Languages and Finite Automata

1
Finite Automata
2
Finite Automaton

Input
String
Output
Finite Automaton
String
3
Finite Accepter

Input
String
Output
Accept or Reject
Finite Automaton
4
Transition Graph

abba - Finite Accepter
initial state
final state accept
transition
state
5
Initial Configuration

Input String
6
Reading the Input

7

8

9

10
Input finished
Output accept
11
Rejection

12

13

14

15
Input finished
Output reject
16
Another Rejection

17

Output reject
18
Another Example
19
(No Transcript)
20
(No Transcript)
21
(No Transcript)
22
Input finished
Output accept
23
Rejection
24
(No Transcript)
25
(No Transcript)
26
(No Transcript)
27
Input finished
Output reject
28
Formalities

• Deterministic Finite Accepter (DFA)

(A finite) set of states
(A finite) set of input alphabet
transition function
initial state (an element of Q)
set of final states (a subset of Q)
29
Input Alphabet

30
Set of States

31
Initial State

32
Set of Final States

33
Transition Function

34

35

36
(No Transcript)
37
Transition Function

38
Extended Transition Function

39
(No Transcript)
40
(No Transcript)
41
(No Transcript)
42
Observation There is a walk from to
with label
43
Example There is a walk from to
with label
44
Recursive Definition
45

46
Languages Accepted by DFAs

• Take DFA
• Definition
• The language contains
• all input strings accepted by
• strings that drive to a
  final state

47
Example

accept
48
Another Example

accept
accept
accept
49
Formally

• For a DFA
• Language accepted by

50
Observation

• Language rejected by

51
More Examples

trap state
accept
52

all strings with prefix
accept
53

all strings without substring
54
Regular Languages

• A language is regular if there is
• a DFA such that
• All regular languages form a language family

55
Examples of regular languages
all strings with prefix
all strings with suffix
all strings without substring
There exist automata that accept these Languages
(see previous slides).
56
Another Example

• The language
• is regular

57
There exist languages which are not Regular
Example
There is no DFA that accepts such a language
(we will prove this later!)