Le rning Regular. A Languages via Alt rn ting. A E Autom ta

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2 Le rning Regular A Languages via Alt rn ting A E Autom ta A YALE MIT PENN IJCAI 15, Buenos Aires 2

3 4 The Problem Learn an unknown regular language L using MQ and EQ data mining neural networks geometry verification synthesis Learner membership queries w L? yes / no Equivalence queries A = L? yes / no: counterexample L Teacher

4 5 Motivation and Goal questions [Angluin, 1987] proposed L* which uses deterministic finite automata DFA [Bollig et al., 2009], proposed NL* which uses non-deterministic finite automata NFA AFA DFA NFA

5 6 Motivation and Goal questions Succinctness [Angluin, 1987] proposed L* which uses deterministic finite automata DFA [Bollig et al., 2009], proposed NL* which uses non-deterministic finite automata NFA Can we generalize this to AL* which uses alternating finite automata AFA? AFA DFA NFA AFA

6 7 Motivation and Goal questions Succinctness [Angluin, 1987] proposed L* which uses deterministic finite automata DFA [Bollig et al., 2009], proposed NL* which uses non-deterministic finite automata NFA Can we generalize this to AL* which uses alternating finite automata AFA? AFA DFA NFA AFA [Bollig et al., 2009] showed that NL* outperforms L* on random RE (concatenation, iteration, union)

7 8 Motivation and Goal questions Succinctness [Angluin, 1987] proposed L* which uses deterministic finite automata DFA [Bollig et al., 2009], proposed NL* which uses non-deterministic finite automata NFA Can we generalize this to AL* which uses alternating finite automata AFA? AFA DFA NFA AFA [Bollig et al., 2009] showed that NL* outperforms L* on random RE intersection (concatenation, iteration, union)

8 Motivation and Goal questions Succinctness [Angluin, 1987] proposed L* which uses deterministic finite automata DFA [Bollig et al., 2009], proposed NL* which uses non-deterministic finite automata NFA Can we generalize this to AL* which uses alternating finite automata AFA? AFA DFA NFA AFA Understanding [Bollig et al., 2009] showed that NL* outperforms L* on random RE intersection (concatenation, iteration, union) 9 What are the tradeoffs between L*, NL* & AL*?

9 10 Automata Types Deterministic (DFA) Non-deterministic (NFA) Universal (UFA) Alternating (AFA)

10 Transition Types Transition Type Type Non- Deterministic s1 s1 c c s3 or s4 ic Universal s1 c s3 and s4 Universal s1 c s3 and s4 Alternating s1 c (s3 or s4) and s2 from fromupon upon stat statereading reading e Deterministic s1 s1 c c s2 ic to state(s) 1 c 2 c 1 c

11 Transition Types Transition Type Type Non- Deterministic s1 s1 c c s3 or s4 ic Universal s1 c s3 and s4 Universal s1 c s3 and s4 from fromupon upon stat statereading reading e Deterministic s1 s1 c c s2 ic to state(s) 1 c 2 Alternating s1 c (s3 or s4) and s2 c c 1 c c 1 c

12 Transition Types Transition Type Type Non- Deterministic s1 s1 c c s3 or s4 ic Universal s1 c s3 and s4 Universal s1 c s3 and s4 from fromupon upon stat statereading reading e Deterministic s1 s1 c c s2 ic to state(s) 1 c 2 Alternating s1 c (s3 or s4) and s2 c 1 c

13 Transition Types Transition Type Type Non- Deterministic s1 s1 c c s3 or s4 ic Universal s1 c s3 and s4 Universal s1 c s3 and s4 from fromupon upon stat statereading reading e Deterministic s1 s1 c c s2 ic to state(s) 1 c 2 Alternating s1 c (s3 or s4) and s2 c 1 c 1 c

14 15 Σ = {a,b} Alternating Automaton Ex. a a AND b b Accepts the language Σ*aaΣ* Σ*bbΣ*

15 Succinctness Well known: [Meyer & Fischer, 1971] [Chandra & Stockmeyer, 1976] [Kozen, 1976] AFA NFA UFA DFA May be exponentially bigger than May be doubly exponentially bigger than 16

16 17 Residuality To be precise [Bollig et al.] used residual NFA (NRFA), introduced by [Denis et al.] We thus first extended the idea of residual NFA to residual AFA (ARFA) Details in the paper and poster

17 Succinctness & Residuality AFA ARFA NFA UFA NRFA [ ] [ ] [ ] URFA [Denis et al. 2001] [Duality] DFA DRFA May be exponentially bigger than May be doubly exponentially bigger than 18

18 19 Residuality L*, NL*, UL*, AL* Observation tables Closed and minimal tables Monotone, union and intersection bases Finding an adequate basis

19 An Observation Table Strings: experiments to distinguish states Strings: candidate state representatives M s 1 s 2 s 3 s 4 s 5 e 1 e 2 e 3 e 4 e 5 e M i j = 1 if s i e j 2 L 0 otherwise 20

20 The Complete Observation Table Enumeration of all strings 21 ε a b aa ab aaa aab aba abb baa ε a b aa ab ab bb aaa aab aba abb baa By Myhill-Nerode the ba number Enumeration of distinct rows is of finite. all strings bb We call it the row index

21 The Complete Observation Table 22 The number of distinct columns is also finite. We call it the column index. Enumeration of all strings ε a b aa ab ba bb aaa aab aba abb baa Enumeration of all strings ε a b aa ab ab bb aaa aab aba abb baa

22 Closed Table An observation table T = (S,E,M) is closed w.r.t a subset B S B S ε a b ab aa aaa aab e 1 e 2 e 3 e 4 e 5 e If it satisfies 1) Initialization: ε B 2) Consecution: BΣ S 3) Coverage: all rows not in B are covered by B 23

23 Closed Table An observation table T = (S,E,M) is closed w.r.t a subset B S B S ε a b ab aa aaa aab e 1 e 2 e 3 e 4 e 5 e If it satisfies 1) Initialization: ε B 2) Consecution: BΣ S 3) Coverage: all rows not in B are x-covered by B x {D,N,U,A} That is, the definition differs for L*, NL*, UL* and AL*. 24

24 D-Covered According to L* i.e. when using DFAs S ε a b ab aa aaa aab e 1 e 2 e 3 e 4 e 5 e

25 26 N-Covered According to NL* i.e. when using NFAs S ε a b ab aa aaa aab e 1 e 2 e 3 e 4 e 5 e b = (ε a)

26 27 U-Covered According to UL* i.e. when using UFAs S ε a b ab aa aaa aab e 1 e 2 e 3 e 4 e 5 e b = (ε a)

27 A-Covered According to AL* i.e. when using AFAs S ε a b ab aa aaa aab e 1 e 2 e 3 e 4 e 5 e b = (ε a) ab = (ε a) aa 28

28 29 Minimal Table S ε a b ab aa aaa aab closed e 1 e 2 e 3 e 4 e 5 e S ε a b ab aa aaa aab trivially closed e 1 e 2 e 3 e 4 e 5 e Let T be closed w.r.t. B. T is x-minimal w.r.t. B if forall b B, b is not x- covered by any subset of B\{b}.

29 30 From Tables to Automata Closed and Minimal S ε a b ab aa aaa aab ε e 2 e 3 e 4 e 5 e = (ε a) = (ε a) aa = ε = a b a b a a b

30 31 Need to solve How to decide Is row s a union of rows in B? Poly time [Bollig et al.] Is row s an intersection of rows in B? Poly time [Duality] Is s a monotone combination of rows in B? Poly time [new] Given a set of Boolean vectors S, find a minimal union basis Poly time [Bollig et al.] intersection basis Poly time [Duality] monotone basis NP-complete [new]

31 32 The Learning Alg. Start with table : ε ε 0

32 33 The Learning Alg. Start with basis: ε

33 34 The Learning Alg. If the table is not closed, e.g. s 1 is missing, then add it. Until the table is closed.

34 35 The Learning Alg. If the table is not minimal, e.g. s 2 is redundant, then remove it. Until the table is minimal.

35 36 The Learning Alg. The table is now closed and minimal. Extract the respective AFA.

36 The Learning Alg. 37 Ask an equivalence query. If true, return. Otherwise, use the given counterexample to find some columns to add, and add them.

37 Find Columns to Add Closed and Minimal Counter example w = abcde ε a b ab aa bb ba aaa aab ε e 2 e 3 e 4 e 5 e 6 abcde bcde cde de e B = {ε, a, aa } BΣ = {b, ab, aaa,, aab }

38 Find Columns to Add Closed and Minimal Counter example w = abcde ε a b ab aa bb ba aaa aab ε e 2 e 3 e 4 e 5 e 6 abcde bcde cde de e B = {ε, a, aa } BΣ = {b, ab, aaa,, aab }

39 Find Columns to Add Closed and Minimal Counter example w = abcde ε a b ab aa bb ba aaa aab ε e 2 e 3 e 4 e 5 e 6 bcde cde de e B = {ε, a, aa } BΣ = {b, ab, aaa,, aab }

40 Find Columns to Add Closed and Minimal Counter example w = abcde ε a b ab aa bb ba aaa aab ε e 2 e 3 e 4 e 5 e 6 bcde cde e B = {ε, a, aa } BΣ = {b, ab, aaa,, aab }

41 42 The Learning Alg. Theorem Every counterexample yields at least one new column

42 43 The Learning Alg. Theorem The algorithm AL* returns an AFA for the unknown language after at most m equivalence queries O( Σ mnc) membership queries poly(m, n, c, Σ ) time L* NL* AL* EQ n O(n 2 ) m MQ O( Σ cn 2 ) O( Σ cn 3 ) O( Σ cnm) where n = row index m = column index c = length of longest c.e.

43 44 Residuality Empirical results Comparison of L*, NL*, UL*, AL* in learning random DFAs, NFAs, UFAs, AFAs along # states # MQ # EQ

44 45 Empirical Results

45 46 Rough Summary: Empirical Results In terms of #states generated, AL* is always preferable In terms of #MQ, xl* outperforms the others when targets are xfas In terms of #EQ, L* is always preferable

46 47 Future Work Generalization to Boolean Automata ( ) Heuristics combining xl* s Understanding of Residual AFAs Theorem: The algorithm AL* returns an AFA for the unknown language Conjecture: The algorithm AL* returns an ARFA for the unknown language

47 48 The End Thank you for your attention!

48 # of membership queries Random DFA targets Random NFA targets L*, NL*, L*, NL*, 51

49 # of equivalence queries Random DFA targets Random NFA targets L*, NL*, L*, NL*, 52

50 Resulting # of states Random AFA targets Random UFA targets L*, NL*, UL*, AL* 300 L*, NL*, UL*, AL*

51 # of membership queries Random AFA targets Random UFA targets L*, NL*, UL*, AL* L*, NL*, UL*, AL* 54

52 # of equivalence queries Random AFA targets Random UFA targets L*, NL*, UL*, AL* L*, NL*, UL*, AL*