Weakly-unambiguous Parikh automata and their link with holonomic - PowerPoint PPT Presentation

Weakly-unambiguous Parikh automata and their link with holonomic series Alin Bostan, Arnaud Carayol, Florent Koechlin, Cyril Nicaud LIGM UMR 8049 CNRS May 2020, 12th 1/39

Link between languages and combinatorics � � x | w | = ℓ n x n L ( x ) = ℓ n : number of words of length n w ∈ L n ∈ N Formal languages Generating series − → L ( x ) L 2/39

Link between languages and combinatorics � � x | w | = ℓ n x n L ( x ) = ℓ n : number of words of length n w ∈ L n ∈ N Formal languages Generating series − → L ( x ) L Regular − → rational L ( x ) = P ( x ) / Q ( x ) b a b � q 0 ( x ) = xq 0 ( x ) + xq 1 ( x ) 0 1 q 1 ( x ) = 1 + xq 1 ( x ) + xq 0 ( x ) a x L ( x ) = 1 − 2 x 2/39

Link between languages and combinatorics � | w | a 1 . . . x | w | ar L ( x 1 , . . . , x r ) = x Σ = { a 1 , . . . , a r } r 1 w ∈ L Formal languages Generating series − → L ( x ) L Regular − → rational L ( x ) = P ( x ) / Q ( x ) a b b � q 0 ( x a , x b ) = x a q 0 ( x a , x b ) + x b q 1 ( x a , x b ) 0 1 q 1 ( x a , x b ) = 1 + x b q 1 ( x a , x b ) + x a q 0 ( x a , x b ) a x b L ( x a , x b ) = 1 − ( x a + x b ) 2/39

Link between languages and combinatorics � � x | w | = ℓ n x n L ( x ) = ℓ n : number of words of length n w ∈ L n ∈ N Formal languages Generating series − → L ( x ) L Regular − → rational L ( x ) = P ( x ) / Q ( x ) Unambiguous context-free − → algebraic P ( x , L ( x )) = 0 � � S → aSB | ε S ( x ) = xS ( x ) B ( x ) + 1 B → cB | bS B ( x ) = xB ( x ) + xS ( x ) x 2 S ( x ) 2 − ( 1 − x ) S ( x ) + 1 − x = 0 2/39

Link between languages and combinatorics � | w | a 1 . . . x | w | ar L ( x 1 , . . . , x r ) = x Σ = { a 1 , . . . , a r } r 1 w ∈ L Formal languages Generating series − → L ( x ) L Regular − → rational L ( x ) = P ( x ) / Q ( x ) Unambiguous context-free − → algebraic P ( x , L ( x )) = 0 � � S → aSB | ε S ( � x ) = x a S ( � x ) B ( � x ) + 1 B → cB | bS B ( � x ) = x c B ( � x ) + x b S ( � x ) x a x b S ( x a , x b , x c ) 2 − ( 1 − x c ) S ( x a , x b , x c ) + 1 − x c = 0 2/39

Link between languages and combinatorics � | w | a 1 . . . x | w | ar L ( x 1 , . . . , x r ) = x Σ = { a 1 , . . . , a r } r 1 w ∈ L Formal languages Generating series − → L ( x ) L Regular − → rational L ( x ) = P ( x ) / Q ( x ) ambiguous context-free − → ? � � S → aSB | ε S ( � x ) = x a S ( � x ) B ( � x ) + 1 B → cB | bS B ( � x ) = x c B ( � x ) + x b S ( � x ) x a x b S ( x a , x b , x c ) 2 − ( 1 − x c ) S ( x a , x b , x c ) + 1 − x c = 0 2/39

Link between languages and combinatorics � � x | w | = ℓ n x n L ( x ) = ℓ n : number of words of length n w ∈ L n ∈ N Formal languages Generating series − → L ( x ) L Regular − → rational L ( x ) = P ( x ) / Q ( x ) Unambiguous context-free − → algebraic P ( x , L ( x )) = 0 1 − 2 x + 225 x 2 ( 1 − 25 x )( 625 x 2 + 14 x + 1 ) = 1 + 9 x + 49 x 2 + . . . [ Bousquet-Mélou 08 ] G ( x ) = 1 + 2 x + 11 x 2 + . . . [ Bostan & Kauers 10, Drmota & Banderier 13 ] 2/39

Analytic criteria for inherent ambiguity Theorem (Chomsky and Schützenberger 63) The generating series of an unambiguous context-free language is algebraic. Contraposition If the generating series of a context-free language is not algebraic, then it is inherently ambiguous. 3/39

Detailed Example Example (Flajolet 87) D = { a n 1 b a n 2 b . . . a n k b : k ∈ N ∗ , n 1 = 1 and ∃ j < k , n j + 1 � = 2 n j } is inherently ambiguous. aab / ∈ D abaabaaab ∈ D abaabaaaab / ∈ D ab a 2 b a 4 b . . . a 2 k − 1 b / ∈ D 4/39

Detailed Example Example (Flajolet 87) D = { a n 1 b a n 2 b . . . a n k b : k ∈ N ∗ , n 1 = 1 and ∃ j < k , n j + 1 � = 2 n j } is inherently ambiguous. By contradiction, suppose D is unambiguous. Then D ( x ) is algebraic Aim: build from D ( x ) a series that is not algebraic and use closure properties 5/39

Detailed Example Example (Flajolet 87) D = { a n 1 b a n 2 b . . . a n k b : k ∈ N ∗ , n 1 = 1 and ∃ j < k , n j + 1 � = 2 n j } is inherently ambiguous. By contradiction, suppose D is unambiguous. Then D ( x ) is algebraic Aim: build from D ( x ) a series that is not algebraic and use closure properties B = ab ( ab ∗ ) ∗ \ D = { ab a 2 b a 4 b . . . a 2 k − 1 b : k ∈ N ∗ } x 2 B ( x ) = 1 − x − D ( x ) = algebraic x 1 − So B ( x ) = � k ≥ 1 x 2 k − 1 + k , which is lacunary So B ( x ) is not algebraic. Contradiction 5/39

Remarks on this method Analytic criteria for solving some instances of an undecidable problem It can avoid technical proofs on automata based on pumping techniques. L = { a n b m c p : n = m or m = p } is inherently ambiguous as a 2 1 CF language yet L ( x ) = ( 1 − x 2 )( 1 − x ) − 1 − x 3 is rational Specific about inherent ambiguity questions. → language of primitive words L P aabb ∈ L P , abab / ∈ L P CFL: open not unambiguous CFL: [Peterson 96] 6/39

Hierarchy of languages and series Language Generating series − → L ( x ) L Regular − → rational Q ( x ) L ( x ) = P ( x ) � � Unambiguous context-free − → algebraic P ( x , L ( x )) = 0 � ? − → holonomic P ( x , ∂ x ) · L ( x ) = 0 7/39

Holonomic series in one variable (Stanley 80) n a n x n is holonomic (or D-finite) if it satisfies a A series f ( x ) = � differential equation of the form: P k ( x ) f ( k ) ( x ) + . . . + P 0 ( x ) f ( x ) = 0 with P i ( x ) ∈ Q [ x ] Equivalently a n satisfies a linear recurrence of the form p r ( n ) a n + r + . . . + p 0 ( n ) a n = 0 with p i ( n ) ∈ Q [ n ] Closed by sum, product, composition with algebraic series, Hadamard product... 8/39

Example of holonomic series rational series F = P / Q : ( PQ ) F ′ + ( PQ ′ − P ′ Q ) F = 0 → Linear recurrence with constant coefficients algebraic series (the proof is however not straightforward) F ( x ) = √ 1 − x := � 4 − n � 2 n � x n satisfies F 2 − 1 − x = 0 1 − 2 n n 2 ( 1 − x ) F ′ − F = 0 2 ( n + 1 ) u n + 1 − ( 2 n + 1 ) u n = 0 F ( x ) = e x := � x n / n ! is holonomic but is not algebraic F ′ − F = 0 ( n + 1 ) u n + 1 − u n = 0 9/39

Holonomic series in several variables (Lipshitz 89) A series f ( x 1 , . . . , x n ) is holonomic (or D-finite) if it satisfies a system of partial derivative equations of the form:  x ) ∂ r 1 A 1 , r 1 ( � x 1 f ( � x ) + . . . + A 1 , 1 ( � x ) ∂ x 1 f ( � x ) + A 1 , 0 ( � x ) f ( � x ) = 0      . . .     x ) ∂ r n  A n , r n ( � x n f ( � x ) + . . . + A n , 1 ( � x ) ∂ x n f ( � x ) + A n , 0 ( � x ) f ( � x ) = 0 with A i , j ( � x ) ∈ Q [ � x ] , and � x = ( x 1 , . . . , x n ) . We only use closure properties rather than the definition 10/39

Holonomic series in several variables Theorem (Lipshitz 1988, 1989) Holonomic series are closed under : 1 arithmetic operations + , × , − 2 specialization to 1, when it is well-defined: if f ( x 1 , . . . , x n ) is holonomic, then f ( x , 1 , . . . , 1 ) is holonomic too 3 Hadamard’s product ⊙ � a ( i 1 , . . . , i n ) x i 1 1 . . . x i n f ( x 1 , . . . , x n ) = n i ∈ N n � b ( i 1 , . . . , i n ) x i 1 1 . . . x i n g ( x 1 , . . . , x n ) = n i ∈ N n � a ( i 1 , . . . , i n ) b ( i 1 , . . . , i n ) x i 1 1 . . . x i n f ⊙ g ( x 1 , . . . , x n ) = n i ∈ N n 11/39

Crucial particular case: support series Let S ⊆ N n . The support series of S is � x i 1 1 . . . x i n g ( x 1 , . . . , x n ) = n ( i 1 ,..., i n ) ∈S � a ( i 1 , . . . , i n ) x i 1 1 . . . x i n Let f ( x 1 , . . . , x n ) = n . Then: ( i 1 ,..., i n ) ∈ N n � a ( i 1 , . . . , i n ) x i 1 1 . . . x i n ( f ⊙ g )( x 1 , . . . , x n ) = n ( i 1 ,..., i n ) ∈S 12/39

Example of Hadamard’s product Example Ω 3 = { w ∈ ( a + b + c ) ∗ : | w | a � = | w | b or | w | b � = | w | c } . abbca ∈ Ω 3 , abbcca / ∈ Ω 3 . Ω 3 is context-free, inherently ambiguous as a CFL. 1 1 1 Ω 3 ( x a , x b , x c ) = ⊙ ( ( 1 − x a )( 1 − x b )( 1 − x c ) − 1 − x a x b x c ) 1 − ( x a + x b + x c ) � �� � � �� � ( a + b + c ) ∗ | w | a � = | w | b or | w | b � = | w | c 1 1 1 = 1 − ( x a + x b + x c ) − 1 − ( x a + x b + x c ) ⊙ 1 − x a x b x c 13/39

Example of Hadamard’s product Example Ω 3 = { w ∈ ( a + b + c ) ∗ : | w | a � = | w | b or | w | b � = | w | c } . 1 1 1 1 1 1 − ( x a + x b + x c ) ⊙ 1 − x a x b x c = [ y − 1 a y − 1 b y − 1 ] ya + xb 1 − ( xa yb + xc c y a y b y c 1 − y a y b y c yc ) Mgfun [Chyzak] and gfun [Salvy and Zimmermann] give: x ) ∂ 3 x ) ∂ 2 p 3 ( � x a Ω 3 ( � x ) + p 2 ( � x a Ω 3 ( � x ) + p 1 ( � x ) ∂ x a Ω 3 ( � x ) + p 0 ( � x )Ω 3 ( � x ) = 0 with � p i � ∞ ≤ 7344 and deg( p i ) ≤ 9. 14/39