Regular realizability problems and context-free languages

Transcription

1 Regular realizability problems and context-free languages Alexander Rubtsov 23 Mikhail Vyalyi Computing Centre of Russian Academy of Sciences 2 Moscow Institute of Physics and Technology 3 National Research University Higher School of Economics 26 June 2015

2 1 Regular realizability problems Definition Examples Properties 2 Relation with -theory Rational cones Complexity of RR-Problems 3 Rational index

3 Regular realizability problems Relation with -theory Rational index Definition of the problems Filter We fix language F Σ called filter. A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

4 Regular realizability problems Relation with -theory Rational index Definition of the problems Filter We fix language F Σ called filter. L(A) REG input of the problem, where A is NFA. Regular realizability problem NRR(F) = {A A NFA, L(A) F } A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

5 Regular realizability problems Relation with -theory Rational index Examples Periodic filters Per 1 = {(1 k #) n k, n N} 111#111# #111# Per 1 A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

6 Regular realizability problems Relation with -theory Rational index Examples Periodic filters Per 1 = {(1 k #) n k, n N} 111#111# #111# Per 1 The problem NRR(Per 1 ) is NP-complete A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

7 Regular realizability problems Relation with -theory Rational index Examples Periodic filters Per 1 = {(1 k #) n k, n N} 111#111# #111# Per 1 The problem NRR(Per 1 ) is NP-complete Per 2 = {(w#) n w Σ, Σ = 2, n N} w # w # # w # Per 2 A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

8 Regular realizability problems Relation with -theory Rational index Examples Periodic filters Per 1 = {(1 k #) n k, n N} 111#111# #111# Per 1 The problem NRR(Per 1 ) is NP-complete Per 2 = {(w#) n w Σ, Σ = 2, n N} w # w # # w # Per 2 The problem NRR(Per 2 ) is PSPACE-complete A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

9 Regular realizability problems Relation with -theory Rational index Examples Periodic filters Per 1 = {(1 k #) n k, n N} 111#111# #111# Per 1 The problem NRR(Per 1 ) is NP-complete Per 2 = {(w#) n w Σ, Σ = 2, n N} w # w # # w # Per 2 The problem NRR(Per 2 ) is PSPACE-complete Anderson T., Loftus J., Rampersad N., Santean N., Shallit J. Detecting palindromes, patterns and borders in regular languages. Information and Computation. Vol. 207, P M.V. Paper in Russian, A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

10 Regular realizability problems Relation with -theory Rational index Rational dominance Definition A language L A is rationally dominated by L B if there exists a rational relation R such that L = {u A v L (v, u) R} L rat L A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

11 Regular realizability problems Relation with -theory Rational index Rational transductions Definition A finite state transducer (FST) T(x) : Γ is a (nondeterministic) automaton with output tape T = (, Γ, Q, q 0, δ, F), where input alphabet; Γ output alphabet; Q set of states; δ : Q {ε} Γ {ε} Q transitions relation; q 0 initial state; F set of accepting states. If transducer T on input x has no path to accepting state, then T(x) =. A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

12 Regular realizability problems Relation with -theory Rational index Rational transductions In other words L rat L T FST : L = T(L ) A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

13 Regular realizability problems Relation with -theory Rational index Reduction on filters Proposition F 1 rat F 2 NRR(F 1 ) log NRR(F 2 ) A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

14 1 Regular realizability problems Definition Examples Properties 2 Relation with -theory Rational cones Complexity of RR-Problems 3 Rational index

15 Regular realizability problems Relation with -theory Rational index Rational cone Definition A rational cone is a class of languages closed under rational dominance. Denote by T (L) the least rational cone that includes language L and call it rational cone generated by L. A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

16 Regular realizability problems Relation with -theory Rational index Rational cone Definition A rational cone is a class of languages closed under rational dominance. Denote by T (L) the least rational cone that includes language L and call it rational cone generated by L. Theorem (Chomsky, Schützenberger) T (D 2 ) = D n = S SS a 1 Sā 1 a nsā n ε. A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

17 Regular realizability problems Relation with -theory Rational index J. Berstel s book cover A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

18 D 2 GRE QRT FCL S D 1 LIN ROCL REG

19 D 2 REG

20 D 2 T (D 2 ) = REG

21 D 2 T (D 2 ) = NRR(D 2 ) is P-complete REG

22 D 2 T (D 2 ) = NRR(D 2 ) is P-complete L NRR(L) P P REG

23 D 2 T (Σ ) = REG P REG

24 D 2 T (Σ ) = REG NRR(Σ ) is -complete P REG

25 D 2 T (Σ ) = REG NRR(Σ ) is -complete L REG NRR(L) P REG

26 D 2 P GRE QRT FCL S D 1 LIN ROCL REG

27 D 2 P GRE QRT FCL S D 1 LIN ROCL

28 D 2 LIN is generated by Symmetric language S a 1 Sā 1 a nsā n ε P GRE QRT FCL S D 1 LIN ROCL

29 D 2 LIN is generated by Symmetric language S a 1 Sā 1 a nsā n ε NRR(S) P GRE QRT FCL S D 1 LIN ROCL

30 Th. (Anderson, Loftus, Rampersad, Santean, Shallit, 2009) Similar theorem about palindromes: NRR(PAL). D 2 P LIN is generated by Symmetric language S a 1 Sā 1 a nsā n ε NRR(S) GRE QRT FCL S D 1 LIN ROCL

31 D 2 P LIN is generated by Symmetric language S a 1 Sā 1 a nsā n ε NRR(S) QRT = T σ (LIN) GRE S QRT FCL D 1 LIN ROCL

32 Lemma If L, L a for all a A, are easy languages then σ(l) is also easy. D 2 P LIN is generated by Symmetric language S a 1 Sā 1 a nsā n ε NRR(S) QRT = T σ (LIN) GRE S QRT FCL D 1 LIN ROCL

33 D 2 P LIN is generated by Symmetric language S a 1 Sā 1 a nsā n ε NRR(S) QRT = T σ (LIN) GRE S QRT FCL D 1 LIN ROCL

34 D 2 ROCL is generated by D 1 P GRE QRT FCL S D 1 LIN ROCL

35 Lemma If L c recognizable by a counter automaton, then NRR(L c). Thanks to Abuzer Yakaryilmaz for the key-lemma. D 2 P ROCL is generated by D 1 NRR(D 1 ) GRE QRT FCL S D 1 LIN ROCL

36 D 2 ROCL is generated by D 1 NRR(D 1 ) FCL = T σ (ROCL) P GRE S QRT FCL D 1 LIN ROCL

37 D 2 ROCL is generated by D 1 NRR(D 1 ) FCL = T σ (ROCL) P GRE S LIN QRT FCL D 1 ROCL

38 D 2 NRR(FCL) P GRE S LIN QRT FCL D 1 ROCL

39 D 2 NRR(FCL) NRR(QRT) P GRE S LIN QRT FCL D 1 ROCL

40 D 2 NRR(FCL) NRR(QRT) GRE = T σ (LIN ROCL) P GRE QRT FCL S LIN D 1 ROCL

41 D 2 P NRR(FCL) NRR(QRT) GRE = T σ (LIN ROCL) NRR(GRE) GRE S QRT FCL D 1 LIN ROCL

42 D 2 P GRE S QRT FCL D 1 LIN ROCL

43 D 2 S # P GRE S QRT FCL D 1 LIN ROCL

44 D 2 Theorem NRR(S ) is P-complete under # deterministic log space reductions. S # P GRE S QRT FCL D 1 LIN ROCL

45 1 Regular realizability problems Definition Examples Properties 2 Relation with -theory Rational cones Complexity of RR-Problems 3 Rational index

46 Regular realizability problems Relation with -theory Rational index Rational index Definition The rational index ρ L (n) of a language L is a function that returns the maximum length of the shortest word from the intersection of the language L and a language L(A) recognizing by an automaton A with n states provided L(A) L : ρ L (n) = max (min{ w : w L(A) L }) A: Q A =n w A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

47 Regular realizability problems Relation with -theory Rational index Rational index Definition The rational index ρ L (n) of a language L is a function that returns the maximum length of the shortest word from the intersection of the language L and a language L(A) recognizing by an automaton A with n states provided L(A) L : ρ L (n) = max (min{ w : w L(A) L }) A: Q A =n w Theorem (Boasson, Courcelle, Nivat, 1981) If L rat L then there exists a constant c such that ρ L (n) cn(ρ L (cn) + 1). A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

48 Regular realizability problems Relation with -theory Rational index Properties of rational index Proposition Rational index of an arbitrary context-free language is bounded from below by a linear function. A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

49 Regular realizability problems Relation with -theory Rational index Properties of rational index Proposition Rational index of an arbitrary context-free language is bounded from below by a linear function. Theorem (Pierre 1992) The rational index of any generator of the rational cone of belongs to exp(θ(n 2 / log n)). A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

50 Regular realizability problems Relation with -theory Rational index Properties of rational index Proposition Rational index of an arbitrary context-free language is bounded from below by a linear function. Theorem (Pierre 1992) The rational index of any generator of the rational cone of belongs to exp(θ(n 2 / log n)). Theorem (Pierre, Farrinone, 1990) For a positive algebraic number γ > 1 there exists a context-free language with the rational index Θ(n γ ). A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

51 Regular realizability problems Relation with -theory Rational index Complexity of RR problems Theorem Let F be a context-free filter with polynomially bounded rational index, then the problem NRR(F) belongs to NSPACE(log 2 n). A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

52 Regular realizability problems Relation with -theory Rational index Complexity of RR problems Theorem Let F be a context-free filter with polynomially bounded rational index, then the problem NRR(F) belongs to NSPACE(log 2 n). Conjecture Let F be a context-free filter with polynomially bounded rational index, then the problem NRR(F) belongs to. A. Rubtsov, M. Vyalyi RR-problems and 26 June / 16

53 Thank You!

54 D 2 S # P GRE S QRT FCL D 1 LIN ROCL REG

55 1 Regular realizability problems Definition Examples Properties 2 Relation with -theory Rational cones Complexity of RR-Problems 3 Rational index