[ ] A Nominal Techniques Warsaw, 10-15 September, 2019 FoPSS - PowerPoint PPT Presentation

Computer Science 101 Theorem: Every algorithm to sort numbers must work n in time . Ω ( n log n ) in the comparison model Here, numbers are atoms accessible via relations: < = This amounts to restricting the class of legal atom renamings. FoPSS, Warsaw, 10-11/09/19 18

[ ] A Nominal Sets: Basic Defnitions

[ ] A or: Sets with Atoms Nominal Sets: Basic Defnitions

Atoms Let be an infinite, countable set of atoms. A a, b, c, d, e, . . . ∈ A FoPSS, Warsaw, 10-11/09/19 20

Atoms Let be an infinite, countable set of atoms. A a, b, c, d, e, . . . ∈ A Aut( A ) - the group of all bijections of A FoPSS, Warsaw, 10-11/09/19 20

Atoms Let be an infinite, countable set of atoms. A a, b, c, d, e, . . . ∈ A Aut( A ) - the group of all bijections of A ( π · σ ) · ρ = π · ( σ · ρ ) π · π − 1 = id π · id = π = id · π FoPSS, Warsaw, 10-11/09/19 20

Atoms Let be an infinite, countable set of atoms. A a, b, c, d, e, . . . ∈ A Aut( A ) - the group of all bijections of A ( π · σ ) · ρ = π · ( σ · ρ ) the dot omitted π · π − 1 = id frow now on π · id = π = id · π FoPSS, Warsaw, 10-11/09/19 20

Atoms Let be an infinite, countable set of atoms. A a, b, c, d, e, . . . ∈ A Aut( A ) - the group of all bijections of A ( π · σ ) · ρ = π · ( σ · ρ ) the dot omitted π · π − 1 = id frow now on π · id = π = id · π ( a b ) ∈ Aut( A ) - the swap of and b a FoPSS, Warsaw, 10-11/09/19 20

Atoms Let be an infinite, countable set of atoms. A a, b, c, d, e, . . . ∈ A Aut( A ) - the group of all bijections of A ( π · σ ) · ρ = π · ( σ · ρ ) the dot omitted π · π − 1 = id frow now on π · id = π = id · π ( a b ) ∈ Aut( A ) - the swap of and b a For example: ( a b )( b c )( c a ) = ( b c ) ( a b ) − 1 = ( a b ) FoPSS, Warsaw, 10-11/09/19 20

Von Neumann hierarchy A hierarchy of universes: U 0 = ∅ U α +1 = PU α U β = S α < β U α defined for every ordinal number. FoPSS, Warsaw, 10-11/09/19 21

Von Neumann hierarchy A hierarchy of universes: U 0 = ∅ U α +1 = PU α U β = S α < β U α defined for every ordinal number. Elements of sets are other sets, in a well founded way FoPSS, Warsaw, 10-11/09/19 21

Von Neumann hierarchy A hierarchy of universes: U 0 = ∅ U α +1 = PU α U β = S α < β U α defined for every ordinal number. Elements of sets are other sets, in a well founded way Every set sits somewhere in this hierarchy. FoPSS, Warsaw, 10-11/09/19 21

Sets with atoms A - a countable set of atoms FoPSS, Warsaw, 10-11/09/19 22

Sets with atoms A - a countable set of atoms A hierarchy of universes: U 0 = ∅ U α +1 = PU α + A U β = S α < β U α FoPSS, Warsaw, 10-11/09/19 22

Sets with atoms A - a countable set of atoms A hierarchy of universes: U 0 = ∅ U α +1 = PU α + A U β = S α < β U α Elements of sets with atoms are atoms or other sets with atoms, in a well founded way FoPSS, Warsaw, 10-11/09/19 22

Renaming atoms A canonical renaming action: : U × Aut( A ) → U · FoPSS, Warsaw, 10-11/09/19 23

Renaming atoms A canonical renaming action: : U × Aut( A ) → U · a · π = π ( a ) X · π = { x · π | x ∈ X } FoPSS, Warsaw, 10-11/09/19 23

Renaming atoms A canonical renaming action: : U × Aut( A ) → U · a · π = π ( a ) X · π = { x · π | x ∈ X } This is a group action of : Aut( A ) x · ( πσ ) = ( x · π ) · σ x · id = x FoPSS, Warsaw, 10-11/09/19 23

Renaming atoms A canonical renaming action: : U × Aut( A ) → U · a · π = π ( a ) X · π = { x · π | x ∈ X } This is a group action of : Aut( A ) x · ( πσ ) = ( x · π ) · σ x · id = x Fact: For every , the function π · π is a bijection on . U FoPSS, Warsaw, 10-11/09/19 23

Finite support S ⊆ A supports if x implies ∀ a ∈ S. π ( a ) = a x · π = x FoPSS, Warsaw, 10-11/09/19 24

Finite support S ⊆ A supports if x implies ∀ a ∈ S. π ( a ) = a x · π = x π ∈ Aut S ( A ) FoPSS, Warsaw, 10-11/09/19 24

Finite support S ⊆ A supports if x implies ∀ a ∈ S. π ( a ) = a x · π = x π ∈ Aut S ( A ) A legal set with atoms, or nominal set: - has a finite support, - every element of it has a finite support, - and so on. FoPSS, Warsaw, 10-11/09/19 24

Finite support S ⊆ A supports if x implies ∀ a ∈ S. π ( a ) = a x · π = x π ∈ Aut S ( A ) A legal set with atoms, or nominal set: - has a finite support, - every element of it has a finite support, - and so on. A set is equivariant if it has empty support. FoPSS, Warsaw, 10-11/09/19 24

Examples is supported by { a } a ∈ A FoPSS, Warsaw, 10-11/09/19 25

Examples is supported by { a } a ∈ A is equivariant A FoPSS, Warsaw, 10-11/09/19 25

Examples is supported by { a } a ∈ A is equivariant A is supported by S ⊆ A S FoPSS, Warsaw, 10-11/09/19 25

Examples is supported by { a } a ∈ A is equivariant A is supported by S ⊆ A S is supported by A \ S S FoPSS, Warsaw, 10-11/09/19 25

Examples is supported by { a } a ∈ A is equivariant A is supported by S ⊆ A S is supported by A \ S S Fact: is fin. supp. iff it is finite or co-finite S ⊆ A FoPSS, Warsaw, 10-11/09/19 25

Examples is supported by { a } a ∈ A is equivariant A is supported by S ⊆ A S is supported by A \ S S Fact: is fin. supp. iff it is finite or co-finite S ⊆ A A (2) = { ( d, e ) | d, e 2 A , d 6 = e } is equivariant FoPSS, Warsaw, 10-11/09/19 25

Examples is supported by { a } a ∈ A is equivariant A is supported by S ⊆ A S is supported by A \ S S Fact: is fin. supp. iff it is finite or co-finite S ⊆ A A (2) = { ( d, e ) | d, e 2 A , d 6 = e } is equivariant ✓ A ◆ = {{ d, e } | d, e 2 A , d 6 = e } is equivariant 2 FoPSS, Warsaw, 10-11/09/19 25

[ ] A Basic Properties

Closure properties Fact: if and are legal sets then X Y , , , , are legal. X + Y X \ Y X × Y X ∪ Y X ∩ Y FoPSS, Warsaw, 10-11/09/19 27

Closure properties Fact: if and are legal sets then X Y , , , , are legal. X + Y X \ Y X × Y X ∪ Y X ∩ Y Indeed: if S supports and supports T X Y then S ∪ T supports , , ... X ∪ Y X ∩ Y FoPSS, Warsaw, 10-11/09/19 27

Closure properties Fact: if and are legal sets then X Y , , , , are legal. X + Y X \ Y X × Y X ∪ Y X ∩ Y Indeed: if S supports and supports T X Y then S ∪ T supports , , ... X ∪ Y X ∩ Y (But: does not support !) S ∩ T X ∩ Y FoPSS, Warsaw, 10-11/09/19 27

Closure properties Fact: if and are legal sets then X Y , , , , are legal. X + Y X \ Y X × Y X ∪ Y X ∩ Y Indeed: if S supports and supports T X Y then S ∪ T supports , , ... X ∪ Y X ∩ Y (But: does not support !) S ∩ T X ∩ Y Fact: if is legal and is finitely supported X Y ⊆ X then is legal. Y FoPSS, Warsaw, 10-11/09/19 27

Powersets Fact: is not legal (though it is equivariant). P A FoPSS, Warsaw, 10-11/09/19 28

Powersets Fact: is not legal (though it is equivariant). P A Define: is finitely supported P fs X = { Y ⊆ X | Y } FoPSS, Warsaw, 10-11/09/19 28

Powersets Fact: is not legal (though it is equivariant). P A Define: is finitely supported P fs X = { Y ⊆ X | Y } Fact: if is legal then is legal. X P fs X FoPSS, Warsaw, 10-11/09/19 28

Powersets Fact: is not legal (though it is equivariant). P A Define: is finitely supported P fs X = { Y ⊆ X | Y } Fact: if is legal then is legal. X P fs X Key step : if supports X S then supports . S · π X · π FoPSS, Warsaw, 10-11/09/19 28

Powersets Fact: is not legal (though it is equivariant). P A Define: is finitely supported P fs X = { Y ⊆ X | Y } Fact: if is legal then is legal. X P fs X Key step : if supports X S then supports . S · π X · π πσπ − 1 ∈ Aut S ( A ) σ ∈ Aut S · π ( A ) = ⇒ FoPSS, Warsaw, 10-11/09/19 28

Powersets Fact: is not legal (though it is equivariant). P A Define: is finitely supported P fs X = { Y ⊆ X | Y } Fact: if is legal then is legal. X P fs X Key step : if supports X S then supports . S · π X · π πσπ − 1 ∈ Aut S ( A ) σ ∈ Aut S · π ( A ) = ⇒ X · π = ( X · πσπ − 1 ) · π = ( X · π ) · σ FoPSS, Warsaw, 10-11/09/19 28

Actions and supports Fact: if supports and π | S = σ | S X S then . X · π = X · σ FoPSS, Warsaw, 10-11/09/19 29

Actions and supports Fact: if supports and π | S = σ | S X S then . X · π = X · σ Proof: if then π | S = σ | S πσ − 1 ∈ Aut S ( A ) so X · σ = ( X · πσ − 1 ) · σ = X · π FoPSS, Warsaw, 10-11/09/19 29

Actions and supports Fact: if supports and π | S = σ | S X S then . X · π = X · σ Proof: if then π | S = σ | S πσ − 1 ∈ Aut S ( A ) so X · σ = ( X · πσ − 1 ) · σ = X · π NB. these proofs are “easy”. FoPSS, Warsaw, 10-11/09/19 29

Equivariant relations A (binary) relation is a set of pairs. Let’s see what equivariance means for such sets: iff ( x, y ) ∈ R = ⇒ ( x, y ) · π ∈ R R · π = R FoPSS, Warsaw, 10-11/09/19 30

Equivariant relations A (binary) relation is a set of pairs. Let’s see what equivariance means for such sets: iff ( x, y ) ∈ R = ⇒ ( x, y ) · π ∈ R R · π = R is equivariant iff R ⊆ X × Y implies for all π ( x · π ) R ( y · π ) xRy FoPSS, Warsaw, 10-11/09/19 30

Equivariant relations A (binary) relation is a set of pairs. Let’s see what equivariance means for such sets: iff ( x, y ) ∈ R = ⇒ ( x, y ) · π ∈ R R · π = R is equivariant iff R ⊆ X × Y implies for all π ( x · π ) R ( y · π ) xRy Similarly for -supported relations, but for S π ∈ Aut S ( A ) FoPSS, Warsaw, 10-11/09/19 30

Equivariant function A function is a binary relation. is equivariant iff R ⊆ X × Y implies for all π ( x · π ) R ( y · π ) xRy FoPSS, Warsaw, 10-11/09/19 31

Equivariant function A function is a binary relation. is equivariant iff R ⊆ X × Y implies for all π ( x · π ) R ( y · π ) xRy f : X → Y is equivariant iff f ( x · π ) = f ( x ) · π for all π FoPSS, Warsaw, 10-11/09/19 31

Equivariant function A function is a binary relation. is equivariant iff R ⊆ X × Y implies for all π ( x · π ) R ( y · π ) xRy f : X → Y is equivariant iff f ( x · π ) = f ( x ) · π for all π Similarly for -supported functions, but for S π ∈ Aut S ( A ) FoPSS, Warsaw, 10-11/09/19 31

Examples For fixed : 2 , 5 ∈ A FoPSS, Warsaw, 10-11/09/19 32

Examples For fixed : 2 , 5 ∈ A R = { (5 , 2) } ⇥ { (2 , d ) | d � = 5 } ⇥ { ( d, d ) } 5 R 2 2 5 FoPSS, Warsaw, 10-11/09/19 32

Examples For fixed : 2 , 5 ∈ A R = { (5 , 2) } ⇥ { (2 , d ) | d � = 5 } ⇥ { ( d, d ) } 5 5 R ∗ R 2 2 2 2 5 5 FoPSS, Warsaw, 10-11/09/19 32