Separating regular languages by piecewise testable and unambiguous languages

Transcription

1 Separating regular languages by piecewise testable and unambiguous languages Thomas Place, Lorijn van Rooijen, Marc Zeitoun LaBRI Univ. Bordeaux CNRS September, 2013 Highlights of Logic, Games and Automata Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 1/14

2 Separation problem S is a fixed class of languages. In this talk, S = BΣ 1 (<), FO 2 (<). Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 2/14

3 Separation problem S is a fixed class of languages. In this talk, S = BΣ 1 (<), FO 2 (<). L 1, L 2 Reg are S-separable iff there exists K S such that L 1 L 2 Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 2/14

4 Separation problem S is a fixed class of languages. In this talk, S = BΣ 1 (<), FO 2 (<). L 1, L 2 Reg are S-separable iff there exists K S such that K S L 1 L 2 A \ K Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 2/14

5 Separation problem Decision problem: Given L 1, L 2 Reg, are they S-separable? Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 3/14

6 Separation problem Decision problem: Given L 1, L 2 Reg, are they S-separable? Can we compute a separator? Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 3/14

7 Separation problem Decision problem: Given L 1, L 2 Reg, are they S-separable? Can we compute a separator? What is the complexity of the decision problem? And of computing a separator? Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 3/14

8 Motivation Captures discriminative power of logics Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 4/14

9 Motivation Captures discriminative power of logics Generalization of membership problem Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 4/14

10 Motivation Captures discriminative power of logics Generalization of membership problem L 1 L 2 = A \ L 1 Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 4/14

11 Motivation Captures discriminative power of logics Generalization of membership problem K S L 1 L 2 = A \ L 1 Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 4/14

12 Classes of separators Piecewise testable languages: definable by BΣ 1 (<) formulas (recall previous talk). Unambiguous languages: definable by FO 2 (<) formulas. Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 5/14

13 Classes of separators Piecewise testable languages: definable by BΣ 1 (<) formulas (recall previous talk). Unambiguous languages: definable by FO 2 (<) formulas. Membership is decidable for both BΣ 1 (<) and FO 2 (<) languages. Simon 75 Schützenberger 76, Thérien, Wilke 98 Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 5/14

14 Stratification of S In general, there is no smallest separator in S. Restriction of S to quantifier depth k: S[k]. Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 6/14

15 Stratification of S In general, there is no smallest separator in S. Restriction of S to quantifier depth k: S[k]. If two languages are S[k]-separable, there is a smallest separator in S[k]. Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 6/14

16 Stratification of S In general, there is no smallest separator in S. Restriction of S to quantifier depth k: S[k]. If two languages are S[k]-separable, there is a smallest separator in S[k]. Eg. (a 2 ) and b(b 2 ) have no smallest BΣ 1 (<)-separator, while a \ {a, a 3, a 5 } is the smallest BΣ 1 (<)[6]-separator. Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 6/14

17 Computing smallest separator for S[k] Quantifier depth provides indexed equivalence relations for BΣ 1 (<) resp. FO 2 (<) languages: Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 7/14

18 Computing smallest separator for S[k] Quantifier depth provides indexed equivalence relations for BΣ 1 (<) resp. FO 2 (<) languages: w 1 k w 2 w 1 and w 2 satisfy the same BΣ 1 (<) resp. FO 2 (<) formulas up to depth k. Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 7/14

19 Computing smallest separator for S[k] Quantifier depth provides indexed equivalence relations for BΣ 1 (<) resp. FO 2 (<) languages: w 1 k w 2 w 1 and w 2 satisfy the same BΣ 1 (<) resp. FO 2 (<) formulas up to depth k. L is a BΣ 1 (<) resp. FO 2 (<) language L is a union of k -classes for some k N. Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 7/14

20 Computing smallest separator for S[k] Increasing k refines the smallest potential separator: Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 8/14

21 Computing smallest separator for S[k] Increasing k refines the smallest potential separator: L L Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 8/14

22 Computing smallest separator for S[k] Increasing k refines the smallest potential separator: [L] 1 L L Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 8/14

23 Computing smallest separator for S[k] Increasing k refines the smallest potential separator: [L] 1 L L Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 8/14

24 Computing smallest separator for S[k] Increasing k refines the smallest potential separator: [L] 2 L L Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 8/14

25 Computing smallest separator for S[k] Increasing k refines the smallest potential separator: [L] 3 L L Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 8/14

26 Computing smallest separator for S[k] Increasing k refines the smallest potential separator: [L] 3 L L Refining [L] k gives a semi-algorithm for separability. Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 8/14

27 Computing smallest separator for S[k] Increasing k refines the smallest potential separator: [L] 3 L L Refining [L] k gives a semi-algorithm for separability. From which level of refinement can we conclude non-separability? Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 8/14

28 Testing non-separability A pair of words w 1 L 1, w 2 L 2 is a k-witness of non-separability provided w 1 k w 2. Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 9/14

29 Testing non-separability A pair of words w 1 L 1, w 2 L 2 is a k-witness of non-separability provided w 1 k w 2. Abstraction: tuples (i 1, f 1, i 2, f 2 ) A 1 A 2 i 1 f 1 k i 2 f 2 Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 9/14

30 Testing non-separability A pair of words w 1 L 1, w 2 L 2 is a k-witness of non-separability provided w 1 k w 2. Abstraction: tuples (i 1, f 1, i 2, f 2 ) A 1 A 2 i 1 f 1 k i 2 f 2 L 1, L 2 are k-separable {k-witnesses} =. {k + 1-witnesses} {k-witnesses}. Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 9/14

31 Testing non-separability A pair of words w 1 L 1, w 2 L 2 is a k-witness of non-separability provided w 1 k w 2. Abstraction: tuples (i 1, f 1, i 2, f 2 ) A 1 A 2 i 1 f 1 k i 2 f 2 L 1, L 2 are k-separable {k-witnesses} =. {k + 1-witnesses} {k-witnesses}. Limit behaviour? Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 9/14

32 Testing non-separability A pair of words w 1 L 1, w 2 L 2 is a k-witness of non-separability provided w 1 k w 2. Abstraction: tuples (i 1, f 1, i 2, f 2 ) A 1 A 2 i 1 f 1 k i 2 f 2 L 1, L 2 are k-separable {k-witnesses} =. {k + 1-witnesses} {k-witnesses}. Limit behaviour? We can compute K such that limit = {K-witnesses} = Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 9/14

33 Testing non-separability 1 Verify that the languages are not separable by any language of a sufficient, computable, index K. 2 Find a pattern in the automata producing k-witnesses for arbitrarily large k. (For BΣ 1 (<): same as in previous talk) Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 10/14

34 Testing non-separability 1 Verify that the languages are not separable by any language of a sufficient, computable, index K. 2 Find a pattern in the automata producing k-witnesses for arbitrarily large k. (For BΣ 1 (<): same as in previous talk) Patterns in the automata, independently of the bound K, provide - Better complexity - A yes/no answer for separability - But no separator Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 10/14

35 Main result Theorem For S = BΣ 1 (<), FO 2 (<), we can compute K N such that TFAE i. L 1, L 2 are S-separable. ii. L 1, L 2 are S[K]-separable. iii. [L 1 ] K separates L 1 from L 2. iv. A 1, A 2 do not contain a pattern witnessing non-separability. Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 11/14

36 Main result Condition iv. yields the following complexity results: Theorem Given NFAs A 1, A 2, one can determine whether the languages L(A 1 ) and L(A 2 ) are BΣ 1 (<)-separable in PTIME, FO 2 (<)-separable in EXPTIME, with respect to Q 1, Q 2, A. Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 12/14

37 Future work Obtain tight bounds on the size of BΣ 1 (<) resp. FO 2 (<) - separators Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 13/14

38 Future work Obtain tight bounds on the size of BΣ 1 (<) resp. FO 2 (<) - separators Efficient computation of the separators Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 13/14

39 Future work Obtain tight bounds on the size of BΣ 1 (<) resp. FO 2 (<) - separators Efficient computation of the separators Consider other classes S Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 13/14

40 Thank you Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 14/14

41 BΣ 1 (<): Witnesses in automata L(A 1 ) and L(A 2 ) are not BΣ 1 (<)-separable iff both A 1 and A 2 have a ( u, B)-path: Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 15/14

42 BΣ 1 (<): Witnesses in automata L(A 1 ) and L(A 2 ) are not BΣ 1 (<)-separable iff both A 1 and A 2 have a ( u, B)-path: A 1 B 1 B p u 0 u 1 u p 1 u p Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 15/14

43 BΣ 1 (<): Witnesses in automata L(A 1 ) and L(A 2 ) are not BΣ 1 (<)-separable iff both A 1 and A 2 have a ( u, B)-path: A 1 B 1 B p u 0 u 1 u p 1 u p A 2 B 1 B p u 0 u 1 u p 1 u p Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 15/14

44 BΣ 1 (<): Witnesses in automata L(A 1 ) and L(A 2 ) are not BΣ 1 (<)-separable iff both A 1 and A 2 have a ( u, B)-path: A 1 B 1 B p u 0 u 1 u p 1 u p A 2 B 1 B p u 0 u 1 u p 1 u p This can be determined in PTIME( Q 1, Q 2, A ). Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 15/14

45 Detecting forbidden patterns One can determine in PTIME( Q 1, Q 2, A ) whether ( u, B) such that both A 1 and A 2 have a ( u, B)-path. Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 16/14

46 Detecting forbidden patterns One can determine in PTIME( Q 1, Q 2, A ) whether ( u, B) such that both A 1 and A 2 have a ( u, B)-path. By adding meta-transitions in A i, finding a ( u, B)-witness reduces to: Given states p i, q i, r i of A i, is there B A st. the paths = B = B B B B B p 1 q 1 r 1 p 2 q 2 r 2 occur in A 1 resp. A 2? Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 16/14

47 Detecting forbidden patterns One can determine in PTIME( Q 1, Q 2, A ) whether ( u, B) such that both A 1 and A 2 have a ( u, B)-path. By adding meta-transitions in A i, finding a ( u, B)-witness reduces to: Given states p i, q i, r i of A i, is there B A st. the paths = B = B B B B B p 1 q 1 r 1 p 2 q 2 r 2 occur in A 1 resp. A 2? This is in Ptime: iteratively use Tarjan s algorithm. Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 16/14

48 Detecting forbidden patterns If B exists, B C 1 def = alph scc(q 1, A 1 ) alph scc(q 2, A 2 ) (linear) Restrict the automata to alphabet C 1, and repeat the process: C i+1 def = alph scc(q 1, A 1 Ci ) alph scc(q 2, A 2 Ci ). After n A iterations, C n = C n+1. If C n =, the answer is no. If C n, it is the maximal possible B with (= B)-loops around q 1, q 2. Then, determine the remaining paths. (linear) Overall linear algorithm. Thomas Place, Lorijn van Rooijen, Marc Zeitoun Separation by PTL and UL 17/14