representing exponentiation in nominal setsober:wissmann.pdf · questionpreliminariespart 1:...

Question Preliminaries Part 1: Positive Results Part 2: Negative Results Application References

Representing Exponentiation in Nominal Sets

Stefan Milius, Lutz Schröder, Thorsten Wißmann

May 31, 2016

Thorsten Wißmann May 31, 2016 1 / 16

Exponentiation

Automata as coalgebrasX → 2 & X × A→ X

X → 2× XA

Syntactic description

In sets ↔ A finite: (−)A = |A|-tupleIn nominal sets ↔ we want an infinite A: (−)A = |A|-tuple

? How to describe (−)A in nominal sets?Finitary representation? ?

Exponentiation

X → 2× XA

In sets ↔ A finite: (−)A = |A|-tuple

In nominal sets ↔ we want an infinite A: (−)A = |A|-tuple

Exponentiation

X → 2× XA

Exponentiation

X → 2× XA

Exponentiation

X → 2× XA

Main Theorem

For a nominal set A . . .

(−)A is the quotient ofa polynomial functor

Constants,Finite Products,

Infinite Coproducts

Part 1⇐==⇒Part 2

A is orbit-finite& strong

Relative order ofatoms is fixed

Milius, Schröder, Wißmann’16

Main Theorem

Infinite Coproducts

Main Theorem

Infinite Coproducts

The Framework of Nominal Sets

Support for a Sf(A)

Finite permutations on A

-action · : Sf(A)× X → X

“S ⊆ A supports x ∈ X ”, if for all π ∈ Sf(A)

π fixes S︸︷︷︸π(v)=v ∀v∈S

=⇒ π fixes x︸︷︷︸π·x=x

(X , ·) a Nominal Set“·” a Sf(A)-action & every x ∈ X finitely supported

x , y in the same orbit of (X , ·)if there is σ with σ · x = y .

A2 + 1 ∼=

Either infinite

(a, b)

(b, a) (c , a)

(a, d)

or singleton

π (a b)

Support for a Sf(A)

A2 + 1 ∼=

Either infinite

(a, b)

(b, a) (c , a)

(a, d)

or singleton

π (a b)

Support for a Sf(A)

A2 + 1 ∼=

Either infinite

(a, b)

(b, a) (c , a)

(a, d)

or singleton

π (a b)

Support for a Sf(A)

A2 + 1 ∼=

Either infinite

(a, b)

(b, a) (c , a)

(a, d)

or singleton

π (a b)

Morphisms do not introduce information

Information carried by x ∈ X

Which atoms: a, b?∈ supp(x)

Which order: (a b) · x ?= x

A A× A Pf(A)

a (a, b) {a, b}outl

Definition: f : X → Y is equivariant

f (π · x) = π · f (x) ∀x ∈ X , π ∈ Sf(A)

f does not introduce new atoms: supp(f (x)) ⊆ supp(x)

f does not introduce the order of atoms.

A A× A Pf(A)

a (a, b) {a, b}outl

f (π · x) = π · f (x) ∀x ∈ X , π ∈ Sf(A)

A A× A Pf(A)

a (a, b) {a, b}outl

f (π · x) = π · f (x) ∀x ∈ X , π ∈ Sf(A)

A A× A Pf(A)

a (a, b) {a, b}outl

f (π · x) = π · f (x) ∀x ∈ X , π ∈ Sf(A)

Exponentiation in Nominal Sets

Exponentiation = finitely supported maps

Y X = maps f : X → Y where f finitely supported w.r.t.

(π ? f )(x) = π · f (π−1 · x)

Nom(X ,Y ) ⊆

6=, if Y 6∈ Set

Y X ⊆

6=, if Y 6∈ Setand X infinite

Set(X ,Y )

X × (−) a (−)X

f : S × X → Y

fc : S → Y X

supp(fc (s)) ⊆ supp(s)︷︸︸︷fc(s)(x) = f (s, x)

Counit = evalY : X × Y X → Y

(π ? f )(x) = π · f (π−1 · x)

Nom(X ,Y ) ⊆

6=, if Y 6∈ Set

Y X ⊆

Set(X ,Y )

X × (−) a (−)X

f : S × X → Y

fc : S → Y X

(π ? f )(x) = π · f (π−1 · x)

Nom(X ,Y ) ⊆

6=, if Y 6∈ Set

Y X ⊆

Set(X ,Y )

X × (−) a (−)X

f : S × X → Y

fc : S → Y X

(π ? f )(x) = π · f (π−1 · x)

Nom(X ,Y ) ⊆

6=, if Y 6∈ Set

Y X ⊆

Set(X ,Y )

X × (−) a (−)X

f : S × X → Y

fc : S → Y X

(π ? f )(x) = π · f (π−1 · x)

Nom(X ,Y ) ⊆

6=, if Y 6∈ Set

Y X ⊆

Set(X ,Y )

X × (−) a (−)X

f : S × X → Y

fc : S → Y X

Part 1: Positive Results

⇐=A is orbit-finite

& strong

Thorsten Wißmann May 31, 2016 6.∞ / 16

Exponentiation by the atoms A

FX = A× X ×∐n∈N

An × X n

qX : FX × A→ X

qX (a, d , ~v , ~x , b) =

{xi where i is minimal s.t. vi = b

(a b) · d if no such i exists.

Lemma qX equivariant & natural in X

Lemma qX : FX → XA surjective

An × X n

qX : FX × A→ X

qX (a, d , ~v , ~x , b) =

An × X n

qX : FX × A→ X

qX (a, d , ~v , ~x , b) =

An × X n

qX : FX × A→ X

qX (a, d , ~v , ~x , b) =

Exponentiation by tuples of atoms An

n times the previous construction

F F · · ·FX F n

(−)A(XA · · · )A (−)A

q q∗···∗q

Example for the construction

If a is contained, then swap:

g(a, (x , y)) =

(y , a) if x = a

(a, x) if y = a

(x , y) otherwise

g : A× A2 → A2

f := g(a) ∈ (A2)A2

supp(f ) = {a}

t ∈ FFA2 7→ f

(c , d)

(a, c)

(c, c)

(d , a)

(a, a)

g(a, (x , y)) =

(y , a) if x = a

(a, x) if y = a

(x , y) otherwise

g : A× A2 → A2

f := g(a) ∈ (A2)A2

supp(f ) = {a}

t ∈ FFA2 7→ f

(c , d)

(a, c)

(c, c)

(d , a)

(a, a)

g(a, (x , y)) =

(y , a) if x = a

(a, x) if y = a

(x , y) otherwise

g : A× A2 → A2

f := g(a) ∈ (A2)A2

supp(f ) = {a}

t ∈ FFA2 7→ f

(c , d)

(a, c)

(c, c)

(d , a)

(a, a)

Exponentiation by tuples of distinct atoms An 6=

uniq : X n �∐

1≤k≤n

X k 6=

Removes duplicates:uniq(~x) =

(vi | 1 ≤ i ≤ n,∀j < i : vj 6= vi

)X n 6=

X n∐

1≤k≤nX k 6=

fill : X n × X 2n 6= → X n 6=

Removes duplicates and fills the gaps:fill(~v , ~w) = out1...n(uniq(~v ~w))

X n 6= × X 2n 6=

X n × X 2n 6= X n 6=

m×X 2n 6= outl

The restriction map rX : XAn → XAn 6=is:

equivariant, surjective, natural in X .

F n (−)An

(−)An 6=q(n) r

uniq : X n �∐

1≤k≤n

X k 6=

(vi | 1 ≤ i ≤ n,∀j < i : vj 6= vi

)X n 6=

X n∐

1≤k≤nX k 6=

fill : X n × X 2n 6= → X n 6=

X n 6= × X 2n 6=

X n × X 2n 6= X n 6=

m×X 2n 6= outl

F n (−)An

(−)An 6=q(n) r

uniq : X n �∐

1≤k≤n

X k 6=

(vi | 1 ≤ i ≤ n,∀j < i : vj 6= vi

)X n 6=

X n∐

1≤k≤nX k 6=

fill : X n × X 2n 6= → X n 6=

X n 6= × X 2n 6=

X n × X 2n 6= X n 6=

m×X 2n 6= outl

F n (−)An

(−)An 6=q(n) r

Strong nominal sets

X a strong nominal set

π fixes supp(x)︸︷︷︸π(v)=v ∀v∈supp(x)

⇐==⇒ π fixes x︸︷︷︸

π·x=x

⇔ “Order of atoms is fixed”

Universal Property

O ⊆ X one element per orbit of Xf0(supp(x)) ⊆ supp(x)

X strongf0 : O → Y a map

imply: f0 extends uniquely to an equivariant map f : X → Y .

Kurz, Petrisan, Velebil’10

Strong nominal sets

X a strong nominal set

π fixes supp(x)︸︷︷︸π(v)=v ∀v∈supp(x)

⇐==⇒ π fixes x︸︷︷︸

π·x=x

⇔ “Order of atoms is fixed”

Universal Property

O ⊆ X one element per orbit of Xf0(supp(x)) ⊆ supp(x)

X strongf0 : O → Y a map

imply: f0 extends uniquely to an equivariant map f : X → Y .

Kurz, Petrisan, Velebil’10

Exponentiation by strong & orbit-finite E

E strong & single-orbit

E ∼= An 6=

with n = | supp(e)|, e ∈ E

E strong & orbit-finite

E = disjoint union of its orbits= An1 6= + . . .+ Ank 6=

(k orbits)

By power laws

(−)E ∼= (−)An1 6= × . . .× (−)A

F n1 . . . F nk

(−)An1 . . . (−)A

(−)An1 6= . . . (−)A

n1︷︸︸︷q∗...∗q

× ×nk︷︸︸︷

q∗...∗q

···

∼= (−)An1 6=+...+Ank 6=∼= (−)E

E ∼= An 6=

(k orbits)

By power laws

(−)E ∼= (−)An1 6= × . . .× (−)A

F n1 . . . F nk

(−)An1 . . . (−)A

(−)An1 6= . . . (−)A

n1︷︸︸︷q∗...∗q

× ×nk︷︸︸︷

q∗...∗q

···

∼= (−)An1 6=+...+Ank 6=∼= (−)E

E ∼= An 6=

(k orbits)

By power laws

(−)E ∼= (−)An1 6= × . . .× (−)A

F n1 . . . F nk

(−)An1 . . . (−)A

(−)An1 6= . . . (−)A

n1︷︸︸︷q∗...∗q

× ×nk︷︸︸︷

q∗...∗q

···

∼= (−)An1 6=+...+Ank 6=∼= (−)E

E ∼= An 6=

(k orbits)

By power laws

(−)E ∼= (−)An1 6= × . . .× (−)A

F n1 . . . F nk

(−)An1 . . . (−)A

(−)An1 6= . . . (−)A

n1︷︸︸︷q∗...∗q

× ×nk︷︸︸︷

q∗...∗q

···

∼= (−)An1 6=+...+Ank 6=∼= (−)E

Part 2: Negative Results

=⇒ A is orbit-finite& strong

Thorsten Wißmann May 31, 2016 12.∞ / 16

E necessarily orbit-finite

Proposition

If (−)E finitary, then E is orbit-finite

Generally, in a cartesian closed lfp-category:

Proposition

If (−)E finitary and 1 is finitely generated, then E is finitelygenerated.

ContrapositionE not orbit-finite

⇒ (−)E not finitary⇒ (−)E not the quotient of a polynomial Nom-functor

Proposition

Sub-strength of a Nom-functor

DefinitionX < Y := {(x , y) ∈ X × Y | supp(x) ⊆ supp(y)}

Sub-strength sX ,Y : GX < Y → G (X < Y ),

not necessarily natural, but GX < Y G (X < Y )

outl G outl

Example for functors with a sub-strength1 Identity, Constant functors, Finitary Powerset Pf2 Closed under finite products, arbitrary coproducts, composition⇒ Polynomial functors

outl G outl

E necessarily strong

Proposition

Let E ⊆ A2 be the nominal set of unordered pairs

not strong

. Then (−)E isnot the quotient of any Nom-functor with a sub-strength.

Generalization: E an arbitrary non-strong nominal set.

Corollary

E not strong ⇒ (−)E not the quotient of a polynomial functor

Proposition

not strong

Corollary

Proposition

not strong

Corollary

Application

TheoremFor any functor G : Nom→ Nom built by

constants, identity, finitary powersetfinite products, arbitrary coproducts,exponentiation (−)E by a orbit-finite & strong nominal set E

The rational fixpoint of G is just:1 The rational fixpoint of the raw set functor . . .2 . . . modulo the equations by q : F → (−)E

Alexander Kurz, Daniela Petrisan, Jiri Velebil.“Algebraic Theories over Nominal Sets”. In: CoRRabs/1006.3027 (2010).

Stefan Milius, Lutz Schröder, Thorsten Wißmann.“Regular Behaviours with Names: On RationalFixpoints of Endofunctors on Nominal Sets”.submitted; available at http://www8.cs.fau.de/ext/thorsten/nomliftings.pdf. 2016.

Thorsten Wißmann May 31, 2016 ∞ / 16

Examples for exponentials

Y n = Set(n,Y ) if n is finite, i.e. |n|-tuples.constx(y) = x ∈ Y X , supp(constx) = supp(x)const = outlc .Y ω ( Set(ω,Y ) = finitely supported streams2X = Pfs(X ) finitely supported subsets

Thorsten Wißmann May 31, 2016 ∞ / 16

Checklist

Steps of the constructionExponentiation by. . .

1 . . . finite sets.2 . . . the atoms A3 . . . n-tuples of atoms An

4 . . . distinct n-tuples of atoms An 6=

5 . . . single-orbit strong nominal sets X ∼= An 6=

6 . . . orbit-finite strong nominal sets E ∼=∐

i∈O(E)Ani 6=

Exponentiation (−)E quotient of a polynomial functorThen E is necessarily. . .

1 . . . orbit-finite2 . . . strongThorsten Wißmann May 31, 2016 ∞ / 16

representing exponentiation in nominal setsober:wissmann.pdf · questionpreliminariespart 1:...

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