Shape-Representation and Shape Similarity CIS 601 by Rolf Lakaemper modified by Longin Jan Latecki.
Language Recognition (12.4) Longin Jan Latecki Temple University
description
Transcript of Language Recognition (12.4) Longin Jan Latecki Temple University
![Page 1: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/1.jpg)
1
Language Recognition (12.4) Longin Jan LateckiTemple University
Based on slides by Costas Busch from the coursehttp://www.cs.rpi.edu/courses/spring05/modcomp/and …
![Page 2: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/2.jpg)
2
Three Equivalent Representations
Finite automata
Regularexpressions
Regular languages
Each can
describethe others
Kleene’s Theorem:
For every regular expression, there is a deterministic finite-state automaton that defines the same language, and vice versa.
![Page 3: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/3.jpg)
3
EXAMPLE 1
Consider the language { ambn | m, n N}, which is represented by the regular expression a*b*.
A regular grammar for this language can
be written as follows:
S | aS | B B b | bB.
![Page 4: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/4.jpg)
4
Regular Expression
Regular Grammar
a* S | aS
(a+b)* S | aS | bS
a* + b* S | A | BA a | aAB b | bB
a*b S b | aS
ba* S bAA | aA
(ab)* S | abS
![Page 5: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/5.jpg)
5
NFAs Regular grammarsThus, the language recognized by FSA
is a regular language
Every NFA can be converted into a corresponding regular grammar and vice versa.
Each symbol A of the grammar is associated with a non-terminal node of the NFA sA, in particular, start symbol
S is associated with the start state sS.
Every transition is associated with a grammar production:
T(sA,a) = sB A aB.
Every production B is associated with final state sB.
![Page 6: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/6.jpg)
6
Equivalent FSA and regular grammar, Ex. 4, p. 772.
G=(V,T,S,P)
V={S, A, B, 0, 1} with
S=s0, A=s1, and B=s2,
T={0,1}, and productions are
S 0A | 1B | 1 | λ
A 0A | 1B | 1
B 0A | 1B | 1 | λ
![Page 7: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/7.jpg)
7
Kleene’s Theorem
LanguagesGenerated byRegular Expressions
LanguagesRecognizedby FSA
![Page 8: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/8.jpg)
8
LanguagesGenerated byRegular Expressions
LanguagesRecognizedby FSA
LanguagesGenerated byRegular Expressions
LanguagesRecognizedby FSA
We will show:
![Page 9: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/9.jpg)
9
Proof - Part 1
r)(rL
For any regular expression
the language is recognized by FSA (= is a regular language)
LanguagesGenerated byRegular Expressions
LanguagesRecognizedby FSA
Proof by induction on the size of r
![Page 10: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/10.jpg)
10
Induction BasisPrimitive Regular Expressions: a,,
NFAs
)()( 1 LML
)(}{)( 2 LML
)(}{)( 3 aLaML
regularlanguages
a
![Page 11: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/11.jpg)
11
Inductive Hypothesis
Assume for regular expressions andthat and are regular languages
1r 2r
)( 1rL )( 2rL
![Page 12: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/12.jpg)
12
Inductive StepWe will prove:
1
1
21
21
*
rL
rL
rrL
rrL
Are regular Languages
![Page 13: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/13.jpg)
13
By definition of regular expressions:
11
11
2121
2121
**
rLrL
rLrL
rLrLrrL
rLrLrrL
![Page 14: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/14.jpg)
14
)( 1rL )( 2rLBy inductive hypothesis we know: and are regular languages
Regular languages are closed under:
*1
21
21
rL
rLrL
rLrL Union
Concatenation
Star
We need to show:
This fact is illustrated in Fig. 2 on p. 821.
![Page 15: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/15.jpg)
15
Therefore:
** 11
2121
2121
rLrL
rLrLrrL
rLrLrrL
Are regularlanguages
And trivially: ))(( 1rL is a regular language
![Page 16: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/16.jpg)
16
Proof - Part 2
LanguagesGenerated byRegular Expressions
LanguagesRecognizedby FSA
Lr LrL )(
For any regular language there is a regular expression with
Proof by construction of regular expression
![Page 17: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/17.jpg)
17
Since is regular take the NFA that accepts it
LM
LML )(
Single final state
![Page 18: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/18.jpg)
18
From construct the equivalentGeneralized Transition Graph in which transition labels are regular
expressions
M
Example:
a
ba,
cM
a
ba
c
![Page 19: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/19.jpg)
19
Another Example:
ba a
b
b
0q 1q 2q
ba,a
b
b
0q 1q 2q
b
b
![Page 20: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/20.jpg)
20
Reducing the states:
ba a
b
b
0q 1q 2q
b
0q 2q
babb*
)(* babb
![Page 21: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/21.jpg)
21
Resulting Regular Expression:
0q 2q
babb*
)(* babb
*)(**)*( bbabbabbr
LMLrL )()(
![Page 22: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/22.jpg)
22
In GeneralRemoving states:
iq q jqa b
cde
iq jq
dae* bce*dce*
bae*
![Page 23: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/23.jpg)
23
The final transition graph:
0q fq
1r
2r
3r4r
*)*(* 213421 rrrrrrr
LMLrL )()(
The resulting regular expression:
![Page 24: Language Recognition (12.4) Longin Jan Latecki Temple University](https://reader035.fdocuments.in/reader035/viewer/2022081603/568139ce550346895da17dcf/html5/thumbnails/24.jpg)
24
Three Equivalent Representations
Finite automata
Regularexpressions
Regular languages
Each can
describethe others