Learning Regular Languages over Large Alphabets Oded Maler Irini - PowerPoint PPT Presentation

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) ǫ a • Σ = { a, b } ǫ − + a + + • S ⊆ Σ ∗ prefixes b − − − − • R = S · Σ \ S boundary ba + + aa • E ⊆ Σ ∗ suffixes ab + + − − bb • f : S ∪ R × E → { + , −} classification baa − − function − − bab

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) ǫ a • Σ = { a, b } ǫ − + a + + • S ⊆ Σ ∗ prefixes b − − − − • R = S · Σ \ S boundary ba + + aa • E ⊆ Σ ∗ suffixes ab + + − − bb • f : S ∪ R × E → { + , −} classification baa − − function − − bab • f s : E → { + , −} for all s ∈ S ∪ R f s ( e ) = f ( s · e )

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

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) ǫ a q 1 ǫ − + a + + b − − q 0 − − ba + + aa q 2 ab + + − − bb baa − − − − bab

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) ǫ a q 1 ǫ − + a a + + b − − q 0 − − ba b + + aa q 2 ab + + − − bb baa − − − − bab

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) ǫ a a, b q 1 ǫ − + a a + + b − − q 0 − − ba b + + aa q 2 ab + + − − bb baa − − − − bab

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) ǫ a a, b q 1 ǫ − + a a + + b − − q 0 − − ba b + + aa a, b q 2 ab + + − − bb baa − − − − bab

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) a, b q 1 ǫ a a ǫ − + q 0 b a + + q 2 a, b b − − − − ba + + aa ab + + − − bb baa − − − − bab

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) a, b q 1 ǫ a a ǫ − + q 0 • S ∪ R is prefix-closed b a + + q 2 a, b b − − − − ba + + aa ab + + − − bb baa − − − − bab

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) a, b q 1 ǫ a a ǫ − + q 0 • S ∪ R is prefix-closed b a + + q 2 a, b b − − • E is suffix-closed − − ba + + aa ab + + − − bb baa − − − − bab

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) a, b q 1 ǫ a a ǫ − + q 0 • S ∪ R is prefix-closed b a + + q 2 a, b b − − • E is suffix-closed − − ba + + aa ab + + • T closed − − bb ∀ r ∈ R , ∃ s ∈ S, f r = f s baa − − − − bab

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) a, b q 1 ǫ a a ǫ − + q 0 • S ∪ R is prefix-closed b a + + q 2 a, b b − − • E is suffix-closed − − ba + + aa ab + + • T closed − − bb ∀ r ∈ R , ∃ s ∈ S, f r = f s baa − − + − bab

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) a, b q 1 ǫ a a ǫ − + q 0 q 3 • S ∪ R is prefix-closed b b a + + q 2 a, b b − − • E is suffix-closed − − ba bab + − aa + + • T closed + + ab ∀ r ∈ R , ∃ s ∈ S, f r = f s bb − − − − baa : : :

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) a, b q 1 ǫ a a ǫ − + q 0 q 3 • S ∪ R is prefix-closed b b a + + q 2 a, b b − − • E is suffix-closed − − ba bab + − aa + + • T closed + + ab ∀ r ∈ R , ∃ s ∈ S, f r = f s bb − − • T consistent − − baa ∀ s, s ′ ∈ S, ∀ a ∈ Σ, f s = f s ′ ⇒ f s · a = f s ′ · a : : :

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) a, b q 1 ǫ a b a ǫ − + q 0 q 3 • S ∪ R is prefix-closed b b a + + q 2 a, b b − − • E is suffix-closed − − ba bab + − aa + + • T closed + + ab ∀ r ∈ R , ∃ s ∈ S, f r = f s bb − − • T consistent − − baa ∀ s, s ′ ∈ S, ∀ a ∈ Σ, f s = f s ′ ⇒ f s · a = f s ′ · a : : :

Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Σ , S, R, E, f ) a, b q 1 ǫ a b a a, b ǫ − + : q 0 q 4 ? • S ∪ R is prefix-closed a + + : b b q 2 q 3 b − − − a • E is suffix-closed − − + ba a b bab + − : aa + + : • T closed + + : ab ∀ r ∈ R , ∃ s ∈ S, f r = f s bb − − : • T consistent − − : baa ∀ s, s ′ ∈ S, ∀ a ∈ Σ, f s = f s ′ ⇒ f s · a = f s ′ · a : : : :

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } )

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a, b q 0 ? ǫ − a b

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a, b q 0 ? ǫ − a + − b

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a a, b q 0 q 1 ? ǫ − + a b − b

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a a, b q 0 q 1 ? ǫ − + a b − b aa ab

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a a, b q 0 q 1 ? ǫ − + a b − b aa − + ab

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a q 0 q 1 b ǫ − a + a b − b aa − + ab

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a q 0 q 1 b ǫ − a + a b Ask Equivalence Query: − b counterexample: − ba aa − + ab

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a q 0 q 1 b ǫ − a + a b ba − counterexample: − ba b − − aa ab +

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a q 0 q 1 b ǫ − a + a b b − − ba counterexample: − ba − aa ab +

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a q 0 q 1 b ǫ − a + a b b − − ba − aa ab + bb baa bab

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a q 0 q 1 b ǫ − a + a b b − − ba − aa ab + bb + baa − bab +

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a q 0 q 1 b ǫ − a + a b b − Table is not consistent! − ba − aa ab + bb + baa − bab +

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a q 0 q 1 b ǫ − a + a b b − Table is not consistent! − b a − aa ab + bb + baa − bab +

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a a q 0 q 1 b ǫ − a + a b b − Table is not consistent! − b a − aa ab + bb + baa − bab +

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a a b q 0 q 1 ǫ − + + − b a b a b − − q 2 − − b a − aa ab + bb + baa − bab +

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a a b q 0 q 1 ǫ − + + − b a b a b − − q 2 − − b a − + aa ab + − bb + − baa − − bab + −

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a a b q 0 q 1 ǫ − + a + − b a b a b − − q 2 − − ba − + aa ab + − bb + − baa − − bab + −

Learning Symbolic Automata Symbolic Learning Conclusion L ∗ Example (Σ = { a, b } ) observation table hypothesis automaton ǫ a a b q 0 q 1 ǫ − + a + − b a b a b − − q 2 − − ba Ask Equivalence Query: − + aa True ab + − bb + − baa − − bab + −

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

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Σ = [1 , 100] ⊆ N [1] [1 , 50] , [71 , 100] q 1 q 3 [73 , 100] [1 , 50] [51 , 70] q 0 [2 , 72] [51 , 100] [21 , 100] [1 , 20] q 2 q 4 Σ

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Σ = [1 , 100] ⊆ N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 [1 , 50] a 12 [51 , 70] a 32 q 0 [2 , 72] [51 , 100] a 02 a 21 a 22 [21 , 100] [1 , 20] q 2 q 4 a 41 Σ

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Σ = [1 , 100] ⊆ N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 Σ = { a 01 , a 02 , . . . } [1 , 50] a 12 [51 , 70] a 32 q 0 [2 , 72] [51 , 100] a 02 a 21 a 22 [21 , 100] [1 , 20] q 2 q 4 a 41 Σ

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Σ = [1 , 100] ⊆ N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 Σ = { a 01 , a 02 , . . . } [1 , 50] Σ q 0 = { a 01 , a 02 } a 12 [51 , 70] a 32 q 0 Σ q 1 = { a 11 , a 12 , a 13 } [2 , 72] . . . [51 , 100] a 02 a 21 a 22 [21 , 100] [1 , 20] q 2 q 4 a 41 Σ

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Σ = [1 , 100] ⊆ N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 Σ = { a 01 , a 02 , . . . } [1 , 50] Σ q 0 = { a 01 , a 02 } a 12 [51 , 70] a 32 q 0 Σ q 1 = { a 11 , a 12 , a 13 } [2 , 72] . . . [51 , 100] a 02 � a = Σ a 21 a 22 [21 , 100] [1 , 20] a ∈ Σ q q 2 q 4 a 41 Σ

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Σ = [1 , 100] ⊆ N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 Σ = { a 01 , a 02 , . . . } [1 , 50] Σ q 0 = { a 01 , a 02 } a 12 [51 , 70] a 32 q 0 Σ q 1 = { a 11 , a 12 , a 13 } [2 , 72] . . . [51 , 100] a 02 � a = Σ a 21 a 22 [21 , 100] [1 , 20] a ∈ Σ q q 2 q 4 ψ q : Σ → Σ q , ∀ q ∈ Q a 41 Σ

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Σ = [1 , 100] ⊆ N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 Σ = { a 01 , a 02 , . . . } [1 , 50] Σ q 0 = { a 01 , a 02 } a 12 [51 , 70] a 32 q 0 Σ q 1 = { a 11 , a 12 , a 13 } [2 , 72] . . . [51 , 100] a 02 � a = Σ a 21 a 22 [21 , 100] [1 , 20] a ∈ Σ q q 2 q 4 ψ q : Σ → Σ q , ∀ q ∈ Q a 41 Σ ψ q 1 (10) = a 12

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Σ = [1 , 100] ⊆ N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 Σ = { a 01 , a 02 , . . . } [1 , 50] Σ q 0 = { a 01 , a 02 } a 12 [51 , 70] a 32 q 0 Σ q 1 = { a 11 , a 12 , a 13 } [2 , 72] . . . [51 , 100] a 02 � a = Σ a 21 a 22 [21 , 100] [1 , 20] a ∈ Σ q q 2 q 4 ψ q : Σ → Σ q , ∀ q ∈ Q a 41 Σ ψ q 1 (10) = a 12 δ : Q × Σ → Q

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Σ = [1 , 100] ⊆ N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 Σ = { a 01 , a 02 , . . . } [1 , 50] Σ q 0 = { a 01 , a 02 } a 12 [51 , 70] a 32 q 0 Σ q 1 = { a 11 , a 12 , a 13 } [2 , 72] . . . [51 , 100] a 02 � a = Σ a 21 a 22 [21 , 100] [1 , 20] a ∈ Σ q q 2 q 4 ψ q : Σ → Σ q , ∀ q ∈ Q a 41 Σ ψ q 1 (10) = a 12 δ : Q × Σ → Q δ ( q, a ) = q ′ iff δ ( q, a ) = q ′ , a ∈ a

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton A = (Σ , Σ , ψ, Q, δ, q 0 , F )

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton A = (Σ , Σ , ψ, Q, δ, q 0 , F ), where • Σ is the input alphabet, • Q is a finite set of states, • q 0 is the initial state, • F is the set of accepting states,

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton A = (Σ , Σ , ψ, Q, δ, q 0 , F ), where • Σ is the input alphabet, • Q is a finite set of states, • q 0 is the initial state, • F is the set of accepting states, • Σ is a finite alphabet, decomposable into Σ = � q ∈ Q Σ q ,

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton A = (Σ , Σ , ψ, Q, δ, q 0 , F ), where • Σ is the input alphabet, • Q is a finite set of states, • q 0 is the initial state, • F is the set of accepting states, • Σ is a finite alphabet, decomposable into Σ = � q ∈ Q Σ q , • ψ = { ψ q : q ∈ Q } is a family of total surjective functions ψ q : Σ → Σ q ,

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton A = (Σ , Σ , ψ, Q, δ, q 0 , F ), where • Σ is the input alphabet, • Q is a finite set of states, • q 0 is the initial state, • F is the set of accepting states, • Σ is a finite alphabet, decomposable into Σ = � q ∈ Q Σ q , • ψ = { ψ q : q ∈ Q } is a family of total surjective functions ψ q : Σ → Σ q , • δ : Q × Σ → Q is a partial transition function decomposable into a family of total functions δ q : { q } × Σ q → Q ,

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton A = (Σ , Σ , ψ, Q, δ, q 0 , F ), where • Σ is the input alphabet, • Q is a finite set of states, • q 0 is the initial state, • F is the set of accepting states, • Σ is a finite alphabet, decomposable into Σ = � q ∈ Q Σ q , • ψ = { ψ q : q ∈ Q } is a family of total surjective functions ψ q : Σ → Σ q , • δ : Q × Σ → Q is a partial transition function decomposable into a family of total functions δ q : { q } × Σ q → Q , w = a 1 a 2 . . . a n accepted δ ∗ ( q 0 , a 1 a 2 . . . a n ) = δ ∗ ( δ ( q 0 , ψ q ( a 1 )) , a 2 . . . a n ) ∈ F

Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton A = (Σ , Σ , ψ, Q, δ, q 0 , F ), where • Σ is the input alphabet, • Q is a finite set of states, • q 0 is the initial state, • F is the set of accepting states, • Σ is a finite alphabet, decomposable into Σ = � q ∈ Q Σ q , • ψ = { ψ q : q ∈ Q } is a family of total surjective functions ψ q : Σ → Σ q , • δ : Q × Σ → Q is a partial transition function decomposable into a family of total functions δ q : { q } × Σ q → Q , L ( A ) = { w ∈ Σ ∗ | δ ∗ ( q 0 , w ) ∈ F } w = a 1 a 2 . . . a n accepted δ ∗ ( q 0 , a 1 a 2 . . . a n ) = δ ∗ ( δ ( q 0 , ψ q ( a 1 )) , a 2 . . . a n ) ∈ F

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

Learning Symbolic Automata Symbolic Learning Conclusion Symbolic Algorithm Evidences a 11 [1] [1 , 50] , [71 , 100] a 31 , a 33 q 1 q 3 [73 , 100] a 13 a 01 [1 , 50] a 12 [51 , 70] a 32 q 0 [2 , 72] [51 , 100] a 02 a 21 a 22 [21 , 100] [1 , 20] q 2 q 4 a 41 Σ

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,

Learning Learning Re Regular gular Languages Languages over er Lar Large ge Alphabets

Review Languages and Grammars Alphabets, strings, languages Regular Languages CS 301

Review Languages and Grammars Alphabets, strings, languages Regular Languages

Review Languages and Grammars Alphabets, strings, languages Regular Languages

Turing Machines Context-Free Languages n b n R a ww Regular Languages a * a * b * 1

Review Languages and Grammars Alphabets, strings, languages Regular Languages CS

Review Languages and Grammars Alphabets, strings, languages Regular Languages CS 301 -

Review Languages and Grammars CS 301 - Lecture 5 Alphabets, strings, languages Regular

Recursively Enumerable and Recursive Languages 1 L Definition: Let be a recursively

Decidability M Does Machine have three states ? Is string a binary number?

Learning Automata over Large Alphabets Oded Maler Irini Eleftheria Mens CNRS-V ERIMAG University

Computing Functions Deterministic with Turing Machines Infinite tape in both directions

Review Languages and Grammars Alphabets, strings, languages CS 301 - Lecture 6 Regular

= Languages Accepted by NPDAs Accept (Grammars) NPDAs Context-Free Languages 1 Proof - Step

More Applications for any string w L , | w | m of The Pumping Lemma w =

( ) ( ) if = M DFA 1 2 0 , 1 2 0 L ( M ) { 10 } * 2 = 1 q q q 1 2 0 1

( M ) { abba } = M Take DFA M a , b Definition: L M ( ) The language

Positive Properties are closed under: Union of Context-Free languages L is context free 1 L

Pushdown Automata Stack PDAs States 1 a , b c The States q 1 q 2 Pop Input Push symbol

The Pumping Lemma of different strings for Context-Free Languages Example: S AB A aBb

CS 301 Lecture 08 Regular languages recap Stephen Checkoway February 12, 2018 1 / 39

U i 0 1 2 3 4 L L L L L L L ... = = language and: i 0 =

MA/CSSE 474 Theory of Computation How many regular/non-regular languages are there? Closure

CFLs and Regular Languages We can show that every RL is also a CFL CFLs and Regular Languages