The document provides an overview of automata theory, detailing the definition, characteristics, and various types of automata, including finite automata, deterministic finite automata (DFA), and non-deterministic finite automata (NDFA). It explains the basic components of automata such as states, transition functions, input, and output relations, as well as examples like vending machines and traffic lights. The document also covers related terminologies, including alphabet, strings, Kleene star, and language sets.

1. Mr. Sudip Kambli, Asst. Prof.
Automata

2. Mr. Sudip Kambli, Asst. Prof.`
Defination of Automaton
Theory of Computation1

3. Mr. Sudip Kambli, Asst. Prof.
Definition of an Automaton
 Automaton are abstract model of machine that perform computational on an input
by moving through a series of state and configurations.
 At each state of the computation a transition function determines the next
configuration on the basis of a finite portion of the present configuration .
 As a result ones the computation reaches an accepting configuration it accept the
input.
 The most general and powerful automata is Turing Machine.
 The major objective of an automata theory is to develop method by which
computer scientist can describe and analyse the dynamics behaviours of discrete
system ,in which the signals are sampled periodically.
 The behaviour of these discrete system is determined by the way that the system is
constructed from storage and combinational elements.
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4. Mr. Sudip Kambli, Asst. Prof.
Model of discrete automaton
X1
X2
Xn
Y1
Y2
Yn
AUTOMATON
Q1,Q2……….
Qn
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5. Mr. Sudip Kambli, Asst. Prof.
Characteristics of an Automata
 Input: At each of the discrete instance of time t1, t2, t3, ….., tn the input values are as
X1, X2, X3, …., Xn, each of which can take a finite number of fixed values from the input
alphabet ∑, are applied to the input side of the model.
 Output: Y1, Y2, Y3, …., Yn, are the output of the discrete automata model, each of
which can take a finite number of fixed values form an output Y.
 States: An state is an condition of processing the inputs. At any instant of time the
automaton can be in one of the states q1, q2, q3, ….. qn.
 State relation: The next state of an automaton at any instant of time is determined by
the present state and the present input.
 Output relation: The output is related to either state only or to both the input and the
current state.
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6. Mr. Sudip Kambli, Asst. Prof.
Example of Automata:
 Sequential machine: A sequential machine is a mathematical model of a certain type of simple computational
structure.
 Vending Machines: A vending machine is an automated machine that dispenses numerous items such as cold
drinks, snacks, beverages, alcohol etc. to sales automatically, after a buyer inserts currency or credit into the
machine. Vending machine is works on finite state automate to control the functions process.
 Traffic Lights: The optimization of traffic light controllers in a city is a systematic representation of handling the
instructions of traffic rules. Its process depends on a set of instruction works in a loop with switching among
instruction to control traffic.
⊳ Regular Expression Matching: It is a technique to checking the two or more regular expression are similar to
each other or not. The finite state machine is useful to checking out that the expressions are acceptable or not
by a machine or not.
⊳ Speech Recognition: Speech recognition via machine is the technology enhancement that is capable to
identify words and phrases in spoken language and convert them to a machine-readable format. Receiving
words and phrases from real world and then converted it into machine readable language automatically is
effectively solved by using finite state machine.
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7. Mr. Sudip Kambli, Asst. Prof.
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).
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8. Mr. Sudip Kambli, Asst. Prof.
Formal definition of a Finite Automaton
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).
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9. Mr. Sudip Kambli, Asst. Prof.
Related Terminologies
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 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)

10. Mr. Sudip Kambli, Asst. Prof.
Related Terminologies
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 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,………..}

11. Mr. Sudip Kambli, Asst. Prof.
Related Terminologies
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 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 }

12. Mr. Sudip Kambli, Asst. Prof.
Finite Automaton
Finite Automaton
Finite Automaton (Output) Finite Automaton with (Output)
Mealy machineMoore machine
NNon-deterministic Finite
Automaton (NDFA / NFA)
Deterministic Finite
Automaton (DFA)
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13. Mr. Sudip Kambli, Asst. Prof.
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. 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).
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14. Mr. Sudip Kambli, Asst. Prof.
A DFA is represented by digraphs called
state diagram.
The vertices represent the states.
The arcs labelled 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.
Graphical Representation of a DFA
• Example
Let a deterministic finite automaton be →
Q = {a, b, c},
∑ = {0, 1},
q0 = {a},
F = {c}, and
Transition function δ
• Transition function δ as shown by the
following table −
Present
state
Next State
for input 0
Next State for
input 1
a a b
b c a
c b c
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15. Mr. Sudip Kambli, Asst. Prof.
Graphical Representation…
b
0 1
START
a c
1 0
1 0
Present
state
Next State
for input 0
Next State for
input 1
a a b
b c a
c b c
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16. Mr. Sudip Kambli, Asst. Prof.
Non-deterministic Finite Automaton
 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. As it has finite number of states, the machine is
called Non-deterministic Finite Machine or Non-deterministic Finite Automaton.
 Formal Definition of a DFA
 A 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 alphabet.
 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). 16

17. Mr. Sudip Kambli, Asst. Prof.
 Graphical Representation of an NDFA: (same as
DFA)
 An NDFA is represented by digraphs called state
diagram.
 The vertices represent the states.
 The arcs labelled 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.
Graphical Representation of a NDFA
• Example
Let a Non-deterministic finite automaton be
Q = {a, b, c},
∑ = {0, 1},
q0 = {a},
F = {c}, and
Transition function δ
Present
state
Next State
for input 0
Next State for
input 1
a a,b b
b c a,c
c b,c c
• Transition function δ as shown by the
following table −
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18. Mr. Sudip Kambli, Asst. Prof.
Graphical Representation…
b
0 0,1
START
a c
1 0
0,1 0,1
Present
state
Next State
for input 0
Next State for
input 1
a a,b b
b c a,c
c b,c c
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19. Mr. Sudip Kambli, Asst. Prof.
Thank You
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