Theory of Computation : Automata Theory

Automata Theory is a branch of computer science that deals with designing abstract selfpropelled computing devices that follow a predetermined sequence of operations automatically. An automaton with a finite number of states is called a Finite Automaton. This is a brief and concise tutorial that introduces the fundamental concepts of Finite Automata, Regular Languages, and Pushdown Automata before moving onto Turing machines and Decidability.

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. 
Finite Automata: It is used to recognize patterns of specific type input. It is the most restricted type of automata which can accept only regular languages (languages which can be expressed by regular expression using OR (+), Concatenation (.), Kleene Closure(*) like a*b*, (a+b) etc.)

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 : 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 : 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 : 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 : 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 : 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 : 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, bb, ba, bb}
Finite Automaton can be classified into two types −
  • Deterministic Finite Automaton (DFA)
  • Non-deterministic Finite Automaton (NDFA / NFA)

Deterministic Finite Automaton (DFA)

In DFA, for each input symbol, one can determine the state to which the machine will move. Hence, it is called Deterministic Automaton. In deterministic FA, there is only one move from every state on every input symbol. As it has a finite number of states, the machine is called Deterministic Finite Machine or Deterministic Finite Automaton.

Formal Definition of a DFA

A DFA 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.
  • δ is the transition function where δ: Q × ∑ → Q
  • 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).
Graphical Representation of a DFA

A DFA is represented by digraphs called state diagram.
  • The vertices represent the states.
  • The arcs labeled with an input alphabet show the transitions.
  • The initial state is denoted by an empty single incoming arc.
  • The final state is indicated by double circles.
Example

Let a deterministic finite automaton be →
Q = {a, b, c},
∑ = {0, 1},
q0 = {a},
F = {c}


Transition function δ as shown by the following table −

Present StateNext State for Input 0Next State for Input 1
aab
bca
cbc
Its graphical representation would be as follows −


Non-deterministic Finite Automaton (NDFA / NFA)

 In NDFA, for a particular input symbol, the machine can move to any combination of the states in the machine. In other words, the exact state to which the machine moves cannot be determined. Hence, it is called Non-deterministic Automaton. In Non-Deterministic FA, there can be zero or more than one move from one state for an input symbol. As it has finite number of states, the machine is called Non-deterministic Finite Machine or Non-deterministic Finite Automaton.

Formal Definition of an NDFA


An NDFA 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 alphabets.
  • δ is the transition function where δ: Q × ∑ → 2Q
  • (Here the power set of Q (2Q) has been taken because in case of NDFA, from a state, transition can occur to any combination of Q states)
  • 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).
Graphical Representation of a NDFA

A NDFA is represented by digraphs called state diagram.
  • The vertices represent the states.
  • The arcs labeled with an input alphabet show the transitions.
  • The initial state is denoted by an empty single incoming arc.
  • The final state is indicated by double circles.
Example

Let a deterministic finite automaton be →
Q = {a, b, c},
∑ = {0, 1},
q0 = {a},
F = {c}

Transition function δ as shown by the following table −
Present StateNext State for Input 0Next State for Input 1
aa, bb
bca, c
cb, cc
Its graphical representation would be as follows −

Note:
  • Language accepted by NDFA and DFA are same.
  • Power of NDFA and DFA is same.
  • No. of states in NDFA is less than or equal to no. of states in equivalent DFA.
  • For NFA with n-states, in worst case, the maximum states possible in DFA is 2n
  • Every NFA can be converted to corresponding DFA.
DFA vs NDFA

The following table lists the differences between DFA and NDFA.
DFANDFA
The transition from a state is to a single particular next state for each input symbol. Hence it is called deterministic.The transition from a state can be to multiple next states for each input symbol. Hence it is called non-deterministic.
Empty string transitions are not seen in DFA.NDFA permits empty string transitions.
Backtracking is allowed in DFAIn NDFA, backtracking is not always possible.
Requires more space.Requires less space.
A string is accepted by a DFA, if it transits to a final state.A string is accepted by a NDFA, if at least one of all possible transitions ends in a final state.


Next Article - Theory of Computation NFA to DFA Conversion

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