learning automata over large alphabets
play

Learning Automata over Large Alphabets Oded Maler Irini Eleftheria - PowerPoint PPT Presentation

Dec 11, 2023 •116 likes •1.2k views

Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Learning Automata over Large Alphabets Oded Maler Irini Eleftheria Mens CNRS-V ERIMAG University of Grenoble EQINOCS Workshop,

  1. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Laguages over Large Alphabets - Traditionally automata theory is flat, based on small alphabets, e.g. { a , b } - In verification, for example, we have sequences over a huge state-space like B n for very large n - Or we want to have languages over numbers or vectors - We use symbolic automata with a modest number of states - We do not want to enumerate all transitions but represent them symbolically using predicates on the alphabet - We will use inequalities (intervals) for numbers or Boolean functions for Boolean vectors V ERIMAG O Maler - IE Mens 10 / 22

  2. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Symbolic Automata a 11 [ 0 , 10 ) [ 0 , 50 ) , [ 70 , 100 ) a 31 , a 33 q 1 q 3 a 13 A = (Σ , Σ , ψ, Q , δ, q 0 , F ) [ 10 , 30 ) a 01 - Q finite set of states, [ 0 , 50 ) [ 30 , 100 ) [ 50 , 70 ) - q 0 initial state, a 12 a 32 q 0 - F accepting states, [ 50 , 100 ) a 02 - Σ large concrete alphabet, [ 20 , 100 ) - δ ⊆ Q × 2 Σ × Q a 21 a 22 [ 0 , 20 ) q 2 q 4 a 41 - Σ finite alphabet (symbols) Σ - ψ q : Σ → Σ q , q ∈ Q [ [ a 01 ] ] = [ 0 , 50 ) Σ = [ 0 , 100 ) ⊆ R A is complete and deterministic [ [ a ] ] = { a ∈ Σ | ψ ( a ) = a } if ∀ q ∈ Q { [ [ a ] ] | a ∈ Σ q } forms a partition of Σ w = 20 · 40 · 60 + V ERIMAG O Maler - IE Mens 11 / 22

  3. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Learning using Evidences and Representatives Let Σ be a subset of R - To characterize continuations of u , ask queries about u · a for a finite sample of Σ (evidence) u - Evidence can be a fixed set, random, or a result of binary search ? ? ? ? - Form evidence compatible partitions - All evidences within a partition block a 1 a 2 ... a 3 a k behave the same | Σ - Estimate boundaries using split , binary [ [ a ] ] µ ( a ) ˆ µ ( b ) ˆ [ [ b ] ] p search ,. . . evidences - Associate a symbol to each partition block µ ( u ) · a i | a i ∈ [ µ ( u · a ) = { ˆ [ a ] ] } - Choose one evidence as the representative for representatives each new symbol µ ( u · a ) = ˆ ˆ µ ( u ) · ˆ µ ( a ) V ERIMAG O Maler - IE Mens 12 / 22

  4. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Symbolic Observation Table - T = (Σ , Σ , S , R , ψ, E , f , µ, ˆ µ ) - Prefixes are symbolic words E - Symbols represent sets of letters (“fat” edges) ε a - Suffixes are concrete words (distinguish states) ε − + S a 1 + − - Fill in the table according to the representatives − − a 2 a 1 a 3 + − ε a 1 a 4 − + a 1 a 2 R a 1 a 5 − − a 1 a 2 a 2 a 6 + − a 3 a 5 a 6 a 4 a 1 a 3 a 1 a 4 a 1 a 5 a 2 a 6 V ERIMAG O Maler - IE Mens 13 / 22

  5. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Symbolic Observation Table - T = (Σ , Σ , S , R , ψ, E , f , µ, ˆ µ ) E - ψ = { ψ s } s ∈ S , ψ s : Σ → Σ s semantics ε a - [ [ a ] ] = { a ∈ Σ | ψ ( a ) = a } ε − + - µ : Σ → 2 Σ evidences S a 1 + − - µ ( ε ) = { ε } , µ ( s · a ) = ˆ µ ( s ) · µ ( a ) − − a 2 - ˆ µ : Σ → Σ representative a 1 a 3 + − - ˆ µ ( ε ) = ε, ˆ µ ( s · a ) = ˆ µ ( s ) · ˆ µ ( a ) a 1 a 4 − + R - f : ( S ∪ R ) · E → {− , + } classif. function a 1 a 5 − − - f ( s · e ) = f (ˆ µ ( s ) · e ) , f s ( e ) = f ( s · e ) a 2 a 6 + − V ERIMAG O Maler - IE Mens 13 / 22

  6. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Counter-example Treatment (Symbolic Breakpoint) Proposition If w is a counter-example to A T then there exists an i-factorization of w such that either f (ˆ µ ( s i − 1 · a i ) · v i ) � = f (ˆ µ ( s i ) · v i ) (1) or f (ˆ µ ( s i − 1 ) · a i · v i ) � = f (ˆ µ ( s i − 1 ) · ˆ µ ( a i ) · v i ) (2) • If (1), then v i is a new distinguishing word vertical expansion - Table not closed → new state • If (2), then a i is a new evidence for a i . horizontal expansion - Evidence incompatibility → new transition / refinement V ERIMAG O Maler - IE Mens 14 / 22

  7. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Counter-example Treatment (Symbolic Breakpoint) Let w = a 1 · · · · a i · · · a | w | = u i · a i · v i ) be a counter-example. f (ˆ µ ( s i − 1 · a i ) · v i ) � = f (ˆ µ ( s i ) · v i ) s i = δ ( ε , u i · a i ) ε µ ( u i ) ˆ s s ′ µ ( a i ) ˆ v i v i � = V ERIMAG O Maler - IE Mens 15 / 22

  8. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Counter-example Treatment (Symbolic Breakpoint) Let w = a 1 · · · · a i · · · a | w | = u i · a i · v i ) be a counter-example. f (ˆ µ ( s i − 1 · a i ) · v i ) � = f (ˆ µ ( s i ) · v i ) s i = δ ( ε , u i · a i ) s · a i is a ε new state µ ( u i ) ˆ s s ′ ε ˆ µ ( a i ) ˆ µ ( u i ) v i s s ′ v i � = µ ( a i ) ˆ � = v i new v i � = V ERIMAG O Maler - IE Mens 15 / 22

  9. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Counter-example Treatment (Symbolic Breakpoint) Let w = a 1 · · · · a i · · · a | w | = u i · a i · v i ) be a counter-example. f (ˆ µ ( s i − 1 · a i ) · v i ) � = f (ˆ µ ( s i ) · v i ) f (ˆ µ ( s i − 1 ) · a i · v i ) � = f (ˆ µ ( s i − 1 ) · ˆ µ ( a i ) · v i ) s i = δ ( ε , u i · a i ) s · a i is a ε ε new state µ ( u i ) ˆ ˆ µ ( u i ) s s ′ s ε µ ( a i ) ˆ a i µ ( a i ) ˆ ˆ µ ( u i ) v i � = s s ′ v i v i v i � = µ ( a i ) ˆ � = v i � = new v i � = V ERIMAG O Maler - IE Mens 15 / 22

  10. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Counter-example Treatment (Symbolic Breakpoint) Let w = a 1 · · · · a i · · · a | w | = u i · a i · v i ) be a counter-example. f (ˆ µ ( s i − 1 · a i ) · v i ) � = f (ˆ µ ( s i ) · v i ) f (ˆ µ ( s i − 1 ) · a i · v i ) � = f (ˆ µ ( s i − 1 ) · ˆ µ ( a i ) · v i ) s i = δ ( ε , u i · a i ) s · a i is a ε ε new state refine [ [ a i ] ] µ ( u i ) ˆ µ ( u i ) ˆ s s ′ s ε µ ( a i ) ˆ a i ε µ ( a i ) ˆ µ ( u i ) ˆ v i � = µ ( u i ) ˆ s s ′ v i v i s v i ˆ µ ( a i ) a i � = µ ( a i ) ˆ � = v i � = new v i v i v i � = � = V ERIMAG O Maler - IE Mens 15 / 22

  11. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Learning with a Teacher p a i a i + 1 . . . . . . | Σ error in the partitions - Equivalence is checked by an oracle (teacher) returning a minimal counter-examples (in length and lexicographically) - Choose as evidence the min element of the interval ( Σ has min) - The counter-example indicates the minimal element of a new transition (in horizontal expansion) - The partition bounds are exact and no error is introduced p [ [ a ] ] [ [ b ] ] | Σ µ ( a ) ˆ µ ( b ) ˆ V ERIMAG O Maler - IE Mens 16 / 22

  12. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton ε ε V ERIMAG O Maler - IE Mens 17 / 22

  13. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton ε ε ε − V ERIMAG O Maler - IE Mens 17 / 22

  14. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton ε ε ε − [ [ a 0 ] ] = [ 1 , 100 ) 1 a 0 V ERIMAG O Maler - IE Mens 17 / 22

  15. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton ε ε ε − [ [ a 0 ] ] = [ 1 , 100 ) 1 + a 0 V ERIMAG O Maler - IE Mens 17 / 22

  16. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton ε ε ε − [ [ a 0 ] ] = [ 1 , 100 ) 1 + a 0 V ERIMAG O Maler - IE Mens 17 / 22

  17. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton Σ ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 100 ) 1 a 0 + a 0 V ERIMAG O Maler - IE Mens 17 / 22

  18. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton Σ ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 100 ) 1 a 0 + a 0 [ [ a 1 ] ] = [ 1 , 100 ) 1 1 a 0 a 1 V ERIMAG O Maler - IE Mens 17 / 22

  19. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton Σ ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 100 ) Σ 1 a 0 + a 0 [ [ a 1 ] ] = [ 1 , 100 ) 1 1 − a 0 a 1 Ask Equivalence Query: V ERIMAG O Maler - IE Mens 17 / 22

  20. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton Σ ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 100 ) Σ 1 a 0 + a 0 [ [ a 1 ] ] = [ 1 , 100 ) 1 1 − a 0 a 1 Ask Equivalence Query: counterexample − 24 24 ∈ [ [ a 0 ] ] but 1 �∼ 24 − → refine a 0 V ERIMAG O Maler - IE Mens 17 / 22

  21. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) Σ [ [ a 2 ] ] = [ 24 , 100 ) 1 + a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 100 ) − a 0 a 1 24 a 2 Ask Equivalence Query: counterexample − 24 24 ∈ [ [ a 0 ] ] but 1 �∼ 24 − → refine a 0 V ERIMAG O Maler - IE Mens 17 / 22

  22. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) Σ [ [ a 2 ] ] = [ 24 , 100 ) 1 + a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 100 ) − a 0 a 1 24 − a 2 Ask Equivalence Query: counterexample − 24 24 ∈ [ [ a 0 ] ] but 1 �∼ 24 − → refine a 0 V ERIMAG O Maler - IE Mens 17 / 22

  23. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) Σ [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 100 ) − a 0 a 1 24 − a 2 Ask Equivalence Query: V ERIMAG O Maler - IE Mens 17 / 22

  24. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) Σ [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 100 ) − a 0 a 1 24 − a 2 Ask Equivalence Query: counterexample + 1 · 66 66 ∈ [ [ a 1 ] ] but 1 �∼ 66 − → refine a 1 V ERIMAG O Maler - IE Mens 17 / 22

  25. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) Σ [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 100 ) − a 0 a 1 24 − a 2 Ask Equivalence Query: counterexample + 1 · 66 66 ∈ [ [ a 1 ] ] but 1 �∼ 66 − → refine a 1 V ERIMAG O Maler - IE Mens 17 / 22

  26. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 66 ) − a 0 a 1 [ [ a 3 ] ] = [ 66 , 100 ) 24 − a 2 Ask Equivalence Query: 1 66 counterexample + 1 · 66 a 0 a 3 66 ∈ [ [ a 1 ] ] but 1 �∼ 66 − → refine a 1 V ERIMAG O Maler - IE Mens 17 / 22

  27. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 66 ) − a 0 a 1 [ [ a 3 ] ] = [ 66 , 100 ) 24 − a 2 Ask Equivalence Query: 1 66 counterexample + 1 · 66 + a 0 a 3 66 ∈ [ [ a 1 ] ] but 1 �∼ 66 − → refine a 1 V ERIMAG O Maler - IE Mens 17 / 22

  28. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 66 ) − a 0 a 1 [ [ a 3 ] ] = [ 66 , 100 ) 24 − a 2 Ask Equivalence Query: 1 66 + a 0 a 3 V ERIMAG O Maler - IE Mens 17 / 22

  29. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 66 ) − a 0 a 1 [ [ a 3 ] ] = [ 66 , 100 ) 24 − a 2 Ask Equivalence Query: 1 66 − 24 · 1 counterexample + a 0 a 3 1 �∼ 24 · 1 − → add distinguishing suffix 1 V ERIMAG O Maler - IE Mens 17 / 22

  30. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 66 ) − a 0 a 1 [ [ a 3 ] ] = [ 66 , 100 ) 24 − a 2 Ask Equivalence Query: 1 66 − 24 · 1 counterexample + a 0 a 3 1 �∼ 24 · 1 − → add distinguishing suffix 1 V ERIMAG O Maler - IE Mens 17 / 22

  31. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − [ 24 , 100 ) a 0 a 0 1 1 [ [ a 1 ] ] = [ 1 , 66 ) − + a 0 a 1 [ [ a 3 ] ] = [ 66 , 100 ) 24 − − a 2 Ask Equivalence Query: 1 66 − 24 · 1 counterexample + − a 0 a 3 1 �∼ 24 · 1 − → add distinguishing suffix 1 V ERIMAG O Maler - IE Mens 17 / 22

  32. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − a 0 a 3 − 24 · 1 counterexample 1 �∼ 24 · 1 − → add distinguishing suffix 1 V ERIMAG O Maler - IE Mens 17 / 22

  33. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 100 ) a 0 a 3 − 24 · 1 counterexample 24 1 a 2 a 4 1 �∼ 24 · 1 − → add distinguishing suffix 1 V ERIMAG O Maler - IE Mens 17 / 22

  34. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 100 ) a 0 a 3 − 24 · 1 counterexample 24 1 − − a 2 a 4 1 �∼ 24 · 1 − → add distinguishing suffix 1 V ERIMAG O Maler - IE Mens 17 / 22

  35. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 100 ) a 0 a 3 − 24 · 1 counterexample 24 1 − − a 2 a 4 1 �∼ 24 · 1 − → add distinguishing suffix 1 V ERIMAG O Maler - IE Mens 17 / 22

  36. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 Σ [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 100 ) a 0 a 3 24 1 − − a 2 a 4 V ERIMAG O Maler - IE Mens 17 / 22

  37. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 Σ [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 100 ) a 0 a 3 + 24 · 51 counterexample 24 1 − − a 2 a 4 V ERIMAG O Maler - IE Mens 17 / 22

  38. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 Σ [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 100 ) a 0 a 3 + 24 · 51 counterexample 24 1 − − a 2 a 4 51 ∈ [ [ a 4 ] ] but 1 �∼ 51 − → refine a 4 V ERIMAG O Maler - IE Mens 17 / 22

  39. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ 1 , 51 ) [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 51 ) a 0 a 3 + 24 · 51 counterexample [ [ a 5 ] ] = [ 51 , 100 ) 24 1 − − a 2 a 4 51 ∈ [ [ a 4 ] ] but 1 �∼ 51 24 51 a 2 a 5 − → refine a 4 V ERIMAG O Maler - IE Mens 17 / 22

  40. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ 1 , 51 ) [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 51 ) a 0 a 3 + 24 · 51 counterexample [ [ a 5 ] ] = [ 51 , 100 ) 24 1 − − a 2 a 4 51 ∈ [ [ a 4 ] ] but 1 �∼ 51 24 51 + − a 2 a 5 − → refine a 4 V ERIMAG O Maler - IE Mens 17 / 22

  41. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − [ 51 , 100 ) a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ 1 , 51 ) [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 51 ) a 0 a 3 + 24 · 51 counterexample [ [ a 5 ] ] = [ 51 , 100 ) 24 1 − − a 2 a 4 51 ∈ [ [ a 4 ] ] but 1 �∼ 51 24 51 + − a 2 a 5 − → refine a 4 V ERIMAG O Maler - IE Mens 17 / 22

  42. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − [ 51 , 100 ) a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ 1 , 51 ) [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 51 ) a 0 a 3 + 24 · 51 counterexample [ [ a 5 ] ] = [ 51 , 100 ) 24 1 − − a 2 a 4 51 ∈ [ [ a 4 ] ] but 1 �∼ 51 24 51 + − a 2 a 5 − → refine a 4 V ERIMAG O Maler - IE Mens 17 / 22

  43. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − [ 51 , 100 ) a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ 1 , 51 ) [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 51 ) a 0 a 3 True [ [ a 5 ] ] = [ 51 , 100 ) 24 1 − − a 2 a 4 24 51 + − a 2 a 5 V ERIMAG O Maler - IE Mens 17 / 22

  44. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − [ 51 , 100 ) a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ 1 , 51 ) [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 51 ) a 0 a 3 True [ [ a 5 ] ] = [ 51 , 100 ) 24 1 − − a 2 a 4 24 51 + − a 2 a 5 M = { ε, 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 V ERIMAG O Maler - IE Mens 17 / 22

  45. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example with Teacher ( Σ = [ 1 , 100 ) ) Teacher returns minimal counterexamples observation table semantics hypothesis automaton [ 1 , 24 ) [ 66 , 100 ) ε 1 ε ε a 0 ε − + [ [ a 0 ] ] = [ 1 , 24 ) [ 1 , 66 ) [ [ a 2 ] ] = [ 24 , 100 ) 1 + − [ 51 , 100 ) a 0 a 0 24 − − [ 24 , 100 ) a 2 [ [ a 1 ] ] = [ 1 , 66 ) a 2 [ 1 , 51 ) [ [ a 3 ] ] = [ 66 , 100 ) 1 1 − + a 0 a 1 a 2 Ask Equivalence Query: 1 66 + − [ [ a 4 ] ] = [ 1 , 51 ) a 0 a 3 True [ [ a 5 ] ] = [ 51 , 100 ) 24 1 − − a 2 a 4 24 51 + − a 2 a 5 M = { ε, 1 , 24 , 1 1 , 1 66 , 24 1 , 24 51 , 1 1 1 , 1 66 1 , 24 1 1 , 24 51 1 } L ∗ over (Σ ∩ N ) → | M | = 790 , | MQ | = 789 , | EQ | = 2 , | S | = 4 , | R | = 396 | M | = 11 , | MQ | = 7 , | EQ | = 5 , | S | = 3 , | R | = 4 V ERIMAG O Maler - IE Mens 17 / 22

  46. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Learning without a Teacher p a i a i + 1 . . . . . . | Σ error in the partitions - Equivalence is checked by testing random words selected using a probability distribution D - Counter-examples are not minimal we may have errors in the boundaries - Counter-examples may be missed terminate algorithm and return hypothesis after r ( ε, δ, i ) random words have been tested, none of which is a counter-example - The final hypothesis A is a good approximation of the target language L with high probability P ( d ( L , L A ) < ε ) ≥ 1 − δ (PAC learnability) V ERIMAG O Maler - IE Mens 18 / 22

  47. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton V ERIMAG O Maler - IE Mens 19 / 22

  48. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε V ERIMAG O Maler - IE Mens 19 / 22

  49. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε ε − V ERIMAG O Maler - IE Mens 19 / 22

  50. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε ε − 13 42 68 78 92 ˆ µ ( a 1 ) µ ( a 2 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22

  51. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε 27 ε − 13 42 68 78 92 | ˆ µ ( a 1 ) µ ( a 2 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22

  52. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε 27 ε − 13 42 68 78 92 | 13 µ ( a 1 ) ˆ µ ( a 2 ) ˆ + a 1 68 − a 2 V ERIMAG O Maler - IE Mens 19 / 22

  53. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε 27 ε − 13 42 68 78 92 | 13 µ ( a 1 ) ˆ µ ( a 2 ) ˆ + a 1 68 − a 2 V ERIMAG O Maler - IE Mens 19 / 22

  54. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε 27 ε − 13 42 68 78 92 | 13 + ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 a 1 68 − a 2 V ERIMAG O Maler - IE Mens 19 / 22

  55. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε 27 ε − 13 42 68 78 92 | 13 + ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 a 1 68 − a 2 2 18 26 46 54 µ ( a 3 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22

  56. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton ε ε 27 ε − 13 42 68 78 92 | 13 + µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 a 1 68 − a 2 2 18 26 46 54 13 18 − a 1 a 3 µ ( a 3 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22

  57. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε ε ε a 1 27 ε − 13 42 68 78 92 Σ | 13 x ≥ 27 + ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 a 1 68 − a 2 2 18 26 46 54 13 18 − a 1 a 3 µ ( a 3 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22

  58. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε ε a 1 ε 27 ε − 13 42 68 78 92 Σ | 13 x ≥ 27 + ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 a 1 68 − a 2 2 18 26 46 54 13 18 − a 1 a 3 Ask Equivalence Query: µ ( a 3 ) ˆ counterexample − 12 · 73 · 11 add distinguishing string 11 − → new state V ERIMAG O Maler - IE Mens 19 / 22

  59. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε a 1 ε 27 ε − 13 42 68 78 92 Σ | 13 x ≥ 27 + ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 a 1 68 − a 2 2 18 26 46 54 13 18 − a 1 a 3 Ask Equivalence Query: µ ( a 3 ) ˆ counterexample − 12 · 73 · 11 add distinguishing string 11 − → new state V ERIMAG O Maler - IE Mens 19 / 22

  60. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε a 1 ε 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 27 + − ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 a 1 68 − − a 2 2 18 26 46 54 13 18 − + a 1 a 3 Ask Equivalence Query: µ ( a 3 ) ˆ counterexample − 12 · 73 · 11 add distinguishing string 11 − → new state V ERIMAG O Maler - IE Mens 19 / 22

  61. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε a 1 ε 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 27 + − µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 − − a 1 a 2 2 18 26 46 54 13 18 − + a 1 a 3 Ask Equivalence Query: µ ( a 3 ) ˆ counterexample − 12 · 73 · 11 a 2 add distinguishing string 11 − → new state V ERIMAG O Maler - IE Mens 19 / 22

  62. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 27 + − µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 − − a 1 a 2 2 18 26 46 54 13 18 − + a 1 a 3 µ ( a 3 ) ˆ a 2 17 27 64 72 94 µ ( a 4 ) ˆ µ ( a 5 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22

  63. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 27 + − ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 68 − − a 1 a 2 2 18 26 46 54 13 18 − + a 1 a 3 µ ( a 3 ) ˆ a 2 45 17 27 64 72 94 | µ ( a 4 ) ˆ µ ( a 5 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22

  64. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 27 + − µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 − − a 1 a 2 2 18 26 46 54 13 18 − + a 1 a 3 µ ( a 3 ) ˆ 68 17 a 2 a 4 a 2 68 72 a 2 a 5 45 17 27 64 72 94 | µ ( a 4 ) ˆ µ ( a 5 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22

  65. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 27 + − ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 68 − − a 1 a 2 2 18 26 46 54 13 18 − + a 1 a 3 µ ( a 3 ) ˆ 68 17 − − a 2 a 4 a 2 68 72 + − a 2 a 5 45 17 27 64 72 94 | µ ( a 4 ) ˆ µ ( a 5 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22

  66. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 27 + − ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 68 − − a 1 a 2 2 18 26 46 54 13 18 − + a 1 a 3 µ ( a 3 ) ˆ 68 17 − − a 2 a 4 a 2 68 72 + − a 2 a 5 45 17 27 64 72 94 | µ ( a 4 ) ˆ µ ( a 5 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22

  67. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 + − x ≥ 45 µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 2 18 26 46 54 13 18 − + a 1 a 3 µ ( a 3 ) ˆ 68 17 − − a 2 a 4 a 2 68 72 + − a 2 a 5 45 17 27 64 72 94 | ˆ µ ( a 4 ) µ ( a 5 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22

  68. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 45 + − ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 2 18 26 46 54 13 18 − + a 1 a 3 µ ( a 3 ) ˆ Ask Equivalence Query: 68 17 − − a 2 a 4 counterexample − 12 · 73 · 11 a 2 68 72 + − 45 a 2 a 5 17 27 64 72 94 add 73 as evidence of a 1 | ˆ µ ( a 4 ) µ ( a 5 ) ˆ − → new transition V ERIMAG O Maler - IE Mens 19 / 22

  69. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 45 + − µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 2 18 26 46 54 73 13 18 − + a 1 a 3 µ ( a 3 ) ˆ ˆ µ ( a 6 ) Ask Equivalence Query: 68 17 − − a 2 a 4 counterexample − 12 · 73 · 11 a 2 68 72 + − 45 a 2 a 5 17 27 64 72 94 add 73 as evidence of a 1 | µ ( a 4 ) ˆ µ ( a 5 ) ˆ − → new transition V ERIMAG O Maler - IE Mens 19 / 22

  70. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 45 + − µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 63 2 18 26 46 54 73 13 18 − + | a 1 a 3 µ ( a 3 ) ˆ ˆ µ ( a 6 ) Ask Equivalence Query: 68 17 − − a 2 a 4 counterexample − 12 · 73 · 11 a 2 68 72 + − 45 a 2 a 5 17 27 64 72 94 add 73 as evidence of a 1 | µ ( a 4 ) ˆ µ ( a 5 ) ˆ − → new transition V ERIMAG O Maler - IE Mens 19 / 22

  71. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x < 27 ε 11 ε ε a 1 27 ε − + 13 42 68 78 92 Σ | 13 x ≥ 45 + − µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 63 2 18 26 46 54 73 13 18 − + | a 1 a 3 µ ( a 3 ) ˆ ˆ µ ( a 6 ) Ask Equivalence Query: 13 73 + − a 1 a 6 counterexample − 12 · 73 · 11 a 2 68 17 − − 45 a 2 a 4 17 27 64 72 94 add 73 as evidence of a 1 | 68 72 + − a 2 a 5 µ ( a 4 ) ˆ µ ( a 5 ) ˆ − → new transition V ERIMAG O Maler - IE Mens 19 / 22

  72. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x ≥ 63 x < 27 ε 11 ε ε a 1 27 ε − + 68 78 92 13 42 | x < 63 13 + − x ≥ 45 µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 63 2 18 26 46 54 73 13 18 | − + a 1 a 3 µ ( a 3 ) ˆ µ ( a 6 ) ˆ 13 73 + − a 1 a 6 a 2 68 17 − − 45 a 2 a 4 17 27 64 72 94 | 68 72 + − a 2 a 5 ˆ µ ( a 4 ) µ ( a 5 ) ˆ V ERIMAG O Maler - IE Mens 19 / 22

  73. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x ≥ 63 x < 27 ε 11 ε a 1 ε 27 ε − + 13 42 68 78 92 | x < 63 13 + − x ≥ 45 ˆ µ ( a 1 ) µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 63 2 18 26 46 54 73 13 18 − + | a 1 a 3 µ ( a 3 ) ˆ ˆ µ ( a 6 ) Ask Equivalence Query: 13 73 + − a 1 a 6 − 52 · 47 counterexample a 2 68 17 − − a 2 a 4 45 17 27 64 72 94 add 47 as evidence of a 2 | 68 72 + − a 2 a 5 ˆ µ ( a 4 ) µ ( a 5 ) ˆ − → refine existing transition V ERIMAG O Maler - IE Mens 19 / 22

  74. Learning Languages The L* Algorithm Learning over Large Alphabets Learning with/without a Teacher Conclusions Example without Teacher ( Σ = [ 1 , 100 ) ) Counterexamples are not minimal observation table semantics hypothesis automaton x ≥ 63 x < 27 ε 11 ε a 1 ε 27 ε − + 13 42 68 78 92 | x < 63 13 + − x ≥ 45 µ ( a 1 ) ˆ µ ( a 2 ) ˆ a 1 68 x ≥ 27 − − a 1 a 2 a 2 x < 45 63 2 18 26 46 54 73 13 18 − + | a 1 a 3 µ ( a 3 ) ˆ ˆ µ ( a 6 ) Ask Equivalence Query: 13 73 + − a 1 a 6 − 52 · 47 counterexample a 2 68 17 − − a 2 a 4 45 17 27 47 64 72 94 add 47 as evidence of a 2 | 68 72 + − a 2 a 5 µ ( a 4 ) ˆ µ ( a 5 ) ˆ − → refine existing transition V ERIMAG O Maler - IE Mens 19 / 22

Recommend

Learning Regular Languages over Large Alphabets Oded Maler Irini Eleftheria Mens CNRS-VERIMAG

Learning Regular Languages over Large Alphabets Oded Maler Irini Eleftheria Mens CNRS-VERIMAG

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,

1.62k views • 144 slides
Learning Learning Re Regular gular Languages Languages over er Lar Large ge Alphabets

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

Irini-Eleftheria Irini-Eleftheria Mens Mens V ERIMAG , University of Grenoble-Alpes Learning Learning Re Regular gular Languages Languages over er Lar Large ge Alphabets Alphabets 10 October 2017 Jury Members Oded Maler Directeur de

896 views • 68 slides
Logical Foundations of Data Science, UCL, Nov 2015 Supported by a Royal Academy of Engineering

Logical Foundations of Data Science, UCL, Nov 2015 Supported by a Royal Academy of Engineering

Automata over infjnite alphabets: Automata over infjnite alphabets: Investigations in Fresh-Register Automata Investigations in Fresh-Register Automata Nikos Tzevelekos, Queen Mary University of London Andrzej Murawski, University of Warwick

956 views • 74 slides
Review Languages and Grammars Alphabets, strings, languages Regular Languages CS 301 -

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

Review Languages and Grammars Alphabets, strings, languages Regular Languages CS 301 - Lecture 7 Deterministic Finite Automata Nondeterministic Finite Automata More Pumping Lemma Equivalence of NFA and DFA Minimizing a

381 views • 11 slides
Introduction to Finite Automata Languages Deterministic Finite Automata Representations of

Introduction to Finite Automata Languages Deterministic Finite Automata Representations of

Introduction to Finite Automata Languages Deterministic Finite Automata Representations of Automata 1 Alphabets An alphabet is any finite set of symbols. Examples: ASCII, Unicode, { 0,1} ( binary alphabet ), { a,b,c} . 2 Strings

397 views • 37 slides
Review Languages and Grammars Alphabets, strings, languages CS 301 - Lecture 6 Regular

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

Review Languages and Grammars Alphabets, strings, languages CS 301 - Lecture 6 Regular Languages Deterministic Finite Automata Nonregular Languages and Nondeterministic Finite Automata Equivalence of NFA and DFA the

435 views • 15 slides
Compression and Estimation Over Large Alphabets Alon Orlitsky Narayana P. Santhanam

Compression and Estimation Over Large Alphabets Alon Orlitsky Narayana P. Santhanam

Compression and Estimation Over Large Alphabets Alon Orlitsky Narayana P. Santhanam Krishnamurthy Viswanathan Junan Zhang UCSD 1 Univeral Compression [Sh 48] [Fi 66, Da 73] Setup: A alphabet p collection of p.d.s over A n random

528 views • 31 slides
( ) ( ) if = M DFA 1 2 0 , 1 2 0 L ( M ) { 10 } * 2 = 1 q q q 1 2 0 1

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

Review Languages and Grammars Alphabets, strings, languages CS 301 - Lecture 3 Regular Languages NFA DFA Equivalence Deterministic Finite Automata Regular Expressions Nondeterministic Finite Automata Today: Fall 2008

444 views • 11 slides
Review Languages and Grammars Alphabets, strings, languages Regular Languages CS

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

Review Languages and Grammars Alphabets, strings, languages Regular Languages CS 301 - Lecture 22 Deterministic Finite and Nondeterministic Automata Equivalence of NFA and DFA and Minimizing a DFA Non-Determinism,

267 views • 15 slides
Review Languages and Grammars Alphabets, strings, languages Regular Languages CS 301

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

Review Languages and Grammars Alphabets, strings, languages Regular Languages CS 301 - Lecture 10 Deterministic Finite and Nondeterministic Automata Equivalence of NFA and DFA Chomsky and Greibach Regular Expressions

476 views • 9 slides
Review Languages and Grammars CS 301 - Lecture 5 Alphabets, strings, languages Regular

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

Review Languages and Grammars CS 301 - Lecture 5 Alphabets, strings, languages Regular Languages Regular Grammars, Regular Deterministic Finite Automata Languages, and Properties of Nondeterministic Finite Automata Regular

576 views • 16 slides
( M ) { abba } = M Take DFA M a , b Definition: L M ( ) The language

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

Review Languages and Grammars CS 301 - Lecture 2 Alphabets, strings, languages Deterministic Finite Automata (DFA) Regular Languages and Definition and Use Nondeterministic Finite Today: Automata Regular Languages

493 views • 23 slides
Pushdown Automata Stack PDAs States 1 a , b c The States q 1 q 2 Pop Input Push symbol

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

Review Languages and Grammars Alphabets, strings, languages Regular Languages CS 301 - Lecture 11 Deterministic Finite and Nondeterministic Automata Equivalence of NFA and DFA Nondeterministic Pushdown Regular Expressions

901 views • 16 slides
CALF: Categorical Automata Learning Framework Matteo Sammartino Alexandra Silva Gerco van

CALF: Categorical Automata Learning Framework Matteo Sammartino Alexandra Silva Gerco van

CALF: Categorical Automata Learning Framework Matteo Sammartino Alexandra Silva Gerco van Heerdt May 23, 2017 1 / 61 Active automata learning Active automata learning algorithms learn an automaton describing the behaviour of a system by

768 views • 61 slides
Review Languages and Grammars Alphabets, strings, languages Regular Languages

Review Languages and Grammars Alphabets, strings, languages Regular Languages

Review Languages and Grammars Alphabets, strings, languages Regular Languages Deterministic Finite and Nondeterministic Automata Equivalence of NFA and DFA Regular Expressions and Regular Grammars CS 301 - Lecture 27

299 views • 9 slides
Review Languages and Grammars Alphabets, strings, languages Regular Languages

Review Languages and Grammars Alphabets, strings, languages Regular Languages

Review Languages and Grammars Alphabets, strings, languages Regular Languages Deterministic Finite and Nondeterministic Automata CS 301 - Lecture 29 Equivalence of NFA and DFA Regular Expressions and Regular Grammars

971 views • 8 slides
Learning Symbolic Automata Samuel Drews Loris DAntoni University of Wisconsin-Madison

Learning Symbolic Automata Samuel Drews Loris DAntoni University of Wisconsin-Madison

Learning Symbolic Automata Samuel Drews Loris DAntoni University of Wisconsin-Madison Motivation Model System Automata Learning Java API code Classic Automata 1,3,5,7 1,3,5,7 0,2,4,6 0,2,4,6 Alphabet = {0,1,2,3,4,5,6,7}

846 views • 68 slides
Learning of Automata Models Learning of Automata Models Extended with Data B Bengt Jonsson t J

Learning of Automata Models Learning of Automata Models Extended with Data B Bengt Jonsson t J

Learning of Automata Models Learning of Automata Models Extended with Data B Bengt Jonsson t J Uppsala University Uppsala University Acknowledgments Fid Fides Aarts A t M ik M Maik Mertens t Therese Bohlin Therese Bohlin Harald

1.19k views • 59 slides
Pushdown Automata 7-0 Pushdown Automata The automata we saw so far were

Pushdown Automata 7-0 Pushdown Automata The automata we saw so far were

Griffith University 3130CIT Theory of Computation (Based on slides by Harald Sndergaard of The University of Melbourne) Pushdown Automata 7-0 Pushdown Automata The automata we saw so far were limited by their

368 views • 15 slides
Automata Learning Borja Balle Amazon Research Cambridge 1 Foundations of Programming Summer

Automata Learning Borja Balle Amazon Research Cambridge 1 Foundations of Programming Summer

Automata Learning Borja Balle Amazon Research Cambridge 1 Foundations of Programming Summer School (Oxford) July 2018 1 Based on work completed before joining Amazon Brief History of Automata Learning 1967 Gold: Regular languages are

865 views • 61 slides
Automata Learning: An Algebraic Approach Henning Urbat joint work with Lutz Schr oder

Automata Learning: An Algebraic Approach Henning Urbat joint work with Lutz Schr oder

Automata Learning: An Algebraic Approach Henning Urbat joint work with Lutz Schr oder Oberseminar October 22, 2019 Automata Learning Black Box System Formal Model s 0 learn ejectGrog insertCoin s 1

1.17k views • 77 slides
For students from 4 to 8 years How to they introduce STEM areas? What are automata? What are

For students from 4 to 8 years How to they introduce STEM areas? What are automata? What are

AutoSTEM automata to teach STEM subjects to young learners Erasmus KA201 large-scale project to create innovative resources For students from 4 to 8 years How to they introduce STEM areas? What are automata? What are they made from?

534 views • 9 slides
Learning I/O Automata Fides Aarts and Frits Vaandrager ICIS, Radboud Universiteit Nijmegen

Learning I/O Automata Fides Aarts and Frits Vaandrager ICIS, Radboud Universiteit Nijmegen

Introduction I/O Behavior and Determinism Interface Automata Learning IOAs Conclusions and Future Work Learning I/O Automata Fides Aarts and Frits Vaandrager ICIS, Radboud Universiteit Nijmegen CONCUR 2010, Paris Fides Aarts and Frits

1.15k views • 56 slides
Automata Theory Why Study Automata? What the Course is About 1 Why Study Automata? A survey of

Automata Theory Why Study Automata? What the Course is About 1 Why Study Automata? A survey of

Automata Theory Why Study Automata? What the Course is About 1 Why Study Automata? A survey of Stanford grads 5 years out asked which of their courses did they use in their job. Basics like intro-programming took the top spots, of course.

584 views • 32 slides

More recommend