2. Terminologies
• Symbols:
– Symbols are an entity or individual objects, which can be any letter,
alphabet or any picture.
– Example: 1, a, b, #
• Alphabets:
– Alphabets are a finite set of symbols. It is denoted by ∑.
– Examples: ∑ = {a, b} , ∑ = {A, B, C, D} , ∑ = {0, 1, 2}, ∑ = {#, β, Δ}
• String:
• It is a finite collection of symbols from the alphabet. The string is
denoted by w.
• Example 1:
• If ∑ = {a, b}, various string that can be generated from ∑ are {ab,
aa, aaa, bb, bbb, ba, aba.....}.
• A string with zero occurrences of symbols is known as an empty
string. It is represented by ε.
• The number of symbols in a string w is called the length of a
string. It is denoted by |w|.
3. Definition
An automaton is defined as a system where energy, materials and
information are transformed, transmitted and used for
performing some functions without direct participation of man.
4. Definition
• An automaton is an abstract model of a digital computer.
• Every automaton includes some essential features.
• It has a mechanism for reading input.
– This input mechanism can read the input from left to right
– One symbol at a time
– Can also detect the end of the input string
• An automaton can produce output of some sort.
• It may have a temporary storage device.
• An automaton has a control unit which can be in any one of
finite number of internal states and which can change state in
some defined manner.
6. Formal Definition
• Finite Automaton (FA)
F
q
Q
M ,
,
,
, 0
• Finite Automaton (FA)
F
q
Q
M ,
,
,
, 0
Q
0
q
F
: set of states
: input alphabet
: transition function
: initial state
: set of accepting states
12. • The transition function which maps
Q x ∑* into Q
(i.e. maps a state and a string of input symbols
including the empty string into a state) is called the
indirect transition function.
38. Languages Accepted by FAs
FA
Definition:
The language contains
all input strings accepted by
= { strings that bring
to an accepting state}
M
M
L
M
M
M
L
39. 0
q 1
q 2
q 3
q 4
q
a b b a
5
q
a a b
b
b
a,
b
a,
abba
M
L M
accept
40. Example ( being null string)
0
q 1
q 2
q 3
q 4
q
a b b a
5
q
a a b
b
b
a,
b
a,
abba
ab
M
L ,
,
M
accept
accept
accept
43. 1
0, q
a
q
2
q 3
q 4
q
a b b a
5
q
a a b
b
b
a,
b
a,
0
q 1
q
44. 5
0, q
b
q
1
q 2
q 3
q 4
q
a b b a
5
q
a a b
b
b
a,
b
a,
0
q
45. 0
q 1
q 2
q 3
q 4
q
a b b a
5
q
a a b
b
b
a,
b
a,
3
2, q
b
q
46. Transition Function
0
q 1
q 2
q 3
q 4
q
a b b a
5
q
a a b
b
b
a,
a b
0
q
1
q
2
q
3
q
4
q
5
q
1
q 5
q
5
q 2
q
5
q 3
q
4
q 5
q
b
a,
5
q
5
q
5
q
5
q
48. 2
0,
* q
ab
q
3
q 4
q
a b b a
5
q
a a b
b
b
a,
b
a,
0
q 1
q 2
q
49. 4
0,
* q
abba
q
0
q 1
q 2
q 3
q 4
q
a b b a
5
q
a a b
b
b
a,
b
a,
50. 5
0,
* q
abbbaa
q
1
q 2
q 3
q 4
q
a b b a
5
q
a a b
b
b
a,
b
a,
0
q
51. Deterministic Final Automata
• DFA refers to deterministic finite automata.
– Deterministic refers to the uniqueness of the computation. The finite
automata are called deterministic finite automata if the machine reads
an input string one symbol at a time.
• In DFA, given the current state we know what the next step
will be.
• In DFA, there is only one path for specific input from the
current state to the next state.
• DFA does not accept the null move, i.e., the DFA cannot
change state without any input character.
• DFA accepts zero length string i.e. {}
• DFA can contain multiple final states. It is used in Lexical
Analysis in Compiler.
52. QUESTIONS
1. Draw a DFA which accept 00 and 11 at the end
of a string containing 0, 1 in it, e.g., 01010100
but not 000111010.
2. Construct a DFA which accepts set of all
strings over Σ={a,b} of length ≤2
3. Create a DFA which accepts strings of even
length.
4. Design a DFA in which start and end symbol
must be different
Given: Input alphabet, Σ={a, b}
53. QUESTIONS
5. Design a DFA in which every 'a' should be
followed by 'b'
Given: Input alphabet, Σ={a, b}
6. Design a DFA such that:
L = {anbm | n,m ≥ 1} Given: Input alphabet, Σ={a,
b}
Language L = {ab, aab, aaab, abbb, aabb,
aaaabbbb, ...}
