2.
Automata are abstract mathematical models
of machines that perform computations on
an input by moving through a series of states
or configurations.
3.
Automaton
An automaton is represented formally by the 5-
tuple of elements {Q,Σ,δ,q0,A}, where:
▪ Q is a finite set of states.
▪ Σ is a finite set of symbols.
4.
Automaton
An automaton is represented formally by the 5-
tuple of elements {Q,Σ,δ,q0,A}, where:
▪ δ is the transition function.
▪ q0 is the start state, where q0∈ Q.
▪ A is a set of states of Q (i.e. A⊆Q) called accept states.
5.
Input word
An automaton reads a finite string of symbols
a1,a2,...., an , where ai ∈ Σ, which is called a input
word.
Accepting word
A word w ∈ Σ* is accepted by the automaton if
qn ∈ A.
6.
Run
A run of the automaton on an input word w =
a1,a2,...., an ∈ Σ*, is a sequence of states q0,q1,q2,....,
qn, where qi ∈ Q such that q0 is the start state and
qi = δ(qi-1,ai) for 0 < i ≤ n.
7.
A finite automaton (FA) is a simple idealized
machine used to recognize patterns within
input taken from some character set (or
alphabet) C.
8.
A finite automaton consists of:
a finite set S of N states
a special start state
a set of final (or accepting) states
a set of transitions T from one state to another,
labeled with chars in C
As noted above, we can represent a FA
graphically, with nodes for states, and arcs for
transitions.
9.
We execute our FA on an input sequence as
follows:
Begin in the start state
If the next input char matches the label on a
transition from the current state to a new state,
go to that new state
Continue making transitions on each input char
▪ If no move is possible, then stop
▪ If in accepting state, then accept
10.
It is a finite state machine which accepts the
word "nice". In this FSM the only accepting
state is number 7.
11.
For each input symbol in Σ, there is exactly
one transition of each state (possibly back to
the state itself).
It do not accept empty strings.
12.
A deterministic finite automaton (DFA)
consists of
A finite set of states (often denoted Q)
A finite set Σ of symbols (alphabet)
A transition function that takes as argument a
state and a symbol and returns a state (often
denoted δ)
13.
A start state often denoted q0
A set of ﬁnal or accepting states (often denoted F)
We have q0 ∈ Q and F ⊆ Q
14.
So a DFA is mathematically represented as a
5-uple
(Q, Σ, δ, q0, F )
The transition function δ is a function in
Q×Σ→Q
Q × Σ is the set of 2-tuples (q, a) with q ∈ Q and a ∈
Σ
15.
How to present a
DFA? With a
transition table
16.
How to present a
DFA? With a
transition table
Inputs
0
1
17.
How to present a
DFA? With a
transition table
Inputs
States
0
q0
q1
q2
1
18.
How to present a
DFA? With a
transition table
The → indicates the
start state
0
q0
q1
q2
1
19.
How to present a
DFA? With a
transition table
The → indicates the
start state
The ∗ indicates the
ﬁnal state(s)
0
q0
q1
* q2
1
20.
How to present a
DFA? With a
transition table
δ (q0,0) = q1
δ (q0,0) = q0
0
q0
q1
* q2
1
q1
q0
21.
How to present a
DFA? With a
transition table
δ (q0,0) = q1
δ (q0,0) = q0
δ (q1,0) = q2
δ (q1,0) = q1
0
1
q0
q1
q0
q1
q2
q1
* q2
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