Toward a Curry-Howard Correspondence for Linear, Reversible - - PowerPoint PPT Presentation

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toward a curry howard correspondence for linear

Toward a Curry-Howard Correspondence for Linear, Reversible - - PowerPoint PPT Presentation

Sep 30, 2022 144 likes •642 views

Toward a Curry-Howard Correspondence for Linear, Reversible Computation Reversible Computation 2020 Kostia Chardonnet 1 , 2 Alexis Saurin 2 Benot Valiron 1 1 Universit Paris Saclay 2 Universit de Paris Classical vs Quantum Control Known :

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Toward a Curry-Howard Correspondence for Linear, Reversible Computation

Reversible Computation 2020

Kostia Chardonnet1,2 Alexis Saurin2 Benoît Valiron1

1Université Paris Saclay 2Université de Paris

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Classical vs Quantum Control

Computer Known :

Rich Type System
Classical Control

QRAM

⟲

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Classical vs Quantum Control

Computer Known :

Rich Type System
Classical Control

QRAM Elementary gates

⟳

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Classical vs Quantum Control

Computer Known :

Rich Type System
Classical Control

QRAM Elementary gates Outcome of measurement

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Classical vs Quantum Control

Computer Known :

Rich Type System
Classical Control

QRAM Elementary gates Outcome of measurement

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Classical vs Quantum Control

Computer Known :

Rich Type System
Classical Control

QRAM Elementary gates Outcome of measurement Missing :

No Rich Type System
No Quantum Control

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The Curry-Howard Correspondence

Types

Types describe data, structure programs.
“Well-typed Programs Cannot Go Wrong” - Robin Milner

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The Curry-Howard Correspondence

Types

Types describe data, structure programs.
“Well-typed Programs Cannot Go Wrong” - Robin Milner

Example : toString : nat → string toString(5) = "five".

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The Curry-Howard Correspondence

Types

Types describe data, structure programs.
“Well-typed Programs Cannot Go Wrong” - Robin Milner

Example : toString : nat → string toString(5) = "five". toString(5) Well typed ! toString("toto") Ill typed !

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The Curry-Howard Correspondence

toString(5) Well typed ! toString("toto") Ill typed ! toString : nat → string 5 : nat toString(5) : string

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The Curry-Howard Correspondence

toString(5) Well typed ! toString("toto") Ill typed ! toString : nat → string 5 : nat toString(5) : string A → B A B

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The Curry-Howard Correspondence

toString(5) Well typed ! toString("toto") Ill typed ! nat → string nat string A → B A B

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The Curry-Howard Correspondence

toString(5) Well typed ! toString("toto") Ill typed ! nat → string nat string A → B A B Curry-Howard Correspondence !

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The Curry-Howard Correspondence

Curry-Howard Correspondence ! λ-calculus Logic & Proofs Types Formulas Typed terms Proofs Evaluation Cut Elimination Formal Program Verification

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Our Work

Based on [Sabry, Valiron, Vizzotto] and [Baelde, Doumane, Saurin] Sabry et al. Baelde et al. This Work Linear ✓ ✓ ✓ Reversible ✓ ✗ ✓ (Co)-Inductive ✗ ✓ ✓ Curry-Howard ✗ ✗ ✓ Quantum Case ✓ ✗ WIP

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Syntax

(Base types) A, B ::= 1 | X | A ⊕ B | A ⊗ B | µX.A | νX.A (Isos, first-order) α ::= A ↔ B (Isos, higher-order) T ::= α1 → · · · → αn → α inv

let in let in

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Syntax

(Base types) A, B ::= 1 | X | A ⊕ B | A ⊗ B | µX.A | νX.A (Isos, first-order) α ::= A ↔ B (Isos, higher-order) T ::= α1 → · · · → αn → α inv

nat = µX.1 ⊕ X
lists(A) = [A] = µX. 1 ⊕ (A ⊗ X)
streams(A) = νX. A ⊗ X

let in let in

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Syntax

(Base types) A, B ::= 1 | X | A ⊕ B | A ⊗ B | µX.A | νX.A (Isos, first-order) α ::= A ↔ B (Isos, higher-order) T ::= α1 → · · · → αn → α (Isos) ω ::= {e1 ↔ e′

1 | . . . | en ↔ e′ n} | λf .ω |

µf .ω | f | ω1 ω2 | inv ω

λg.µf .      [ ] ↔ [ ] h :: t ↔ let x = g h in let y = f t in x :: y      : A ↔ B → [A] ↔ [B]

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Syntax

(Base types) A, B ::= 1 | X | A ⊕ B | A ⊗ B | µX.A | νX.A (Isos, first-order) α ::= A ↔ B (Isos, higher-order) T ::= α1 → · · · → αn → α (Isos) ω ::= {e1 ↔ e′

1 | . . . | en ↔ e′ n} | λf .ω |

µf .ω | f | ω1 ω2 | inv ω

λg.µf .      [ ] ↔ [ ] h :: t ↔ let x = g h in let y = f t in x :: y      : A ↔ B → [A] ↔ [B]

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Syntax

(Base types) A, B ::= 1 | X | A ⊕ B | A ⊗ B | µX.A | νX.A (Isos, first-order) α ::= A ↔ B (Isos, higher-order) T ::= α1 → · · · → αn → α (Isos) ω ::= {e1 ↔ e′

1 | . . . | en ↔ e′ n} | λf .ω |

µf .ω | f | ω1 ω2 | inv ω

λg.µf .      [ ] ↔ [ ] h :: t ↔ let x = g h in let y = f t in x :: y      : A ↔ B → [A] ↔ [B]

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Properties

Syntax

Language comes with a rewriting system and a type system.
Ensuring exhaustivity and non-overlapping of clauses.
Ensuring productivity.

Semantic

Isos denote computations from A → B and B → A.

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Syntax - Example 2

map = λg.µf.      [ ] ↔ [ ] h :: t ↔ let x = g h in let y = f t in x :: y      : A ↔ B → [A] ↔ [B] map g f let in let in A B B A map g f let inv in let in A B B A

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Syntax - Example 2

map = λg.µf.      [ ] ↔ [ ] h :: t ↔ let x = g h in let y = f t in x :: y      : A ↔ B → [A] ↔ [B] map⊥ = λg.µf.      [ ] ↔ [ ] let x = g h in ↔ h :: t let y = f t in x :: y      : A ↔ B → [B] ↔ [A] map g f let inv in let in A B B A

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Syntax - Example 2

map = λg.µf.      [ ] ↔ [ ] h :: t ↔ let x = g h in let y = f t in x :: y      : A ↔ B → [A] ↔ [B] map⊥ = λg.µf.      [ ] ↔ [ ] let x = g h in ↔ h :: t let y = f t in x :: y      : A ↔ B → [B] ↔ [A] map⊥ = λg.µf.      [ ] ↔ [ ] x :: y ↔ let h = (inv (g)) x in let t = f y in h :: t      : A ↔ B → [B] ↔ [A]

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Results

Confluence

t1 t2 t3 t4 ∗ ∗ ∗ ∗

Type Preservation

If ⊢ t : A and t → t′ then ⊢ t′ : A.

Progress

If ⊢ t : A either t → t′ or t is a value.

Isos

For each ω we have ω ◦ (inv ω) = id = (inv ω) ◦ ω.

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Words on µMALL

Linear Logic with Induction (µX.A) and Co-Induction (νX.A) A ⊢ B[X ← µX.B] A ⊢ µX.B µR

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Words on µMALL

Linear Logic with Induction (µX.A) and Co-Induction (νX.A) 0 = ⊢ µX.1 ⊕ X

nat

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Words on µMALL

Linear Logic with Induction (µX.A) and Co-Induction (νX.A) 0 = ⊢ 1 ⊕ µX.1 ⊕ X µR ⊢ µX.1 ⊕ X

nat

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Words on µMALL

Linear Logic with Induction (µX.A) and Co-Induction (νX.A) 0 = ⊢ 1 ⊕1 ⊢ 1 ⊕ µX.1 ⊕ X µR ⊢ µX.1 ⊕ X

nat

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Words on µMALL

Linear Logic with Induction (µX.A) and Co-Induction (νX.A) 0 = 1R ⊢ 1 ⊕1 ⊢ 1 ⊕ µX.1 ⊕ X µR ⊢ µX.1 ⊕ X

nat

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Words on µMALL

Linear Logic with Induction (µX.A) and Co-Induction (νX.A) 0 = 1R ⊢ 1 ⊕1 ⊢ 1 ⊕ µX.1 ⊕ X µR ⊢ µX.1 ⊕ X

nat

n + 1 = ⊢ nat

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Words on µMALL

Linear Logic with Induction (µX.A) and Co-Induction (νX.A) 0 = 1R ⊢ 1 ⊕1 ⊢ 1 ⊕ µX.1 ⊕ X µR ⊢ µX.1 ⊕ X

nat

n + 1 = ⊢ 1 ⊕ nat µR ⊢ nat

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Words on µMALL

Linear Logic with Induction (µX.A) and Co-Induction (νX.A) 0 = 1R ⊢ 1 ⊕1 ⊢ 1 ⊕ µX.1 ⊕ X µR ⊢ µX.1 ⊕ X

nat

n + 1 = ⊢ nat ⊕2 ⊢ 1 ⊕ nat µR ⊢ nat

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Words on µMALL

Linear Logic with Induction (µX.A) and Co-Induction (νX.A) 0 = 1R ⊢ 1 ⊕1 ⊢ 1 ⊕ µX.1 ⊕ X µR ⊢ µX.1 ⊕ X

nat

n + 1 = n ⊢ nat ⊕2 ⊢ 1 ⊕ nat µR ⊢ nat

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Words on µMALL

Linear Logic with Induction (µX.A) and Co-Induction (νX.A) 0 = 1R ⊢ 1 ⊕1 ⊢ 1 ⊕ µX.1 ⊕ X µR ⊢ µX.1 ⊕ X

nat

n + 1 = n ⊢ nat ⊕2 ⊢ 1 ⊕ nat µR ⊢ nat 1 = 1R ⊢ 1 ⊕1 ⊢ 1 ⊕ nat µR ⊢ nat ⊕2 ⊢ 1 ⊕ nat µR ⊢ nat

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Words on µMALL

Linear Logic with Induction (µX.A) and Co-Induction (νX.A) Stream0 = ⊢ nat ⊢νX.nat ⊗ X ⊗ ⊢ nat ⊗ (νX.nat ⊗ X) νR ⊢νX.nat ⊗ X

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Words on µMALL

Linear Logic with Induction (µX.A) and Co-Induction (νX.A) Stream0 = ⊢ nat ⊢νX.nat ⊗ X ⊗ ⊢ nat ⊗ (νX.nat ⊗ X) νR ⊢νX.nat ⊗ X

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Words on µMALL

Linear Logic with Induction and Co-Induction . . . µR ⊢ µX.X µR ⊢ µX.X ⊢µX.X µR ⊢ µX.X Infinite derivations represented as graphs

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Words on µMALL

A derivation is valid if in every infinite branch :

Infinity of rules µL
Infinity of rules νR

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From Type Derivation To Proofs

Typed Terms Proofs ω : A ↔ B ω⊥ : B ↔ A π : A ⊢ B π⊥ : B ⊢ A

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From Derivations To Proofs - Example

Let us take the recursive identity on lists µf .      [ ] ↔ [ ] h :: t ↔ let t′ = f t in h :: t′      : [A] ↔ [A] let in

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From Derivations To Proofs - Example

let in µf .      [ ] ↔ [ ] h :: t ↔ let t′ = f t in h :: t′      ⊢ [A] 1L 1 ⊢ [A] A, [A] ⊢ [A] ⊗L A ⊗ [A] ⊢ [A] ⊕L 1 ⊕ (A ⊗ [A]) ⊢ [A] µL [A] ⊢ [A]

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From Derivations To Proofs - Example

let in µf .      [ ] ↔ [ ] h :: t ↔ let t′ = f t in h :: t′      1R ⊢ 1 ⊕1

R

⊢ 1 ⊕ (A ⊗ [a]) µR ⊢ [A] 1L 1 ⊢ [A] A, [A] ⊢ [A] ⊗L A ⊗ [A] ⊢ [A] ⊕L 1 ⊕ (A ⊗ [A]) ⊢ [A] µL [A] ⊢ [A]

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From Derivations To Proofs - Example

let in µf .      [ ] ↔ [ ] h :: t ↔ let t′ = f t in h :: t′      1R ⊢ 1 ⊕1

R

⊢ 1 ⊕ (A ⊗ [a]) µR ⊢ [A] 1L 1 ⊢ [A] [A] ⊢ [A] A, [A] ⊢ [A] cut A, [A] ⊢ [A] ⊗L A ⊗ [A] ⊢ [A] ⊕L 1 ⊕ (A ⊗ [A]) ⊢ [A] µL [A] ⊢ [A]

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From Derivations To Proofs - Example

let in µf .      [ ] ↔ [ ] h :: t ↔ let t′ = f t in h :: t′      1R ⊢ 1 ⊕1

R

⊢ 1 ⊕ (A ⊗ [a]) µR ⊢ [A] 1L 1 ⊢ [A] id [A] ⊢ [A] [A] ⊢ [A] cut [A] ⊢ [A] A, [A] ⊢ [A] cut A, [A] ⊢ [A] ⊗L A ⊗ [A] ⊢ [A] ⊕L 1 ⊕ (A ⊗ [A]) ⊢ [A] µL [A] ⊢ [A]

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From Derivations To Proofs - Example

let in µf .      [ ] ↔ [ ] h :: t ↔ let t′ = f t in h :: t′      1R ⊢ 1 ⊕1

R

⊢ 1 ⊕ (A ⊗ [a]) µR ⊢ [A] 1L 1 ⊢ [A] id [A] ⊢ [A] [A] ⊢ [A] cut [A] ⊢ [A] id a ⊢ a id [a] ⊢ [a] ⊗R A, [A] ⊢ A ⊗ [A] ⊕2

R

A, [A] ⊢ 1 ⊕ A ⊗ [A] µR A, [A] ⊢ [A] cut A, [A] ⊢ [A] ⊗L A ⊗ [A] ⊢ [A] ⊕L 1 ⊕ (A ⊗ [A]) ⊢ [A] µL [A] ⊢ [A]

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Conclusion

Results

Confluence.
Type Preservation.
Progress.
Isos are isomorphisms.

In Progress

Show that derivations built isos are valid.
Show that our reduction simulates cut-elimination.
Show that π, π⊥ are isomorphisms.
Consider the quantum case.

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