Automata Theory Introduction

Automata – What is it?

The term "Automata" is derived from the Greek word "αὐτόματα" which means "self-acting". An automaton (Automata in plural) is an abstract self-propelled computing device which follows a predetermined sequence of operations automatically.

An automaton with a finite number of states is called a Finite Automaton (FA) or Finite State Machine (FSM).

Formal definition of a Finite Automaton

An automaton can be represented by a 5-tuple (Q, ∑, δ, q₀, F), where −

Q is a finite set of states.

∑ is a finite set of symbols, called the alphabet of the automaton.

δ is the transition function.

q₀ is the initial state from where any input is processed (q₀ ∈ Q).

F is a set of final state/states of Q (F ⊆ Q).

Related Terminologies

Alphabet

Definition − An alphabet is any finite set of symbols.

Example − ∑ = {a, b, c, d} is an alphabet set where ‘a’, ‘b’, ‘c’, and ‘d’ are symbols.

String

Definition − A string is a finite sequence of symbols taken from ∑.

Example − ‘cabcad’ is a vapd string on the alphabet set ∑ = {a, b, c, d}

Length of a String

Definition − It is the number of symbols present in a string. (Denoted by |S|).

Examples −

If S = ‘cabcad’, |S|= 6

If |S|= 0, it is called an empty string (Denoted by λ or ε)

Kleene Star

Definition − The Kleene star, ∑*, is a unary operator on a set of symbols or strings, ∑, that gives the infinite set of all possible strings of all possible lengths over ∑ including λ.

Representation − ∑* = ∑₀ ∪ ∑₁ ∪ ∑₂ ∪……. where ∑_p is the set of all possible strings of length p.

Example − If ∑ = {a, b}, ∑* = {λ, a, b, aa, ab, ba, bb,………..}

Kleene Closure / Plus

Definition − The set ∑⁺ is the infinite set of all possible strings of all possible lengths over ∑ excluding λ.

Representation − ∑⁺ = ∑₁ ∪ ∑₂ ∪ ∑₃ ∪…….

∑⁺ = ∑* − { λ }

Example − If ∑ = { a, b } , ∑⁺ = { a, b, aa, ab, ba, bb,………..}

Language

Definition − A language is a subset of ∑* for some alphabet ∑. It can be finite or infinite.

Example − If the language takes all possible strings of length 2 over ∑ = {a, b}, then L = { ab, aa, ba, bb }