1. Learning from Text Colin de la Higuera University of Nantes

2. Outline • Motivations, definition and difficulties • Some negative results • Learning k-testable languages from text • Learning k-reversible languages from text • Conclusions http://pagesperso.lina.univ-nantes.fr/~cdlh/slides/ Chapters 8 and 11

3. 1 Identification in the limit yields A class of languages L Pres  ℕX a L A learner The naming function G A class of grammars L(a())=yields() (ℕ)=(ℕ) yields()=yields()

4. Learning from text • Only positive examples are available • Danger of over-generalization: why not return *? • The problem is “basic”: • Negative examples might not be available • Or they might be heavily biased: near-misses, absurd examples… • Base line: all the rest is learning with help

5. GI as a search problem PTA ? 

6. From the sample to the PTA PTA(S+) 7 a 11 a a 4 8 b 2 a b a 14 a b 9 12 5 1 a 15 b 13 3 b a a 10 6 S+={, aaa, aaba, ababa, bb, bbaaa}

7. Unanswered Questions • Data is unlabelled… • Is this a clustering problem? • Is this a problem posed in other settings?

8. 2 The theory • Gold 67: No super-finite class can be identified from positive examples (or text) only • Necessary and sufficient conditions for learning exist • Literature: • inductive inference, • ALT series, …

9. Limit point • A class L of languages has a limit pointif there exists an infinite sequence Lnnℕ of languages in L such that L0  L1  … Lk  …, and there exists another language LL such that L= nℕLn • L is called a limit point of L

10. L is a limit point L0 L1 Li L2 L L3

11. Theorem If L admits a limit point, then L is not learnable from text Proof: Letsibe a presentation in length-lex order for Li,and s be a presentation in length-lex order for L. Thennℕi: knsik= sk Note: having a limit point is a sufficient condition for non learnability; not a necessary condition

12. Mincons classes • A class is mincons if there is an algorithm which, given a sample S, builds a grammar GG such that S L L(G) L =L(G) • Ie there is a unique minimum (for inclusion) consistent grammar

13. Accumulation point (Kapur 91) A class L of languages has an accumulation pointif there exists an infinite sequence Sn nℕ of sets such that S0  S1  … Sn  …, and L= nℕSn  L …and for any nℕ there exists a language Ln’ in L such that Sn  Ln’  L. The language L is called an accumulation point of L

14. Ln’ S0 S1 S2 S3 L is an accumulation point Sn L

15. Theorem (for Mincons classes) L admits an accumulation point iff L is not learnable from text

16. Infinite Elasticity • If a class of languages has a limit point there exists an infinite ascending chain of languages L0 L1  …  Ln  …. • This property is called infinite elasticity

17. Infinite Elasticity x0 x1 xi Xi+1 Xi+2 Xi+3 Xi+4 x2 x3

18. Finite elasticity L has finite elasticity if it does not have infinite elasticity

19. Theorem (Wright) If L(G) has finite elasticity and is mincons, then G is learnable.

20. L(G’) TG x1 x3 x2 x4 Tell tale sets Forbidden L(G)

21. Theorem (Angluin) G is learnable iff there is a computable partial function : Gℕ* such that: • nℕ, (G,n) is defined iffGG and L(G) • GG, TM={(G,n): nℕ} is a finite subset of L(G) called a tell-tale subset • G,G’M, if TM L(G’) then L(G’) L(G)

22. Proposition (Kapur 91) A language L in L has a tell-tale subsetiffL is not an accumulation point. (for mincons)

23. Summarizing • Many alternative ways of proving that identification in the limit is not feasible • Methodological-philosophical discussion • We still need practical solutions

24. 3 Learning k-testable languages P. García and E. Vidal. Inference of K-testable languages in the strict sense and applications to syntactic pattern recognition. Pattern Analysis and Machine Intelligence, 12(9):920–925, 1990 P. García, E. Vidal, and J. Oncina. Learning locally testable languages in the strict sense. In Workshop on Algorithmic Learning Theory (ALT 90), pages 325–338, 1990

25. Definition Let k0, a k-testable language in the strict sense (k-TSS) is a 5-tuple Zk=(, I, F, T, C) with: •  a finite alphabet • I, Fk-1(allowed prefixes of length k-1 and suffixes of length k-1) • Tk (allowed segments) • C<k contains all strings of length less than k • Note that I∩F=C∩Σk-1

26. The k-testable language is L(Zk)=I*  *F - *(k-T)*C • Strings (of length at least k) have to use a good prefix and a good suffix of length k-1, and all sub-strings have to belong to T. Strings of length less than k should be in C • Or: k-T defines the prohibited segments • Key idea: use a window of size k

27. a b a a  a b An example (2-testable) I={a} F={a} T={aa, ab, ba} C={,a}

28. Window language • By sliding a window of size 2 over a string we can parse • ababaaababababaaaa OK • aaabbaaaababab not OK

29. The hierarchy of k-TSS languages • k-TSS()={L*: L is k-TSS} • All finite languages are in k-TSS() if k is large enough! • k-TSS()  [k+1]-TSS() • (bak)* [k+1]-TSS() • (bak)* k-TSS()

30. a a a  b b a A language that is not k-testable

31. K-TSS inference Given a sample S, L(ak-TSS(S))= Zk where Zk=((S), I(S), F(S), T(S), C(S) ) and • (S) is the alphabet used in S • C(S)=(S)<kS • I(S)=(S)k-1Pref(S) • F(S)= (S)k-1Suff(S) • T(S)=(S)k {v: uvwS}

32. Example • S={a, aa, abba, abbbba} • Let k=3 • (S)={a, b} • I(S)= {aa, ab} • F(S)= {aa, ba} • C(S)= {a , aa} • T(S)={abb, bbb, bba} • L(a3-TSS(S))= ab*a+a

33. Building the corresponding automaton • Each string in IC and PREF(IC) is a state • Each substring of length k-1 of strings in T is a state •  is the initial state • Add a transition labeled b from u to ub for each state ub • Add a transition labeled b from au to ub for each aub in T • Each state/substring that is in F is a final state • Each state/substring that is in C is a final state

34. a aa aa a a a I={aa, ab}   b F={aa, ba} ab ab T={abb, bbb, bba} C={a, aa} b bb a bb ba ba b Running the algorithm S={a, aa, abba, abbbba}

35. Properties (1) • S L(ak-TSS(S)) • L(ak-TSS(S)) is the smallest k-TSS language that contains S • If there is a smaller one, some prefix, suffix or substring has to be absent

36. Properties (2) • ak-TSS identifies any k-TSS language in the limit from polynomial data • Once all the prefixes, suffixes and substrings have been seen, the correct automaton is returned • If YS, L(ak-TSS(Y)) L(ak-TSS(S))

37. Properties (3) • L(ak+1-TSS(S)) L(ak-TSS(S)) In Ik+1 (resp. Fk+1 and Tk+1) there are less allowed prefixes (resp. suffixes or substrings) than in Ik (resp. Fk and Tk) • k>maxxSx, L(ak-TSS(S))= S • Because for a large k, Tk(S)=

38. 4 Learning k-reversible languages from text D. Angluin. Inference of reversible languages. Journal of the Association for Computing Machinery, 29(3):741–765, 1982

39. The k-reversible languages • The class was proposed by Angluin (1982) • The class is identifiable in the limit from text • The class is composed by regular languages that can be accepted by a DFA such that its reverse is deterministic with a look-ahead of k

40. Let A=(, Q, , I, F) be a NFA, we denote by AT=(, Q, T, F, I) the reversal automaton with: T(q,a)={q’Q: q(q’,a)}

41. a b A a 2 b 1 0 a 4 a a 3 a AT b a 2 b 1 0 a 4 a a 3

42. Some definitions • u is a k-successor of q if │u│=k and (q,u) • u is a k-predecessor of q if │u│=k and T(q,uT) •  is 0-successor and 0-predecessor of any state

43. a b A a 2 b • aa is a 2-successor of 0 and 1 but not of 3 • a is a 1-successor of 3 • aa is a 2-predecessor of 3 but not of 1 1 0 a 4 a a 3

44. A NFA is deterministic with look-ahead kifq,q’Q: qq’ (q,q’I)  (q,q’(q”,a))  (u is a k-successor of q)  (v is a k-successor of q’)  uv

45. Prohibited: u 1 │u│=k a a u 2

46. Example a b This automaton is not deterministic with look-ahead 1 but is deterministic with look-ahead 2 a 2 b 1 0 a 4 a a 3

47. K-reversible automata • A is k-reversible if A is deterministic and AT is deterministic with look-ahead k • Example b b a a b a 2 b a 2 1 0 1 0 b b deterministic with look-ahead 1 deterministic

48. Notations • RL(, k) is the set of all k reversible languages over alphabet  • RL() is the set of all k-reversible languages over alphabet  (ie for all values of k) • ak-RL is the learning algorithm we describe

49. Properties • There are some regular languages that are not in RL() • RL(,k)RL(,k-1) check

50. Violation of k-reversibility Two states q, q’ violate the k-reversibility condition if • they violate the deterministic condition: q,q’(q”,a) or • they violate the look-ahead condition: • q,q’F, uk: u is k-predecessor of both q and q’ • uk, (q,a)=(q’,a) and u is k-predecessor of both q and q’