TOC

Theory of Computation

Automata
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).
Related Terminologies
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}

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