Cartesian Cubical Computational Type Theory Favonia Inst for Adv - - PowerPoint PPT Presentation

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Cartesian Cubical Computational Type Theory Favonia Inst for Adv - - PowerPoint PPT Presentation

Mar 20, 2024 188 likes •672 views

Cartesian Cubical Computational Type Theory Favonia Inst for Adv Study U of Minnesota Joint work with Carlo Angiuli, Evan Cavallo, Daniel R. Grayson, Robert Harper and Jonathan Sterling (a shameless rip-o ff of Carlo's previous talks)

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Cartesian Cubical Computational Type Theory

Inst for Adv Study ↦ U of Minnesota

Favonia

Bonn, Germany, 2018/6/5 Joint work with Carlo Angiuli, Evan Cavallo, Daniel R. Grayson, Robert Harper and Jonathan Sterling (a shameless rip-off of Carlo's previous talks)

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AHW 2016: CHTT Part I AH 2017: CHTT Part II AFH 2017: CHTT Part III CH 2018: CHTT Part IV Cartesian cubical + computational Dependent types Univalent Kan universes Higher inductive types Identiication types

Some History

Coquand's notes BL 14

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AHW 2016: CHTT Part I AH 2017: CHTT Part II AFH 2017: CHTT Part III ABCFHL 2017: CCTT CH 2018: CHTT Part IV Cartesian cubical + computational Dependent types Univalent Kan universes Higher inductive types Identiication types Cartesian cubical

Some History

Coquand's notes BL 14

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AHW 2016: CHTT Part I AH 2017: CHTT Part II AFH 2017: CHTT Part III ABCFHL 2017: CCTT CH 2018: CHTT Part IV Cartesian cubical + computational Dependent types Univalent Kan universes Higher inductive types Identiication types Cartesian cubical

Some History

Coquand's notes BL 14

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New Features of HoTT

Univalence

inductive types with path generators if e is an equivalence between types A and B, then ua(E):A=B

Higher Inductive Types

circle sphere torus

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Equality and Paths

Deinitional Equality

Silent in theory 2 + 3 ≡ 5 fst ⟨M,N⟩ ≡ M

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Equality and Paths

Deinitional Equality

Silent in theory If A ≡ B and M : A then M : B 2 + 3 ≡ 5 fst ⟨M,N⟩ ≡ M

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Equality and Paths

Deinitional Equality

Visible in theory Silent in theory

Paths

If P : Path(A, B) and M : A then transport(M,P) : B If A ≡ B and M : A then M : B 2 + 3 ≡ 5 fst ⟨M,N⟩ ≡ M

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Not Math Equality!

Deinitional Equality Issue #1

Not very extensional x : ℕ, y : ℕ ⊦ x + y ≢ y + x : ℕ

(various reasonable trade-os)

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winding

π1(S1)

ℤ winding(loop) ≢ any numeral

Deinitional Equality Issue #2

Not Math Equality!

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winding

π1(S1)

ℤ winding(loop) ≢ any numeral

Deinitional Equality Issue #2

Not Math Equality!

For any M : ℕ, there is a numeral N* such that ⊦ M ≡ N* : ℕ

Canonicity

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Restore Canonicity

Canonicity for ℕ means canonicity for every type

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Restore Canonicity

Canonicity for ℕ means canonicity for every type

M : ℕ × A fst(M) ≡ ??? : ℕ

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Restore Canonicity

Canonicity for ℕ means canonicity for every type

M : ℕ × A fst(M) ≡ ??? : ℕ By canonicity of pair types, M ≡ ⟨P,Q⟩ and fst(M) ≡ fst ⟨P,Q⟩ ≡ P ≡ some numeral

(rule) (by i.h.) (by i.h.)

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Restore Canonicity

But canonicity fails for paths!

Path-elim[p.C](M, x.N) : C[M/p] Path-elim(rel(M), x.N) ≡ N[M/x] : C[rel(M)/p] rel(M) : Path(M,M)

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Restore Canonicity

But canonicity fails for paths!

Path-elim[p.C](M, x.N) : C[M/p] Path-elim(rel(M), x.N) ≡ N[M/x] : C[rel(M)/p] rel(M) : Path(M,M) Path-elim(ua(E), x.N) ≡ ??? Path-elim(loop, x.N) ≡ ???

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Restore Canonicity

Can we have canonicity + univalence?

Yes with De Morgan cubes (CCHM) Yes with Cartesian cubes (Part III by AFH)

And higher inductive types?

Important examples with De Morgan cubes (CHM) Yes with cartesian cubes (Part IV by CH)

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Cubes

Idea: each type manages its own paths

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Cubes

Idea: each type manages its own paths loop : base = base

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Cubes

Idea: each type manages its own paths loopx: a genuine constructor of the circle loop : base = base x:𝕁 ⊦ loopx : S1 loop0 ≡ base : S1 loop1 ≡ base : S1

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Cartesian Cubes

Introducing 𝕁 the formal interval

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Cartesian Cubes

Γ, x:𝕁, Γ' ⊦ x:𝕁 Γ ⊦ 0:𝕁 Γ ⊦ 1:𝕁 Introducing 𝕁 the formal interval

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Cartesian Cubes

Γ, x:𝕁, Γ' ⊦ x:𝕁

⬄ M is an n-cube in A

Γ ⊦ 0:𝕁 Γ ⊦ 1:𝕁 Introducing 𝕁 the formal interval x1:𝕁, x2:𝕁, ..., xn:𝕁 ⊦ M : A

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Cartesian Cubes

Γ, x:𝕁, Γ' ⊦ x:𝕁 Γ ⊦ 0:𝕁 Γ ⊦ 1:𝕁 Introducing 𝕁 the formal interval Cartesian: works as normal contexts M⟨0/x⟩ M⟨1/x⟩ M⟨y/x⟩

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Ordinary Types

Γ ⊦ λa.M : (a:A) → B Γ, a:A ⊦ M : B Ordinary typing rules hold uniformly with any number of 𝕁 in the Γ

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Ordinary Types

Γ ⊦ λa.M : (a:A) → B Γ, a:A ⊦ M : B Ordinary typing rules hold uniformly with any number of 𝕁 in the Γ apF(M) F(Mx) F(Mx⟨0/x⟩) F(Mx⟨1/x⟩)

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New Path Types

⟨x⟩M : Pathx.A(M⟨0/x⟩,M⟨1/x⟩) x:𝕁 ⊦ M : A Dimension abstraction

(⟨x⟩M)@r ≡ M⟨r/x⟩ : A⟨r/x⟩ x:𝕁 ⊦ M : A P@0 ≡ N0 : A⟨0/x⟩ P : Pathx.A(N0,N1) P@r : A⟨r/x⟩ P : Pathx.A(N0,N1) P@1 ≡ N1 : A⟨1/x⟩ P : Pathx.A(N0,N1)

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Kan 1/2: Coercion

coex.A[r ↝ r'](M) : A⟨r'/x⟩ M : A⟨r/x⟩

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Kan 1/2: Coercion

coex.A[r ↝ r'](M) : A⟨r'/x⟩ A⟨0/x⟩ A⟨1/x⟩ M : A⟨r/x⟩ M

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Kan 1/2: Coercion

coex.A[r ↝ r'](M) : A⟨r'/x⟩ A⟨0/x⟩ A⟨1/x⟩ M : A⟨r/x⟩ M coex.A[0↝1](M)

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Kan 1/2: Coercion

coex.A[r ↝ r'](M) : A⟨r'/x⟩ A⟨0/x⟩ A⟨1/x⟩ M : A⟨r/x⟩ M coex.A[0↝1](M) coex.A[0↝x](M)

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Kan 1/2: Coercion

coex.A[r ↝ r'](M) : A⟨r'/x⟩ A⟨0/x⟩ A⟨1/x⟩ M : A⟨r/x⟩ M coex.A[0↝1](M) coex.A[0↝x](M) coex.A[r ↝ r](M) ≡ M : A⟨r/x⟩

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Kan 2/2: Homogeneous Comp

M x y N0 N1

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Kan 2/2: Homogeneous Comp

M x y N0 N1 hcomA[0↝1](M; x=0 ↪ y.N0, x=1 ↪ y.N1]

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Kan 2/2: Homogeneous Comp

M x y N0 N1 hcomA[0↝1](M; x=0 ↪ y.N0, x=1 ↪ y.N1]

hcomA[r ↝ r](M; ...) ≡ M : A hcomA[r ↝ r'](M; ..., ri=ri ↪ y.Ni, ...) ≡ Ni⟨r'/y⟩ : A

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Kan 2/2: Homogeneous Comp

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Fibrant Replacement

S1: hcom as the third constructor (the cubical, syntactical way)

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Fibrant Replacement

S1: hcom as the third constructor (the cubical, syntactical way) add only homogeneous ones ⇨ compat with base changes ⇨ no size blow-up! (category-theoretic argument for sizes mostly learnt from Dan Grayson)

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Univalent Kan Universes

  • 0. Vx(A,B,E) type

A line between A⟨0/x⟩ and B⟨1/x⟩ witnessed by the equivalence E

  • 1. hcomU[r ↝ r'](A; ...) type

Make the universes Kan

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Oh, Diagonals!

coex.hcom[s↝s'](A; ...)[r↝r'](M)

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Oh, Diagonals!

coex.hcom[s↝s'](A; ...)[r↝r'](M) s=s' ↦ coex.A[r↝r'](M) r=r' ↦ M

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Oh, Diagonals!

coex.hcom[s↝s'](A; ...)[r↝r'](M) s=s' ↦ coex.A[r↝r'](M) r=r' ↦ M

hcom...[s↝s'](..., r=r' ↪ ...)

diagonals for coherence conditions

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Computational Semantics

A computation system for closed terms

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Computational Semantics

A computation system for closed terms

(λa.M)N ↦ M[N/a] (⟨x⟩M)@r ↦ M⟨r/x⟩ coex.A[r ↝ r'](M) ↦ coex.A'[r ↝ r'](M)

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Computational Semantics

A computation system for closed terms

(λa.M)N ↦ M[N/a]

Computational semantics: values

(⟨x⟩M)@r ↦ M⟨r/x⟩ coex.A[r ↝ r'](M) ↦ coex.A'[r ↝ r'](M)

Canonicity as a corollary

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Computational Semantics

Directly usable as a type theory

x : ℕ, y : ℕ ⪢ x + y ≐ y + x ∈ ℕ

with all the extensionalities

See our Part III paper for details

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Implementations

RedPRL

Our irst try, in PRL (Nuprl) style

yacc red

redprl.org

(stay tuned for Anders' talk!) (work in progress)

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Summary of Cartesian Cubes

Canonicity + univalence?

Yes! (Part III by AFH)

And higher inductive types?

Yes! (Part IV by CH)

And the full HoTT?

Yes! (Parts III & IV)