A state machine is any device storing the status of something at a given time. The status changes based on inputs, providing the resulting output for the implemented changes. A finite state machine has finite internal memory.

1. Formalization
Machines and States
D.K.S.Hasaranga & A.G.A.L.Dhanushka

2. Finite State Machine
• Symbol
 Ex: a, b, c, 0, 1, 2, 3, ….
• Alphabet – Collection of symbols (Ɛ)
 Ex: {a,b}, {0,1,2}
• String – Sequence of symbols
 Ex: a, b, 0, 1, aa, bb, ab, 01, ….

3. Finite State Machine …
• Language – Set of Strings
 Ex: Ɛ = {0,1}
L1 = set of all strings of length 2
= {00, 01, 10, 11}
L2 = Set of all strings of length 3
= {000, 001, 010, 011, 100, 101, 110, 111}
L3 = Set of all strings that begin with “0”
= {0, 00, 01, 000, 001, ……}
Finite
Infinite

4. Finite State Machine …
Finite Automata
F A without output F A with output
DFA NFA Moore
Machine
Mealy
Machine
Ɛ - NFA

5. DFA (Deterministic Finite Automata)
 It is the simplest model of the computation
 It has a very limited memory
Every DFA can define using 5 tuples. They are,
 Q = Set of states
 Ɛ = Inputs
 q0 = Start state or initial state
 F = Set of final states
 ƍ = Transition function from Q x Ɛ Q

6. DFA (Deterministic Finite Automata) …
• Q = {A, B, C, D}
• Ɛ = {0, 1}
• q0 = A
• F = {D}
A B
C D
1
1
1
1
00 0 0
A C B
B D A
C A D
D B C
0 1

7. NFA (Non-Deterministic Finite Automata)
Every NFA can define using 5 tuples. They are,
 Q = Set of states - {A, B}
 Ɛ = Inputs - {0, 1}
 q0 = Start state or initial state - A
 F = Set of final states - B
 ƍ = Transition function from Q x Ɛ ?
A B0
0, 1
• The machine can move to any combination of the states in the machine (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.
L = {Set of all strings ends with 0}

8. NFA (Non-Deterministic Finite Automata)
A B0
0, 1
ƍ = Transition function from Q x Ɛ 2Q
Little Exercise
A x 0 A , B , A B
A x 1 A
B x 0 ø
B x 1 ø
Possibilities = A,B,AB, ø
= 4
= 22

9. Mealy Machine
Every Mealy Machine can define using 6 tuples. They are,
 Q = Finite set of state
 Ɛ = Finite non-empty set of Input Alphabets
 Δ = The set of Output Alphabets
 ƍ = Transition function from Q x Ɛ Q
 λ = Output function : Ɛ x Q Δ
 q0 = Initial state / Start state
A Mealy Machine is an FSM whose output depends on the present state as well as the
present input.
a/0
a/1
b/0
b/0
b/1
A
B
a/0
C
L = {Set of all strings ends with aa or bb}

10. Moore Machine
Every Moore Machine can define using 6 tuples. They are,
 Q = Finite set of state
 Ɛ = Finite non-empty set of Input Alphabets
 Δ = The set of Output Alphabets
 ƍ = Transition function from Q x Ɛ Q
 λ = Output function : Q Δ
 q0 = Initial state / Start state
Moore machine is a finite state machine in which the next state is decided by the current
state and current input symbol. The output symbol at a given time depends only on the
present state of the machine.

11. Moore Machine …
A/b B/b
1
01
Example
C/a
0
Construct a moore machine that prints ‘a’ whenever the sequence ‘01’ is encountered
in any input binary
Ɛ = {0,1}
Δ = {a,b}
Input String 0 1 0 1
A B C B C
b b a b a
0 1