Finite Automata

1
Finite Automata
2
Introductory Example

• An automaton that accepts all legal Pascal
  identifiers

Letter
2
2
1
"yes"
Digit
Letter or Digit
3
"no"
Letter or Digit
3
Deterministic Finite Automata (DFA)

• M (Q, ?, ?, q0, F)
• Q finite set of internal states
• ? finite set of symbols - input alphabet
• ? Q ? ? ? Q transition function
• q0?Q initial state
• F?Q set of final states

4
Operational Manner
Input file
Control unit q0
yes/no
5
Transition Graphs

• M (Q, ?, ?, q0, F)
• Q vertices (circles)
• Edge (qi, qj) labelled a for ?(qi, a) qj
• Initial vertice q0
• Final vertices (double circles) in F

6
Example 1

• M (q0, q1, q2, 0, 1, ?, q0, q1)
• ?(q0, 0) q0 ?(q0, 1) q1
• ?(q1, 0) q0 ?(q1, 1) q2
• ?(q2, 0) q2 ?(q2, 1) q1

1
0
0
0
q0
q1
q2
1
1
7
Extended Transition Function

• ?(q0, a) q1 ?(q1, b) q2 ? ?(q0, ab)
  q2

8
Extended Transition Function

• ?(q0, a) q1 ?(q1, b) q2 ? ?(q0, ab)
  q2
• ?(q, ?) q
• ?(q, wa) ?(?(q, w), a)

9
Languages and DFAs

• M (Q, ?, ?, q0, F)
• L(M) w?? ?(q0, w)?F
• L(M) w?? ?(q0, w)?F

10
Example 1

• M (q0, q1, q2, a, b, ?, q0, q1)
• L(M) ?

a
a, b
a, b
b
q0
q1
q2
11
Example 1

• M (q0, q1, q2, a, b, ?, q0, q1)
• L(M) anb n ? 0

a
a, b
a, b
b
q0
q1
q2
trap state
12
Theorem

• M (Q, ?, ?, q0, F)
• GM associated transition graph
• w??
• ?(qi, w) qj iff there is a walk labelled w
  from qi to qj

13
Example 2

• L(M) w?a, b w starts with ab

14
Example 2

• L(M) w?a, b w starts with ab

a, b
b
a
q0
q1
q2
a
b
q3
trap state
a, b
15
Example 3

• L(M) w?0, 1 w does not contain 001

16
Example 3

• L(M) w?0, 1 w does not contain 001

0, 1
1
0
1
0
1
?
0
001
00
trap state
0
17
Regular Languages

• L is regular iff L L(M) for some DFA M

18
Example 4

• L awa w?a, b

19
Example 4

• L awa w?a, b

b
a
b
a
q0
q2
q3
b
a
q1
trap state
a, b
20
Example 5

• L awa w?a, b
• L2 aw1aaw2a w1, w2?a, b

21
Example 5

• L2 aw1aaw2a w1, w2?a, b

b
a
b
b
b
a
a
q4
q5
q0
q2
q3
b
a
b
a
q1
trap state
a, b
22
Nondeterministic Finite Automata (NFA)

• M (Q, ?, ?, q0, F)
• Q finite set of internal states
• ? finite set of symbols - input alphabet
• ? Q ? (? ? ?) ? 2Q transition function
• q0?Q initial state
• F?Q set of final states

23
Example 6
a
a
q1
q2
q3
a
q0
a
a
q4
q5
a
24
Example 7
0
0, 1
q0
q1
q2
1
?
25
Extended Transition Function

• ?(qi, w) Qj

26
Extended Transition Function

• ?(qi, w) contains qj iff there is a walk
  labelled w
• from qi to qj

27
Example 8

• ?(q1, a) ?
• ?(q2, ?) ?

?
?
a
q0
q1
q2
28
Example 8

• ?(q1, a) q0, q1, q2
• ?(q2, ?) q0, q2

?
?
a
q0
q1
q2
29
Example 8

• ?(qi, w) ?

?
?
a
q0
q1
q2
30
Example 8

• ?(qi, w) ?
• Evaluate all walks of length at most ? (1
  ?)w
• ? is the number of ?-edges

?
?
a
q0
q1
q2
31
Languages and NFAs

• M (Q, ?, ?, q0, F)
• L(M) w?? ?(q0, w) ? F ? ?

32
Languages and NFAs
0
0, 1
q0
q1
q2
1
?
L(M) ?
33
Languages and NFAs
0
0, 1
q0
q1
q2
1
?
L(M) (10)n n ? 0
34
Languages and NFAs
0
0, 1
q0
q1
q2
1
?
?(q0, 110) ?(q2, 0) ? dead
configuration
35
Homework

• Exercises 1, 5, 6, 11, 14, 17, 15, 17 of Section
  2.1 - Linzs book.
• Exercises 3, 4, 6, 7, 9, 10 of Section 2.2 -
  Linzs book.
• Reading Why nondeterminism - Linzs book.
• Presentation Section 2.4 - Linzs book
  (procedures mark and reduce).

36
Equivalence of DFAs and NFAs

• A class of automata may be more powerful than
  another.
• DFA is a restricted kind of NFA.
• DFA ?(qi, a) qj
• NFA ?(qi, a) qj

37
Equivalence of DFAs and NFAs

• DFA and NFA are equally powerful.
• NFA ?(q, w) qi, qj, ..., qk

a label of one state
38
Example 9

• 2Q ?, q0, q1, q0, q1

b
?
a
q0
q1
q2
a
39
Example 9

• ?(q0, a) q1, q2 ?(q0, b) ?

b
?
a
q0
q1
q2
a
40
Example 9

• ?(q0, a) q1, q2 ?(q0,
  b) ?
• ?(q1, q2, a) q1, q2 ?(q1, q2, b)
  q0

b
?
a
q0
q1
q2
a
41
Example 9
b
a
q1,q2
q0
a
b
?
a, b
42
Theorem

• Given MN (QN, ?, ?N, q0N, FN)
• there exists MD (QD, ?, ?D, q0D, FD)
• such that L(MD) L(MN)

43
Procedure NFA?DFA

• Create GD with vertex q0D q0N.
• Repeat
• Take any vertex qi, qj, ..., qk of GD that has
  a missing edge for a??.
• Compute ?(qi, a), ?(qj, a), ..., ?(qk, a).
• Create new ql, qm, ..., qn ?(qi, a)??(qj,
  a)? ... ??(qk, a).
• Add qi, qj, ..., qk ? a ? ql, qm, ..., qn.
• Every state of GD containing qf ? FN is a final
  vertex.
• If ? ? L(MN) then q0N is also a final vertex.

44
Example 10

0
1
0, 1
0, 1
q0
q1
q2
45
Homework

• Exercises 3, 8, 11, 12 of Section 2.3 - Linzs
  book.
• Exercises 1, 4 of Section 2.4 - Linzs book.
• Programming Implement procedures mark and reduce
  (Section 2.4)
• (Submission 15/October).