Automata Theory Introduction

Automata What is it?

The term "Automata" is derived from the Greek word "" 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 valid 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 }