Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
1
LearningRegular
ω-Languages
The Goal
Device an algorithm for learning
2
an unknown regularω-language L
The Goal
Device an algorithm for learning
3
an unknown regularω-language L
set of infinite words (ω-words)
The Goal
Device an algorithm for learning
4
an unknown regularω-language L
accepted by a finite automaton
The Goal
Device an algorithm for learning
5
Learner
an unknown regularω-language L
The Goal
Device an algorithm for learning
6
Learner
an unknown regularω-language L
L
Teacher
The Goal
Device an algorithm for learning
7
Learner
an unknown regularω-language L
L
Teacher
membership queries
equivalencequeries
Coping with ω-words
8
Learner
L
Teacher
Isabcccccabdebbbaaaabcdaa…
in L?
Coping with ω-words
9
Learner
L
Teacher
Isabcccccabdebbbaaaabcdaa…
in L?
Coping with ω-words
10
Learner
L
Teacher
Iswingardium laviosaω in L?
Coping with ω-words
11
Learner
L
Teacher
Iswingardium laviosaω in L?
wingardium laviosa laviosa laviosa laviosa laviosa …
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
The Goal
13
Is uvω in L ?
Learner
L
Teacher
The Goal
14
Is uvω in L ?
Yes / No
Learner
L
Teacher
The Goal
15
Is uvω in L ?
Yes / No
Is H same as L ?
Learner
L
Teacher
The Goal
16
Is uvω in L ?
Yes / No
Is H same as L ?
Yes / No, c.e: uvω
Learner
L
Teacher
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
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
Previous Work on Learning ω-Langs.
19
allregular
ω-languages
Safe
DP / DMDS / DR
DWP
DCDB
DWCDWB
strictly less expressive
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
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
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
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
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
However …
25
L$
SyntacticFORC
RecurrentFDFA
[Calbrix, Nivat & Podelski ‘93]
[Maler & Staiger ‘93]
However …
26
L$
SyntacticFORC
RecurrentFDFA
May be exp. bigger
than
[Calbrix, Nivat & Podelski ‘93]
[Maler & Staiger ‘93]
However …
27
L$
SyntacticFORC
RecurrentFDFA
May be exp. bigger
than
May be quadr.
bigger than
[Calbrix, Nivat & Podelski ‘93]
[Maler & Staiger ‘93]
Main Contribution
28
Learner
Lω –
a learning algorithm for all 3 representations
(periodic, syntactic, recurrent)
Why is it difficult?
29
ω-aut.
L*
1962
1987
2014
1993
1994
2005
2012
2008
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
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
For finite words
The Syntactic Right Congruence
32
x »L y iff v2*. xv2L , yv2L
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
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
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
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
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}
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.
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.
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.
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.
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
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
Family of DFAs (FDFA)
44
Leading DFAM
(M, P1, P2, P3, P4, P5 )
ProgressLeading
Family of DFAs (FDFA)
45
Leading DFAM
11P1
(M, P1, P2, P3, P4, P5 )
ProgressLeading
Family of DFAs (FDFA)
46
Leading DFAM
11P1
2
2P2
(M, P1, P2, P3, P4, P5 )
ProgressLeading
Family of DFAs (FDFA)
47
Leading DFAM
11P1
2
2P2
33P3
(M, P1, P2, P3, P4, P5 )
ProgressLeading
Family of DFAs (FDFA)
48
Leading DFAM
11P1
2
2P2
33P3
4
4P4
(M, P1, P2, P3, P4, P5 )
ProgressLeading
Family of DFAs (FDFA)
49
Leading DFAM
11P1
2
2P2
33P3
4
4P4
5
P5
(M, P1, P2, P3, P4, P5 )
ProgressLeading
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.
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!
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!
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!
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!
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
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
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
4-state automaton
58
0
1
2
2
1
2
1
00,1,2
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
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
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
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
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
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 !
Saturation (Consistency)
How can we use this without losing saturation?
65
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
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
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 ?
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 ?
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 ?
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 ?
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 ?
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 ?
More generally
74
SyntacticFDFA
RecurrentFDFA
May be exp. bigger
than
Periodic FDFA (L$)
Generalizing to arbitrary n we get the following lower bound
The Learner Lω
75
Same scheme for all 3
canonical FDFAs
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
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
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
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
81
THE ENDThank you for your
attention!
Comments or questions?
Learning
Regular
Languagesvia
Alternating
Automata
IJCAI’15
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