Finite Automata (FA) is a fundamental concept in Theory of Computation (TOC) and plays a key role in recognizing patterns and designing compilers. It is the simplest model of computation used to recognize Regular Languages.

 Definition of Finite Automata (FA)

A Finite Automaton (FA) is formally defined as a 5-tuple (Q, Σ, δ, q₀, F), where:

• Q → Finite set of states.

• Σ → Finite set of input symbols (Alphabet).

• δ (Transition Function) → Mapping from (State × Input) → State.

• q₀ (Start State) → The initial state of the automaton.

• F (Final States) → Set of accepting states.

Types of Finite Automata

Finite Automata are classified into two types:

DFA and NFA are equivalent in power (both recognize the same set of languages).

 Example of DFA

Language: Strings over {0,1} that end with 1.

DFA Representation:

States: {q0, q1}
Alphabet: {0,1}
Start State: q0
Final State: q1
Transitions:

Explanation:

• Start at q0.

• If 1 is encountered, move to q1.

• Stay in q1 if more 1s are encountered.

• Any string ending in 1 is accepted.

 Example of NFA

Language: Strings containing 10 as a substring.

NFA Representation:

States: {q0, q1, q2}
Alphabet: {0,1}
Start State: q0
Final State: q2
Transitions:

Explanation:

• Start at q0.

• If 1 appears, move to q1.

• If 0 follows, move to q2 (accepting state).

• Once in q2, stay there for any input.

 Key Differences Between DFA & NFA

DFA is always more structured but may require more states than an equivalent NFA.

 Applications of Finite Automata

• Lexical Analysis in Compilers (Token Recognition).

• Pattern Matching (Regular Expressions).

• Text Searching Algorithms (e.g., KMP Algorithm).

• Hardware Circuit Design (Sequential Circuits).

 GATE 1992 Question on Finite Automata

Q: Which of the following is true?
(A) Every DFA is an NFA.
(B) Every NFA is a DFA.
(C) NFA can recognize more languages than DFA.
(D) DFA and NFA recognize different languages.

Correct Answer: (A) Every DFA is an NFA.
Explanation: DFA is a special case of NFA where each state has only one transition per input. However, both DFA and NFA recognize the same set of Regular Languages.

 Summary

Finite Automata is a basic computational model that recognizes Regular Languages.
DFA has unique transitions, while NFA allows multiple transitions.
Both DFA and NFA are equivalent in power but differ in complexity.
Used in compilers, text searching, and pattern recognition.

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GATE 1992 Theory Of Computation Topic -Finite Automata

In the GATE 1992 Computer Science Engineering (CSE) exam, the Theory of Computation section featured questions on Finite Automata, focusing on regular expressions and their identities.

Sample Question from GATE 1992

Question:
Which of the following regular expression identities are true?

A. r∗=r∗r^* = r^*r∗=r∗
B. (r∗s∗)∗=(r+s)∗(r^* s^*)^* = (r + s)^*(r∗s∗)∗=(r+s)∗
C. (r+s)∗=r∗+s∗(r + s)^* = r^* + s^*(r+s)∗=r∗+s∗
D. r∗s∗=r∗+s∗r^* s^* = r^* + s^*r∗s∗=r∗+s∗ExamSIDE

Answer:
Option B is true. This identity holds because the Kleene star applied to the concatenation of two starred expressions encompasses all combinations of strings from both languages, which is equivalent to the union of the two languages starred.

Options C and D are false. The equality (r+s)∗=r∗+s∗(r + s)^* = r^* + s^*(r+s)∗=r∗+s∗ does not hold because the left-hand side includes all strings formed by any number of concatenations of strings from either $r$ or $s$, including mixed sequences, whereas the right-hand side includes only strings formed entirely from $r$ or entirely from $s$. Similarly, r∗s∗=r∗+s∗r^* s^* = r^* + s^*r∗s∗=r∗+s∗ is not generally true for regular expressions.

Video Resource

For a comprehensive understanding of Finite Automata concepts and GATE questions from 1989 to 1992, you can refer to the following video:

• GATE 1989 TO 1992 QUESTIONS OF TOC
  This video covers various Theory of Computation questions, including those on Finite Automata, providing detailed explanations and solutions.

If you need further assistance with specific topics or additional resources, feel free to ask!