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

FoPSS 2019

3rd Summer School

• n Foundations of Programming

and Software Systems

Warsaw, 10-15 September, 2019

Nominal Techniques

[ ]

A

FoPSS

Summer Schools on Foundations of Programming and Software Systems

Supported by: This time also by:

FoPSS

2017: Braga (Portugal)

Probabilistic Programming

FoPSS

2017: Braga (Portugal)

Probabilistic Programming

2018: Oxford (UK)

Logic and Learning

FoPSS

2017: Braga (Portugal)

Probabilistic Programming

2018: Oxford (UK)

Logic and Learning

2019: Warsaw (Poland)

Nominal Techniques

• Andrzej Murawski:

Nominal game semantics

FoPSS 2019

Our lecturers:

• Andrew M. Pitts:

Nominal sets and functional programming

• Mikołaj Bojańczyk:

Computation theory with atoms

• Maribel Fernández:

Nominal rewriting and unification

• Johannes Borgström:

Nominal process calculi and modal logics

• Murdoch J. Gabbay:

Advanced nominal techniques

• Sławomir Lasota:

Computation theory with atoms II

FoPSS 2019

Warsaw, 10-11 September, 2019

Basic Nominal Techniques

[ ]

A

Bartek Klin University of Warsaw

What is it all about?

local names and name dependence mathematics of, and computation with: Nominal techniques:

What is it all about?

local names and name dependence mathematics of, and computation with: highly symmetrical structures Nominal techniques:

What is it all about?

local names and name dependence mathematics of, and computation with: highly symmetrical structures “slightly infinite” structures Nominal techniques:

What is it all about?

local names and name dependence mathematics of, and computation with: highly symmetrical structures “slightly infinite” structures structures acessible via limited interfaces Nominal techniques:

What is it all about?

local names and name dependence mathematics of, and computation with: highly symmetrical structures “slightly infinite” structures structures acessible via limited interfaces Nominal techniques:

Concrete and abstract syntax

2 ∗ 3 + 3 ∗ (7 − 2) +

−

∗ 2 3 7 ∗ 3 2

parsing

Concrete and abstract syntax

2 ∗ 3 + 3 ∗ (7 − 2) +

−

∗ 2 3 7 ∗ 3 2

parsing Algebraic features:

• definitions by recursion
• proofs by induction
• ...

Complications with local names

3

parsing

let x = 3 in let x = x + 1 in x + 5 let

x

let

x

+

x 1

+

x 5

Complications with local names

3

parsing

let x = 3 in let x = x + 1 in x + 5 let

x

let

x

+

x 1

+

x 5

Complications with local names

3

parsing

let x = 3 in let x = x + 1 in x + 5 let

x

let

x

+

x 1

+

x 5

Complications with local names

3

parsing

let x = 3 in let x = x + 1 in x + 5 let

x

let

x

+

x 1

+

x 5

Expressions depend

• n names!

Name dependence

Idea: Let every expression come equipped (or: atoms) that occur in it with an explicit dependence on some names

Name dependence

Idea: Let every expression come equipped (or: atoms) that occur in it with an explicit dependence on some names nominal expressions

Name dependence

Idea: Let every expression come equipped (or: atoms) that occur in it with an explicit dependence on some names nominal expressions More ambitious idea: Let everything come equipped (or: atoms) that occur in it with an explicit dependence on some names

Name dependence

Idea: Let every expression come equipped (or: atoms) that occur in it with an explicit dependence on some names nominal expressions More ambitious idea: Let everything come equipped (or: atoms) that occur in it with an explicit dependence on some names nominal sets

Name dependence

What does it mean to depend on a name? Q:

Name dependence

What does it mean to depend on a name? Q: A: depends on a name

X

a a

if renaming to any other name would alter X

Name dependence

What does it mean to depend on a name? Q: A: depends on a name

X

a a

if renaming to any other name would alter X Idea revisited: Let everything come equipped affects it with information on how renaming names nominal sets

What is it all about?

local names and name dependence mathematics of, and computation with: highly symmetrical structures “slightly infinite” structures structures acessible via limited interfaces Nominal techniques:

A graph built of atoms

atomic names:

• nodes:
• edges:

ab a 6= b a 6= c ab bc

a, b, c, d, e, . . .

A graph built of atoms

ab ac ad ba bc bd ca cb cd da db dc

atomic names:

• nodes:
• edges:

ab a 6= b a 6= c ab bc

a, b, c, d, e, . . .

A graph built of atoms

ab ac ad ba bc bd ca cb cd da db dc

atomic names:

• nodes:
• edges:

ab a 6= b a 6= c ab bc

a, b, c, d, e, . . .

atom renaming:

a

b

c

A graph built of atoms

ab ac ad ba bc bd ca cb cd da db dc

atomic names:

• nodes:
• edges:

ab a 6= b a 6= c ab bc

a, b, c, d, e, . . .

atom renaming:

a

b

c

A graph built of atoms

ab ac ad ba bc bd ca cb cd da db dc

atomic names:

• nodes:
• edges:

ab a 6= b a 6= c ab bc

a, b, c, d, e, . . .

atom renaming:

a

b

c

A graph built of atoms

ab ac ad ba bc bd ca cb cd da db dc

atomic names:

• nodes:
• edges:

ab a 6= b a 6= c ab bc

a, b, c, d, e, . . .

atom renaming:

a

b

c

A graph built of atoms

ab ac ad ba bc bd ca cb cd da db dc

atomic names:

• nodes:
• edges:

ab a 6= b a 6= c ab bc

a, b, c, d, e, . . .

atom renaming:

a

b

c

A graph built of atoms

ab ac ad ba bc bd ca cb cd da db dc

atomic names:

• nodes:
• edges:

ab a 6= b a 6= c ab bc

a, b, c, d, e, . . .

atom renaming:

a

b

c

A graph built of atoms

ab ac ad ba bc bd ca cb cd da db dc

atomic names:

• nodes:
• edges:

ab a 6= b a 6= c ab bc

a, b, c, d, e, . . .

atom renaming:

a

b

c

What is it all about?

local names and name dependence mathematics of, and computation with: highly symmetrical structures “slightly infinite” structures structures acessible via limited interfaces Nominal techniques:

Slightly infinite

• nodes:
• edges:

ab a 6= b a 6= c ab bc

The same graph:

Slightly infinite

• nodes: {(a, b) : a, b 2 A : a 6= b}
• edges:
• {(a, b), (b, c)} : a, b, c ∈ A

: a 6= b ^ b 6= c ^ a 6= c

• nodes:
• edges:

ab a 6= b a 6= c ab bc

The same graph:

Slightly infinite

• nodes: {(a, b) : a, b 2 A : a 6= b}
• edges:
• {(a, b), (b, c)} : a, b, c ∈ A

: a 6= b ^ b 6= c ^ a 6= c

• nodes:
• edges:

ab a 6= b a 6= c ab bc

The same graph: Infinite, but presented by finite means

An example problem

• nodes:
• edges:

ab a 6= b a 6= c ab bc

An example problem

Is it 3-colorable?

• nodes:
• edges:

ab a 6= b a 6= c ab bc

An example problem

Is it 3-colorable?

• nodes:
• edges:

ab a 6= b a 6= c ab bc

No.

ab ad bc be ca cd db de ea ec

An example problem

Is it 3-colorable? Is 3-colorability decidable?

• nodes:
• edges:

ab a 6= b a 6= c ab bc

No.

ab ad bc be ca cd db de ea ec

What is it all about?

local names and name dependence mathematics of, and computation with: highly symmetrical structures “slightly infinite” structures structures acessible via limited interfaces Nominal techniques:

Computer Science 101

Theorem: Every algorithm to sort numbers must work in time .

n

Ω(n log n)

Computer Science 101

Theorem: Every algorithm to sort numbers must work in time .

n

Ω(n log n)

in the comparison model

Computer Science 101

Theorem: Every algorithm to sort numbers must work in time .

n

Ω(n log n)

in the comparison model Here, numbers are atoms accessible via relations:

= <

Computer Science 101

Theorem: Every algorithm to sort numbers must work in time .

n

Ω(n log n)

in the comparison model Here, numbers are atoms accessible via relations:

= <

This amounts to restricting the class

• f legal atom renamings.

Nominal Sets: Basic Defnitions

[ ]

A

Nominal Sets: Basic Defnitions

[ ]

A

• r: Sets with Atoms

Atoms

Let be an infinite, countable set of atoms.

A a, b, c, d, e, . . . ∈ A

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

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

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

the dot omitted frow now on

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 (a b) ∈ Aut(A) - the swap of and

a

b

(π · σ) · ρ = π · (σ · ρ) π · π−1 = id π · id = π = id · π

the dot omitted frow now on

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 (a b) ∈ Aut(A) - the swap of and

a

b

(a b)(b c)(c a) = (b c)

For example:

(a b)−1 = (a b) (π · σ) · ρ = π · (σ · ρ) π · π−1 = id π · id = π = id · π

the dot omitted frow now on

Von Neumann hierarchy

U0 = ∅ Uβ = S

α<β Uα

A hierarchy of universes:

Uα+1 = PUα

defined for every ordinal number.

Von Neumann hierarchy

U0 = ∅ Uβ = S

α<β Uα

A hierarchy of universes:

Uα+1 = PUα

defined for every ordinal number.

Elements of sets are other sets, in a well founded way

Von Neumann hierarchy

U0 = ∅ Uβ = S

α<β Uα

A hierarchy of universes:

Uα+1 = PUα

defined for every ordinal number.

Elements of sets are other sets, in a well founded way

Every set sits somewhere in this hierarchy.

Sets with atoms

A - a countable set of atoms

U0 = ∅ Uα+1 = PUα + A Uβ = S

α<β Uα

A hierarchy of universes:

Sets with atoms

A - a countable set of atoms

U0 = ∅ Uα+1 = PUα + A Uβ = S

α<β Uα

A hierarchy of universes:

Sets with atoms

A - a countable set of atoms

Elements of sets with atoms are atoms

• r other sets with atoms, in a well founded way

Renaming atoms

A canonical renaming action:

· : U × Aut(A) → U

Renaming atoms

A canonical renaming action:

· : U × Aut(A) → U

a · π = π(a) X · π = {x · π | x ∈ X}

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

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

Finite support

S ⊆ A supports if ∀a ∈ S.π(a) = a x · π = x

implies

x

Finite support

S ⊆ A supports if ∀a ∈ S.π(a) = a x · π = x

implies

π ∈ AutS(A)

x

Finite support

S ⊆ A supports if ∀a ∈ S.π(a) = a x · π = x

implies A legal set with atoms, or nominal set:

• has a finite support,
• every element of it has a finite support,
• and so on.

π ∈ AutS(A)

x

Finite support

S ⊆ A supports if ∀a ∈ S.π(a) = a x · π = x

implies 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.

π ∈ AutS(A)

x

Examples

{a} a ∈ A

is supported by

Examples

{a} a ∈ A

is supported by

A

is equivariant

Examples

{a} a ∈ A

is supported by

S S ⊆ A

is supported by

A

is equivariant

Examples

{a} a ∈ A

is supported by

S S ⊆ A

is supported by

A \ S S

is supported by

A

is equivariant

Examples

{a} a ∈ A

is supported by

S S ⊆ A

is supported by

A \ S S

is supported by

A

is equivariant Fact: is fin. supp. iff it is finite or co-finite

S ⊆ A

Examples

{a} a ∈ A

is supported by

S S ⊆ A

is supported by

A \ S S

is supported by

A

is equivariant 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

Examples

{a} a ∈ A

is supported by

S S ⊆ A

is supported by

A \ S S

is supported by

A

is equivariant 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 2 ◆ = {{d, e} | d, e 2 A, d 6= e} is equivariant

Basic Properties

[ ]

A

Closure properties

Fact: if and are legal sets then , , , , are legal. X Y X ∪ Y X ∩ Y X + Y X \ Y X × Y

Closure properties

Fact: if and are legal sets then , , , , are legal. X Y X ∪ Y X ∩ Y X + Y X \ Y X × Y Indeed: if

S supports and supports X

T Y then S ∪ T supports , , ... X ∪ Y X ∩ Y

Closure properties

Fact: if and are legal sets then , , , , are legal. X Y X ∪ Y X ∩ Y X + Y X \ Y X × Y Indeed: if

S supports and supports X

T Y then S ∪ T supports , , ... X ∪ Y X ∩ Y (But: does not support !) S ∩ T X ∩ Y

Closure properties

Fact: if and are legal sets then , , , , are legal. X Y X ∪ Y X ∩ Y X + Y X \ Y X × Y Indeed: if

S supports and supports X

T 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 then is legal. X Y ⊆ X Y

Powersets

Fact: is not legal (though it is equivariant).

PA

Powersets

Fact: is not legal (though it is equivariant).

PA

Define: is finitely supported

PfsX = {Y ⊆ X | Y }

Powersets

Fact: is not legal (though it is equivariant).

PA

Define: is finitely supported

PfsX = {Y ⊆ X | Y }

Fact: if is legal then is legal. X PfsX

Powersets

Fact: is not legal (though it is equivariant).

PA

Define: is finitely supported

PfsX = {Y ⊆ X | Y }

Fact: if is legal then is legal. X PfsX Key step: if supports then supports .

S X S · π X · π

Powersets

Fact: is not legal (though it is equivariant).

PA

Define: is finitely supported

PfsX = {Y ⊆ X | Y }

Fact: if is legal then is legal. X PfsX Key step: if supports then supports .

S X S · π X · π σ ∈ AutS·π(A) = ⇒ πσπ−1 ∈ AutS(A)

Powersets

Fact: is not legal (though it is equivariant).

PA

Define: is finitely supported

PfsX = {Y ⊆ X | Y }

Fact: if is legal then is legal. X PfsX Key step: if supports then supports .

S X S · π X · π σ ∈ AutS·π(A) = ⇒ πσπ−1 ∈ AutS(A) X · π = (X · πσπ−1) · π = (X · π) · σ

Actions and supports

Fact: if supports and then .

S X π|S = σ|S X · π = X · σ

Actions and supports

Fact: if supports and then .

S X π|S = σ|S X · π = X · σ

Proof: if then

π|S = σ|S πσ−1 ∈ AutS(A)

so X · σ = (X · πσ−1) · σ = X · π

Actions and supports

Fact: if supports and then .

S X π|S = σ|S X · π = X · σ

Proof: if then

π|S = σ|S πσ−1 ∈ AutS(A)

so X · σ = (X · πσ−1) · σ = X · π

• NB. these proofs are “easy”.

Equivariant relations

A (binary) relation is a set of pairs. Let’s see what equivariance means for such sets: R · π = R

(x, y) ∈ R = ⇒ (x, y) · π ∈ R

iff

Equivariant relations

A (binary) relation is a set of pairs.

R ⊆ X × Y

is equivariant iff

xRy (x · π)R(y · π)

implies for all π

Let’s see what equivariance means for such sets: R · π = R

(x, y) ∈ R = ⇒ (x, y) · π ∈ R

iff

Equivariant relations

A (binary) relation is a set of pairs.

R ⊆ X × Y

is equivariant iff

xRy (x · π)R(y · π)

implies for all π

Let’s see what equivariance means for such sets: R · π = R

(x, y) ∈ R = ⇒ (x, y) · π ∈ R

iff Similarly for -supported relations, but for

S π ∈ AutS(A)

Equivariant function

A function is a binary relation.

R ⊆ X × Y

is equivariant iff

xRy (x · π)R(y · π)

implies for all π

Equivariant function

A function is a binary relation.

R ⊆ X × Y

is equivariant iff

xRy (x · π)R(y · π)

implies for all π

π f : X → Y is equivariant iff f(x · π) = f(x) · π for all

Equivariant function

A function is a binary relation.

R ⊆ X × Y

is equivariant iff

xRy (x · π)R(y · π)

implies for all π

Similarly for -supported functions, but for

S π ∈ AutS(A)

π f : X → Y is equivariant iff f(x · π) = f(x) · π for all

Examples

For fixed :

2, 5 ∈ A

Examples

R = {(5, 2)} ⇥ {(2, d) | d = 5} ⇥ {(d, d)}

2 5 5

2

R

For fixed :

2, 5 ∈ A

Examples

R = {(5, 2)} ⇥ {(2, d) | d = 5} ⇥ {(d, d)}

2 5 5

2

R

2 5 5 2

R∗ For fixed :

2, 5 ∈ A

Examples

R = {(5, 2)} ⇥ {(2, d) | d = 5} ⇥ {(d, d)}

2 5 5

2

R

2 5 5 2

R∗ For fixed :

2, 5 ∈ A

R , are supported by

R∗ {2, 5}

Examples ctd.

Equivariant binary relations on :

A

Examples ctd.

Equivariant binary relations on :

A

• empty
• total
• equality
• inequality

Examples ctd.

Equivariant binary relations on :

A

• empty
• total
• equality
• inequality

No equivariant function from to , but

A

2

• A

{({a, b}, a) | a, b ∈ A}

is an equivariant relation.

Examples ctd.

Equivariant binary relations on :

A

• empty
• total
• equality
• inequality

No equivariant function from to , but

A

2

• A

{({a, b}, a) | a, b ∈ A}

is an equivariant relation. Only equiv. functions from to are projections

A2

Only equiv. function from to is the diagonal

A A

A2

Intuition

A relation/function/... is equivariant iff it only “checks” equality of atoms, and does not mention specific atoms.

Intuition

A relation/function/... is equivariant iff it only “checks” equality of atoms, and does not mention specific atoms. A relation/function/... supported by , may additionally mention specific atoms from .

S S

Equivariant functions preserve supports

Fact: if supports then supports .

S

and supports T

f : X → Y S ∪ T f(x) x ∈ X

Equivariant functions preserve supports

Fact: if supports then supports .

S

and supports T

f : X → Y S ∪ T f(x) x ∈ X

Proof: AutS∪T (A) = AutS(A) ∩ AutT (A) so if π ∈ AutS∪T (A) then f(x) · π = f(x · π) = f(x)

Equivariant functions preserve supports

Fact: if supports then supports .

S

and supports T

f : X → Y S ∪ T f(x) x ∈ X

Proof: AutS∪T (A) = AutS(A) ∩ AutT (A) so if π ∈ AutS∪T (A) then f(x) · π = f(x · π) = f(x)

• NB. another “easy” proof.

Least supports

Fact: for finite and ,

S

T if supports and supports X

S

T X then supports . S ∩ T X

Least supports

Fact: for finite and ,

S

T if supports and supports X

S

T X then supports . S ∩ T X So: every legal has the least support . X supp(X)

Least supports

Fact: for finite and ,

S

T if supports and supports X

S

T X then supports . S ∩ T X

• NB. This is harder to prove!

One way: induction on . |S4T| So: every legal has the least support . X supp(X)

Proof

Assume and support .

S

T X

S

T

Proof

Assume and support .

S

T X

S

T

a

Proof

Assume and support .

S

T X

S

T

a

Goal: supports . X S \ a

Proof

Assume and support .

S

T X

S

T

a

Goal: supports . X S \ a Take any . π ∈ AutS\a(A)

Proof

Assume and support .

S

T X

S

T

a

Goal: supports . X S \ a Take any . π ∈ AutS\a(A) b

π(b)

Pick a fresh : . b

b, π(b) 62 S [ T

Proof

Assume and support .

S

T X

S

T

a

Goal: supports . X S \ a Take any . π ∈ AutS\a(A) b

π(b)

Pick a fresh : . b

b, π(b) 62 S [ T

Put , . Then:

σ = (a b) θ = (a π(b)) σ, θ = AutT (A)

Proof

Assume and support .

S

T X

S

T

a

Goal: supports . X S \ a Take any . π ∈ AutS\a(A) b

π(b)

Pick a fresh : . b

b, π(b) 62 S [ T

Put , . Then:

σ = (a b) θ = (a π(b)) σ, θ = AutT (A) σπθ = AutS(A)

Proof

Assume and support .

S

T X

S

T

a

Goal: supports . X S \ a Take any . π ∈ AutS\a(A) b

π(b)

Pick a fresh : . b

b, π(b) 62 S [ T

Put , . Then:

σ = (a b) θ = (a π(b)) σ, θ = AutT (A) σπθ = AutS(A)

so:

X · π = ((X · σ) · σπθ) · θ = X

Name abstraction

For an (equivariant) set , X define a relation on so: ≈ A × X

(a, x) ≈ (b, y) ⇐ ⇒ x · (a c) = y · (b c)

for fresh :

c c 62 {a, b} [ supp(x, y)

Name abstraction

For an (equivariant) set , X define a relation on so: ≈ A × X

(a, x) ≈ (b, y) ⇐ ⇒ x · (a c) = y · (b c)

for fresh :

c c 62 {a, b} [ supp(x, y)

Fact: is an equivariant equivalence relation. ≈

Name abstraction

For an (equivariant) set , X define a relation on so: ≈ A × X

(a, x) ≈ (b, y) ⇐ ⇒ x · (a c) = y · (b c)

for fresh :

c c 62 {a, b} [ supp(x, y)

Fact: is an equivariant equivalence relation. ≈ Define:

[A]X = (A × X)/≈

Name abstraction

For an (equivariant) set , X define a relation on so: ≈ A × X

(a, x) ≈ (b, y) ⇐ ⇒ x · (a c) = y · (b c)

for fresh :

c c 62 {a, b} [ supp(x, y)

Fact: is an equivariant equivalence relation. ≈ Define:

[A]X = (A × X)/≈

Fact: is an equivariant set.

[A]X supp([a, x]≈) = supp(x) \ {a}

Name abstraction

For an (equivariant) set , X define a relation on so: ≈ A × X

(a, x) ≈ (b, y) ⇐ ⇒ x · (a c) = y · (b c)

for fresh :

c c 62 {a, b} [ supp(x, y)

Fact: is an equivariant equivalence relation. ≈ Define:

[A]X = (A × X)/≈

Fact: is an equivariant set.

[A]X supp([a, x]≈) = supp(x) \ {a} α-equivalence

Exercises

• 1. If supports and supports

f : X → Y

then supports .

S T g : Y → Z

S ∪ T

f; g : X → Z

Exercises

• 1. If supports and supports

f : X → Y

then supports .

• 2. For an equivariant set , the transitive closure

X

function is equivariant.

(−)∗ : Pfs(X × X) → Pfs(X × X)

S T g : Y → Z

S ∪ T

f; g : X → Z

Exercises

• 1. If supports and supports

f : X → Y

then supports .

• 3. For an equivariant set , the least support

is equivariant. function supp : X → PfinA

X

• 2. For an equivariant set , the transitive closure

X

function is equivariant.

(−)∗ : Pfs(X × X) → Pfs(X × X)

S T g : Y → Z

S ∪ T

f; g : X → Z

Exercises

• 1. If supports and supports

f : X → Y

then supports .

• 3. For an equivariant set , the least support

is equivariant. function supp : X → PfinA

X

• 2. For an equivariant set , the transitive closure

X

function is equivariant.

(−)∗ : Pfs(X × X) → Pfs(X × X)

• 4. In a finite equivariant set, every element

is equivariant.

S T g : Y → Z

S ∪ T

f; g : X → Z