsilon (∈) transition A Transition that does not consume any input symbol is called an ε-transitions. Epsilon NFA (∈-NFA) is also called a null NFA or an NFA lambda.

• ∈-NFA state diagrams usually label Epsilon with the Greek letter “∈”. It is also called lambda.
• ∈-transitions give a convenient way of designing the machine systems.
• Due to ε-transitions, the first string of language may be empty.

Note: ∈ab∈a = aba, where ∈ is empty. Epsilon (∈) transition is also known as an empty move or empty transition.

💡 Think of ε-transitions as "magical moves" that let the automaton change state spontaneously, without reading any input symbol.

Applications of ε-NFA

• Regex to Automata Conversion – ε-NFA helps in converting regular expressions to automata easily using ε-transitions.
• Modular Automaton Design – Breaks complex problems into smaller NFAs and connects them using ε-transitions.
• Compiler Design – Used in building lexical analyzers during the token generation phase.
• Parallel Path Execution – Allows exploring multiple paths without consuming input, useful in theoretical modeling.
• Simplifying NFA Construction – Makes it easier to construct NFAs with fewer transitions for complex patterns.

Examples of Epsilon-NFA

There are various examples of epsilon-NFA. Let’s explain some of them.

 ∈ – NFA: Example 

Draw an epsilon NFA that accepts the string “a or b”.

The transition table for the above epsilon NFA diagram is given below

∈ – NFA: Example 

Draw an epsilon NFA that accepts the string “a, b, or c”.

The transition table for the above epsilon NFA diagram is given below

∈ – NFA: Example 

Draw an epsilon NFA that accepts the string “ab” or "ac"

Note:

ε -NFAs add a convenient feature but (in a sense) they bring us nothing new. They do not extend the class of languages that can be represented.

Both NFAs and ε -NFAs recognize exactly the same languages.