Formal Definition of Automata | FSM | 5-Tuple Machine
Ø An automaton is
Quintuple or 5-tuple machine.
Ø It is represented by
M= (Q, ∑, δ, q0, F), where −
·
Q is a finite set of states.
·
∑ is a finite set of symbols,
called the input alphabet of the automaton.
·
δ is the transition function/mapping
function which maps
Q x ∑ à Q (Here
x is Cartesian Product)
Above
mapping is usually represented by
1.
Transition
function/Mapping function
2.
Transition
diagram / Transition graph / Transition State diagram/ State diagram
3.
Transition
table/Transition state table/Mapping table
·
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).
Ø Example of lift
control:
SFL____________________2___q2
FFL____________________1___q1
GFL____________________0___q0
Q={
q0 , q1 , q2}, ∑={0,1,2},
q0=Initial
State, F={ q2 } and δ is given by
1)Transition function:
By using the below
formula we can easily write the transition functions
δ(Present State, Input Symbol)=Next
State
δ( q0, 0)= q0 ,δ
(q0,1)= q1
,δ ( q0,2 )= q2
δ( q1, 0)= q0 ,δ
(q1,1)= q1 ,δ (
q1,2 )= q2
δ( q2, 0)= q0 ,δ
(q2,1)= q1 ,δ (
q2,2 )= q2
2)Transition diagram:
3)Transition Table:
Present State
|
Next State
|
||
Input 0
|
Input 1
|
Input 2
|
|
àq0
|
q0
|
q1
|
q2
|
q1
|
q0
|
q1
|
q2
|
*q2
|
q0
|
q1
|
q2
|
Note:
1)Final state/Accepting state in transition table can be represented by either * or simple circle
2)An Automaton with finite number of states is called Finite State Machine or Finite Automata.
1)Final state/Accepting state in transition table can be represented by either * or simple circle
2)An Automaton with finite number of states is called Finite State Machine or Finite Automata.
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