learning regular - university of pennsylvania...(deterministic finite automaton) and the syntactic...
TRANSCRIPT
![Page 1: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/1.jpg)
1
LearningRegular
ω-Languages
![Page 2: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/2.jpg)
The Goal
Device an algorithm for learning
2
an unknown regularω-language L
![Page 3: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/3.jpg)
The Goal
Device an algorithm for learning
3
an unknown regularω-language L
set of infinite words (ω-words)
![Page 4: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/4.jpg)
The Goal
Device an algorithm for learning
4
an unknown regularω-language L
accepted by a finite automaton
![Page 5: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/5.jpg)
The Goal
Device an algorithm for learning
5
Learner
an unknown regularω-language L
![Page 6: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/6.jpg)
The Goal
Device an algorithm for learning
6
Learner
an unknown regularω-language L
L
Teacher
![Page 7: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/7.jpg)
The Goal
Device an algorithm for learning
7
Learner
an unknown regularω-language L
L
Teacher
membership queries
equivalencequeries
![Page 8: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/8.jpg)
Coping with ω-words
8
Learner
L
Teacher
Isabcccccabdebbbaaaabcdaa…
in L?
![Page 9: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/9.jpg)
Coping with ω-words
9
Learner
L
Teacher
Isabcccccabdebbbaaaabcdaa…
in L?
![Page 10: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/10.jpg)
Coping with ω-words
10
Learner
L
Teacher
Iswingardium laviosaω in L?
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Coping with ω-words
11
Learner
L
Teacher
Iswingardium laviosaω in L?
wingardium laviosa laviosa laviosa laviosa laviosa …
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Coping with ω-words
12
Learner
L
Teacher
Iswingardium laviosaω in L?
wingardium laviosa laviosa laviosa laviosa laviosa …
THM:
Two regular ω-languages are equivalent iff
they agree on the set of ultimately periodic words
![Page 13: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/13.jpg)
The Goal
13
Is uvω in L ?
Learner
L
Teacher
![Page 14: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/14.jpg)
The Goal
14
Is uvω in L ?
Yes / No
Learner
L
Teacher
![Page 15: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/15.jpg)
The Goal
15
Is uvω in L ?
Yes / No
Is H same as L ?
Learner
L
Teacher
![Page 16: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/16.jpg)
The Goal
16
Is uvω in L ?
Yes / No
Is H same as L ?
Yes / No, c.e: uvω
Learner
L
Teacher
![Page 17: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/17.jpg)
The Goal
17
Is uvω in L ?
Yes / No
Is H same as L ?
Yes / No, c.e: uvω
H s.t. H=L
Learner
L
Teacher
![Page 18: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/18.jpg)
Motivation
Same problem for regular (finitary) languages is solved by the L* algorithm [Angluin]
L* has found applications in many areas including AI, neural networks, geometry, data mining, verificationand synthesis and many more.
Reactive systems concerns ω-languages, using L* for this limits application to safety (and excludes liveness, fairness)
18
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Previous Work on Learning ω-Langs.
19
allregular
ω-languages
Safe
DP / DMDS / DR
DWP
DCDB
DWCDWB
strictly less expressive
![Page 20: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/20.jpg)
Previous Work on Learning ω-Langs.
20
allregular
ω-languages
Safe
DP / DMDS / DR
DWP
DCDB
DWCDWB
strictly less expressive
• [de la Higuera & Janodet, 2004]
• [Jayasrirani et al, 2012]
• [Saoudi & Yokomori, 1993]
From Prefixes
From Lassos
![Page 21: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/21.jpg)
Previous Work on Learning ω-Langs.
21
allregular
ω-languages
Safe
DP / DMDS / DR
DWP
DCDB
DWCDWB
strictly less expressive
• [de la Higuera & Janodet, 2004]
• [Jayasrirani et al, 2012]
• [Saoudi & Yokomori, 1993]
• [Maler & Pnueli ,1995]
From Prefixes
From Lassos
From Lassos
![Page 22: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/22.jpg)
Previous Work on Learning ω-Langs.
22
allregular
ω-languages
Safe
DP / DMDS / DR
DWP
DCDB
DWCDWB
strictly less expressive
• [de la Higuera & Janodet, 2004]
• [Jayasrirani et al, 2012]
• [Saoudi & Yokomori, 1993]
• [Maler & Pnueli ,1995]
From Prefixes
From Lassos
From Lassos
goal
![Page 23: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/23.jpg)
Previous Work on Learning ω-Langs.
23
allregular
ω-languages
Safe
DP / DMDS / DR
DWP
DCDB
DWCDWB
strictly less expressive
• [de la Higuera & Janodet, 2004]
• [Jayasrirani et al, 2012]
• [Saoudi & Yokomori, 1993]
• [Maler & Pnueli ,1995]
From Prefixes
From Lassos
From Lassos
goal
![Page 24: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/24.jpg)
Previous Work on Learning ω-Langs.
24
allregular
ω-languages
Safe
DP / DMDS / DR
DWP
DCDB
DWCDWB
strictly less expressive
• [de la Higuera & Janodet, 2004]
• [Jayasrirani et al, 2012]
• [Saoudi & Yokomori, 1993]
• [Maler & Pnueli ,1995]
From Prefixes
From Lassos
From Lassos
goal
![Page 25: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/25.jpg)
However …
25
L$
SyntacticFORC
RecurrentFDFA
[Calbrix, Nivat & Podelski ‘93]
[Maler & Staiger ‘93]
![Page 26: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/26.jpg)
However …
26
L$
SyntacticFORC
RecurrentFDFA
May be exp. bigger
than
[Calbrix, Nivat & Podelski ‘93]
[Maler & Staiger ‘93]
![Page 27: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/27.jpg)
However …
27
L$
SyntacticFORC
RecurrentFDFA
May be exp. bigger
than
May be quadr.
bigger than
[Calbrix, Nivat & Podelski ‘93]
[Maler & Staiger ‘93]
![Page 28: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/28.jpg)
Main Contribution
28
Learner
Lω –
a learning algorithm for all 3 representations
(periodic, syntactic, recurrent)
![Page 29: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/29.jpg)
Why is it difficult?
29
ω-aut.
L*
1962
1987
2014
1993
1994
2005
2012
2008
![Page 30: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/30.jpg)
Why is it difficult?
30
states of the minimal DFA (deterministic finite automaton)
and the syntactic right congruence ~ of a language.
L* works due to the Myhill-NerodeTheorem, stating a 1-to-1 relationship between
1
2
53
4
a
b
c
a
bc
a
12
3 54
ω-aut.
L*
1962
1987
2014
1993
1994
2005
2012
2008
![Page 31: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/31.jpg)
Why is it difficult?
31
states of the minimal DFA (deterministic finite automaton)
and the syntactic right congruence ~ of a language.
L* works due to the Myhill-NerodeTheorem, stating a 1-to-1 relationship between
1
2
53
4
a
b
c
a
bc
a
12
3 54
But for ω-langs, the syntactic right congruence is not informative enough!
ω-aut.
L*
1962
1987
2014
1993
1994
2005
2012
2008
![Page 32: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/32.jpg)
For finite words
The Syntactic Right Congruence
32
x »L y iff v2*. xv2L , yv2L
![Page 33: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/33.jpg)
For finite words
Example:
L = { w w has an even number of a’s}
The Syntactic Right Congruence
33
x »L y iff v2*. xv2L , yv2L
![Page 34: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/34.jpg)
For finite words
Example:
L = { w w has an even number of a’s}
Then »L has two equivalence classes:
words with an odd number of a’s and
words with an even number of a’s
The Syntactic Right Congruence
34
x »L y iff v2*. xv2L , yv2L
a
a
b b10
![Page 35: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/35.jpg)
For finite words:
For ω-words:
The Syntactic Right Congruence
35
x »L y iff v2*. xv2L , yv2L
x »L y iff w2!. xw2L, yw2L
![Page 36: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/36.jpg)
For finite words:
For ω-words:
The Syntactic Right Congruence
36
x »L y iff v2*. xv2L , yv2L
x »L y iff w2!. xw2L, yw2Lx »L y iff u,v2*. xuv!2L, yuv!2L
![Page 37: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/37.jpg)
For ω-words:
The Syntactic Right Congruence
37
x »L y iff u,v2*. xuv!2L, yuv!2L
Example:
L = { w | w has finitely many a’s}
![Page 38: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/38.jpg)
For ω-words:
The Syntactic Right Congruence
38
x »L y iff u,v2*. xuv!2L, yuv!2L
Example:
L = { w | w has finitely many a’s}
Then xuv!2L iff uv! has finitely may a’s.
![Page 39: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/39.jpg)
For ω-words:
The Syntactic Right Congruence
39
x »L y iff u,v2*. xuv!2L, yuv!2L
Example:
L = { w | w has finitely many a’s}
Then xuv!2L iff uv! has finitely may a’s.
Regardless of x. Thus »L has just one equivalence class.
![Page 40: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/40.jpg)
For ω-words:
The Syntactic Right Congruence
40
x »L y iff u,v2*. xuv!2L, yuv!2L
Example:
L = { w | w has finitely many a’s}
Then xuv!2L iff uv! has finitely may a’s.
Regardless of x. Thus »L has just one equivalence class.
![Page 41: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/41.jpg)
For ω-words:
The Syntactic Right Congruence
41
x »L y iff u,v2*. xuv!2L, yuv!2L
Example:
L = { w | w has finitely many a’s}
Then xuv!2L iff uv! has finitely may a’s.
Regardless of x. Thus »L has just one equivalence class.
But clearly an ω-automaton for L requires at least two states.
![Page 42: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/42.jpg)
Solution Foundation
Families of DFAs (FDFA)
A non-traditional representation of ω-automata
inspired by Maler & Staiger’s ’97 families of right congruences (FORC)
and their canonical representation of a Syntactic FORC
for which they have shown a Myhill-Nerode Theorem
42
![Page 43: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/43.jpg)
Family of Right Congruences [MS97]
43
Leading Right Congruence
2
2
1
1
4
4
5
5
Plus some restriction (details ommited)
33
(»,¼1,¼2,¼3,¼4,¼5)
ProgressLeading
![Page 44: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/44.jpg)
Family of DFAs (FDFA)
44
Leading DFAM
(M, P1, P2, P3, P4, P5 )
ProgressLeading
![Page 45: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/45.jpg)
Family of DFAs (FDFA)
45
Leading DFAM
11P1
(M, P1, P2, P3, P4, P5 )
ProgressLeading
![Page 46: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/46.jpg)
Family of DFAs (FDFA)
46
Leading DFAM
11P1
2
2P2
(M, P1, P2, P3, P4, P5 )
ProgressLeading
![Page 47: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/47.jpg)
Family of DFAs (FDFA)
47
Leading DFAM
11P1
2
2P2
33P3
(M, P1, P2, P3, P4, P5 )
ProgressLeading
![Page 48: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/48.jpg)
Family of DFAs (FDFA)
48
Leading DFAM
11P1
2
2P2
33P3
4
4P4
(M, P1, P2, P3, P4, P5 )
ProgressLeading
![Page 49: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/49.jpg)
Family of DFAs (FDFA)
49
Leading DFAM
11P1
2
2P2
33P3
4
4P4
5
P5
(M, P1, P2, P3, P4, P5 )
ProgressLeading
![Page 50: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/50.jpg)
Family of DFAs (FDFA)
50
Leading DFAM
11P1
2
2P2
33P3
4
4P4
5
P5
(M, P1, P2, P3, P4, P5 )
ProgressLeading
That restriction removed.
![Page 51: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/51.jpg)
FDFA Exact Acceptance
(u,v)2 M, P1, P2, P3, P4, P5
P1P1P1
22
44
33 55
MM
2211
44
33 552
4
3 5
M
21
4
3 5
P4P4P4 P5P5P5
P3P3P3
P1P1P1
?uv!
![Page 52: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/52.jpg)
FDFA Exact Acceptance
(u,v)2 M, P1, P2, P3, P4, P5
P1P1P1
22
44
33 55
MM
2211
44
33 552
4
3 5
M
21
4
3 5
P4P4P4 P5P5P5
P3P3P3
P1P1P1
u
?uv!
![Page 53: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/53.jpg)
FDFA Exact Acceptance
(u,v)2 M, P1, P2, P3, P4, P5
P1P1P1
22
44
33 55
MM
2211
44
33 552
4
3 5
M
21
4
3 5
P4P4P4 P5P5P5
P3P3P3
P1P1P1
u
v
?uv!
![Page 54: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/54.jpg)
FDFA Exact Acceptance
(u,v)2 M, P1, P2, P3, P4, P5
P1P1P1
22
44
33 55
MM
2211
44
33 552
4
3 5
M
21
4
3 5
P4P4P4 P5P5P5
P3P3P3
P1P1P1
u
v
?uv!
![Page 55: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/55.jpg)
FORC and FDFA – Example
L = { w | w has finitely many a’s}
1
a,b
1
No a’s
Some a’s
FORC FDFA
Leading
Progress
All prefixes are equally good
![Page 56: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/56.jpg)
FORC and FDFA – Example
L = { w | w has finitely many a’s}
1
a,b
a,ba
P1
b
1
1
No a’s
Some a’s
FORC FDFA
Leading
Progress
All prefixes are equally good
![Page 57: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/57.jpg)
FORC and FDFA – Example
L = { w | w has finitely many a’s}
1
a,b
a,ba
P1
b
1
1
No a’s
Some a’s
FORC FDFA
(abba,bbb)
(abba,bab)
Leading
Progress
All prefixes are equally good
![Page 58: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/58.jpg)
4-state automaton
58
0
1
2
2
1
2
1
00,1,2
![Page 59: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/59.jpg)
4-state automaton
59
0
1
2
2
1
2
1
00,1,2
Some periods of are easy to find:
11
22
100201
10020122
![Page 60: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/60.jpg)
4-state automaton
60
0
1
2
2
1
2
1
00,1,2
Some periods of are easy to find:
11
22
100201
10020122
Some are hard:
1012
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4-state automaton
61
0
1
2
2
1
2
1
00,1,2
Some periods of are easy to find:
11
22
100201
10020122
Some are hard:
10121012 1012
1012 1012
![Page 62: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/62.jpg)
4-state automaton
62
0
1
2
2
1
2
1
00,1,2
Some periods of are easy to find:
11
22
100201
10020122
Some are hard:
10121012 1012
1012 1012
The DFA for periods of has 23 states
=> L$ > 23
![Page 63: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/63.jpg)
4-state automaton
63
0
1
2
2
1
2
1
00,1,2
All periods that loop back are easy to find:
11
22
100201
10020122
![Page 64: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/64.jpg)
4-state automaton
64
0
1
2
2
1
2
1
00,1,2
All periods that loop back are easy to find:
11
22
100201
10020122
The DFA that accepts only periods of that loop back has only 5 states !
![Page 65: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/65.jpg)
Saturation (Consistency)
How can we use this without losing saturation?
65
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Saturation (Consistency)
How can we use this without losing saturation?
66
We say that a language is saturated if
uv!=xy! implies (u,v)2 L, (x,y)2 L
![Page 67: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/67.jpg)
Saturation (Consistency)
How can we use this without losing saturation?
To achieve saturation, we change the definition of acceptance.
67
We say that a language is saturated if
uv!=xy! implies (u,v)2 L, (x,y)2 L
![Page 68: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/68.jpg)
FDFA Normalized Acceptance
P1P1P1 22
44
33 55
MM
2211
44
33 552
4
3 5
M
21
4
3 5
P4 P5P5P5
P3P3P3
P1P1P1
(u,v)2 M, P1, P2, P3, P4, P5 ?
![Page 69: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/69.jpg)
FDFA Normalized Acceptance
P1P1P1 22
44
33 55
MM
2211
44
33 552
4
3 5
M
21
4
3 5
P4 P5P5P5
P3P3P3
P1P1P1
Normalization seeks for the smallest repetition
of the period that loops back
(u,v)2 M, P1, P2, P3, P4, P5 ?
![Page 70: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/70.jpg)
FDFA Normalized Acceptance
P1P1P1 22
44
33 55
MM
2211
44
33 552
4
3 5
M
21
4
3 5
P4 P5P5P5
P3P3P3
P1P1P1
Normalization seeks for the smallest repetition
of the period that loops back
(u,v)2 M, P1, P2, P3, P4, P5 ?
![Page 71: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/71.jpg)
FDFA Normalized Acceptance
P1P1P1 22
44
33 55
MM
2211
44
33 552
4
3 5
M
21
4
3 5
P4 P5P5P5
P3P3P3
P1P1P1
Normalization seeks for the smallest repetition
of the period that loops back
(u,v)2 M, P1, P2, P3, P4, P5 ?
![Page 72: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/72.jpg)
FDFA Normalized Acceptance
P1P1P1 22
44
33 55
MM
2211
44
33 552
4
3 5
M
21
4
3 5
P4 P5P5P5
P3P3P3
P1P1P1
Normalization seeks for the smallest repetition
of the period that loops back
(u,v)2 M, P1, P2, P3, P4, P5 ?
![Page 73: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/73.jpg)
FDFA Normalized Acceptance
P1P1P1 22
44
33 55
MM
2211
44
33 552
4
3 5
M
21
4
3 5
P4 P5P5P5
P3P3P3
P1P1P1
Normalization seeks for the smallest repetition
of the period that loops back
We term Recurrent FDFA the FDFA where progress DFA recognize only periods that loop back. It is saturated under normalized acceptance!
(u,v)2 M, P1, P2, P3, P4, P5 ?
![Page 74: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/74.jpg)
More generally
74
SyntacticFDFA
RecurrentFDFA
May be exp. bigger
than
Periodic FDFA (L$)
Generalizing to arbitrary n we get the following lower bound
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The Learner Lω
75
Same scheme for all 3
canonical FDFAs
![Page 76: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/76.jpg)
Minimality?
The Recurrent FDFA for a given L is not necessarilythe minimal FDFA for L.
According to the normalized acceptance criterion a progress DFA Pu should give correct results only to extensions that close a cycle to u. On other extensions it has freedom.
This has the flavor of minimization with unknowns, which is NP complete.
The Recurrent FDFA chooses to treat all don’t cares as rejecting.
76
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Time Complexity
Interestingly, [Klarlund, 1994] has shown that choosing a leading automaton which is more refined (bigger) than the syntactic right congruence, may yield an overall smaller FDFA.
If we are given such a leading automaton, we can feedit to the learning algorithm, in which case it will yield the smaller FDFA.
However, the same phenomenon can cause the learning algorithm for the syntactic/recurrent FDFAs to work as hard as the one for the periodic FDFA
77
![Page 78: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/78.jpg)
Time Complexity
The worst case time complexity for all 3 families is thus polynomial in the size of the periodic FDFA.
Some positive results on sizes obtained via recurrent FDFA learner vs. periodic FDFA learner on randomly generated Muller automata.
78
![Page 79: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/79.jpg)
Future Direction
Find smaller canonical representations ?
Find smallest FDFA ?
Learning the leading first ?
L*-learning for (traditional) ω-automata ?
Other types of learning for ω-languages ?
80
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81
THE ENDThank you for your
attention!
Comments or questions?
Learning
Regular
Languagesvia
Alternating
Automata
IJCAI’15
![Page 81: Learning Regular - University of Pennsylvania...(deterministic finite automaton) and the syntactic right congruence ~ of a language. L* works due to the Myhill-NerodeTheorem, stating](https://reader034.vdocument.in/reader034/viewer/2022051902/5ff181c3da92ff7633576fb9/html5/thumbnails/81.jpg)
Time Complexity
The worst case time complexity for all 3 families is thus polynomial in the size of the periodic FDFA.
Some positive results on sizes obtained via recurrent FDFA learner vs. periodic FDFA learner on randomly generated Muller automata.
83