# Automata Theory

## 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 −

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, ∑*, 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 }