On the commutative equivalence of bounded context-free and regular - PowerPoint PPT Presentation

On the commutative equivalence of bounded context-free and regular languages F. D’Alessandro 1 B. Intrigila 2 1 Dipartimento di Matematica “G. Castelnuovo” Universit` a di Roma “La Sapienza” 2 Dipartimento di Matematica Universit` a di Roma “Tor Vergata” F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 1/30

Main result Every bounded context-free language L 1 is commutatively equivalent to a regular language L 2 F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 2/30

Main result Every bounded context-free language L 1 is commutatively equivalent to a regular language L 2 There exists a bijection f : L 1 − → L 2 such that, for every u ∈ L 1 , u and f ( u ) have the same Parikh vector F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 2/30

Overview of the presentation ◮ Bounded and sparse context-free languages ◮ The problem ◮ Outline of the solution F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 3/30

Bounded languages Definition Let L ⊆ A ∗ . L is called n -bounded if there exist n words u 1 , u 2 , . . . , u n such that L ⊆ u ∗ 1 u ∗ 2 · · · u ∗ n . L is called bounded if it is n -bounded for some n F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 4/30

Sparse languages L ⊆ A ∗ The counting function of L is the map c L : N − → N such that c L ( n ) = Card( L ∩ A n ) F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 5/30

Sparse and bounded languages Definition L is sparse or poly-slender if c L ( n ) is upper bounded by a polynomial F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 6/30

Sparse and bounded languages Definition L is sparse or poly-slender if c L ( n ) is upper bounded by a polynomial Theorem (Latteux and Thierrin 1984; Ibarra and Ravikumar, 1986; Raz 1997; Ilie, Rozenberg and Salomaa 2000) A context-free language is sparse if and only if it is bounded F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 6/30

Sparse and bounded languages Theorem (D’Alessandro, Intrigila, and Varricchio, 2006) Let L be a bounded context-free language over the alphabet A Then there exists a regular language L ′ over an alphabet B such that, for all n ≥ 0 , c L ( n ) = c L ′ ( n ) F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 7/30

The problem ◮ Commutative Equivalence of languages ◮ Our problem ◮ Some classical theorems on bounded context-free languages F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 8/30

The Parikh morphism ◮ A = { a 1 , . . . , a t } ◮ ψ : A ∗ − → N t ◮ ∀ u ∈ A ∗ , ψ ( u ) = ( | u | a 1 , | u | a 2 , . . . , | u | a t ) F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 9/30

Commutative Equivalence Let L 1 , L 2 ⊆ A ∗ L 1 is commutatively equivalent to L 2 if there exists a bijection f : L 1 − → L 2 such that, for every u ∈ L 1 , ψ ( u ) = ψ ( f ( u )) F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 10/30

Main result Theorem (D.I. 2011) Let L 1 ⊆ u ∗ 1 · · · u ∗ k be bounded context-free language. Then L 1 is commutatively equivalent to a regular language L 2 F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 11/30

Main result Theorem (D.I. 2011) Let L 1 ⊆ u ∗ 1 · · · u ∗ k be bounded context-free language. Then L 1 is commutatively equivalent to a regular language L 2 Obstruction: ◮ inherently ambiguity of bounded context-free languages ◮ ambiguity of the product u ∗ 1 · · · u ∗ k in the free monoid A ∗ F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 11/30

Some classical theorems on bounded context-free languages F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 12/30

Parikh Theorem ◮ Definition Given languages L 1 , L 2 ⊆ A ∗ , L 1 is letter-equivalent (or Parikh equivalent) to L 2 if ψ ( L 1 ) = ψ ( L 2 ) . ◮ Theorem (Parikh, 1966) Given a context-free language L 1 , there exists a regular language L 2 which is letter-equivalent to L 1 F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 13/30

Parikh Theorem ◮ L 1 = ( ab ) ∗ ∪ ( ba ) ∗ , L 2 = ( ab ) ∗ ◮ ψ ( L 1 ) = ψ ( L 2 ) = { ( n, n ) : n ∈ N } ◮ L 1 cannot be commutatively equivalent to L 2 F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 14/30

Ginsburg Theorems Given words u 1 , . . . , u k ∈ A + , we define the function: φ : N k − → u ∗ 1 u ∗ 2 · · · u ∗ k , such that, for every ( n 1 , . . . , n k ) ∈ N k , φ ( n 1 , . . . , n k ) = u n 1 1 u n 2 2 · · · u n k k F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 15/30

Ginsburg Theorems φ : N k − → u ∗ 1 u ∗ 2 · · · u ∗ k , φ ( n 1 , . . . , n k ) = u n 1 1 u n 2 2 · · · u n k k Theorem (Ginsburg 1966) L ⊆ u ∗ 1 u ∗ 2 · · · u ∗ k L is context-free iff φ − 1 ( L ) is a finite union of linear sets, each having a stratified sets of periods F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 16/30

Ginsburg Theorems Theorem (Ginsburg, 1966) L ⊆ u ∗ 1 u ∗ 2 · · · u ∗ context-free k L is unambiguous iff φ − 1 ( L ) is a finite union of disjoint linear sets, each with stratified and linearly independent periods L = { a i b j c k | i, j, k ∈ N , i = j or j = k } is ambiguous F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 17/30

Outline of the solution F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 18/30

Inherent ambiguity of L 1. Faithful representation of L by a semilinear set 2. “Geometrical decomposition of semi-linear sets” [ D’Alessandro, Intrigila, and Varricchio, 2010, Quasi-polynomials, linear Diophantine equations and semi-linear sets, to appear in Theoret. Comput. Sci. ] Ambiguity of u ∗ 1 · · · u ∗ n 3. Arguments of Combinatorics of variable-length codes 4. Arguments of elementary number theory F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 19/30

Faithful representation by semilinear set Theorem ( Eilenberg Cross-section, 1974) Let α : A ∗ → B ∗ be a morphism and let L be a rational language of A ∗ . There exists a rational subset L ′ of L such that α maps bijectively L ′ of α ( L ) Theorem (Eilenberg and Sch¨ utzenberger, 1969) Every semi-linear set is represented as a finite and disjoint union of unambiguous linear sets F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 20/30

Faithful representation by semilinear set Theorem ( Eilenberg Cross-section, 1974) Let α : A ∗ → B ∗ be a morphism and let L be a rational language of A ∗ . There exists a rational subset L ′ of L such that α maps bijectively L ′ of α ( L ) Theorem (Eilenberg and Sch¨ utzenberger, 1969) Every semi-linear set is represented as a finite and disjoint union of unambiguous linear sets Every semi-linear set is semi-simple set F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 20/30

Faithful representation by semilinear set φ : N k − → u ∗ 1 u ∗ 2 · · · u ∗ k , φ ( n 1 , . . . , n k ) = u n 1 1 u n 2 2 · · · u n k k Theorem If L is bounded context-free, then there exists a semi-simple set B of N k such that φ ( B ) = L and φ is injective on B F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 21/30

The basic case φ : N k − → u ∗ 1 u ∗ 2 · · · u ∗ k , φ ( B ) = L B = { b 0 + b 1 n 1 + · · · + b m n m : n i ∈ N } b 0 , b 1 , . . . , b m ∈ N k F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 22/30

The basic case φ ( B ) = L B = { b 0 + b 1 n 1 + · · · + b m n m : n i ∈ N } F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 23/30

The basic case φ ( B ) = L B = { b 0 + b 1 n 1 + · · · + b m n m : n i ∈ N } u = φ ( b 0 + n 1 b 1 + · · · + n m b m ) F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 23/30

The basic case φ ( B ) = L B = { b 0 + b 1 n 1 + · · · + b m n m : n i ∈ N } u = φ ( b 0 + n 1 b 1 + · · · + n m b m ) Because of some elementary properties of φ and ψ , one has: F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 23/30

The basic case φ ( B ) = L B = { b 0 + b 1 n 1 + · · · + b m n m : n i ∈ N } u = φ ( b 0 + n 1 b 1 + · · · + n m b m ) Because of some elementary properties of φ and ψ , one has: ψ ( u ) = ψ ( φ ( b 0 )) + n 1 ψ ( φ ( b 1 )) + · · · + n m ψ ( φ ( b m )) F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 23/30

The basic case The latter formula suggests that a natural candidate for the commutative equivalence of L is: L ′ = φ ( b 0 ) φ ( b 1 ) ∗ · · · φ ( b m ) ∗ F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 24/30

The basic case Indeed, taking L ′ = φ ( b 0 ) φ ( b 1 ) ∗ · · · φ ( b m ) ∗ one defines the function → L ′ f : L − as: f ( u ) = f ( φ ( b 0 + n 1 b 1 + · · · + n m b m )) = F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 25/30

The basic case Indeed, taking L ′ = φ ( b 0 ) φ ( b 1 ) ∗ · · · φ ( b m ) ∗ one defines the function → L ′ f : L − as: f ( u ) = f ( φ ( b 0 + n 1 b 1 + · · · + n m b m )) = φ ( b 0 ) φ ( b 1 ) n 1 · · · φ ( b m ) n m F. D’Alessandro, B. Intrigila Prague, September 12-16, 2011 25/30