Learning Regular Languages over Large Alphabets

Transcription

1 Irini-Eleftheri Mens VERIMAG, University of Grenoble-Alpes Lerning Regulr Lnguges over Lrge Alphbets 10 October 2017 Jury Members Oded Mler Directeur de thèse Lurent Fribourg Exminteur Dn Angluin Rpporteur Eric Gussier Exminteur Peter Hbermehl Rpporteur Frits Vndrger Exminteur

2 Blck Box Lerning Model Lnguge Identifiction System Identifiction Inductive Inference 1 / 31

3 A Short Prehistory nd History of Automton Lerning Edwrd F Moore. Gednken-experiments on sequentil mchines. Defines the problem s blck box model inference. E. Mrk Gold. Lnguge identifiction in the limit. E. Mrk Gold. System identifiction vi stte chrcteriztion. Lerning finite utomt is possible in finite time. He first uses the bsic ide tht underlies tble-bsed methods. E. Mrk Gold. Complexity of utomton identifiction from given dt. Finding the miniml utomton comptible with given smple is NP-hrd. Dn Angluin. Lerning regulr sets from queries nd counter-exmples. The L ctive lerning lgorithm with membership nd equivlence queries. Polynomil in the utomton size. Ronld L. Rivest nd Robert E. Schpire. Inference of finite utomt using homing sequences. An improved version of the L lgorithm using the brekpoint method to tret counter-exmples. 2 / 31

4 Mchine Lerning smll smple M = {(x, y) : x X, y Y} Lern Lerning Regulr Lnguges over lrge or infinite lphbets Σ n lphbet X = Σ set of words Y = {+, } Lern Model f : X Y f (x) = y, (x, y) M predict or identify f (x) for ll x X Model f is lnguge L Σ The model is n symbolic utomton 3 / 31

5 The smple M is known before the lerning procedure strts. The smple M is given. Types of Lerning Off-line vs Online Pssive vs Active The smple M is updted during lerning. The smple M is chosen by the lerning lgorithm. Lerning using Queries The lerning lgorithm cn ccess queries e.g., membership queries, equivlence queries, etc. w? L L(H) L w Σ MQ( ) Yes / No Hypothesis H True / EQ( ) Counter-exmple (cex) 4 / 31

6 Outline Preliminries Regulr Lnguges nd Automt The L Algorithmic Scheme Lrge Alphbets Motivtion Symbolic Representtion of Trnsitions - Symbolic Automt Lerning Symbolic Automt Why L cnnot be pplied? Our Solution The Algorithm Equivlence Queries nd Counter-Exmples Adpttion to the Boolen Alphbet Experimentl Results Conclusion 5 / 31

7 Outline Preliminries Regulr Lnguges nd Automt The L Algorithmic Scheme Lrge Alphbets Motivtion Symbolic Representtion of Trnsitions - Symbolic Automt Lerning Symbolic Automt Why L cnnot be pplied? Our Solution The Algorithm Equivlence Queries nd Counter-Exmples Adpttion to the Boolen Alphbet Experimentl Results Conclusion 5 / 31

8 Regulr Lnguges nd Automt b Σ = {, b} L Σ is lnguge Σ is n lphbet w = 1 n is word Σ is the set of ll words b b prefixes suffixes ε b b b bb... ε b b b bb b bb / 31

9 Regulr Lnguges nd Automt b Σ = {, b} L Σ is lnguge Equivlence reltion u L v iff u w L v w L Nerode s Theorem L is regulr lnguge iff L hs finitely mny equivlence clsses. Q = Σ / (sttes in the miniml representtion of L. b b prefixes suffixes ε b b b bb... ε b b b bb b bb ε b b bb b b 6 / 31

10 Regulr Lnguges nd Automt A sufficient smple tht chrcterizes the lnguge ε b ε b ε + b + + b b + + bb + b b b b b bb b b b b b bb b b 7 / 31

11 Regulr Lnguges nd Automt A sufficient smple tht chrcterizes the lnguge S R E ε b ε + b + + b b + + bb + b b b ε b bb b b S prefixes (sttes) R boundry (R = S Σ \ S) E suffixes (distinguishing strings) f : S R E {+, } clssif. function f s : E {+, } residul functions 7 / 31

12 Regulr Lnguges nd Automt A sufficient smple tht chrcterizes the lnguge S R E ε b ε + b + + b b + + bb + b b b b bb ε b b S prefixes (sttes) R boundry (R = S Σ \ S) E suffixes (distinguishing strings) f : S R E {+, } clssif. function f s : E {+, } residul functions A L = (Σ, Q, q 0, δ, F) - Q = S - q 0 = [ε] - δ([u], ) = [u ] - F = {[u] : (u ε) L} The miniml utomton for L 7 / 31

13 The L Algorithmic Scheme Active lerning using queries Lerner Initilize w? L Techer L Σ b ε b b Fill in Tble +/ MQ( ) b b b bb EQ( ) D. Angluin. Lerning regulr sets from queries nd counter-exmples, / 31

14 strt q0 0, 1 q3 3, 4 0 2, 3, 4 0 0, 2, 3, 4 0, q1 q5 0, 1, 2, 3, 4 3, 4 2 0, 1, 2, 3, 4 q4 1 q2 1, 2, 3, 4 q6 The L Algorithmic Scheme Active lerning using queries Lerner Initilize w? L Techer L Σ b b ε b b Fill in Tble MQ( ) b Mke Hypothesis H +/ L(H)? = L EQ( ) b bb D. Angluin. Lerning regulr sets from queries nd counter-exmples, / 31

15 strt q0 0, 1 q3 3, 4 0 2, 3, 4 0 0, 2, 3, 4 0, q1 q5 0, 1, 2, 3, 4 3, 4 2 0, 1, 2, 3, 4 q4 1 q2 1, 2, 3, 4 q6 The L Algorithmic Scheme Active lerning using queries Lerner Initilize w? L Techer L Σ b ε b b Fill in Tble Mke Hypothesis H +/ L(H)? = L MQ( ) EQ( ) b b b bb Tret cex counter-exmple (cex) True Return H D. Angluin. Lerning regulr sets from queries nd counter-exmples, / 31

16 Outline Preliminries Regulr Lnguges nd Automt The L Algorithmic Scheme Lrge Alphbets Motivtion Symbolic Representtion of Trnsitions - Symbolic Automt Lerning Symbolic Automt Why L cnnot be pplied? Our Solution The Algorithm Equivlence Queries nd Counter-Exmples Adpttion to the Boolen Alphbet Experimentl Results Conclusion 8 / 31

17 Lnguges over Lrge Alphbets Input: x 3 x 2 x 1 x 0 x1 : x2 : x3 : x4 : f UNICODE N Boolen Vectors (B n ) Time Series R 9 / 31

18 Symbolic Automt x < 30 q 0 Σ R x x < 50 x q 2 q 1 x < 20 Σ [ 01 ] = {x Σ : x < 50} (w = , +) w = x < x q 4 33 q x < 50 3 x x < A = (Σ, Σ, ψ, Q, δ, q 0, F) - Q finite set of sttes, - q 0 initil stte, - F ccepting sttes, - Σ lrge concrete lphbet, - δ Q Σ Q - Σ finite lphbet (symbols) - ψ q : Σ Σ q, q Q - [[]] = { Σ ψ() = } A is complete nd deterministic if q Q {[[]] Σ q} forms prtition of Σ. 10 / 31

19 Outline Preliminries Regulr Lnguges nd Automt The L Algorithmic Scheme Lrge Alphbets Motivtion Symbolic Representtion of Trnsitions - Symbolic Automt Lerning Symbolic Automt Why L cnnot be pplied? Our Solution The Algorithm Equivlence Queries nd Counter-Exmples Adpttion to the Boolen Alphbet Experimentl Results Conclusion 10 / 31

20 Lerning over Lrge Alphbets Why L cnnot be pplied? u The lerner sks MQ s for ll continutions of stte ( Σ, sk MQ(u )) Inefficient for lrge finite lphbets Not pplicble to infinite lphbets Σ 11 / 31

21 Lerning over Lrge Alphbets Why L cnnot be pplied? k u Σ The lerner sks MQ s for ll continutions of stte ( Σ, sk MQ(u )) Inefficient for lrge finite lphbets Not pplicble to infinite lphbets Our solution: Use finite smple of evidences to lern the trnsitions Evidences: µ() = { 1, 2 } 11 / 31

22 Lerning over Lrge Alphbets Why L cnnot be pplied? k u b Σ The lerner sks MQ s for ll continutions of stte ( Σ, sk MQ(u )) Inefficient for lrge finite lphbets Not pplicble to infinite lphbets Our solution: Use finite smple of evidences to lern the trnsitions Form evidence comptible prtitions Associte symbol to ech prtition block Evidences: µ() = { 1, 2 } 11 / 31

23 Lerning over Lrge Alphbets Why L cnnot be pplied? k ˆµ() u ˆµ(b) b Evidences: µ() = { 1, 2 } Representtive: ˆµ() = 1 Σ The lerner sks MQ s for ll continutions of stte ( Σ, sk MQ(u )) Inefficient for lrge finite lphbets Not pplicble to infinite lphbets Our solution: Use finite smple of evidences to lern the trnsitions Form evidence comptible prtitions Associte symbol to ech prtition block Ech symbol hs one representtive evidence 11 / 31

24 Lerning over Lrge Alphbets Why L cnnot be pplied? k ˆµ() u ˆµ(b) b Evidences: µ() = { 1, 2 } Representtive: ˆµ() = 1 Σ The lerner sks MQ s for ll continutions of stte ( Σ, sk MQ(u )) Inefficient for lrge finite lphbets Not pplicble to infinite lphbets Our solution: Use finite smple of evidences to lern the trnsitions Form evidence comptible prtitions Associte symbol to ech prtition block Ech symbol hs one representtive evidence The prefixes re symbolic 11 / 31

25 Symbolic Lerning Algorithm Lerner 12 / 31

26 Symbolic Lerning Algorithm Lerner Initilize ε 12 / 31

27 Symbolic Lerning Algorithm Lerner Initilize Fill in Tble prtilly ε Repet for ech new stte q: Smple evidences 12 / 31

28 Symbolic Lerning Algorithm Lerner Initilize Fill in Tble prtilly ε Repet for ech new stte q: Smple evidences Ask MQ s 12 / 31

29 Symbolic Lerning Algorithm Lerner Initilize Fill in Tble prtilly ε Repet for ech new stte q: Smple evidences Ask MQ s Lern prtitions 12 / 31

30 Symbolic Lerning Algorithm Lerner Initilize Fill in Tble prtilly Σ ε = { 1, 2 } ε Repet for ech new stte q: Smple evidences Ask MQ s Lern prtitions Define the symbolic lphbet Σ q 12 / 31

31 Symbolic Lerning Algorithm Lerner Initilize Fill in Tble prtilly Σ ε = { 1, 2 } ε Repet for ech new stte q: Smple evidences Ask MQ s Lern prtitions Define the symbolic lphbet Σ q Select representtive ˆµ(), Σ q 12 / 31

32 Symbolic Lerning Algorithm Lerner Initilize Fill in Tble prtilly Σ ε = { 1, 2 } ε Repet for ech new stte q: Smple evidences Ask MQ s Lern prtitions Define the symbolic lphbet Σ q Select representtive ˆµ(), Σ q 12 / 31

33 strt q0 0, 1 q3 3, 4 0 2, 3, 4 0 0, 2, 3, 4 0, q1 q5 0, 1, 2, 3, 4 3, 4 2 0, 1, 2, 3, 4 q4 1 q2 1, 2, 3, 4 q6 Symbolic Lerning Algorithm Lerner Initilize Fill in Tble prtilly Mke Hypothesis H Σ ε = { 1, 2 } ε ε 1 2 Repet for ech new stte q: Smple evidences Ask MQ s Lern prtitions Define the symbolic lphbet Σ q Select representtive ˆµ(), Σ q 12 / 31

34 strt q0 0, 1 q3 3, 4 0 2, 3, 4 0 0, 2, 3, 4 0, q1 q5 0, 1, 2, 3, 4 3, 4 2 0, 1, 2, 3, 4 q4 1 q2 1, 2, 3, 4 q6 Symbolic Lerning Algorithm Lerner Initilize Fill in Tble prtilly Mke Hypothesis H Tret cex cex Σ ε = { 1, 2 } ε ε 1 2 Repet for ech new stte q: Smple evidences Ask MQ s Lern prtitions Define the symbolic lphbet Σ q Select representtive ˆµ(), Σ q 12 / 31

35 Evidence Comptibility Solve Incomptibility x [ 3 ] x x x x x x x x [ 4 ] [ 1 ] [ 2 ] Boundry Modifiction New Trnsition Evidence Comptibility A stte u is evidence comptible when f u = f u ˆµ() for every evidence [] Evidence incomptibility t stte u v : u ˆµ() + u 13 / 31

36 Counter-exmple Tretment (Symbolic Brekpoint) Let w = 1 i w = u i i v i be counter-exmple. f (ˆµ(s i 1 i) v i) f (ˆµ(s i) v i) f (ˆµ(s i 1) i v i) f (ˆµ(s i 1) ˆµ( i) v i) s i = δ(ε, u i i ) ˆµ(u i ) ε s i is new stte ˆµ(u i ) ε refine [ i ] ˆµ( i ) s s v i ˆµ(u i ) ε ˆµ( i ) s i ˆµ(u i ) ε v i s s ˆµ( i ) v i new v i v i ˆµ( i ) s i v i v i verticl expnsion v i horizontl expnsion 14 / 31

37 Exmple over the lphbet Σ = [1, 100) observtion tble semntics hypothesis utomton ε ε ε Σ ε = { 1, 2 } ˆµ( 1 ) ˆµ( 2 ) 1 Σ 1 = { 3 } ε x 27 x < 27 Σ 1 ˆµ( 3 ) 15 / 31

38 Exmple over the lphbet Σ = [1, 100) observtion tble semntics hypothesis utomton ε 11 ε ε Σ ε = { 1, 2 } ˆµ( 1 ) ˆµ( 2 ) 1 Σ 1 = { 3 } ˆµ( 3 ) ε x 27 x < 27 Σ Ask Equivlence Query: counter-exmple: w = , 1 dd distinguishing string 11 discover new stte (verticl expnsion) 15 / 31

39 Exmple over the lphbet Σ = [1, 100) observtion tble semntics hypothesis utomton ε 11 ε ε Σ ε = { 1, 2 } ˆµ( 1 ) ˆµ( 2 ) 1 Σ 1 = { 3 } ˆµ( 3 ) x < 27 ε Σ x x 43 x < 43 2 Σ 2 = { 4, 5 } ˆµ( 4 ) ˆµ( 5 ) 15 / 31

40 Exmple over the lphbet Σ = [1, 100) observtion tble semntics hypothesis utomton ε 11 ε ε Σ ε = { 1, 2 } ˆµ( 1 ) ˆµ( 2 ) 1 Σ 1 = { 3, 6 } ˆµ( 3 ) ˆµ( 6 ) 2 Σ 2 = { 4, 5 } ˆµ( 4 ) ˆµ( 5 ) ε x 27 x < 27 Σ 1 2 Ask Equivlence Query: counter-exmple: w = , x 43 dd 73 s evidence of 1 x < 43 dd new trnsition (horizontl expnsion) 15 / 31

41 Exmple over the lphbet Σ = [1, 100) observtion tble semntics hypothesis utomton ε 11 ε ε Σ ε = { 1, 2 } ˆµ( 1 ) ˆµ( 2 ) 1 Σ 1 = { 3, 6 } ˆµ( 3 ) ˆµ( 6 ) 2 Σ 2 = { 4, 5 } ˆµ( 4 ) ˆµ( 5 ) x < 27 ε x < 63 x x 63 x 43 x < / 31

42 Exmple over the lphbet Σ = [1, 100) observtion tble semntics hypothesis utomton ε 11 ε ε Σ ε = { 1, 2 } ˆµ( 1 ) ˆµ( 2 ) 1 Σ 1 = { 3, 6 } ˆµ( 3 ) ˆµ( 6 ) 2 Σ 2 = { 4, 5 } ˆµ( 4 ) ˆµ( 5 ) ε x 27 x < 27 x < x 63 x 43 x < 43 Ask Equivlence Query: counter-exmple: w = 52 46, dd 46 s evidence of 2 refine existing trnsition (horizontl expnsion) 15 / 31

43 Exmple over the lphbet Σ = [1, 100) observtion tble semntics hypothesis utomton ε 11 ε ε Σ ε = { 1, 2 } ˆµ( 1 ) ˆµ( 2 ) 1 Σ 1 = { 3, 6 } ˆµ( 3 ) ˆµ( 6 ) 2 Σ 2 = { 4, 5 } ε x 27 x < 27 x < Ask Equivlence Query: True x 63 x 52 return current hypothesis x < 52 return hypothesis ˆµ( 4 ) ˆµ( 5 ) 15 / 31

44 Outline Preliminries Regulr Lnguges nd Automt The L Algorithmic Scheme Lrge Alphbets Motivtion Symbolic Representtion of Trnsitions - Symbolic Automt Lerning Symbolic Automt Why L cnnot be pplied? Our Solution The Algorithm Equivlence Queries nd Counter-Exmples Adpttion to the Boolen Alphbet Experimentl Results Conclusion 15 / 31

45 Equivlence Queries nd Counter-Exmples Wht is the error? L L(H) All w L L(H) re counter-exmples A helpful techer cn compute L L(H) to find counter-exmples. When the techer provides miniml counter-exmples (i.e., miniml in lengthlexicogrphic order), then one evidence per prtition is used the boundries re exctly determined finl hypothesis contins no error The lgorithm termintes with correct conjecture fter sking t most O(mn 2 ) MQ s nd t most O(mn) EQ s, when Σ is totlly-ordered. 16 / 31

46 Equivlence Queries nd Counter-Exmples Wht is the error? L L(H) All w L L(H) re counter-exmples In the bsence of helpful techer nd the lerner cn use only MQ s EQ s re pproximted by testing: select set of words rndomly sk MQ s for them check if the result mtches with H return counter-exmple A hypothesis utomton H is Probbly Approximtely Correct (PAC) iff Pr(P(L L(H)) < ɛ) > 1 δ. Sufficient tests for hypothesis H i to be PAC: r i = 1 ɛ (ln 1 δ + (i + 1) ln 2). [Ang87] 17 / 31

47 Outline Preliminries Regulr Lnguges nd Automt The L Algorithmic Scheme Lrge Alphbets Motivtion Symbolic Representtion of Trnsitions - Symbolic Automt Lerning Symbolic Automt Why L cnnot be pplied? Our Solution The Algorithm Equivlence Queries nd Counter-Exmples Adpttion to the Boolen Alphbet Experimentl Results Conclusion 17 / 31

48 Adpttion to the Boolen Alphbet Prtition of R (or N) into finite number of intervls Prtition of B n into finite number of cubes / 31

49 Adpttion to the Boolen Alphbet Representtions of the Boolen Cube ψ : B 4 { 1, 2, 3 } B n : 3 x 1 x x 1 x 3 x 1 x 2 x 3 x x 3 1, if x 3 ψ() = 2, if x 1 x 3 3, if x 1 x 3 Boolen Function x1x x 3 x Krnugh mp x x Binry Decision Tree 19 / 31

50 Adpttion to the Boolen Alphbet Lerning Prtitions Σ = B 4 Lerning Binry Decision Trees using the Greedy Splitting Algorithm CART u x1x x 3 x r r b g g 0 1 x 3 x x , if x 3 ψ() = 2, if x 1 x 3 3, if x 1 x 3 Best split: x 1 Use Informtion Gin (Entropy) Mesure to find Best Split x 1 x 3 x 1 x 3 Breimn et l. Clssifiction nd regression trees, / 31

51 Adpttion to the Boolen Alphbet Exmple over Σ = B 4 observtion tble semntics hypothesis utomton ε ε ε x 2 3 x 2 q 0 q 1 2 x 2 0 x / 31

52 Adpttion to the Boolen Alphbet Exmple over Σ = B 4 observtion tble semntics hypothesis utomton ε 0000 ε ε x 2 3 x 2 q 0 q 1 0 x 2 2 x 2 Ask Equivlence Query: counter-exmple: w = (1010) (0000), + w = 0 0, dd distinguishing string (0000) discover new stte evidence incomptibility 21 / 31

53 Adpttion to the Boolen Alphbet Exmple over Σ = B 4 observtion tble ε 0000 ε ε semntics hypothesis utomton 0 x 2 x 3 1 x 2 q 0 q 1 q 2 3 x 2 x 2 x x 2 x2 1 x 3 4 x 1 x 3 Ask Equivlence Query: 21 / 31

54 Adpttion to the Boolen Alphbet Exmple over Σ = B 4 observtion tble ε 0000 ε ε semntics hypothesis utomton 0 x 2 x 3 1 x 2 q 0 q 1 q 2 3 x 2 x 2 x x 2 x2 1 x 3 4 x 1 x 3 Ask Equivlence Query: True terminte: Return H 21 / 31

55 Outline Preliminries Regulr Lnguges nd Automt The L Algorithmic Scheme Lrge Alphbets Motivtion Symbolic Representtion of Trnsitions - Symbolic Automt Lerning Symbolic Automt Why L cnnot be pplied? Our Solution The Algorithm Equivlence Queries nd Counter-Exmples Adpttion to the Boolen Alphbet Experimentl Results Conclusion 21 / 31

56 Empiricl Results Comprison to the best L lgorithm #MQs ( 10 3 ) #EQs #sttes lerned Symbolic Algorithm L Reduced lphbet size Σ Experiment: Trget utomton: - Σ N - 10 Σ Q = 15, - Σ q 5, q Q Structure is fixed PAC criterion for ɛ = δ = 0.05 MQ s = MQ s for lerning + MQ s for testing Rivest nd Schpire. Inference of finite utomt using homing sequences, / 31

57 Empiricl Results Comprison to the best L lgorithm #MQs ( 10 3 ) #EQs #sttes lerned Symbolic Algorithm L Reduced number of sttes in trget Experiment: Trget utomton: - Σ N - Σ = Q 45 - Σ q 5, q Q Rndom structure PAC criterion for ɛ = δ = 0.05 MQ s = MQ s for lerning + MQ s for testing Rivest nd Schpire. Inference of finite utomt using homing sequences, / 31

58 #MQs ( 10 3 ) MT ( 10 2 ) #EQs ( 10 ) Empiricl Results Applying the symbolic lgorithm over the Boolens Symbolic Algorithm ( n ) lphbet size sttes in trget Experiment: Trget utomton: Left: Q = Σ 2 15 Right: Σ = B 8 3 Q 50 BDTs depth 4, q Q PAC criterion for ɛ = δ = 0.05 MQ s = MQ s for lerning + MQ s for testing 24 / 31

59 Empiricl Results Vlid psswords over the ASCII chrcters Control Chrcters Numerls Lower-Cse Letters Punctution Symbols Upper-Cse Letters 25 / 31

60 Empiricl Results Vlid psswords over the ASCII chrcters The Symbolic Algorithm, L Reduced: [RS93] 100 Symbolic Algorithm L Reduced Symbolic Algorithm L Reduced #Sttes lerned #MQs ( 10 3 ) A B C D E pssword type 0 A B C D E pssword type A (pin) B (esy) C (medium) D (medium-strong) E (strong) Length: 4 to 8. Contins only numbers. Length: 4 to 8. It contins ny printble chrcter. Length: 6 to 14. Contins ny printble chrcter but punctution chrcters. Length: 6 to 14. Contins t lest 1 number nd 1 lower-cse letter. Punctution chrcters re llowed. Length: 6 to 14. Contins t lest 1 chrcter from ech group. 26 / 31

61 Empiricl Results Vlid psswords over the ASCII chrcters A (pin) B (esy) C (medium) D (medium-strong) E (strong) Length: 4 to 8. Contins only numbers. Length: 4 to 8. It contins ny printble chrcter. Length: 6 to 14. Contins ny printble chrcter but punctution chrcters. Length: 6 to 14. Contins t lest 1 number nd 1 lower-cse letter. Punctution chrcters re llowed. Length: 6 to 14. Contins t lest 1 chrcter from ech group. 26 / 31

62 Empiricl Results Vlid psswords over the ASCII chrcters Σ = {0, 1,..., 127} Σ = B 7 90 SL SLbool 120 SL SLbool 600 SL SLbool #MQs ( 10 3 ) #Sttes #Symbols A B C D E pssword type 0 A B C D E pssword type 0 A B C D E pssword type 27 / 31

63 Outline Preliminries Regulr Lnguges nd Automt The L Algorithmic Scheme Lrge Alphbets Motivtion Symbolic Representtion of Trnsitions - Symbolic Automt Lerning Symbolic Automt Why L cnnot be pplied? Our Solution The Algorithm Equivlence Queries nd Counter-Exmples Adpttion to the Boolen Alphbet Experimentl Results Conclusion 27 / 31

64 Relted Work Ides similr to ours hve been suggested nd explored in series of ppers, which lso dpt utomton lerning nd the L lgorithm to lrge lphbets. F Howr, B Steffen, nd M Merten (2011). Automt lerning with utomted lphbet bstrction refinement. M Isberner, F Howr, nd B Steffen (2013). Inferring utomt with stte-locl lphbet bstrctions. The hypothesis is prtilly defined hypothesis where the trnsition function is not defined outside the observed evidence. T Berg, B Jonsson, nd H Rffelt (2006). Regulr inference for stte mchines with prmeters. Bsed on lphbet refinement tht genertes new symbols indefinitely. 28 / 31

65 Relted Work Ides similr to ours hve been suggested nd explored in series of ppers, which lso dpt utomton lerning nd the L lgorithm to lrge lphbets. S Drews nd L D Antoni (2017). Lerning symbolic utomt. Gives more generl justifiction for lerning scheme like ours by providing tht lernbility is closed under product nd disjoint union. M Botinčn nd D Bbić (2013). Sigm*: Symbolic lerning of input-output specifictions. Weker termintion results tht is relted to the counter-exmple guided bstrction refinement procedure. Hndles trnsducers insted of utomt. 28 / 31

66 Contribution O Mler nd IE Mens. Lerning regulr lnguges over lrge lphbets. In TACAS, vol 8413 of LNCS, pges Springer, O Mler nd IE Mens. Lerning regulr lnguges over lrge ordered lphbets. Logicl Methods in Computer Science (LMCS), 11(3), O Mler nd IE Mens. A Generic Algorithm for Lerning Symbolic Automt from Membership Queries. In Models, Algorithms, Logics nd Tools, vol of LNCS, pges Springer, / 31

67 Conclusions We presented n lgorithm for lerning regulr lnguges over lrge lphbets using symbolic utomt. We decomposed the problem into lerning new sttes (s in stndrd utomton lerning) nd lerning the lphbet prtitions in ech stte. Modifiction of lphbet prtitions re treted in rigorous wy tht does not introduce superfluous symbols. It cn be done s sttic lerning of concepts/prtitions in the lphbet domin. We defined the notion of evidence comptibility which is n invrince of the lgorithm nd extended the brekpoint method to detect its violtion. We explored in detil nd implemented the cses where lphbets re numbers or Boolen vectors. We hndle both helpful nd non-helpful techers. 30 / 31

68 Future Work Extend the lgorithm to lphbets such s R n nd R n B n using regression trees. Explore the use of other deep lerning methods to lern the lphbet prtitions. Study more relistic situtions where the lerner does not hve full control over the smple nd when some noise is present. Mke more experiments nd lgorithmic improvement for the Boolen cse. Find nd explore convincing clss of pplictions. Thnk you! 31 / 31