Introduction to Automata Theory / Automata Language & Computation / Theory of Computation by PATHAN AHMED KHAN
Automata Theory is a branch of computer science that deals with designing abstract self propelled computing devices that follow a predetermined sequence of operations automatically.
An automaton with a finite number of states is called a Finite Automaton.
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,………..}
Prefix and Suffix of a string:
Prefix String means number of leading symbols of that String.
Example:
String is “abc”
Prefix of a String is: ε, a, ab, abc.
Suffix of a String: Number of its trailing symbols of that String.
String is “abc”
Suffix of a String is : ε, c,bc,abc
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 }
Finite Automaton can be classified into two types −
Deterministic Finite Automaton (DFA)
Non-deterministic Finite Automaton (NDFA / NFA)
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