Compatibility relations on codes and free monoids University of - PowerPoint PPT Presentation

Tomi Kärki Compatibility relations on codes and free monoids University of Turku and Turku Centre for Computer Science (TUCS)

Introduction 2

Introduction 2

Introduction 2

Introduction 2

Introduction 2

Outline of Topics • Word relations • Relational codes • Minimal and maximal relations • Relationally free monoids and stability • Hulls • Defect effect 3

Notations an alphabet A empty word ε a set of words over A ∗ X R ⊆ X × X a relation on X ( x, y ) ∈ R x R y { ( x, x ) | x ∈ X } ι X { ( x, y ) | x, y ∈ X } Ω X � R � reflexive and symmetric closure of R R ∩ ( Y × Y ) R Y { u ∈ A ∗ | ∃ x ∈ X : x R u } R ( X ) 4

Word relations • compatibility relation = reflexive and symmetric 5

Word relations • compatibility relation = reflexive and symmetric • word relation R = compatibility relation and a 1 · · · a m R b 1 · · · b n ⇔ m = n and a i R b i for all i = 1 , 2 , . . . , m where a 1 , . . . , a m , b 1 , . . . , b n ∈ A 5

Word relations • compatibility relation = reflexive and symmetric • word relation R = compatibility relation and a 1 · · · a m R b 1 · · · b n ⇔ m = n and a i R b i for all i = 1 , 2 , . . . , m where a 1 , . . . , a m , b 1 , . . . , b n ∈ A • If u R v , then words u and v are R -compatible 5

Word relations • compatibility relation = reflexive and symmetric • word relation R = compatibility relation and a 1 · · · a m R b 1 · · · b n ⇔ m = n and a i R b i for all i = 1 , 2 , . . . , m where a 1 , . . . , a m , b 1 , . . . , b n ∈ A • If u R v , then words u and v are R -compatible u R v, u ′ R v ′ ⇒ uu ′ R vv ′ , � multiplicativity: • uu ′ R vv ′ , | u | = | v | ⇒ u R v, u ′ R v ′ simplifiability: 5

Word relations Example 1. A = { a, b, c } R = �{ ( a, b ) }� = { ( a, a ) , ( b, b ) , ( c, c ) , ( a, b ) , ( b, a ) } 6

Word relations Example 1. A = { a, b, c } R = �{ ( a, b ) }� = { ( a, a ) , ( b, b ) , ( c, c ) , ( a, b ) , ( b, a ) } abba R baab 6

Word relations Example 1. A = { a, b, c } R = �{ ( a, b ) }� = { ( a, a ) , ( b, b ) , ( c, c ) , ( a, b ) , ( b, a ) } abba R baab abc � R cbc 6

Word relations Example 1. A = { a, b, c } R = �{ ( a, b ) }� = { ( a, a ) , ( b, b ) , ( c, c ) , ( a, b ) , ( b, a ) } abba R baab abc � R cbc Example 2. 6

Word relations Example 1. A = { a, b, c } R = �{ ( a, b ) }� = { ( a, a ) , ( b, b ) , ( c, c ) , ( a, b ) , ( b, a ) } abba R baab abc � R cbc Example 2. Partial words 6

Word relations Example 1. A = { a, b, c } R = �{ ( a, b ) }� = { ( a, a ) , ( b, b ) , ( c, c ) , ( a, b ) , ( b, a ) } abba R baab abc � R cbc Example 2. Partial words k n ♦ w l ♦ dg e ♦ n o w ♦♦ dg ♦ k n o w l e dg e 6

Word relations Example 1. A = { a, b, c } R = �{ ( a, b ) }� = { ( a, a ) , ( b, b ) , ( c, c ) , ( a, b ) , ( b, a ) } abba R baab abc � R cbc Example 2. Partial words k n ♦ w l ♦ dg e ♦ n o w ♦♦ dg ♦ k n o w l e dg e R ↑ = �{ ( ♦ , a ) | a ∈ A }� 6

Relational codes • Let R and S be word relations 7

Relational codes • Let R and S be word relations • X ⊆ A ∗ is an ( R, S ) -code if for all n, m ≥ 1 and x 1 , . . . , x m , y 1 , . . . , y n ∈ X , we have x 1 · · · x m R y 1 · · · y n ⇒ n = m and x i S y i for i = 1 , 2 , . . . , m 7

Relational codes • Let R and S be word relations • X ⊆ A ∗ is an ( R, S ) -code if for all n, m ≥ 1 and x 1 , . . . , x m , y 1 , . . . , y n ∈ X , we have x 1 · · · x m R y 1 · · · y n ⇒ n = m and x i S y i for i = 1 , 2 , . . . , m • ( R, S ) -code relational code ( R, ι ) -code strong R -code ( R, R ) -code weak R -code ( ι, ι ) -code code 7

Relational codes Example. A = { a, b, c } X = { ab, c } S = ι R = ι R = �{ ( a, c ) }� R = �{ ( a, c ) , ( b, c ) }� 8

Relational codes Example. A = { a, b, c } X = { ab, c } S = ι R = ι (prefix) code R = �{ ( a, c ) }� R = �{ ( a, c ) , ( b, c ) }� 8

Relational codes Example. A = { a, b, c } X = { ab, c } S = ι R = ι (prefix) code R = �{ ( a, c ) }� ( R, ι ) -code R = �{ ( a, c ) , ( b, c ) }� 8

Relational codes Example. A = { a, b, c } X = { ab, c } S = ι R = ι (prefix) code R = �{ ( a, c ) }� ( R, ι ) -code R = �{ ( a, c ) , ( b, c ) }� ab R c.c 8

Relational codes x 1 · · · x m R y 1 · · · y n ⇒ n = m and x i S y i for i = 1 , 2 , . . . , m 9

Relational codes x 1 · · · x m R y 1 · · · y n ⇒ n = m and x i S y i for i = 1 , 2 , . . . , m Ω Ω . . . . . . R 2 S 2 R 1 S 1 . . . . . . ι ι 9

Relational codes x 1 · · · x m R y 1 · · · y n ⇒ n = m and x i S y i for i = 1 , 2 , . . . , m Ω Ω Theorem 3. Every ( R, S ) -code X is a code. . . . . . . R 2 S 2 R 1 S 1 . . . . . . ι ι 9

Relational codes x 1 · · · x m R y 1 · · · y n ⇒ n = m and x i S y i for i = 1 , 2 , . . . , m Ω Ω Theorem 3. Every ( R, S ) -code X is a code. . . . . . . Theorem 4. Let X be a subset of A ∗ . X R 2 S 2 is an ( R, S ) -code ⇔ X is an ( R, R ) -code and R X ⊆ S X . R 1 S 1 . . . . . . ι ι 9

Modified Sardinas-Patterson algorithm 10

Modified Sardinas-Patterson algorithm • finite X ⊆ A + 10

Modified Sardinas-Patterson algorithm • finite X ⊆ A + • U 1 = R ( X ) − 1 X \ { ε } 10

Modified Sardinas-Patterson algorithm • finite X ⊆ A + • U 1 = R ( X ) − 1 X \ { ε } • U n +1 = R ( X ) − 1 U n ∪ R ( U n ) − 1 X for n ≥ 1 10

Modified Sardinas-Patterson algorithm • finite X ⊆ A + • U 1 = R ( X ) − 1 X \ { ε } • U n +1 = R ( X ) − 1 U n ∪ R ( U n ) − 1 X for n ≥ 1 • Let i ≥ 2 satisfy U i = U i − t for some t > 0 10

Modified Sardinas-Patterson algorithm • finite X ⊆ A + • U 1 = R ( X ) − 1 X \ { ε } • U n +1 = R ( X ) − 1 U n ∪ R ( U n ) − 1 X for n ≥ 1 • Let i ≥ 2 satisfy U i = U i − t for some t > 0 • X is a weak R -code if and only if i − 1 � ε �∈ U j j =1 10

Modified Sardinas-Patterson algorithm Example. A = { a, b, c } X = { abb, ca, c } R = �{ ( a, b ) , ( b, c ) }� 11

Modified Sardinas-Patterson algorithm Example. A = { a, b, c } X = { abb, ca, c } R = �{ ( a, b ) , ( b, c ) }� U 1 = R ( X ) − 1 X \ { ε } = { a } 11

Modified Sardinas-Patterson algorithm Example. A = { a, b, c } X = { abb, ca, c } R = �{ ( a, b ) , ( b, c ) }� U 1 = R ( X ) − 1 X \ { ε } = { a } U 2 = R ( X ) − 1 U 1 ∪ R ( U 1 ) − 1 X = ∅ ∪ { bb } 11

Modified Sardinas-Patterson algorithm Example. A = { a, b, c } X = { abb, ca, c } R = �{ ( a, b ) , ( b, c ) }� U 1 = R ( X ) − 1 X \ { ε } = { a } U 2 = R ( X ) − 1 U 1 ∪ R ( U 1 ) − 1 X = ∅ ∪ { bb } U 3 = R ( X ) − 1 U 2 ∪ R ( U 2 ) − 1 X = { ε, b } ∪ { ε, b } 11

Modified Sardinas-Patterson algorithm Example. A = { a, b, c } X = { abb, ca, c } R = �{ ( a, b ) , ( b, c ) }� U 1 = R ( X ) − 1 X \ { ε } = { a } U 2 = R ( X ) − 1 U 1 ∪ R ( U 1 ) − 1 X = ∅ ∪ { bb } U 3 = R ( X ) − 1 U 2 ∪ R ( U 2 ) − 1 X = { ε, b } ∪ { ε, b } = ⇒ X is not an ( R, R ) -code ca.ca R c.abb 11

Minimal and maximal relations S ∈ S min ( X, R ) : X is an ( R, S ) -code ∀ S ′ ⊂ S : X is not an ( R, S ′ ) -code 12

Minimal and maximal relations S ∈ S min ( X, R ) : X is an ( R, S ) -code ∀ S ′ ⊂ S : X is not an ( R, S ′ ) -code S ∈ S max ( X, R ) : X is an ( R, S ) -code ∀ S ′ ⊃ S : X is not an ( R, S ′ ) -code R ∈ R min ( X, S ) : X is an ( R, S ) -code ∀ R ′ ⊂ R : X is not an ( R ′ , S ) -code R ∈ R max ( X, S ) : X is an ( R, S ) -code ∀ R ′ ⊃ R : X is not an ( R ′ , S ) -code 12

Minimal and maximal relations S ∈ S min ( X, R ) : X is an ( R, S ) -code ∀ S ′ ⊂ S : X is not an ( R, S ′ ) -code S ∈ S max ( X, R ) : X is an ( R, S ) -code ∀ S ′ ⊃ S : X is not an ( R, S ′ ) -code R ∈ R min ( X, S ) : X is an ( R, S ) -code ∀ R ′ ⊂ R : X is not an ( R ′ , S ) -code R ∈ R max ( X, S ) : X is an ( R, S ) -code ∀ R ′ ⊃ R : X is not an ( R ′ , S ) -code • S max ( X, R ) = { Ω } 12

Minimal and maximal relations S ∈ S min ( X, R ) : X is an ( R, S ) -code ∀ S ′ ⊂ S : X is not an ( R, S ′ ) -code S ∈ S max ( X, R ) : X is an ( R, S ) -code ∀ S ′ ⊃ S : X is not an ( R, S ′ ) -code R ∈ R min ( X, S ) : X is an ( R, S ) -code ∀ R ′ ⊂ R : X is not an ( R ′ , S ) -code R ∈ R max ( X, S ) : X is an ( R, S ) -code ∀ R ′ ⊃ R : X is not an ( R ′ , S ) -code • S max ( X, R ) = { Ω } • R min ( X, S ) = { ι } 12

Minimal and maximal relations • S min ( X, R ) is a unique element 13

Minimal and maximal relations • S min ( X, R ) is a unique element • finding S min ( X, R ) easy 13

Minimal and maximal relations • S min ( X, R ) is a unique element • finding S min ( X, R ) easy • R max ( X, S ) can contain relations of different size 13

Minimal and maximal relations • S min ( X, R ) is a unique element • finding S min ( X, R ) easy • R max ( X, S ) can contain relations of different size • finding R max ( X, S ) hard for arbitrary alphabets 13