TOC Automata Definition

The term "Automata" is derived from the Greek word "αὐτόματα" which means "selfacting".
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, Σ, δ, q0, F), where:
* Q is a finite set of states.
* Σ 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).

Alphabet
* Definition: An alphabet is any finite set of symbols.
* Example: Σ = {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 Σ.
* Example: ‘cabcad’ is a valid string on the alphabet set Σ = {a, b, c, d}

Length of a String
* Definition : It is the number of symbols present in a string. (Denoted by |S|).
* Examples:
o If S=‘cabcad’, |S|= 6
o If |S|= 0, it is called an empty string (Denoted by λ or ε)

Kleene Star
 Definition: The Kleene star, Σ*, is a unary operator on a set of symbols or strings, Σ, that gives the infinite set of all possible strings of all possible lengths over Σ including λ.
 Representation: Σ* = Σ0 U Σ1 U Σ2 U……. where Σp is the set of all possible strings of length p.
 Example: If Σ = {a, b}, Σ*= {λ, a, b, aa, ab, ba, bb,………..}

Kleene Closure / Plus
* Definition: The set Σ+ is the infinite set of all possible strings of all possible lengths
over Σ excluding λ.
 Representation: Σ+ = Σ1 U Σ2 U Σ3 U…….
                           Σ+ = Σ* − { λ }
 Example: If Σ = { a, b } , Σ+ ={ a, b, aa, ab, ba, bb,………..}

Language
 Definition : A language is a subset of Σ* for some alphabet Σ. It can be finite or infinite.
 Example : If the language takes all possible strings of length 2 over Σ = {a, b}, then L = { ab, bb, ba, bb}