## Concatenation Process in DFA

The concatenation process in deterministic finite automata (DFA) is explained as follows:

If there are two regular languages, L1 and L2, their union, denoted as L1 ∩ L2, is also a regular language.

For example:

Let L1 = {an | n > 0} (strings of ‘a’ with at least one occurrence) and L2 = {bn | n > 0} (strings of ‘b’ with at least one occurrence).

When we find the intersection L3 = L1 ∩ L2, it results in {an ∩ bn | n > 0}, which is also regular.

In this context, we state that the concatenation of two DFAs is still a DFA.

Problem: Design a DFA over the alphabet {a,b} where the string starts with ‘a’ and ends with ‘b’.

Solution: We are dealing with two types of languages based on the given condition:

L1 = {a, aab, aabab, …} L2 = {b, bbab, bbabab, …}

For L1, the strings start with ‘a’, and for L2, the strings end with ‘b’.

### Language L1

It initiates with ‘a’

L1 = {a, aab, aabab, …}

The state transition diagram for L1 looks like this:

(Initial state) q1 –a–> q3 (Final state, generates {a, ab, aba, aab, abb, …}) –b–> q2 (Dead state)

### Language L2

It concludes with ‘b’.

L2 = {b, bbab, bbabab, …}

The state transition diagram for L2:

(Initial state) q1 –a–> q1 –b–> q2 (Generates {ab, abb, abab, …})

By following these transitions, strings like {ab, abb, abab, …} are generated, which satisfy the condition of ending with ‘b’.

### Concatenation of L1 and L2

When we concatenate L1 and L2, we get:

L = L1 ∩ L2 = L1.L2 = {ab, aab, abb, aaab, …}

The state transition diagram after concatenating L1 and L2:

(Initial state) q1 –a–> q3 (Generates strings like {ab, aab, abb, …}) –b–> q2 (Dead state)

This process demonstrates the concept of concatenating regular languages using deterministic finite automata.

## What is State Transition Diagram

A state diagram is a diagram used in computer science and related fields to depict how systems behave. These diagrams are used when the system in question consists of a limited number of states. This may accurately reflect the system’s structure in some cases, while in others, it serves as a simplified representation.

## What is a Deterministic finite automaton?

In the realm of theoretical computer science, specifically in the theory of computation, a deterministic finite automaton (DFA) is a type of finite-state machine. This machine is designed to either accept or reject a provided string of symbols. It accomplishes this by progressing through a sequence of states that is uniquely determined by the input string.