Regular realizability problems and context-free languages Alexander Rubtsov23
Mikhail Vyalyi123
1 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
1 Regular realizability problems
Definition Examples Properties
2 Relation with CFL-theory
Rational cones Complexity of RR-Problems
3 Rational index
Regular realizability problems
Relation with CFL-theory
Rational index
Definition of the problems
Filter We fix language F ⊆ Σ∗ called filter.
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
1 / 16
Regular realizability problems
Relation with CFL-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 6= ∅}
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
1 / 16
Regular realizability problems
Relation with CFL-theory
Rational index
Examples Periodic filters Per1 = {(1k #)n | k, n ∈ N} 111#111# · · · #111# ∈ Per1
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
2 / 16
Regular realizability problems
Relation with CFL-theory
Rational index
Examples Periodic filters Per1 = {(1k #)n | k, n ∈ N} 111#111# · · · #111# ∈ Per1 The problem NRR(Per1 ) is NP-complete
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
2 / 16
Regular realizability problems
Relation with CFL-theory
Rational index
Examples Periodic filters Per1 = {(1k #)n | k, n ∈ N} 111#111# · · · #111# ∈ Per1 The problem NRR(Per1 ) is NP-complete Per2 = {(w#)n | w ∈ Σ∗ , |Σ| = 2, n ∈ N} w # w # · · · # w # ∈ Per2
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
2 / 16
Regular realizability problems
Relation with CFL-theory
Rational index
Examples Periodic filters Per1 = {(1k #)n | k, n ∈ N} 111#111# · · · #111# ∈ Per1 The problem NRR(Per1 ) is NP-complete Per2 = {(w#)n | w ∈ Σ∗ , |Σ| = 2, n ∈ N} w # w # · · · # w # ∈ Per2 The problem NRR(Per2 ) is PSPACE-complete
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
2 / 16
Regular realizability problems
Relation with CFL-theory
Rational index
Examples Periodic filters Per1 = {(1k #)n | k, n ∈ N} 111#111# · · · #111# ∈ Per1 The problem NRR(Per1 ) is NP-complete Per2 = {(w#)n | w ∈ Σ∗ , |Σ| = 2, n ∈ N} w # w # · · · # w # ∈ Per2 The problem NRR(Per2 ) 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, 2009. P. 1096–1118. M.V. Paper in Russian, 2009.
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
2 / 16
Regular realizability problems
Relation with CFL-theory
Rational index
Rational dominance
Definition A language L ⊆ A∗ is rationally dominated by L0 ⊆ B∗ if there exists a rational relation R such that L = {u ∈ A∗ | ∃v ∈ L0 (v, u) ∈ R} L 6rat L0
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
3 / 16
Regular realizability problems
Relation with CFL-theory
Rational index
Rational transductions
Definition A finite state transducer (FST) T(x) : ∆∗ → Γ∗ is a (nondeterministic) automaton with output tape T = (∆, Γ, Q, q0 , δ, F), where ∆ – input alphabet; Γ – output alphabet; Q – set of states; δ : Q × ∆ ∪ {ε} × Γ ∪ {ε} × Q – transitions relation; q0 – 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 CFL
26 June 2015
4 / 16
Regular realizability problems
Relation with CFL-theory
Rational index
Rational transductions
In other words L 6rat L0 ⇔ ∃T ∈ FST : L = T(L0 )
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
5 / 16
Regular realizability problems
Relation with CFL-theory
Rational index
Reduction on filters
Proposition F1 6rat F2 ⇒ NRR(F1 ) 6log NRR(F2 )
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
6 / 16
1 Regular realizability problems
Definition Examples Properties
2 Relation with CFL-theory
Rational cones Complexity of RR-Problems
3 Rational index
Regular realizability problems
Relation with CFL-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 CFL
26 June 2015
7 / 16
Regular realizability problems
Relation with CFL-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, Sch¨ utzenberger) T (D2 ) = CFL
Dn = hS → SS | a1 S¯ a1 | · · · | an S¯ an | ε i.
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
7 / 16
Regular realizability problems
Relation with CFL-theory
Rational index
J. Berstel’s book cover
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
8 / 16
D2
CFL
GRE
FCL
QRT S
D1
LIN
ROCL
REG
D2
CFL
REG
D2
CFL
REG
T (D2 ) = CFL
D2
CFL
T (D2 ) = CFL NRR(D2 ) is P-complete
REG
D2
CFL
T (D2 ) = CFL NRR(D2 ) is P-complete L ∈ CFL ⇒ NRR(L) ∈ P
P
REG
D2
CFL
P
REG
T (Σ∗ ) = REG
D2
CFL
T (Σ∗ ) = REG NRR(Σ∗ ) is NL-complete
P
REG
D2
CFL
T (Σ∗ ) = REG NRR(Σ∗ ) is NL-complete L ∈ REG ⇒ NRR(L) ∈ NL
P
NL
REG
NL
D2
CFL
P GRE
FCL
QRT S
D1
LIN
NL
ROCL
REG
NL
D2
CFL
P GRE
QRT S
LIN
FCL
D1
ROCL
D2
LIN is generated by Symmetric language hS → a1 S¯ a1 | · · · | an S¯ an | εi
CFL
P GRE
QRT S
LIN
FCL
D1
ROCL
D2
LIN is generated by Symmetric language hS → a1 S¯ a1 | · · · | an S¯ an | εi
CFL
NRR(S) ∈ NL
P GRE
QRT S
LIN
FCL
D1
ROCL
D2
LIN is generated by Symmetric language
Th. (Anderson, Loftus, Rampersad, Santean, Shallit, 2009)
hS → a1 S¯ a1 | · · · | an S¯ an | εi
Similar theorem about palindromes:
CFL
NRR(S) ∈ NL
NRR(PAL) ∈ NL.
P GRE
QRT S
LIN NL
FCL
D1
ROCL
D2
LIN is generated by Symmetric language hS → a1 S¯ a1 | · · · | an S¯ an | εi
CFL
NRR(S) ∈ NL QRT = T σ (LIN)
P GRE
QRT S
LIN
NL
FCL
D1
ROCL
D2 Lemma If L, La for all a ∈ A, are easy languages then σ(L) is also easy.
LIN is generated by Symmetric language hS → a1 S¯ a1 | · · · | an S¯ an | εi
CFL
NRR(S) ∈ NL QRT = T σ (LIN)
P GRE
QRT S
LIN
NL
FCL
D1
ROCL
D2
LIN is generated by Symmetric language hS → a1 S¯ a1 | · · · | an S¯ an | εi
CFL
NRR(S) ∈ NL QRT = T σ (LIN)
P GRE
QRT S
NL LIN
NL
FCL
D1
ROCL
D2 ROCL is generated by D1
CFL
P GRE
QRT S
LIN
FCL
D1
ROCL
D2 Lemma If Lc recognizable by a counter automaton, then NRR(Lc ) ∈ NL.
ROCL is generated by D1 NRR(D1 ) ∈ NL
CFL
Thanks to Abuzer Yakaryilmaz for the key-lemma.
P GRE
QRT S
LIN
FCL
D1
ROCL NL
D2 ROCL is generated by D1
CFL
NRR(D1 ) ∈ NL FCL = T σ (ROCL)
P GRE
QRT S
LIN
FCL D1
ROCL
NL
D2 ROCL is generated by D1
CFL
NRR(D1 ) ∈ NL FCL = T σ (ROCL)
P GRE
QRT S
NL LIN
FCL D1
ROCL
NL
D2
NRR(FCL) ⊆ NL
CFL
P GRE
QRT S
NL LIN
FCL D1
ROCL
NL
D2
NRR(FCL) ⊆ NL NRR(QRT) ⊆ NL
CFL
P GRE
QRT FCL S
NL
D1
LIN
ROCL
NL
NL
D2
NRR(FCL) ⊆ NL NRR(QRT) ⊆ NL
CFL
GRE = T σ (LIN ∪ ROCL)
P GRE QRT S
NL
FCL
D1
LIN
ROCL
NL
NL
D2
NRR(FCL) ⊆ NL NRR(QRT) ⊆ NL
CFL
GRE = T σ (LIN ∪ ROCL)
P
NRR(GRE) ⊆ NL
GRE
FCL
QRT S
LIN
NL D1
ROCL
D2
CFL
P GRE
FCL
QRT S
LIN
NL D1
ROCL
D2
CFL S↑#
P
GRE
FCL
QRT S
LIN
NL D1
ROCL
D2 Theorem NRR(S↑# ) is P-complete under deterministic log space reductions.
CFL S↑#
P
GRE
FCL
QRT S
LIN
NL D1
ROCL
1 Regular realizability problems
Definition Examples Properties
2 Relation with CFL-theory
Rational cones Complexity of RR-Problems
3 Rational index
Regular realizability problems
Relation with CFL-theory
Rational index
Rational index
Definition The rational index ρL (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 6= ∅: ρL (n) =
A. Rubtsov, M. Vyalyi
max A:|QA |=n
(min{|w| : w ∈ L(A) ∩ L 6= ∅}) w
RR-problems and CFL
26 June 2015
13 / 16
Regular realizability problems
Relation with CFL-theory
Rational index
Rational index
Definition The rational index ρL (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 6= ∅: ρL (n) =
max A:|QA |=n
(min{|w| : w ∈ L(A) ∩ L 6= ∅}) w
Theorem (Boasson, Courcelle, Nivat, 1981) If L0 6rat L then there exists a constant c such that ρL0 (n) 6 cn(ρL (cn) + 1).
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
13 / 16
Regular realizability problems
Relation with CFL-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 CFL
26 June 2015
14 / 16
Regular realizability problems
Relation with CFL-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 CFL belongs to exp(Θ(n2 / log n)).
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
14 / 16
Regular realizability problems
Relation with CFL-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 CFL belongs to exp(Θ(n2 / log n)). Theorem (Pierre, Farrinone, 1990) For a positive algebraic number γ > 1 there exists a context-free language with the rational index Θ(nγ ).
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
14 / 16
Regular realizability problems
Relation with CFL-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(log2 n).
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
15 / 16
Regular realizability problems
Relation with CFL-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(log2 n).
Conjecture Let F be a context-free filter with polynomially bounded rational index, then the problem NRR(F) belongs to NL.
A. Rubtsov, M. Vyalyi
RR-problems and CFL
26 June 2015
15 / 16
Thank You!
D2
CFL ↑ S#
P
GRE
FCL
QRT S
NL
LIN
NL
D1
ROCL
REG
NL
1 Regular realizability problems
Definition Examples Properties
2 Relation with CFL-theory
Rational cones Complexity of RR-Problems
3 Rational index