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Home»VLSI Design»Explain the Concatenation Process in DFA
VLSI Design

Explain the Concatenation Process in DFA

siliconvlsiBy siliconvlsiAugust 20, 2023Updated:May 19, 2024No Comments3 Mins Read
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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’.

Explain the Concatenation Process in DFA
Explain the Concatenation Process in DFA

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.

Complementation Process in DFA

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