SLIDE 1
Learning Symbolic Automata Symbolic Learning Conclusion
Learning Symbolic Automata Symbolic Learning Conclusion
Learning Regular Languages over Large Alphabets Oded Maler Irini - - PowerPoint PPT Presentation
Learning Regular Languages over Large Alphabets Oded Maler Irini - - PowerPoint PPT Presentation
Learning Symbolic Automata Symbolic Learning Conclusion Learning Regular Languages over Large Alphabets Oded Maler Irini Eleftheria Mens CNRS-VERIMAG University of Grenoble 2, avenue de Vignate 38610 Gieres France TACAS 2014 April 10,
SLIDE 2
SLIDE 3
Learning Symbolic Automata Symbolic Learning Conclusion
Outline
Learning Regular Languages Learning regular languages L* Algorithm Queries Observation tables Symbolic Automata Symbolic automata Symbolic languages Symbolic Learning Evidences Symbolic Algorithm Example
SLIDE 4
Learning Symbolic Automata Symbolic Learning Conclusion
Machine learning in general
- given M = {(x, y) | x ∈ X, y ∈ Y }
- find f : X → Y such that f(x) = y
- predict or identify f(x) for all x ∈ X
1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5
SLIDE 5
Learning Symbolic Automata Symbolic Learning Conclusion
Machine learning in general
- given M = {(x, y) | x ∈ X, y ∈ Y }
- find f : X → Y such that f(x) = y
- predict or identify f(x) for all x ∈ X
1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5
SLIDE 6
Learning Symbolic Automata Symbolic Learning Conclusion
Machine learning in general
- given M = {(x, y) | x ∈ X, y ∈ Y }
- find f : X → Y such that f(x) = y
- predict or identify f(x) for all x ∈ X
1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5
SLIDE 7
Learning Symbolic Automata Symbolic Learning Conclusion
Machine learning in general
- given M = {(x, y) | x ∈ X, y ∈ Y }
- find f : X → Y such that f(x) = y
- predict or identify f(x) for all x ∈ X
1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5
SLIDE 8
Learning Symbolic Automata Symbolic Learning Conclusion
Machine learning in general
- given M = {(x, y) | x ∈ X, y ∈ Y }
- find f : X → Y such that f(x) = y
- predict or identify f(x) for all x ∈ X
1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1 1 2 3 4 5 6 7 1.5 1.0 0.5 0.0 0.5 1.0 1.5
SLIDE 9
Learning Symbolic Automata Symbolic Learning Conclusion
Learning regular languages
start q0 q4 3, 4 q6 2 q3 0, 1 1, 2, 3, 4 q2 q5 0, 1, 2, 3, 4 3, 4 2 0, 1 0, 1, 2, 3, 4 1 q1 0, 2, 3, 4 2, 3, 4 1
SLIDE 10
Learning Symbolic Automata Symbolic Learning Conclusion
Learning regular languages
- Σ alphabet
- L ⊆ Σ∗ an unknown regular language (target language)
u ∈ Σ∗ − → {+, −}
start q0 q4 3, 4 q6 2 q3 0, 1 1, 2, 3, 4 q2 q5 0, 1, 2, 3, 4 3, 4 2 0, 1 0, 1, 2, 3, 4 1 q1 0, 2, 3, 4 2, 3, 4 1
SLIDE 11
Learning Symbolic Automata Symbolic Learning Conclusion
Learning regular languages
- Σ alphabet
- L ⊆ Σ∗ an unknown regular language (target language)
u ∈ Σ∗ − → {+, −}
start q0 q4 3, 4 q6 2 q3 0, 1 1, 2, 3, 4 q2 q5 0, 1, 2, 3, 4 3, 4 2 0, 1 0, 1, 2, 3, 4 1 q1 0, 2, 3, 4 2, 3, 4 1
Find automaton A, such that L = L(A)
?
SLIDE 12
Learning Symbolic Automata Symbolic Learning Conclusion
Learning regular languages
- Edward F Moore, 1956
Gedanken-experiments on sequential machines
- E. Mark Gold,1972
System Identification via State Characterization
- Dana Angluin, 1987
Learning regular sets from queries and counterexamples
SLIDE 13
Learning Symbolic Automata Symbolic Learning Conclusion
Learning regular languages
- Edward F Moore, 1956
Gedanken-experiments on sequential machines
- E. Mark Gold,1972
System Identification via State Characterization
- Dana Angluin, 1987
Learning regular sets from queries and counterexamples
- learning ω- languages [MP95]
- learning parameterized languages [BJR06]
- learning register automata [HSM11, HSJC12]
SLIDE 14
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
- Active learning using two types of queries
SLIDE 15
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
- Active learning using two types of queries
Membership Queries
MQ
w ∈ L ?
w ∈ Σ∗ yes / no
SLIDE 16
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
- Active learning using two types of queries
Membership Queries
MQ
w ∈ L ?
w ∈ Σ∗ yes / no
Equivalence Queries
EQ
L = L(h) ?
h yes / c-ex
SLIDE 17
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
- Active learning using two types of queries
Membership Queries
MQ
w ∈ L ?
w ∈ Σ∗ yes / no
Equivalence Queries
EQ
L = L(h) ?
h yes / c-ex
- Exactly learns the minimal DFA in polynomial time
- size of automaton (number of states)
- size of alphabet
- length of counterexample
SLIDE 18
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
+/−
- Initialize
SLIDE 19
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
+/−
b a c
- Initialize
- Ask MQs
SLIDE 20
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
+/− +/− +/− +/−
b a c
- Initialize
- Ask MQs
SLIDE 21
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
+/− +/− +/− +/−
a, b c b a
- Initialize
- Ask MQs
SLIDE 22
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
+/− +/−
a, b c
- Initialize
- Ask MQs
SLIDE 23
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
+/− +/−
a, b c a b c
- Initialize
- Ask MQs
SLIDE 24
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
+/− +/− +/− +/− +/−
a, b c a b c
- Initialize
- Ask MQs
SLIDE 25
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
+/− +/− +/− +/− +/−
a, b c a, c b a b c
- Initialize
- Ask MQs
SLIDE 26
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
+/− +/−
a, b c a, c b
- Initialize
- Ask MQs
- When automaton
is well defined Ask EQ
SLIDE 27
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
+/− +/− +/−
a, b c c a b
- Initialize
- Ask MQs
- When automaton
is well defined Ask EQ
- Use the
counterexample to fix the automaton
SLIDE 28
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Algorithm
+/− +/− +/−
. . .
a, b c c a b
- Initialize
- Ask MQs
- When automaton
is well defined Ask EQ
- Use the
counterexample to fix the automaton
- . . .
SLIDE 29
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
- Σ = {a, b}
SLIDE 30
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
- Σ = {a, b}
- S ⊆ Σ∗ prefixes
SLIDE 31
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
- Σ = {a, b}
- S ⊆ Σ∗ prefixes
- R = S · Σ \ S boundary
SLIDE 32
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
- Σ = {a, b}
- S ⊆ Σ∗ prefixes
- R = S · Σ \ S boundary
- E ⊆ Σ∗ suffixes
SLIDE 33
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
- Σ = {a, b}
- S ⊆ Σ∗ prefixes
- R = S · Σ \ S boundary
- E ⊆ Σ∗ suffixes
- f : S ∪ R × E → {+, −} classification
function
SLIDE 34
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
- Σ = {a, b}
- S ⊆ Σ∗ prefixes
- R = S · Σ \ S boundary
- E ⊆ Σ∗ suffixes
- f : S ∪ R × E → {+, −} classification
function
SLIDE 35
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
- Σ = {a, b}
- S ⊆ Σ∗ prefixes
- R = S · Σ \ S boundary
- E ⊆ Σ∗ suffixes
- f : S ∪ R × E → {+, −} classification
function
SLIDE 36
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
- Σ = {a, b}
- S ⊆ Σ∗ prefixes
- R = S · Σ \ S boundary
- E ⊆ Σ∗ suffixes
- f : S ∪ R × E → {+, −} classification
function
- fs : E → {+, −} for all s ∈ S ∪ R
fs(e) = f(s · e)
SLIDE 37
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
- Σ = {a, b}
- S ⊆ Σ∗ prefixes
- R = S · Σ \ S boundary
- E ⊆ Σ∗ suffixes
- f : S ∪ R × E → {+, −} classification
function
- fs : E → {+, −} for all s ∈ S ∪ R
fs(e) = f(s · e)
SLIDE 38
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
- Σ = {a, b}
- S ⊆ Σ∗ prefixes
- R = S · Σ \ S boundary
- E ⊆ Σ∗ suffixes
- f : S ∪ R × E → {+, −} classification
function
- fs : E → {+, −} for all s ∈ S ∪ R
fs(e) = f(s · e)
SLIDE 39
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
- Σ = {a, b}
- S ⊆ Σ∗ prefixes
- R = S · Σ \ S boundary
- E ⊆ Σ∗ suffixes
- f : S ∪ R × E → {+, −} classification
function
- fs : E → {+, −} for all s ∈ S ∪ R
fs(e) = f(s · e)
SLIDE 40
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
SLIDE 41
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
q0 q1 q2
SLIDE 42
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
q0 q1 q2
a b
SLIDE 43
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
q0 q1 q2
a b a, b
SLIDE 44
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
q0 q1 q2
a b a, b a, b
SLIDE 45
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
q0 q1 q2
a b a, b a, b
SLIDE 46
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
- S ∪ R is prefix-closed
q0 q1 q2
a b a, b a, b
SLIDE 47
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
- S ∪ R is prefix-closed
- E is suffix-closed
q0 q1 q2
a b a, b a, b
SLIDE 48
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab − −
- S ∪ R is prefix-closed
- E is suffix-closed
q0 q1 q2
a b a, b a, b
- T closed
∀r ∈ R, ∃s ∈ S, fr = fs
SLIDE 49
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − aa + + ab + + bb − − baa − − bab + −
- S ∪ R is prefix-closed
- E is suffix-closed
q0 q1 q2
a b a, b a, b
- T closed
∀r ∈ R, ∃s ∈ S, fr = fs
SLIDE 50
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − bab + − aa + + ab + + bb − − baa − − : : :
- S ∪ R is prefix-closed
- E is suffix-closed
q0 q1 q2 q3
a b a, b a, b b
- T closed
∀r ∈ R, ∃s ∈ S, fr = fs
SLIDE 51
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − bab + − aa + + ab + + bb − − baa − − : : :
- S ∪ R is prefix-closed
- E is suffix-closed
q0 q1 q2 q3
a b a, b a, b b
- T closed
∀r ∈ R, ∃s ∈ S, fr = fs
- T consistent
∀s, s′ ∈ S, ∀a ∈ Σ, fs = fs′ ⇒ fs·a = fs′·a
SLIDE 52
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − bab + − aa + + ab + + bb − − baa − − : : :
- S ∪ R is prefix-closed
- E is suffix-closed
q0 q1 q2 q3
a b a, b a, b b
- T closed
∀r ∈ R, ∃s ∈ S, fr = fs
- T consistent
∀s, s′ ∈ S, ∀a ∈ Σ, fs = fs′ ⇒ fs·a = fs′·a
SLIDE 53
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a ǫ − + a + + b − − ba − − bab + − aa + + ab + + bb − − baa − − : : :
- S ∪ R is prefix-closed
- E is suffix-closed
q0 q1 q2 q3
a b a, b a, b b
- T closed
∀r ∈ R, ∃s ∈ S, fr = fs
- T consistent
∀s, s′ ∈ S, ∀a ∈ Σ, fs = fs′ ⇒ fs·a = fs′·a
SLIDE 54
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a b ǫ − + a + + b − − ba − − bab + − aa + + ab + + bb − − baa − − : : :
- S ∪ R is prefix-closed
- E is suffix-closed
q0 q1 q2 q3
a b a, b a, b b
- T closed
∀r ∈ R, ∃s ∈ S, fr = fs
- T consistent
∀s, s′ ∈ S, ∀a ∈ Σ, fs = fs′ ⇒ fs·a = fs′·a
SLIDE 55
Learning Symbolic Automata Symbolic Learning Conclusion
Observation Table - T = (Σ, S, R, E, f)
ǫ a b ǫ − + : a + + : b − − − ba − − + bab + − : aa + + : ab + + : bb − − : baa − − : : : : :
- S ∪ R is prefix-closed
- E is suffix-closed
q0 q1 q2 q3 q4
? a b a, b b a a b
a, b
- T closed
∀r ∈ R, ∃s ∈ S, fr = fs
- T consistent
∀s, s′ ∈ S, ∀a ∈ Σ, fs = fs′ ⇒ fs·a = fs′·a
SLIDE 56
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
SLIDE 57
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a b
hypothesis automaton
q0
?
a, b
SLIDE 58
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a + b −
hypothesis automaton
q0
?
a, b
SLIDE 59
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a + b −
hypothesis automaton
q0
?
a, b
SLIDE 60
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a + b −
hypothesis automaton
q0 q1
? a b
a, b
SLIDE 61
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a + b − aa ab
hypothesis automaton
q0 q1
? a b
a, b
SLIDE 62
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a + b − aa − ab +
hypothesis automaton
q0 q1
? a b
a, b
SLIDE 63
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a + b − aa − ab +
hypothesis automaton
q0 q1
? a b
a, b
SLIDE 64
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a + b − aa − ab +
hypothesis automaton
q0 q1
a b b a
SLIDE 65
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a + b − aa − ab +
hypothesis automaton
q0 q1
a b b a
Ask Equivalence Query:
counterexample: −ba
SLIDE 66
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a + ba − b − aa − ab +
hypothesis automaton
q0 q1
a b b a
counterexample: −ba
SLIDE 67
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a + b − ba − aa − ab +
hypothesis automaton
q0 q1
a b b a
counterexample: −ba
SLIDE 68
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a + b − ba − aa − ab + bb baa bab
hypothesis automaton
q0 q1
a b b a
SLIDE 69
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a + b − ba − aa − ab + bb + baa − bab +
hypothesis automaton
q0 q1
a b b a
SLIDE 70
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a + b − ba − aa − ab + bb + baa − bab +
hypothesis automaton
q0 q1
a b b a
Table is not consistent!
SLIDE 71
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ ǫ − a + b − b a − aa − ab + bb + baa − bab +
hypothesis automaton
q0 q1
a b b a
Table is not consistent!
SLIDE 72
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ a ǫ − a + b − b a − aa − ab + bb + baa − bab +
hypothesis automaton
q0 q1
a b b a
Table is not consistent!
SLIDE 73
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ a ǫ − + a + − b − − b a − − aa − ab + bb + baa − bab +
hypothesis automaton
q0 q1 q2
a b b a b
SLIDE 74
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ a ǫ − + a + − b − − b a − − aa − + ab + − bb + − baa − − bab + −
hypothesis automaton
q0 q1 q2
a b b a b
SLIDE 75
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ a ǫ − + a + − b − − ba − − aa − + ab + − bb + − baa − − bab + −
hypothesis automaton
q0 q1 q2
a b b a b a
SLIDE 76
Learning Symbolic Automata Symbolic Learning Conclusion
L∗ Example (Σ = {a, b})
- bservation table
ǫ a ǫ − + a + − b − − ba − − aa − + ab + − bb + − baa − − bab + −
hypothesis automaton
q0 q1 q2
a b b a b a
Ask Equivalence Query:
True
SLIDE 77
Learning Symbolic Automata Symbolic Learning Conclusion
Outline
Learning Regular Languages Learning regular languages L* Algorithm Queries Observation tables Symbolic Automata Symbolic automata Symbolic languages Symbolic Learning Evidences Symbolic Algorithm Example
SLIDE 78
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ
Σ = [1, 100] ⊆ N
SLIDE 79
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
q0 q1 q2 q3 q4 [1] [1, 50], [71, 100] Σ [1, 50] [51, 100] [2, 72] [1, 20] [51, 70] [73, 100] [21, 100]
Σ = [1, 100] ⊆ N
SLIDE 80
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ
Σ = [1, 100] ⊆ N
SLIDE 81
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
q0 q1 q2 q3 q4 [1] [2, 72] [73, 100] [1, 50] [51, 100] [1, 20] [51, 70] [1, 50], [71, 100] [21, 100] Σ
Σ = [1, 100] ⊆ N
SLIDE 82
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ
Σ = [1, 100] ⊆ N
SLIDE 83
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ
Σ = [1, 100] ⊆ N
SLIDE 84
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ
a01 a02 a11 a12 a22 a32 a31, a33 a13 a21 a41
Σ = [1, 100] ⊆ N a01 = [1, 50] a02 = [51, 100] a11 = . . . . . .
SLIDE 85
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ
a01 a02 a11 a12 a22 a32 a31, a33 a13 a21 a41
Σ = [1, 100] ⊆ N a01 = [1, 50] a02 = [51, 100] a11 = . . . . . . Σ = {a01, a02, . . . }
SLIDE 86
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ
a01 a02 a11 a12 a22 a32 a31, a33 a13 a21 a41
Σ = [1, 100] ⊆ N a01 = [1, 50] a02 = [51, 100] a11 = . . . . . . Σ = {a01, a02, . . . } Σq0 = {a01, a02} Σq1 = {a11, a12, a13} . . .
SLIDE 87
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ
a01 a02 a11 a12 a22 a32 a31, a33 a13 a21 a41
Σ = [1, 100] ⊆ N a01 = [1, 50] a02 = [51, 100] a11 = . . . . . . Σ = {a01, a02, . . . } Σq0 = {a01, a02} Σq1 = {a11, a12, a13} . . .
- a∈Σq
a = Σ
SLIDE 88
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ
a01 a02 a11 a12 a22 a32 a31, a33 a13 a21 a41
Σ = [1, 100] ⊆ N a01 = [1, 50] a02 = [51, 100] a11 = . . . . . . Σ = {a01, a02, . . . } Σq0 = {a01, a02} Σq1 = {a11, a12, a13} . . .
- a∈Σq
a = Σ ψq : Σ → Σq, ∀q ∈ Q
SLIDE 89
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ
a01 a02 a11 a12 a22 a32 a31, a33 a13 a21 a41
Σ = [1, 100] ⊆ N a01 = [1, 50] a02 = [51, 100] a11 = . . . . . . Σ = {a01, a02, . . . } Σq0 = {a01, a02} Σq1 = {a11, a12, a13} . . .
- a∈Σq
a = Σ ψq : Σ → Σq, ∀q ∈ Q ψq1(10) = a12
SLIDE 90
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ
a01 a02 a11 a12 a22 a32 a31, a33 a13 a21 a41
Σ = [1, 100] ⊆ N a01 = [1, 50] a02 = [51, 100] a11 = . . . . . . Σ = {a01, a02, . . . } Σq0 = {a01, a02} Σq1 = {a11, a12, a13} . . .
- a∈Σq
a = Σ ψq : Σ → Σq, ∀q ∈ Q ψq1(10) = a12
δ : Q × Σ → Q
SLIDE 91
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ
a01 a02 a11 a12 a22 a32 a31, a33 a13 a21 a41
Σ = [1, 100] ⊆ N a01 = [1, 50] a02 = [51, 100] a11 = . . . . . . Σ = {a01, a02, . . . } Σq0 = {a01, a02} Σq1 = {a11, a12, a13} . . .
- a∈Σq
a = Σ ψq : Σ → Σq, ∀q ∈ Q ψq1(10) = a12
δ : Q × Σ → Q
δ(q, a) = q′ iff δ(q, a) = q′, a ∈ a
SLIDE 92
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
A = (Σ, Σ, ψ, Q, δ, q0, F)
SLIDE 93
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
A = (Σ, Σ, ψ, Q, δ, q0, F), where
- Σ is the input alphabet,
- Q is a finite set of states,
- q0 is the initial state,
- F is the set of accepting states,
SLIDE 94
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
A = (Σ, Σ, ψ, Q, δ, q0, F), where
- Σ is the input alphabet,
- Q is a finite set of states,
- q0 is the initial state,
- F is the set of accepting states,
- Σ is a finite alphabet, decomposable into Σ =
q∈Q Σq,
SLIDE 95
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
A = (Σ, Σ, ψ, Q, δ, q0, F), where
- Σ is the input alphabet,
- Q is a finite set of states,
- q0 is the initial state,
- F is the set of accepting states,
- Σ is a finite alphabet, decomposable into Σ =
q∈Q Σq,
- ψ = {ψq : q ∈ Q} is a family of total surjective functions
ψq : Σ → Σq,
SLIDE 96
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
A = (Σ, Σ, ψ, Q, δ, q0, F), where
- Σ is the input alphabet,
- Q is a finite set of states,
- q0 is the initial state,
- F is the set of accepting states,
- Σ is a finite alphabet, decomposable into Σ =
q∈Q Σq,
- ψ = {ψq : q ∈ Q} is a family of total surjective functions
ψq : Σ → Σq,
- δ : Q × Σ → Q is a partial transition function decomposable into
a family of total functions δq : {q} × Σq → Q,
SLIDE 97
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
A = (Σ, Σ, ψ, Q, δ, q0, F), where
- Σ is the input alphabet,
- Q is a finite set of states,
- q0 is the initial state,
- F is the set of accepting states,
- Σ is a finite alphabet, decomposable into Σ =
q∈Q Σq,
- ψ = {ψq : q ∈ Q} is a family of total surjective functions
ψq : Σ → Σq,
- δ : Q × Σ → Q is a partial transition function decomposable into
a family of total functions δq : {q} × Σq → Q,
w = a1a2 . . . an accepted δ∗(q0, a1a2 . . . an) = δ∗(δ(q0, ψq(a1)), a2 . . . an) ∈ F
SLIDE 98
Learning Symbolic Automata Symbolic Learning Conclusion
Deterministic Symbolic Automaton
A = (Σ, Σ, ψ, Q, δ, q0, F), where
- Σ is the input alphabet,
- Q is a finite set of states,
- q0 is the initial state,
- F is the set of accepting states,
- Σ is a finite alphabet, decomposable into Σ =
q∈Q Σq,
- ψ = {ψq : q ∈ Q} is a family of total surjective functions
ψq : Σ → Σq,
- δ : Q × Σ → Q is a partial transition function decomposable into
a family of total functions δq : {q} × Σq → Q,
L(A) = {w ∈ Σ∗|δ∗(q0, w) ∈ F}
w = a1a2 . . . an accepted δ∗(q0, a1a2 . . . an) = δ∗(δ(q0, ψq(a1)), a2 . . . an) ∈ F
SLIDE 99
Learning Symbolic Automata Symbolic Learning Conclusion
Outline
Learning Regular Languages Learning regular languages L* Algorithm Queries Observation tables Symbolic Automata Symbolic automata Symbolic languages Symbolic Learning Evidences Symbolic Algorithm Example
SLIDE 100
Learning Symbolic Automata Symbolic Learning Conclusion
Symbolic Algorithm
Evidences
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ
a01 a02 a11 a12 a22 a32 a31, a33 a13 a21 a41
SLIDE 101
Learning Symbolic Automata Symbolic Learning Conclusion
Symbolic Algorithm
Evidences
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ 1
a01
51
a02
1
a11
2
a12
1
a22
51
a32
1
a31,
71
a33
73
a13
21
a21
1
a41
µ : Σ → Σ
SLIDE 102
Learning Symbolic Automata Symbolic Learning Conclusion
Symbolic Algorithm
Evidences
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ 1
a01
51
a02
1
a11
2
a12
1
a22
51
a32
1
a31,
71
a33
73
a13
21
a21
1
a41
µ : Σ → Σ
Counterexample
minimal
- length
- lexicographically
SLIDE 103
Learning Symbolic Automata Symbolic Learning Conclusion
Symbolic Algorithm
Evidences
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ 1
a01
51
a02
1
a11
2
a12
1
a22
51
a32
1
a31,
71
a33
73
a13
21
a21
1
a41
µ : Σ → Σ µ(a) = min{a ∈ Σ | ψ(a) = a} µ(a01) = 1 µ(a02) = 51 . . . µ(a) ⊆ [a]
Counterexample
minimal
- length
- lexicographically
SLIDE 104
Learning Symbolic Automata Symbolic Learning Conclusion
Symbolic Algorithm
Evidences
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ 1
a01
51
a02
1
a11
2
a12
1
a22
51
a32
1
a31,
71
a33
73
a13
21
a21
1
a41
µ : Σ → Σ µ(a) = min{a ∈ Σ | ψ(a) = a} µ(a01) = 1 µ(a02) = 51 . . . µ(a) ⊆ [a]
Counterexample
minimal
- length
- lexicographically
- Discover new states
- Refine partitions
SLIDE 105
Learning Symbolic Automata Symbolic Learning Conclusion
Symbolic Algorithm
Evidences
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ 1
a01
51
a02
1
a11
2
a12
1
a22
51
a32
1
a31,
71
a33
73
a13
21
a21
1
a41
µ : Σ → Σ µ(a) = min{a ∈ Σ | ψ(a) = a} µ(a01) = 1 µ(a02) = 51 . . . µ(a) ⊆ [a]
Counterexample
minimal
- length
- lexicographically
- Discover new states
- Refine partitions
- Split a symbolic letter
- Move letters from one
symbolic letter to another
SLIDE 106
Learning Symbolic Automata Symbolic Learning Conclusion
Symbolic Algorithm
Evidences
q0 q1 q2 q3 q4 [1, 50] [51, 100] [1] [2, 72] [1, 20] [51, 70] [1, 50], [71, 100] [73, 100] [21, 100] Σ 1
a01
51
a02
1
a11
2
a12
1
a22
51
a32
1
a31,
71
a33
73
a13
21
a21
1
a41
µ : Σ → Σ µ(a) = min{a ∈ Σ | ψ(a) = a} µ(a01) = 1 µ(a02) = 51 . . . µ(a) ⊆ [a]
Counterexample
minimal
- length
- lexicographically
- Discover new states
- Refine partitions
- Split a symbolic letter
- Move letters from one
symbolic letter to another
All evidences of the same symbolic letter behave the same evidence compatibility
SLIDE 107
Learning Symbolic Automata Symbolic Learning Conclusion
Symbolic algorithm
Repeat
- Create symbolic letters for all states and set evidences
- Ask MQs to complete the observation table
- Find observation table that is closed, consistent and
evidence compatible
- Ask EQ
- Use the counterexample to update the table
- Discover new state (vertical expansion)
- Refine symbolic letters (horizontal expansion)
SLIDE 108
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ
ψ = {ψs}s∈S hypothesis automaton
SLIDE 109
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
ψ = {ψs}s∈S
ψǫ
hypothesis automaton
SLIDE 110
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0
ψ = {ψs}s∈S
ψǫ [a0] = [1, 100]
hypothesis automaton
SLIDE 111
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
ψ = {ψs}s∈S
ψǫ [a0] = [1, 100]
hypothesis automaton
SLIDE 112
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
ψ = {ψs}s∈S
ψǫ [a0] = [1, 100]
hypothesis automaton
q0 q1
Σ
SLIDE 113
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
ψ = {ψs}s∈S
ψǫ [a0] = [1, 100] ψa0
hypothesis automaton
q0 q1
Σ
SLIDE 114
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
1
a0
1
a1
ψ = {ψs}s∈S
ψǫ [a0] = [1, 100] ψa0 [a1] = [1, 100]
hypothesis automaton
q0 q1
Σ
SLIDE 115
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
1
a0
1
a1 −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 100] ψa0 [a1] = [1, 100]
hypothesis automaton
q0 q1
Σ Σ
Ask Equivalence Query:
SLIDE 116
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
1
a0
1
a1 −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 100] ψa0 [a1] = [1, 100]
hypothesis automaton
q0 q1
Σ Σ
Ask Equivalence Query:
counterexample − 24
SLIDE 117
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
1
a0
1
a1 −
24
a2
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 100]
hypothesis automaton
q0 q1
[1, 23] Σ
Ask Equivalence Query:
counterexample − 24
SLIDE 118
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
1
a0
1
a1 −
24
a2 −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 100]
hypothesis automaton
q0 q1
[1, 23] Σ
Ask Equivalence Query:
counterexample − 24
SLIDE 119
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
1
a0
1
a1 −
24
a2 −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 100]
hypothesis automaton
q0 q1
[24, 100] [1, 23] Σ
Ask Equivalence Query:
SLIDE 120
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
1
a0
1
a1 −
24
a2 −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 100]
hypothesis automaton
q0 q1
[24, 100] [1, 23] Σ
Ask Equivalence Query:
counterexample
+ 1 · 66
SLIDE 121
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
1
a0
1
a1 −
24
a2 −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 100]
hypothesis automaton
q0 q1
[24, 100] [1, 23] Σ
Ask Equivalence Query:
counterexample
+ 1 · 66
SLIDE 122
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
1
a0
1
a1 −
24
a2 −
1
a0
66
a3
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100]
hypothesis automaton
q0 q1
[24, 100] [1, 23] [1, 65]
Ask Equivalence Query:
counterexample
+ 1 · 66
SLIDE 123
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
1
a0
1
a1 −
24
a2 −
1
a0
66
a3 +
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100]
hypothesis automaton
q0 q1
[24, 100] [1, 23] [1, 65]
Ask Equivalence Query:
counterexample
+ 1 · 66
SLIDE 124
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
1
a0
1
a1 −
24
a2 −
1
a0
66
a3 +
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100]
hypothesis automaton
q0 q1
[24, 100] [1, 23] [66, 100] [1, 65]
Ask Equivalence Query:
SLIDE 125
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
1
a0
1
a1 −
24
a2 −
1
a0
66
a3 +
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100]
hypothesis automaton
q0 q1
[24, 100] [1, 23] [66, 100] [1, 65]
Ask Equivalence Query:
counterexample
− 24 · 1
SLIDE 126
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
1
a0
1
a1 −
24
a2 −
1
a0
66
a3 +
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100]
hypothesis automaton
q0 q1
[24, 100] [1, 23] [66, 100] [1, 65]
Ask Equivalence Query:
counterexample
− 24 · 1
SLIDE 127
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ ǫ −
1
a0 +
24
a2 −
1
a0
1
a1 −
1
a0
66
a3 +
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2
hypothesis automaton
q0 q1
[24, 100] [1, 23] [66, 100] [1, 65]
Ask Equivalence Query:
counterexample
− 24 · 1
SLIDE 128
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ 1 ǫ −
1
a0 +
24
a2 −
1
a0
1
a1 −
1
a0
66
a3 +
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2
hypothesis automaton
q0 q1
[24, 100] [1, 23] [66, 100] [1, 65]
Ask Equivalence Query:
counterexample
− 24 · 1
SLIDE 129
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ 1 ǫ − +
1
a0 + −
24
a2 − −
1
a0
1
a1 − +
1
a0
66
a3 + −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2
hypothesis automaton
q0 q1 q2
[1, 23] [66, 100] [1, 65] [24, 100]
Ask Equivalence Query:
counterexample
− 24 · 1
SLIDE 130
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ 1 ǫ − +
1
a0 + −
24
a2 − −
1
a0
1
a1 − +
1
a0
66
a3 + −
24
a2
1
a4
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2 [a4] = [1, 100]
hypothesis automaton
q0 q1 q2
[1, 23] [66, 100] [1, 65] [24, 100]
Ask Equivalence Query:
counterexample
− 24 · 1
SLIDE 131
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ 1 ǫ − +
1
a0 + −
24
a2 − −
1
a0
1
a1 − +
1
a0
66
a3 + −
24
a2
1
a4 − −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2 [a4] = [1, 100]
hypothesis automaton
q0 q1 q2
[1, 23] [66, 100] [1, 65] [24, 100]
Ask Equivalence Query:
counterexample
− 24 · 1
SLIDE 132
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ 1 ǫ − +
1
a0 + −
24
a2 − −
1
a0
1
a1 − +
1
a0
66
a3 + −
24
a2
1
a4 − −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2 [a4] = [1, 100]
hypothesis automaton
q0 q1 q2
[1, 23] Σ [66, 100] [1, 65] [24, 100]
Ask Equivalence Query:
SLIDE 133
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ 1 ǫ − +
1
a0 + −
24
a2 − −
1
a0
1
a1 − +
1
a0
66
a3 + −
24
a2
1
a4 − −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2 [a4] = [1, 100]
hypothesis automaton
q0 q1 q2
[1, 23] Σ [66, 100] [1, 65] [24, 100]
Ask Equivalence Query:
counterexample
+ 24 · 51
SLIDE 134
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ 1 ǫ − +
1
a0 + −
24
a2 − −
1
a0
1
a1 − +
1
a0
66
a3 + −
24
a2
1
a4 − −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2 [a4] = [1, 100]
hypothesis automaton
q0 q1 q2
[1, 23] Σ [66, 100] [1, 65] [24, 100]
Ask Equivalence Query:
counterexample
+ 24 · 51
SLIDE 135
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ 1 ǫ − +
1
a0 + −
24
a2 − −
1
a0
1
a1 − +
1
a0
66
a3 + −
24
a2
1
a4 − −
24
a2
51
a5
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2 [a4] = [1, 50] [a5] = [51, 100]
hypothesis automaton
q0 q1 q2
[1, 23] [1, 50] [66, 100] [1, 65] [24, 100]
Ask Equivalence Query:
counterexample
+ 24 · 51
SLIDE 136
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ 1 ǫ − +
1
a0 + −
24
a2 − −
1
a0
1
a1 − +
1
a0
66
a3 + −
24
a2
1
a4 − −
24
a2
51
a5 + −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2 [a4] = [1, 50] [a5] = [51, 100]
hypothesis automaton
q0 q1 q2
[1, 23] [1, 50] [66, 100] [1, 65] [24, 100]
Ask Equivalence Query:
counterexample
+ 24 · 51
SLIDE 137
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ 1 ǫ − +
1
a0 + −
24
a2 − −
1
a0
1
a1 − +
1
a0
66
a3 + −
24
a2
1
a4 − −
24
a2
51
a5 + −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2 [a4] = [1, 50] [a5] = [51, 100]
hypothesis automaton
q0 q1 q2
[1, 23] [1, 50] [51, 100] [66, 100] [1, 65] [24, 100]
Ask Equivalence Query:
counterexample
+ 24 · 51
SLIDE 138
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ 1 ǫ − +
1
a0 + −
24
a2 − −
1
a0
1
a1 − +
1
a0
66
a3 + −
24
a2
1
a4 − −
24
a2
51
a5 + −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2 [a4] = [1, 50] [a5] = [51, 100]
hypothesis automaton
q0 q1 q2
[1, 23] [1, 50] [51, 100] [66, 100] [1, 65] [24, 100]
Ask Equivalence Query:
counterexample
+ 24 · 51
SLIDE 139
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ 1 ǫ − +
1
a0 + −
24
a2 − −
1
a0
1
a1 − +
1
a0
66
a3 + −
24
a2
1
a4 − −
24
a2
51
a5 + −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2 [a4] = [1, 50] [a5] = [51, 100]
hypothesis automaton
q0 q1 q2
[1, 23] [1, 50] [51, 100] [66, 100] [1, 65] [24, 100]
Ask Equivalence Query: True
SLIDE 140
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ 1 ǫ − +
1
a0 + −
24
a2 − −
1
a0
1
a1 − +
1
a0
66
a3 + −
24
a2
1
a4 − −
24
a2
51
a5 + −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2 [a4] = [1, 50] [a5] = [51, 100]
hypothesis automaton
q0 q1 q2
[1, 23] [1, 50] [51, 100] [66, 100] [1, 65] [24, 100]
Ask Equivalence Query: True M = {ǫ, 1, 24, 1 1, 1 66, 24 1, 24 51, 1 1 1, 1 66 1, 24 1 1, 24 51 1} |M| = 11, |MQ| = 7, |EQ| = 5, |S| = 3, |R| = 4
SLIDE 141
Learning Symbolic Automata Symbolic Learning Conclusion
Example (Σ = [1, 100])
- bservation table
ǫ 1 ǫ − +
1
a0 + −
24
a2 − −
1
a0
1
a1 − +
1
a0
66
a3 + −
24
a2
1
a4 − −
24
a2
51
a5 + −
ψ = {ψs}s∈S
ψǫ [a0] = [1, 23] [a2] = [24, 100] ψa0 [a1] = [1, 65] [a3] = [66, 100] ψa2 [a4] = [1, 50] [a5] = [51, 100]
hypothesis automaton
q0 q1 q2
[1, 23] [1, 50] [51, 100] [66, 100] [1, 65] [24, 100]
Ask Equivalence Query: True M = {ǫ, 1, 24, 1 1, 1 66, 24 1, 24 51, 1 1 1, 1 66 1, 24 1 1, 24 51 1}
|M| = 790, |MQ| = 789, |EQ| = 2, |S| = 4, |R| = 396
|M| = 11, |MQ| = 7, |EQ| = 5, |S| = 3, |R| = 4
SLIDE 142
Learning Symbolic Automata Symbolic Learning Conclusion
Conclusion - Future work
Conclusion:
- learning languages over large or infinite alphabets
- finite number of convex intervals
- without using variables
- polynomial time in the number of states and maximum
number of intervals
Future Work:
- PAC-learning (Probably Approximately Correct Learning)
- the counterexample is not smallest
- learn without equivalence queries
- what if we do not have intervals?
- other types of alphabets
- . . .
SLIDE 143
Learning Symbolic Automata Symbolic Learning Conclusion
Conclusion - Future work
Conclusion:
- learning languages over large or infinite alphabets
- finite number of convex intervals
- without using variables
- polynomial time in the number of states and maximum
number of intervals
Future Work:
- PAC-learning (Probably Approximately Correct Learning)
- the counterexample is not smallest
- learn without equivalence queries
- what if we do not have intervals?
- other types of alphabets
- . . .
SLIDE 144
Learning Symbolic Automata Symbolic Learning Conclusion