## Converting a DFA to a Regular Expression using a GNFA### ContextIn this example, we will use a GNFA (Generalized Nondeterministic Finite Automaton) to convert the given DFA (Deterministic Finite Automaton) to a regular expression.### Diagram ExplanationBelow is a state diagram of a DFA with three states: \(q_1\), \(q_2\), and \(q_3\), transitioning between them based on different input symbols.#### States1. **q1:** Initial state2. **q2:** Intermediate state3. **q3:** Final/accepting state#### Transitions- **From q1:** - To q2 on input \(a, c\) - To q1 on input \(b, d\) (self-loop)- **From q2:** - To q3 on input \(a, c\) - To q1 on input \(b, d\)- **From q3:** - To q2 on input \(a, c\) - To q3 on input \(b, d\) (self-loop) - To q1 on input \(a, c\) ### Process OverviewHere’s a step-by-step explanation of how to convert this DFA to a regular expression using a Generalized Nondeterministic Finite Automaton (GNFA):1. **Add Start and End States:** - Introduce a new start state \( q_{start} \) and a new end state \( q_{end} \). - Place \( q_{start} \) with an ε-transition to the initial state \( q_1 \). - Connect the accepting states to the new end state \( q_{end} \) with ε-transitions.2. **Remove Intermediate States:** - Sequentially remove intermediate states, reconstructing transition labels to account for their removal. - For each state \( q \) that is to be removed, adjust the transitions of the other states to combine them into generalized transitions.3. **Construct Regular Expression:** - Continue adjusting transitions until only \( q_{start} \) and \( q_{end} \) remain. - The resulting transition label between \( q_{start} \) and \( q_{end} \) represents the regular expression equivalent of the DFA.###

given DFA will be converted as follows:-

Answered: Use a GNFA to convert the DFA below to…

Engineering

Computer Science

Use a GNFA to convert the DFA below to a regular expression a, c a, c 91 92 93 b, d b, d a, c b, d

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

## Converting a DFA to a Regular Expression using a GNFA

### Context

In this example, we will use a GNFA (Generalized Nondeterministic Finite Automaton) to convert the given DFA (Deterministic Finite Automaton) to a regular expression.

### Diagram Explanation

Below is a state diagram of a DFA with three states: \(q_1\), \(q_2\), and \(q_3\), transitioning between them based on different input symbols.

#### States

1. **q1:** Initial state
2. **q2:** Intermediate state
3. **q3:** Final/accepting state

#### Transitions

- **From q1:**
- To q2 on input \(a, c\)
- To q1 on input \(b, d\) (self-loop)

- **From q2:**
- To q3 on input \(a, c\)
- To q1 on input \(b, d\)

- **From q3:**
- To q2 on input \(a, c\)
- To q3 on input \(b, d\) (self-loop)
- To q1 on input \(a, c\)

### Process Overview

Here’s a step-by-step explanation of how to convert this DFA to a regular expression using a Generalized Nondeterministic Finite Automaton (GNFA):

1. **Add Start and End States:**
- Introduce a new start state \( q_{start} \) and a new end state \( q_{end} \).
- Place \( q_{start} \) with an ε-transition to the initial state \( q_1 \).
- Connect the accepting states to the new end state \( q_{end} \) with ε-transitions.

2. **Remove Intermediate States:**
- Sequentially remove intermediate states, reconstructing transition labels to account for their removal.
- For each state \( q \) that is to be removed, adjust the transitions of the other states to combine them into generalized transitions.

3. **Construct Regular Expression:**
- Continue adjusting transitions until only \( q_{start} \) and \( q_{end} \) remain.
- The resulting transition label between \( q_{start} \) and \( q_{end} \) represents the regular expression equivalent of the DFA.

###

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