costas busch - rpi1 finite automata. costas busch - rpi2 finite automaton input string output string...
TRANSCRIPT
Costas Busch - RPI 1
Finite Automata
Costas Busch - RPI 2
Finite Automaton
Input
String
Output
String
FiniteAutomaton
Costas Busch - RPI 3
Finite Accepter
Input
“Accept” or“Reject”
String
FiniteAutomaton
Output
Costas Busch - RPI 4
Transition Graph
initialstate
final state“accept”state
transition
Abba -Finite Accepter
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
Costas Busch - RPI 5
Initial Configuration
1q 2q 3q 4qa b b a
5q
a a bb
ba,
Input Stringa b b a
ba,
0q
Costas Busch - RPI 6
Reading the Input
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b b a
ba,
Costas Busch - RPI 7
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b b a
ba,
Costas Busch - RPI 8
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b b a
ba,
Costas Busch - RPI 9
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b b a
ba,
Costas Busch - RPI 10
0q 1q 2q 3q 4qa b b a
Output: “accept”
5q
a a bb
ba,
a b b a
ba,
Input finished
Costas Busch - RPI 11
Rejection
1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b a
ba,
0q
Costas Busch - RPI 12
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b a
ba,
Costas Busch - RPI 13
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b a
ba,
Costas Busch - RPI 14
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b a
ba,
Costas Busch - RPI 15
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
Output:“reject”
a b a
ba,
Input finished
Costas Busch - RPI 16
Another Rejection
1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q
Costas Busch - RPI 17
1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q
Output:“reject”
Costas Busch - RPI 18
Another Example
a
b ba,
ba,
0q 1q 2q
a ba
Costas Busch - RPI 19
a
b ba,
ba,
0q 1q 2q
a ba
Costas Busch - RPI 20
a
b ba,
ba,
0q 1q 2q
a ba
Costas Busch - RPI 21
a
b ba,
ba,
0q 1q 2q
a ba
Costas Busch - RPI 22
a
b ba,
ba,
0q 1q 2q
a ba
Output: “accept”
Input finished
Costas Busch - RPI 23
Rejection
a
b ba,
ba,
0q 1q 2q
ab b
Costas Busch - RPI 24
a
b ba,
ba,
0q 1q 2q
ab b
Costas Busch - RPI 25
a
b ba,
ba,
0q 1q 2q
ab b
Costas Busch - RPI 26
a
b ba,
ba,
0q 1q 2q
ab b
Costas Busch - RPI 27
a
b ba,
ba,
0q 1q 2q
ab b
Output: “reject”
Input finished
Costas Busch - RPI 28
FormalitiesDeterministic Finite Accepter (DFA)
FqQM ,,,, 0
Q
0q
F
: set of states
: input alphabet
: transition function
: initial state
: set of final states
Costas Busch - RPI 29
Input Alphabet
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
ba,
Costas Busch - RPI 30
Set of States
Q
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
543210 ,,,,, qqqqqqQ
ba,
Costas Busch - RPI 31
Initial State
0q
1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q
Costas Busch - RPI 32
Set of Final States
F
0q 1q 2q 3qa b b a
5q
a a bb
ba,
4qF
ba,
4q
Costas Busch - RPI 33
Transition Function
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
QQ :
ba,
Costas Busch - RPI 34
10 , qaq
2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q 1q
Costas Busch - RPI 35
50 , qbq
1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q
Costas Busch - RPI 36
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
32 , qbq
Costas Busch - RPI 37
Transition Function
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b
0q
1q
2q
3q
4q
5q
1q 5q
5q 2q
5q 3q
4q 5q
ba,5q5q5q5q
Costas Busch - RPI 38
Extended Transition Function
*
QQ *:*
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
Costas Busch - RPI 39
20 ,* qabq
3q 4qa b b a
5q
a a bb
ba,
ba,
0q 1q 2q
Costas Busch - RPI 40
40 ,* qabbaq
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
Costas Busch - RPI 41
50 ,* qabbbaaq
1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q
Costas Busch - RPI 42
qwq ,*
Observation: There is a walk from to with label
q qw
q qw
q qkw 21
1 2 k
Costas Busch - RPI 43
50 ,* qabbbaaq
1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q
Example: There is a walk from to with label
0qabbbaa
5q
Costas Busch - RPI 44
Recursive Definition )),,(*(,*
,*
wqwq
q qw1q
qwq
),(
,*
1
1
1
,*
),(,*
qwq
qwq
)),,(*(,* wqwq
Costas Busch - RPI 45
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
2
1
0
0
0
0
,
,,
,,,*
),,(*
,*
q
bq
baq
baq
baq
abq
Costas Busch - RPI 46
Languages Accepted by DFAsTake DFA
Definition:The language contains all input strings accepted by
= { strings that drive to a final state}
M
MLM
M ML
Costas Busch - RPI 47
Example
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
abbaML M
accept
Costas Busch - RPI 48
Another Example
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
abbaabML ,, M
acceptacceptaccept
Costas Busch - RPI 49
Formally
For a DFA
Language accepted by :
FqQM ,,,, 0
M
FwqwML ,*:* 0
0q qw Fq
Costas Busch - RPI 50
ObservationLanguage rejected by :
FwqwML ,*:* 0
M
0q qw Fq
Costas Busch - RPI 51
More Examples
a
b ba,
ba,
0q 1q 2q
}0:{ nbaML n
accept trap state
Costas Busch - RPI 52
ML = { all strings with prefix }ab
a b
ba,
0q 1q 2q
accept
ba,3q
ab
Costas Busch - RPI 53
ML = { all strings without substring }001
0 00 001
1
0
1
10
0 1,0
Costas Busch - RPI 54
Regular Languages
A language is regular if there is a DFA such that
All regular languages form a language family
LM MLL
Costas Busch - RPI 55
abba abbaab,, }0:{ nban
{ all strings with prefix }ab
{ all strings with prefix }ab
{ all strings without substring }001
Examples of regular languages:
There exist automata that accept theseLanguages (see previous slides).
Costas Busch - RPI 56
Another ExampleThe languageis regular:
*,: bawawaL
a
b
ba,
a
b
ba
0q 2q 3q
4q
MLL
Costas Busch - RPI 57
There exist languages which are not Regular:
}0:{ nbaL nn
There is no DFA that accepts such a language
(we will prove this later in the class)
Example: