Learning Regular Languages over Large Alphabets

141 Pages·2014·2 MB·English

Learning Angluin’s Symbolic Conclusion
Learning Regular Languages over Large
Alphabets
Oded Maler Irini Eleftheria Mens
CNRS-VERIMAG
UniversityofGrenoble
2,avenuedeVignate
38610Gieres
France
January 10, 2014
Learning Angluin’s Symbolic Conclusion
Outline
About Learning
Machine learning in general
Learning regular languages
Queries
Angluin’s Algorithm
General idea
Observation table
The algorithm
Symbolic latters
symbolic automata
Using evidences
Symbolic algorithm
example
Conclusion
VERIMAG-IEMens EQINOCS 1/11
111...555
111...000
000...555
000...000
000...555
111...000
111...555111 000 111 222 333 444 555 666 777
Learning Angluin’s Symbolic Conclusion
Machine learning in general
• given a sample M = {(x,y) | x ∈ X, y ∈ Y}
• find a representation f : X → Y such that f(x) = y
• predict or identify f(x) for all x ∈ X
VERIMAG-IEMens EQINOCS 2/11
11..55
11..00
00..55
00..00
00..55
11..00
11..5511 00 11 22 33 44 55 66 77
Learning Angluin’s Symbolic Conclusion
Machine learning in general
• given a sample M = {(x,y) | x ∈ X, y ∈ Y}
• find a representation f : X → Y such that f(x) = y
• predict or identify f(x) for all x ∈ X
1.5
1.0
0.5
0.0
0.5
1.0
1.51 0 1 2 3 4 5 6 7
VERIMAG-IEMens EQINOCS 2/11
11..55
11..00
00..55
00..00
00..55
11..00
11..5511 00 11 22 33 44 55 66 77
Learning Angluin’s Symbolic Conclusion
Machine learning in general
• given a sample M = {(x,y) | x ∈ X, y ∈ Y}
• find a representation f : X → Y such that f(x) = y
• predict or identify f(x) for all x ∈ X
1.5
1.0
0.5
0.0
0.5
1.0
1.51 0 1 2 3 4 5 6 7
VERIMAG-IEMens EQINOCS 2/11
11..55
11..00
00..55
00..00
00..55
11..00
11..5511 00 11 22 33 44 55 66 77
Learning Angluin’s Symbolic Conclusion
Machine learning in general
• given a sample M = {(x,y) | x ∈ X, y ∈ Y}
• find a representation f : X → Y such that f(x) = y
• predict or identify f(x) for all x ∈ X
1.5
1.0
0.5
0.0
0.5
1.0
1.51 0 1 2 3 4 5 6 7
VERIMAG-IEMens EQINOCS 2/11
11..55
11..00
00..55
00..00
00..55
11..00
11..5511 00 11 22 33 44 55 66 77
Learning Angluin’s Symbolic Conclusion
Machine learning in general
• given a sample M = {(x,y) | x ∈ X, y ∈ Y}
• find a representation f : X → Y such that f(x) = y
• predict or identify f(x) for all x ∈ X
1.5
1.0
0.5
0.0
0.5
1.0
1.51 0 1 2 3 4 5 6 7
VERIMAG-IEMens EQINOCS 2/11
Find a DFA A such that L=L(A)
2
3, 4
start q0 0, 1 2, 3, 4 0, 2,0 3, 4 q1 30, 4 q14 1, 2, 3, 4 q6
q3 0, 1
1 q5 2 q2
0, 1, 2, 3, 4
0, 1, 2, 3, 4
• Σ alphabet
• L ⊆ Σ∗ an unknown regular language (target language)
Learning Angluin’s Symbolic Conclusion
Learning regular languages
VERIMAG-IEMens EQINOCS 3/11
Find a DFA A such that L=L(A)
2
3, 4
start q0 0, 1 2, 3, 4 0, 2,0 3, 4 q1 30, 4 q14 1, 2, 3, 4 q6
q3 0, 1
1 q5 2 q2
0, 1, 2, 3, 4
0, 1, 2, 3, 4
Learning Angluin’s Symbolic Conclusion
Learning regular languages
• Σ alphabet
• L ⊆ Σ∗ an unknown regular language (target language)
VERIMAG-IEMens EQINOCS 3/11
Learning Angluin’s Symbolic Conclusion
Learning regular languages
• Σ alphabet
• L ⊆ Σ∗ an unknown regular language (target language)
Find a DFA A such that L=L(A)
2
3, 4
? start q0 0, 1 2, 3, 4 0, 2,0 3, 4 q1 30, 4 q14 1, 2, 3, 4 q6
q3 0, 1
1 q5 2 q2
0, 1, 2, 3, 4
0, 1, 2, 3, 4
VERIMAG-IEMens EQINOCS 3/11