Test sets of commutative languages t epn Holub Department of - PowerPoint PPT Presentation

Test sets of commutative languages Štˇ epán Holub Department of Algebra Charles University in Prague JM’06, 29th August, Rennes Štˇ epán Holub Test sets of commutative languages

Commutative languages Commutative closure of a word u = ℓ 1 . . . ℓ n is the language c ( u ) = { ℓ σ ( 1 ) . . . ℓ σ ( n ) | σ ∈ S n } . Commutative closure of a language is � c ( L ) = c ( u ) . u ∈ L Language L is said to be commutative iff L = c ( L ) . Štˇ epán Holub Test sets of commutative languages

Test sets Two morphisms agree on a language L iff g ( u ) = h ( u ) for any u ∈ L . Write g ≡ L h A subset T of the language L is called its test set iff for any two morphisms g , h g ≡ L h ⇔ g ≡ T h . Štˇ epán Holub Test sets of commutative languages

Parikh vectors Parikh vector of a word u = a k 1 1 . . . a k n n is the n -tuple ( k 1 , . . . , k n ) . ◮ A commutative language is given by the set of its Parikh vectors ◮ A basis of the vector space over Q spanned by the Parikh vectors of a language is called Parikh basis of the language ◮ If two morphisms agree lengthwise on a Parikh basis, they agree lengthwise on the whole language. Štˇ epán Holub Test sets of commutative languages

Parikh vectors Parikh vector of a word u = a k 1 1 . . . a k n n is the n -tuple ( k 1 , . . . , k n ) . ◮ A commutative language is given by the set of its Parikh vectors ◮ A basis of the vector space over Q spanned by the Parikh vectors of a language is called Parikh basis of the language ◮ If two morphisms agree lengthwise on a Parikh basis, they agree lengthwise on the whole language. Štˇ epán Holub Test sets of commutative languages

Known bounds Theorem (Hakala, Kortelainen 1997) Any commutative language over n letters has a test of cardinality at most 3 n 2 . There is a commutative language over n letters, the smallest test set of which has cardinality at least 1 9 n 2 . Theorem (Holub, Kortelainen 2001) The commutative language c ( a k 1 1 · · · a k n n ) has a test of cardinality at most 10 n . Each test set of the commutative language c ( a 1 · · · a n ) has cardinality at least n − 1. Štˇ epán Holub Test sets of commutative languages

Known bounds Theorem (Hakala, Kortelainen 1997) Any commutative language over n letters has a test of cardinality at most 3 n 2 . There is a commutative language over n letters, the smallest test set of which has cardinality at least 1 9 n 2 . Theorem (Holub, Kortelainen 2001) The commutative language c ( a k 1 1 · · · a k n n ) has a test of cardinality at most 10 n . Each test set of the commutative language c ( a 1 · · · a n ) has cardinality at least n − 1. Štˇ epán Holub Test sets of commutative languages

Known bounds Theorem (Hakala, Kortelainen 1997) Any commutative language over n letters has a test of cardinality at most 3 n 2 . There is a commutative language over n letters, the smallest test set of which has cardinality at least 1 9 n 2 . Theorem (Holub, Kortelainen 2001) The commutative language c ( a k 1 1 · · · a k n n ) has a test of cardinality at most 10 n . Each test set of the commutative language c ( a 1 · · · a n ) has cardinality at least n − 1. Štˇ epán Holub Test sets of commutative languages

Known bounds Theorem (Hakala, Kortelainen 1997) Any commutative language over n letters has a test of cardinality at most 3 n 2 . There is a commutative language over n letters, the smallest test set of which has cardinality at least 1 9 n 2 . Theorem (Holub, Kortelainen 2001) The commutative language c ( a k 1 1 · · · a k n n ) has a test of cardinality at most 10 n . Each test set of the commutative language c ( a 1 · · · a n ) has cardinality at least n − 1. Štˇ epán Holub Test sets of commutative languages

The Main Question Commutative language can be ◮ simple: it has a Parikh test set of size O ( n ) ◮ complicated: it has only Parikh test sets of size Ω( n 2 ) What is the criterion? Štˇ epán Holub Test sets of commutative languages

The Main Question Commutative language can be ◮ simple: it has a Parikh test set of size O ( n ) ◮ complicated: it has only Parikh test sets of size Ω( n 2 ) What is the criterion? Štˇ epán Holub Test sets of commutative languages

Parikh test set A set T ⊂ L is a Parikh test set of L iff c ( T ) is a test set of c ( L ) . Two sources of the size of a test sets: ◮ Large Parikh test sets ◮ Many words with the same Parikh vector Štˇ epán Holub Test sets of commutative languages

Parikh test set A set T ⊂ L is a Parikh test set of L iff c ( T ) is a test set of c ( L ) . Two sources of the size of a test sets: ◮ Large Parikh test sets ◮ Many words with the same Parikh vector Štˇ epán Holub Test sets of commutative languages

Parikh test set A set T ⊂ L is a Parikh test set of L iff c ( T ) is a test set of c ( L ) . Two sources of the size of a test sets: ◮ Large Parikh test sets ◮ Many words with the same Parikh vector Štˇ epán Holub Test sets of commutative languages

Parikh test sets and classical test sets A set T ⊂ L is a Parikh test set of L iff c ( T ) is a test set of c ( L ) . ◮ Parikh test sets give a lower bound to cardinality of test sets ◮ If T is a Parikh test set of L , then L has a test set of cardinality 10 � T � , where � � T � = | t | . t ∈ T Štˇ epán Holub Test sets of commutative languages

A complicated language ◮ X = { a 1 , . . . , a n , b 1 , . . . , b n , c 1 , . . . , c n } , | X | = 3 n . ◮ L = c ( { a i b j c i | i , j = 1 , . . . , n } ) , | L | = ( 6 ) n 2 | T | < n 2 ⇒ c ( a n b n c n ) ∩ T = ∅ T ⊂ L , g ( a n ) = a 2 h ( c n ) = a 2 g ( b n ) = b g ( a n ) = a g ( b n ) = b g ( c n ) = a h ( a j ) = a h ( b j ) = a g ( c j ) = a h ( a j ) = a h ( b j ) = a h ( c j ) = a Štˇ epán Holub Test sets of commutative languages

A complicated language ◮ X = { a 1 , . . . , a n , b 1 , . . . , b n , c 1 , . . . , c n } , | X | = 3 n . ◮ L = c ( { a i b j c i | i , j = 1 , . . . , n } ) , | L | = ( 6 ) n 2 | T | < n 2 ⇒ c ( a n b n c n ) ∩ T = ∅ T ⊂ L , g ( a n ) = a 2 h ( c n ) = a 2 g ( b n ) = b g ( a n ) = a g ( b n ) = b g ( c n ) = a h ( a j ) = a h ( b j ) = a g ( c j ) = a h ( a j ) = a h ( b j ) = a h ( c j ) = a Štˇ epán Holub Test sets of commutative languages

A complicated language ◮ X = { a 1 , . . . , a n , b 1 , . . . , b n , c 1 , . . . , c n } , | X | = 3 n . ◮ L = c ( { a i b j c i | i , j = 1 , . . . , n } ) , | L | = ( 6 ) n 2 | T | < n 2 ⇒ c ( a n b n c n ) ∩ T = ∅ T ⊂ L , g ( a n ) = a 2 h ( c n ) = a 2 g ( b n ) = b g ( a n ) = a g ( b n ) = b g ( c n ) = a h ( a j ) = a h ( b j ) = a g ( c j ) = a h ( a j ) = a h ( b j ) = a h ( c j ) = a Štˇ epán Holub Test sets of commutative languages

A complicated language L = c ( { a i b j c i | i , j = 1 , . . . , n } ); c ( a n b n c n ) ∩ T = ∅ g ( a n ) = a 2 g ( b n ) = b g ( c n ) = a h ( c n ) = a 2 h ( a n ) = a h ( b n ) = b g ( a j ) = a g ( b j ) = a g ( c j ) = a h ( a j ) = a h ( b j ) = a h ( c j ) = a b = a = c = n n n a = b = c = j j j Many Parikh vectors?? Štˇ epán Holub Test sets of commutative languages

A complicated language L = c ( { a i b j c i | i , j = 1 , . . . , n } ); c ( a n b n c n ) ∩ T = ∅ a = b = c = n n n b = a = c = j j j Many Parikh vectors?? Štˇ epán Holub Test sets of commutative languages

A simple language with many Parikh vectors ◮ X 1 = { a 1 , . . . , a n , b 1 , . . . , b n } , | X | = 2 n . ◮ L 1 = c ( { a i b j | i , j = 1 , . . . , n } ) , | L | = 2 n 2 T = { a n b i | i = 1 , . . . , n } � { a i b n | i = 1 , . . . , n } is a test set of L with | T | = 2 n − 1, and the Parikh size n . Štˇ epán Holub Test sets of commutative languages

A simple language with many Parikh vectors ◮ X 1 = { a 1 , . . . , a n , b 1 , . . . , b n } , | X | = 2 n . ◮ L 1 = c ( { a i b j | i , j = 1 , . . . , n } ) , | L | = 2 n 2 T = { a n b i | i = 1 , . . . , n } � { a i b n | i = 1 , . . . , n } is a test set of L with | T | = 2 n − 1, and the Parikh size n . Štˇ epán Holub Test sets of commutative languages

Coincidence graph Coincidence graph of L : ◮ G(L)=(X,E) ◮ X is alphabet ◮ ( a , b ) ∈ E iff a and b occur together in a word. ◮ Undirected ◮ Loops are allowed Štˇ epán Holub Test sets of commutative languages

Difference support Example: Let a i ∈ L for each letter a i . Then the coincidence graph is irrelevant. We say that D ⊂ X is a difference support of L if there exist morphisms g and h , which are lenght equivavelnt on L , and iff | g ( a ) | � = | h ( a ) | a ∈ D . Štˇ epán Holub Test sets of commutative languages

Difference support Example: Let a i ∈ L for each letter a i . Then the coincidence graph is irrelevant. We say that D ⊂ X is a difference support of L if there exist morphisms g and h , which are lenght equivavelnt on L , and iff | g ( a ) | � = | h ( a ) | a ∈ D . Štˇ epán Holub Test sets of commutative languages

Difference support Example: Let a i ∈ L for each letter a i . Then the coincidence graph is irrelevant. We say that D ⊂ X is a difference support of L if there exist morphisms g and h , which are lenght equivavelnt on L , and iff | g ( a ) | � = | h ( a ) | a ∈ D . Štˇ epán Holub Test sets of commutative languages