Automata Theory Introduction

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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 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 }

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