Cubical Computational Type Carlo Angiuli Theory Evan Cavallo - - PowerPoint PPT Presentation

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cubical computational type

Cubical Computational Type Carlo Angiuli Theory Evan Cavallo - - PowerPoint PPT Presentation

Sep 27, 2023 130 likes •929 views

2018.02.23 Penn Cubical Computational Type Carlo Angiuli Theory Evan Cavallo (*) Favonia & RedPRL Bob Harper Dan Licata >> redprl.org >> Jon Sterling Todd Wilson 1 Vladimir Voevodsky 1966-2017 2 Cubical &

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Carlo Angiuli Evan Cavallo (*) Favonia Bob Harper Dan Licata Jon Sterling Todd Wilson

Cubical Computational Type Theory & RedPRL

2018.02.23 Penn >> redprl.org >>

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Vladimir Voevodsky 1966-2017

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Cubical & Computational

features of computational type theory (equality types, strict quotients, ...) features of homotopy type theory (HoTT)

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Computational Types

programs/ realizers computation

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Computational Types

programs/ realizers computation computational type theory theory of computation <-----

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Computational Types

programs/ realizers computation computational type theory theory of computation meaning explanation Martin-Löf type theory pre-mathematical in M-L's work <----- <----

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A Minimum Example

M := a | bool | true | false | if(M,M,M)

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A Minimum Example

M := a | bool | true | false | if(M,M,M) true val false val if(true,M,_) ↦ M if(false,_,M) ↦ M bool val

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A Minimum Example

M := a | bool | true | false | if(M,M,M) true val false val if(true,M,_) ↦ M if(false,_,M) ↦ M bool val

The Language

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A Minimum Example

M := a | bool | true | false | if(M,M,M) true val false val if(true,M,_) ↦ M if(false,_,M) ↦ M bool val

The Language What are the types in canonical forms? {bool}

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A Minimum Example

M := a | bool | true | false | if(M,M,M) true val false val if(true,M,_) ↦ M if(false,_,M) ↦ M bool val

The Language What are the types in canonical forms? What are the canonical forms of the types? bool: {true, false} {bool}

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A Minimum Example

M := a | bool | true | false | if(M,M,M) true val false val if(true,M,_) ↦ M if(false,_,M) ↦ M bool val

The Language What are the types in canonical forms? What are the canonical forms of the types? bool: {true, false} {bool} How they are equal? syntactic equality

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A Minimum Example

M := a | bool | true | false | if(M,M,M) true val false val if(true,M,_) ↦ M if(false,_,M) ↦ M bool val

The Language What are the types in canonical forms? What are the canonical forms of the types? bool: {true, false} {bool} How they are equal? syntactic equality One Theory

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A Minimum Example

M := a | bool | true | false | if(M,M,M) bool: {true, false} with syntactic equality types: {bool} with syntactic equality

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A Minimum Example

M := a | bool | true | false | if(M,M,M) bool: {true, false} with syntactic equality types: {bool} with syntactic equality

A ≐ B type

A⇓A' B⇓B' and A'≈B'

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A Minimum Example

M := a | bool | true | false | if(M,M,M) bool: {true, false} with syntactic equality types: {bool} with syntactic equality bool ≐ bool type

A ≐ B type

A⇓A' B⇓B' and A'≈B'

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A Minimum Example

M := a | bool | true | false | if(M,M,M) bool: {true, false} with syntactic equality types: {bool} with syntactic equality

M ≐ N ∈ A

A≐A type, M⇓M', N⇓N', A⇓A' and M'≈A'N'

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A Minimum Example

M := a | bool | true | false | if(M,M,M) bool: {true, false} with syntactic equality types: {bool} with syntactic equality false ≐ false ∈ bool

M ≐ N ∈ A

A≐A type, M⇓M', N⇓N', A⇓A' and M'≈A'N'

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A Minimum Example

M := a | bool | true | false | if(M,M,M) bool: {true, false} with syntactic equality types: {bool} with syntactic equality if(true,true,bool) ≐ true ∈ if(true,bool,bool) false ≐ false ∈ bool

M ≐ N ∈ A

A≐A type, M⇓M', N⇓N', A⇓A' and M'≈A'N'

⇓true ⇓bool

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A Minimum Example

M := a | bool | true | false | if(M,M,M) bool: {true, false} with syntactic equality types: {bool} with syntactic equality

a:A >> M ≐ N ∈ B

P ≐ Q ∈ A implies M[P/a] ≐ N[Q/a] ∈ B

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A Minimum Example

M := a | bool | true | false | if(M,M,M) bool: {true, false} with syntactic equality types: {bool} with syntactic equality

b:bool >> b ≐ if(b,true,false) ∈ bool?

a:A >> M ≐ N ∈ B

P ≐ Q ∈ A implies M[P/a] ≐ N[Q/a] ∈ B

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Variables

variables range over closed terms

In Nuprl and friends

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Variables

variables range over closed terms variables are not subject to inductive analysis

In Nuprl and friends In Coq, Agda, and friends

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Variables

variables range over closed terms variables are not subject to inductive analysis

In Nuprl and friends In Coq, Agda, and friends

pen reduction ⇔ indeterminate vars

closed reduction ⇔ vars over closed terms

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A Functional Example

M := a | M1⟶M2 | \a.M | M1 M2 | ... \a.M val (\a.M1)M2 ↦ M1[M2/a] (M1⟶M2) val

Another Language

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A Functional Example

M := a | M1⟶M2 | \a.M | M1 M2 | ... \a.M val (\a.M1)M2 ↦ M1[M2/a] (M1⟶M2) val

Another Language What are the types in canonical forms? What are the canonical forms of the types? A⟶B: {\a.M} the least fixed point of S.({M⟶N | M⇓, N⇓ in S} union ...) How they are equal? A1⟶B1 ≈ A2⟶B2 if A1 ≐ A2 and B1 ≐ B2 \a.M1 ≈A⟶B \a.M2 if a:A >> M1 ≐ M2 ∈ B

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Open-endedness

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Open-endedness

Open to new constructs

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Open-endedness

Open to new constructs Open to new theories for the same language

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Open-endedness

Open to new constructs Open to new theories for the same language Open to new proof theories (rules in proof assistants) for the same theory

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Open-endedness

Canonicity always holds Open to new constructs Open to new theories for the same language Open to new proof theories (rules in proof assistants) for the same theory

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Homotopy Type Theory

a b

points

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Homotopy Type Theory

a b q:a=b

paths points

p:a=b

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Homotopy Type Theory

a b q:a=b

paths

paths between paths

points

p:a=b h:p=q

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Homotopy Type Theory

a b q:a=b

⋮

paths

paths between paths

points

p:a=b h:p=q

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Type Space Function Continuous Mapping Element Point Dependent Type Fibration Identification Path A a : A f : A → B C : A → Type a =A b

[Awodey and Warren] [Voevodsky et al] [van den Berg and Garner]

Homotopy Type Theory

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Homotopy Type Theory

In some case new proofs were discovered and inspired new results.

[Anel, Biedermann, Finster, Joyal]

Numerous results in homotopy theory mechanized through this. Extensive works in category theory and other fields.

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Key Features of HoTT

1. Identifications as paths
2. Univalence: if e is an equivalence

between A and B, then ua(e):A=B

3. Higher inductive types:

generalized inductive types with (higher) path generators

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Key Features of HoTT

1. Identifications as paths
2. Univalence: if e is an equivalence

between A and B, then ua(e):A=B

3. Higher inductive types:

generalized inductive types with (higher) path generators Problems: 1&2&3 give new identifications

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The Poor Eliminator

can only handle reflexivity coe(p:A=B,a:A):B coe(ua(e),a) is stuck elim-path[a.C](refl-case, path)

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The Poor Eliminator

can only handle reflexivity coe(p:A=B,a:A):B coe(ua(e),a) is stuck Solution each motive C handles paths itself elim-path[a.C](refl-case, path)

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The Happy Eliminator

[Bezem, Coquand, Huber] [Cohen, Coquand, Huber, Mörtberg]

each motive handles paths itself each type has cubical Kan structure This work: extend Nuprl by cubical Kan structures

[Angiuli, Harper, Wilson] [Angiuli, Harper] [Angiuli, Favonia, Harper] [Cavallo, Harper]

elim-path[a.C](refl-case, path)

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Cubical Programming

dim expr r := 0 | 1 | x

1 x

somewhere

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Cubical Programming

dim expr r := 0 | 1 | x

1 x

somewhere

=

n

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Circle

base loop{x}

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Circle

dim expr

base loop{x}

M := S1 | base | loop{r} | S1elim(a.M, M, M, x.M) | ...

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Circle

dim expr

base loop{x} S1 val

M := S1 | base | loop{r} | S1elim(a.M, M, M, x.M) | ...

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Circle

dim expr

base loop{x} base val S1 val

M := S1 | base | loop{r} | S1elim(a.M, M, M, x.M) | ...

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Circle

dim expr

loop{x} val loop{0} ↦ base loop{1} ↦ base base loop{x} base val S1 val

M := S1 | base | loop{r} | S1elim(a.M, M, M, x.M) | ...

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Circle

base loop{x} M ↦ M'  S1elim(a.A, M, B, x.L) ↦ S1elim(a.A, M', B, x.L) S1 val

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Circle

S1elim(a.A, base, B, x._) ↦ B base loop{x} M ↦ M'  S1elim(a.A, M, B, x.L) ↦ S1elim(a.A, M', B, x.L) S1 val

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Circle

S1elim(a.A, base, B, x._) ↦ B base loop{x} S1elim(a.A, loop{x}, _, y.L) ↦ L<x/y> M ↦ M'  S1elim(a.A, M, B, x.L) ↦ S1elim(a.A, M', B, x.L) S1 val

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Kan 1/2: Coercion

M x x.A ∈

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Kan 1/2: Coercion

coe{0⟶1} (x.A,M) M x x.A ∈ ∈

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coe{r⟶r'}(x.A, M) ∈ A<r'/x>

Kan 1/2: Coercion

coe{0⟶1} (x.A,M) M A<r/x> ∈ x x.A ∈ ∈

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Kan 2/2: Homogeneous Composition

M x y N0 N1

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Kan 2/2: Homogeneous Composition

M x y N0 N1 hcom{0⟶1}(A,M) [x=0⟶y.N0, x=1⟶y.N1]

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hcom{r⟶r'}(A, M) [..., ri=r'i⟶y.Ni, ...]

Kan 2/2: Homogeneous Composition

M x y N0 N1 hcom{0⟶1}(A,M) [x=0⟶y.N0, x=1⟶y.N1]

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N3 M N1 N2 N0 z x y

Kan 2/2: Homogeneous Composition

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N3 M N1 N2 N0 hcom{0⟶1}(A,M) [x=0⟶z.N0, y=0⟶z.N1, x=1⟶z.N2, y=1⟶z.N3] z x y

Kan 2/2: Homogeneous Composition

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Kan Circle

coe{r⟶r'}(_.S1, M) ↦ M

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Kan Circle

hcom{r⟶r'}(S1, M)[...] ↦ fcom{r⟶r'}(M)[...] coe{r⟶r'}(_.S1, M) ↦ M formal composition

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Kan Circle

hcom{r⟶r'}(S1, M)[...] ↦ fcom{r⟶r'}(M)[...] coe{r⟶r'}(_.S1, M) ↦ M formal composition fcom{r⟶r}(M)[...] ↦ M

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Kan Circle

hcom{r⟶r'}(S1, M)[...] ↦ fcom{r⟶r'}(M)[...]

r!=r' ri=r'i (the first i)  fcom{r⟶r'}(M)[..., ri=r'i⟶y.Ni, ...] ↦ Ni<r'/y>

coe{r⟶r'}(_.S1, M) ↦ M formal composition fcom{r⟶r}(M)[...] ↦ M

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Kan Circle

hcom{r⟶r'}(S1, M)[...] ↦ fcom{r⟶r'}(M)[...]

r!=r' ri=r'i (the first i)  fcom{r⟶r'}(M)[..., ri=r'i⟶y.Ni, ...] ↦ Ni<r'/y> r!=r' ri!=r'i for all i  fcom{r⟶r'}(M)[...] val

coe{r⟶r'}(_.S1, M) ↦ M formal composition fcom{r⟶r}(M)[...] ↦ M

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Kan Circle

S1elim needs to handle fcom

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Kan Circle

r!=r' ri!=r'i  S1elim(a.A, fcom{r⟶r'}(M)[...], B, x.L) ↦ com{r⟶r'}(y.A[fcom{r⟶y}(...).../a], S1elim(M, B, x.L))[...] S1elim(composition) ↦ composition(S1elim)

S1elim needs to handle fcom

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Cubical Stability

Dimension substs. do not commute with evaluation!

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Cubical Stability

S1elim(a.A, loop{x}, B, y.L) |⟶ L<x/y> ⟶ <0/x> L<0/y>

Dimension substs. do not commute with evaluation!

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Cubical Stability

S1elim(a.A, loop{x}, B, y.L) S1elim(a.A, base, B, y.L) |⟶ L<x/y> |> B ⟶ <0/x> ⟶ <0/x> L<0/y>

Dimension substs. do not commute with evaluation!

<=??=>

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Cubical Stability

S1elim(a.A, loop{x}, B, y.L) S1elim(a.A, base, B, y.L) |⟶ L<x/y> |> B ⟶ <0/x> ⟶ <0/x> L<0/y>

Dimension substs. do not commute with evaluation!

<=??=> Restrict our theory to

nly cubically stable parts

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27 stability: consider every substitution

Cubical Type Theory

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27 stability: consider every substitution

A and B stably recognize the same stable values and have stably equal Kan structures

Cubical Type Theory

(see our arXiv and POPL papers)

A ≐ B type [Ψ]

dim context

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27 stability: consider every substitution

A and B stably recognize the same stable values and have stably equal Kan structures A ≐ A type [Ψ], M and N stably eval to M' and N', A stably treats M' and N' as the same

Cubical Type Theory

(see our arXiv and POPL papers)

A ≐ B type [Ψ] M ≐ N ∈ A [Ψ]

dim context

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Variables

variables range over closed terms variables are indeterminate

In Nuprl and friends In Coq, Agda, and friends

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Variables

variables range over closed terms variables are indeterminate

In Nuprl and friends In Coq, Agda, and friends

exp variables range over closed terms

This work

dim variables are indeterminate

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Our arXiv Papers

Part1: stability and Kan Part2: dependent types Part3: univalence and equality Part4: cubical inductive types

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RedPRL

a proof assistant based

n the new type theory

http://redprl.org

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Conclusion

We extended Nuprl semantics by cubical Kan structures which justify key features of HoTT We also built RedPRL as a prototype