[PPT] - Curry-Howard isomorphism for sequent calculus at last Jos e Carlos PowerPoint Presentation

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Introduction Sequent calculus Natural deduction Results

Curry-Howard isomorphism for sequent calculus at last

Jos´ e Carlos Esp´ ırito Santo

Centro de Matem´ atica Universidade do Minho Portugal jes@math.uminho.pt

Estonian-Finnish Logic Meeting Rakvere, Estonia 14 November 2015

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Introduction Sequent calculus Natural deduction Results

Goal

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Introduction Sequent calculus Natural deduction Results

Goal

In the same sense as the simply-typed λ-calculus (resp. combinators) corresponds to natural deduction (Hilbert systems),

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Introduction Sequent calculus Natural deduction Results

Goal

In the same sense as the simply-typed λ-calculus (resp. combinators) corresponds to natural deduction (Hilbert systems), what variant of the simply-typed lambda-calculus does correspond to the sequent calculus for intuitionistic implicational logic?

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Introduction Sequent calculus Natural deduction Results

Goal

In the same sense as the simply-typed λ-calculus (resp. combinators) corresponds to natural deduction (Hilbert systems), what variant of the simply-typed lambda-calculus does correspond to the sequent calculus for intuitionistic implicational logic? Question in the purest form Clear-cut answer required

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Introduction Sequent calculus Natural deduction Results

Goal

In the same sense as the simply-typed λ-calculus (resp. combinators) corresponds to natural deduction (Hilbert systems), what variant of the simply-typed lambda-calculus does correspond to the sequent calculus for intuitionistic implicational logic? Question in the purest form Clear-cut answer required More than a term assignment, a computational interpretation

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Introduction Sequent calculus Natural deduction Results

Goal

In the same sense as the simply-typed λ-calculus (resp. combinators) corresponds to natural deduction (Hilbert systems), what variant of the simply-typed lambda-calculus does correspond to the sequent calculus for intuitionistic implicational logic? Question in the purest form Clear-cut answer required More than a term assignment, a computational interpretation An interpretation of proofs in terms of a functional language, so that, when cut-elimination is rephrased accordingly, a meaningful, perhaps familiar, computational behavior is recognized

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Introduction Sequent calculus Natural deduction Results

An open question

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Introduction Sequent calculus Natural deduction Results

An open question

Girard, 1989: “From an algorithmic viewpoint, the sequent calculus has no Curry-Howard isomorphism, because of the multitude of ways of writing the same proof”

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Introduction Sequent calculus Natural deduction Results

An open question

Girard, 1989: “From an algorithmic viewpoint, the sequent calculus has no Curry-Howard isomorphism, because of the multitude of ways of writing the same proof” Sørensen-Urzykzyn, 2006: sequent calculus corresponds to explicit substitutions, but “this is just the beginning of the story”

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Introduction Sequent calculus Natural deduction Results

An open question

Girard, 1989: “From an algorithmic viewpoint, the sequent calculus has no Curry-Howard isomorphism, because of the multitude of ways of writing the same proof” Sørensen-Urzykzyn, 2006: sequent calculus corresponds to explicit substitutions, but “this is just the beginning of the story” Wikipedia: “The structure of sequent calculus relates to a calculus whose structure is close to the one of some abstract machines”

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Introduction Sequent calculus Natural deduction Results

Basic unanswered questions

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Introduction Sequent calculus Natural deduction Results

Basic unanswered questions

What does a variable stand for in a sequent calculus proof expression? What is the related substitution operation?

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Introduction Sequent calculus Natural deduction Results

Basic unanswered questions

What does a variable stand for in a sequent calculus proof expression? What is the related substitution operation? What is co-control?

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Introduction Sequent calculus Natural deduction Results

Plan

Sequent calculus λ˜ µ

Internal interpretation

Natural deduction λlet

External interpretation Explains co-control, justifies design of λ˜ µ

Results

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Introduction Sequent calculus Natural deduction Results

SEQUENT CALCULUS

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Introduction Sequent calculus Natural deduction Results Preliminaries

Overview of λ˜ µ

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Introduction Sequent calculus Natural deduction Results Preliminaries

Overview of λ˜ µ

System λ˜ µ: formal vector notation with first-class co-control

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Introduction Sequent calculus Natural deduction Results Preliminaries

Overview of λ˜ µ

System λ˜ µ: formal vector notation with first-class co-control Generalization of (the essence of) λ-calculus

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Introduction Sequent calculus Natural deduction Results Preliminaries

Overview of λ˜ µ

System λ˜ µ: formal vector notation with first-class co-control Generalization of (the essence of) λ-calculus Re-interpretation of a 15-years-old result

The λ-calculus has an isomorphic copy inside λ This copy is a formal vector notation

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Introduction Sequent calculus Natural deduction Results Preliminaries

Overview of λ˜ µ

System λ˜ µ: formal vector notation with first-class co-control Generalization of (the essence of) λ-calculus Re-interpretation of a 15-years-old result

The λ-calculus has an isomorphic copy inside λ This copy is a formal vector notation

Dualization of first-class control as found in λµ-calculus

Concept of co-continuation ˜ µx.t (known from λµ˜ µ) becomes a co-control operator

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Introduction Sequent calculus Natural deduction Results Preliminaries

Overview of λ˜ µ

System λ˜ µ: formal vector notation with first-class co-control Generalization of (the essence of) λ-calculus Re-interpretation of a 15-years-old result

The λ-calculus has an isomorphic copy inside λ This copy is a formal vector notation

Dualization of first-class control as found in λµ-calculus

Concept of co-continuation ˜ µx.t (known from λµ˜ µ) becomes a co-control operator

Variables in proof expressions stand for co-continuations

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Introduction Sequent calculus Natural deduction Results Preliminaries

Overview of λ˜ µ

System λ˜ µ: formal vector notation with first-class co-control Generalization of (the essence of) λ-calculus Re-interpretation of a 15-years-old result

The λ-calculus has an isomorphic copy inside λ This copy is a formal vector notation

Dualization of first-class control as found in λµ-calculus

Concept of co-continuation ˜ µx.t (known from λµ˜ µ) becomes a co-control operator

Variables in proof expressions stand for co-continuations λ˜ µ is developed hand-in-hand with isomorphic natural deduction system λlet

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Introduction Sequent calculus Natural deduction Results Preliminaries

Herbelin’s λ-calculus

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Introduction Sequent calculus Natural deduction Results Preliminaries

Herbelin’s λ-calculus

Permutation-free, focused fragment of sequent calculus

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Introduction Sequent calculus Natural deduction Results Preliminaries

Herbelin’s λ-calculus

Permutation-free, focused fragment of sequent calculus Two-sorted syntax

Proof terms t and lists l Sequents Γ ⇒ t : A and Γ|A ⇒ l : B

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Introduction Sequent calculus Natural deduction Results Preliminaries

Herbelin’s λ-calculus

Permutation-free, focused fragment of sequent calculus Two-sorted syntax

Proof terms t and lists l Sequents Γ ⇒ t : A and Γ|A ⇒ l : B

Sketch of two computational interpretations

Internal interpretation

Lists are stacks Cuts tl are states in a rudimentary abstract machine

External interpretation

Lists l = u1 :: · · · :: um :: [] are “evaluation contexts”/continuations [·]N1 · · · Nm Cuts tl denote M filled in [·]N1 · · · Nm

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Introduction Sequent calculus Natural deduction Results Preliminaries

Herbelin’s λ-calculus

Permutation-free, focused fragment of sequent calculus Two-sorted syntax

Proof terms t and lists l Sequents Γ ⇒ t : A and Γ|A ⇒ l : B

Sketch of two computational interpretations

Internal interpretation

Lists are stacks Cuts tl are states in a rudimentary abstract machine

External interpretation

Lists l = u1 :: · · · :: um :: [] are “evaluation contexts”/continuations [·]N1 · · · Nm Cuts tl denote M filled in [·]N1 · · · Nm

S.c. versus n.d.: inversion of associativity of non-abstractions

N.d. is left associative (· · · (MN1) · · · Nm) S.c. is right associative t(u1 :: · · · (um :: []) · · · )

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Introduction Sequent calculus Natural deduction Results Preliminaries

Vector notation

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Introduction Sequent calculus Natural deduction Results Preliminaries

Vector notation

Recall M, N ::= λx.M (1st form) | x N (2nd form) | (λx.M)N N (3rd form)

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Introduction Sequent calculus Natural deduction Results Preliminaries

Vector notation

Recall M, N ::= λx.M (1st form) | x N (2nd form) | (λx.M)N N (3rd form) Relaxed vector notation: 3rd form becomes M N

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Introduction Sequent calculus Natural deduction Results Preliminaries

Vector notation

Recall M, N ::= λx.M (1st form) | x N (2nd form) | (λx.M)N N (3rd form) Relaxed vector notation: 3rd form becomes M N This introduces the need for vector bookkeeping M[] = M (x Q)N N = x( QN N) (P Q)N N = P( QN N)

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Introduction Sequent calculus Natural deduction Results Preliminaries

A variant of Parigot’s λµ-calculus...

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Introduction Sequent calculus Natural deduction Results Preliminaries

A variant of Parigot’s λµ-calculus...

... called λµ-calculus

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Introduction Sequent calculus Natural deduction Results Preliminaries

A variant of Parigot’s λµ-calculus...

... called λµ-calculus ... whose management of sequents’ RHS we want to “dualize”

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Introduction Sequent calculus Natural deduction Results Preliminaries

A variant of Parigot’s λµ-calculus...

... called λµ-calculus ... whose management of sequents’ RHS we want to “dualize” Control points Mˆ a, where a, b, · · · are co-variables Control operator µa.M

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Introduction Sequent calculus Natural deduction Results Preliminaries

A variant of Parigot’s λµ-calculus...

... called λµ-calculus ... whose management of sequents’ RHS we want to “dualize” Control points Mˆ a, where a, b, · · · are co-variables Control operator µa.M Control operation: if C = ([·]N1 · · · Nm)ˆ b C[µa.M] → [C/a]M

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Introduction Sequent calculus Natural deduction Results Preliminaries

A variant of Parigot’s λµ-calculus...

... called λµ-calculus ... whose management of sequents’ RHS we want to “dualize” Control points Mˆ a, where a, b, · · · are co-variables Control operator µa.M Control operation: if C = ([·]N1 · · · Nm)ˆ b C[µa.M] → [C/a]M Context substitution (rather than “structural substitution”) [C/a](Mˆ a) = C[M′] where M′ = [C/a]M

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Syntax of proof expressions

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Syntax of proof expressions

(Terms) t, u, v ::= λx.t (1st form) | xˆk (2nd form, co-control point) | tk (3rd form) (Generalized vectors) k ::= [] (empty vector) | ˜ µx.v (co-control operator) | u :: k (vector constructor)

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Typing

Recall (terms) t, u, v ::= λx.t | xˆk | tk (gen. vectors) k ::= [] | ˜ µx.t | u :: k

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Typing

Recall (terms) t, u, v ::= λx.t | xˆk | tk (gen. vectors) k ::= [] | ˜ µx.t | u :: k Sequents Γ ⊢ t : A and Γ|A ⊢ k : A

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Typing

Recall (terms) t, u, v ::= λx.t | xˆk | tk (gen. vectors) k ::= [] | ˜ µx.t | u :: k Sequents Γ ⊢ t : A and Γ|A ⊢ k : A Typing/inference rules Ax Γ|A ⊢ [] : A Cut Γ ⊢ t : A Γ|A ⊢ k : B Γ ⊢ tk : B Pass Γ, x : A|A ⊢ k : B Γ, x : A ⊢ xˆk : B Act Γ, x : A ⊢ v : B Γ|A ⊢ ˜ µx.v : B ⊃L Γ ⊢ u : A Γ|B ⊢ k : C Γ|A ⊃ B ⊢ u :: k : C ⊃R Γ, x : A ⊢ t : B Γ ⊢ λx.t : A ⊃ B

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Derived syntax

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Derived syntax

Concatenation of two generalized vectors k@k′ []@k′ = k′ (u :: k)@k′ = u :: (k@k′) (˜ µx.t)@k′ = ˜ µx.tk′

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Derived syntax: co-continuations

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Derived syntax: co-continuations

Recall (terms) t, u, v ::= λx.t | xˆk | tk (gen. vectors) k ::= [] | ˜ µx.t | u :: k

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Derived syntax: co-continuations

Recall (terms) t, u, v ::= λx.t | xˆk | tk (gen. vectors) k ::= [] | ˜ µx.t | u :: k Co-continuations H ::= xˆ[·] | t([·]) | H[u :: [·]] Hence co-continuations have the forms

xˆ(u1 :: · · · :: um :: [·]) or t(u1 :: · · · :: um :: [·]) (m ≥ 0) that is, a non-abstraction with a hole at the right end hole expects a vector

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Derived syntax: co-continuations

Recall (terms) t, u, v ::= λx.t | xˆk | tk (gen. vectors) k ::= [] | ˜ µx.t | u :: k Co-continuations H ::= xˆ[·] | t([·]) | H[u :: [·]] Hence co-continuations have the forms

xˆ(u1 :: · · · :: um :: [·]) or t(u1 :: · · · :: um :: [·]) (m ≥ 0) that is, a non-abstraction with a hole at the right end hole expects a vector

Co-continuation substitution [H/x]t and [H/x]k [H/x](xˆk) = H[k′] with k′ = [H/x]k

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Reduction

Recall (terms) t, u, v ::= λx.t | xˆk | tk (gen. vectors) k ::= [] | ˜ µx.t | u :: k

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Reduction

Recall (terms) t, u, v ::= λx.t | xˆk | tk (gen. vectors) k ::= [] | ˜ µx.t | u :: k Reduction rules (β) (λx.t)(u :: k) → (u(˜ µx.t))k (˜ µ) H[˜ µx.t] → [H/x]t (ǫ) t[] → t (π1) (xˆk)k′ → xˆ(k@k′) (π2) (tk)k′ → t(k@k′)

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Reduction

Recall (terms) t, u, v ::= λx.t | xˆk | tk (gen. vectors) k ::= [] | ˜ µx.t | u :: k Reduction rules (β) (λx.t)(u :: k) → (u(˜ µx.t))k (˜ µ) H[˜ µx.t] → [H/x]t (ǫ) t[] → t (π1) (xˆk)k′ → xˆ(k@k′) (π2) (tk)k′ → t(k@k′) Interp: formal vector notation with first-class co-control

β: function call ˜ µ: co-control operation ǫ and πi: vector bookkeeping in relaxed vector notation

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Reduction

Recall the ˜ µ-rule: H[˜ µx.t] → [H/x]t

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Reduction

Recall the ˜ µ-rule: H[˜ µx.t] → [H/x]t Partition of ˜ µ (ρ) yˆ(˜ µx.t) → [yˆ([·])/x]t (σ) t(˜ µx.t) → [t([·])/x]t (τ) H[u :: ˜ µx.t] → [H[u :: [·]]/x]t

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Introduction Sequent calculus Natural deduction Results The λ˜ µ-calculus

Reduction

Recall the ˜ µ-rule: H[˜ µx.t] → [H/x]t Partition of ˜ µ (ρ) yˆ(˜ µx.t) → [yˆ([·])/x]t (σ) t(˜ µx.t) → [t([·])/x]t (τ) H[u :: ˜ µx.t] → [H[u :: [·]]/x]t Logical procedures cut-elimination β, σ, ǫ, π ˜ µ-elimination = co-control ρ, τ, σ permutation τ, σ, ǫ, π focalization ρ, τ

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Introduction Sequent calculus Natural deduction Results

NATURAL DEDUCTION

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Introduction Sequent calculus Natural deduction Results Preliminaries

Developing natural deduction

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Introduction Sequent calculus Natural deduction Results Preliminaries

Developing natural deduction

Bidirectional syntax

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Introduction Sequent calculus Natural deduction Results Preliminaries

Developing natural deduction

Bidirectional syntax Let-expressions (“Lets” vs cuts) Lets vs explicit substitutions

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Introduction Sequent calculus Natural deduction Results Preliminaries

Developing natural deduction

Bidirectional syntax Let-expressions (“Lets” vs cuts) Lets vs explicit substitutions Isomorphic to sequent calculus

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Introduction Sequent calculus Natural deduction Results The system

Syntax of proof terms of λlet

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Introduction Sequent calculus Natural deduction Results The system

Syntax of proof terms of λlet

(Terms) M, N, P ::= λx.M (abstraction) | app(H) (applicative term) | let x := H in P (let-expression) (Heads) H ::= x (variable) | hd(M) (head term) | HN (application)

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Introduction Sequent calculus Natural deduction Results The system

Typing in λlet

Recall (Terms) M, N, P ::= λx.M | app(H) | let x := H in P (Heads) H ::= x | hd(M) | HN

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Introduction Sequent calculus Natural deduction Results The system

Typing in λlet

Recall (Terms) M, N, P ::= λx.M | app(H) | let x := H in P (Heads) H ::= x | hd(M) | HN Sequents Γ ⊢ M : A and Γ ⊲ H : A

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Introduction Sequent calculus Natural deduction Results The system

Typing in λlet

Recall (Terms) M, N, P ::= λx.M | app(H) | let x := H in P (Heads) H ::= x | hd(M) | HN Sequents Γ ⊢ M : A and Γ ⊲ H : A Rules Γ, x : A ⊲ x : A Hyp Γ ⊲ H : A Γ, x : A ⊢ P : B Γ ⊢ let x := H in P : B Let Γ ⊲ H : A Γ ⊢ app(H) : A WCoercion Γ ⊢ M : A Γ ⊲ hd(M) : A SCoercion Γ, x : A ⊢ M : B Γ ⊢ λx.M : A ⊃ B Intro Γ ⊲ H : A ⊃ B Γ ⊢ N : A Γ ⊲ HN : B Elim

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Introduction Sequent calculus Natural deduction Results The system

Derived syntax

Recall (Terms) M, N, P ::= λx.M | app(H) | let x := H in P (Heads) H ::= x | hd(M) | HN

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Introduction Sequent calculus Natural deduction Results The system

Derived syntax

Recall (Terms) M, N, P ::= λx.M | app(H) | let x := H in P (Heads) H ::= x | hd(M) | HN Continuations K ::= app([·]) | let x := [·] in P | K[[·]N] Hence co-continuations have the forms

app([·]N1 · · · Nm) or let x := [·]N1 · · · Nm in P (m ≥ 0) that is, a non-abstraction with a hole at the left end hole expects a head

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Introduction Sequent calculus Natural deduction Results The system

Reduction in λlet

Recall (Terms) M, N, P ::= λx.M | app(H) | let x := H in P (Heads) H ::= x | hd(M) | HN

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Introduction Sequent calculus Natural deduction Results The system

Reduction in λlet

Recall (Terms) M, N, P ::= λx.M | app(H) | let x := H in P (Heads) H ::= x | hd(M) | HN Rules (beta) hd(λx.M)N → hd(let x := hd(N) in M) (let) let x := H in P → [H/x]P (triv) app(hd(M)) → M (head1) hd(app(H)) → H (head2) K[hd(let x := H in P)] → let x := H in K[hd(P)]

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Introduction Sequent calculus Natural deduction Results The system

Reduction in λlet

Recall (Terms) M, N, P ::= λx.M | app(H) | let x := H in P (Heads) H ::= x | hd(M) | HN Rules (beta) hd(λx.M)N → hd(let x := hd(N) in M) (let) let x := H in P → [H/x]P (triv) app(hd(M)) → M (head1) hd(app(H)) → H (head2) K[hd(let x := H in P)] → let x := H in K[hd(P)] Comments: bidirectional, agnostic, computational λ-calculus

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Introduction Sequent calculus Natural deduction Results

Mapping sequent calculus

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Introduction Sequent calculus Natural deduction Results

Mapping sequent calculus

Map Θ : λ˜ µ → λlet xˆ(u1 :: · · · um :: []) → app(xN1 · · · Nm) xˆ(u1 :: · · · um :: ˜ µy.v) → let y := (xN1 · · · Nm) in P t(u1 :: · · · um :: []) → app(hd(M)N1 · · · Nm) t(u1 :: · · · um :: ˜ µy.v) → let y := (hd(M)N1 · · · Nm) in P Replace each left-introduction by an elimination

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Introduction Sequent calculus Natural deduction Results

Mapping sequent calculus

Map Θ : λ˜ µ → λlet xˆ(u1 :: · · · um :: []) → app(xN1 · · · Nm) xˆ(u1 :: · · · um :: ˜ µy.v) → let y := (xN1 · · · Nm) in P t(u1 :: · · · um :: []) → app(hd(M)N1 · · · Nm) t(u1 :: · · · um :: ˜ µy.v) → let y := (hd(M)N1 · · · Nm) in P Replace each left-introduction by an elimination Inversion of the associativity of non-abstractions

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Introduction Sequent calculus Natural deduction Results

Mapping sequent calculus

Map Θ : λ˜ µ → λlet xˆ(u1 :: · · · um :: []) → app(xN1 · · · Nm) xˆ(u1 :: · · · um :: ˜ µy.v) → let y := (xN1 · · · Nm) in P t(u1 :: · · · um :: []) → app(hd(M)N1 · · · Nm) t(u1 :: · · · um :: ˜ µy.v) → let y := (hd(M)N1 · · · Nm) in P

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Introduction Sequent calculus Natural deduction Results

Mapping sequent calculus

Map Θ : λ˜ µ → λlet xˆ(u1 :: · · · um :: []) → app(xN1 · · · Nm) xˆ(u1 :: · · · um :: ˜ µy.v) → let y := (xN1 · · · Nm) in P t(u1 :: · · · um :: []) → app(hd(M)N1 · · · Nm) t(u1 :: · · · um :: ˜ µy.v) → let y := (hd(M)N1 · · · Nm) in P Proof expressions of λ˜ µ as instructions to build λlet proofs tk → K[hd(M)] where t → M and k → K xˆk → K[x] where k → K u :: k → K[[·]N] where u → N and k → K ˜ µx.v → let x := [·] in P where v → P [] → app([·])

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Introduction Sequent calculus Natural deduction Results

RESULTS

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Introduction Sequent calculus Natural deduction Results Results

Isomorphism

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Introduction Sequent calculus Natural deduction Results Results

Isomorphism

Isomorphism Θ : λ˜ µ ∼ = λlet

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Introduction Sequent calculus Natural deduction Results Results

Isomorphism

Isomorphism Θ : λ˜ µ ∼ = λlet Bijections of syntactic classes terms t M terms

gen. vectors

k K continuations co-continuations H H heads

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Introduction Sequent calculus Natural deduction Results Results

Isomorphism

Isomorphism Θ : λ˜ µ ∼ = λlet Bijections of syntactic classes terms t M terms

gen. vectors

k K continuations co-continuations H H heads Remark: exchange of primitive/derived concepts

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Introduction Sequent calculus Natural deduction Results Results

Isomorphism

Isomorphism Θ : λ˜ µ ∼ = λlet Bijections of syntactic classes terms t M terms

gen. vectors

k K continuations co-continuations H H heads Remark: exchange of primitive/derived concepts Isomorphism of reduction relations β beta ˜ µ let ǫ triv π head

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Introduction Sequent calculus Natural deduction Results Results

Curry Howard at last

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Introduction Sequent calculus Natural deduction Results Results

Curry Howard at last

The two faces of the coin:

Internal interpretation: formal vector notation with first-class co-control External interpretation: bidirectional, agnostic, computational λ-calculus

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Introduction Sequent calculus Natural deduction Results Results

Curry Howard at last

The two faces of the coin:

Internal interpretation: formal vector notation with first-class co-control External interpretation: bidirectional, agnostic, computational λ-calculus

Example: expression xˆk

Internal interpretation: co-control point, fill k in the hole of the co-continuation H that will substitute x

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Introduction Sequent calculus Natural deduction Results Results

Curry Howard at last

The two faces of the coin:

Internal interpretation: formal vector notation with first-class co-control External interpretation: bidirectional, agnostic, computational λ-calculus

Example: expression xˆk

Internal interpretation: co-control point, fill k in the hole of the co-continuation H that will substitute x External interpretation: fill head x in the hole of continuation K, where K = Θk - an instruction for the construction of a λlet derivation (fill x in k better notation for this reading)

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Introduction Sequent calculus Natural deduction Results Results

Curry Howard at last

The two faces of the coin:

Internal interpretation: formal vector notation with first-class co-control External interpretation: bidirectional, agnostic, computational λ-calculus

Example: expression xˆk

Internal interpretation: co-control point, fill k in the hole of the co-continuation H that will substitute x External interpretation: fill head x in the hole of continuation K, where K = Θk - an instruction for the construction of a λlet derivation (fill x in k better notation for this reading)

Example: β-rule (λx.t)(u :: k) → (u(˜ µx.t))k

Internal interpretation: β-rule of a relaxed vector notation

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Introduction Sequent calculus Natural deduction Results Results

Curry Howard at last

The two faces of the coin:

Internal interpretation: formal vector notation with first-class co-control External interpretation: bidirectional, agnostic, computational λ-calculus

Example: expression xˆk

Internal interpretation: co-control point, fill k in the hole of the co-continuation H that will substitute x External interpretation: fill head x in the hole of continuation K, where K = Θk - an instruction for the construction of a λlet derivation (fill x in k better notation for this reading)

Example: β-rule (λx.t)(u :: k) → (u(˜ µx.t))k

Internal interpretation: β-rule of a relaxed vector notation External interpretation: rule beta of λlet written in the notation of instructions that λ˜ µ is

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More Results

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Introduction Sequent calculus Natural deduction Results Results

More Results

Studying forgetful/desugaring maps λlet → λ one gets

strong normalization of λlet, hence of λ˜ µ

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Introduction Sequent calculus Natural deduction Results Results

More Results

Studying forgetful/desugaring maps λlet → λ one gets

strong normalization of λlet, hence of λ˜ µ termination of the let rule, hence of ˜ µ rule of λ˜ µ

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Introduction Sequent calculus Natural deduction Results Results

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Studying forgetful/desugaring maps λlet → λ one gets

strong normalization of λlet, hence of λ˜ µ termination of the let rule, hence of ˜ µ rule of λ˜ µ the latter gives a focalization result: every t ∈ λ˜ µ has a unique ˜ µ-nf (which is a LJT proof, if t is typable)

SLIDE 92

Introduction Sequent calculus Natural deduction Results Results

More Results

Studying forgetful/desugaring maps λlet → λ one gets

strong normalization of λlet, hence of λ˜ µ termination of the let rule, hence of ˜ µ rule of λ˜ µ the latter gives a focalization result: every t ∈ λ˜ µ has a unique ˜ µ-nf (which is a LJT proof, if t is typable)

Agnosticism theorem

SLIDE 93

Introduction Sequent calculus Natural deduction Results Results

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Studying forgetful/desugaring maps λlet → λ one gets

strong normalization of λlet, hence of λ˜ µ termination of the let rule, hence of ˜ µ rule of λ˜ µ the latter gives a focalization result: every t ∈ λ˜ µ has a unique ˜ µ-nf (which is a LJT proof, if t is typable)

Agnosticism theorem

take the s.c. characterization of CBN and CBV...

SLIDE 94

Introduction Sequent calculus Natural deduction Results Results

More Results

Studying forgetful/desugaring maps λlet → λ one gets

strong normalization of λlet, hence of λ˜ µ termination of the let rule, hence of ˜ µ rule of λ˜ µ the latter gives a focalization result: every t ∈ λ˜ µ has a unique ˜ µ-nf (which is a LJT proof, if t is typable)

Agnosticism theorem

take the s.c. characterization of CBN and CBV...

btain the isomorphic characterization in n.d...

SLIDE 95

Introduction Sequent calculus Natural deduction Results Results

More Results

Studying forgetful/desugaring maps λlet → λ one gets

strong normalization of λlet, hence of λ˜ µ termination of the let rule, hence of ˜ µ rule of λ˜ µ the latter gives a focalization result: every t ∈ λ˜ µ has a unique ˜ µ-nf (which is a LJT proof, if t is typable)

Agnosticism theorem

take the s.c. characterization of CBN and CBV...

btain the isomorphic characterization in n.d...

and see how CBN and CBV are superimposed in λlet

SLIDE 96

Introduction Sequent calculus Natural deduction Results Results

More Results

Studying forgetful/desugaring maps λlet → λ one gets

strong normalization of λlet, hence of λ˜ µ termination of the let rule, hence of ˜ µ rule of λ˜ µ the latter gives a focalization result: every t ∈ λ˜ µ has a unique ˜ µ-nf (which is a LJT proof, if t is typable)

Agnosticism theorem

take the s.c. characterization of CBN and CBV...

btain the isomorphic characterization in n.d...

and see how CBN and CBV are superimposed in λlet

“Co-control theorem”

given that the let rule is isomorphic to the ˜ µ rule...

SLIDE 97

Introduction Sequent calculus Natural deduction Results Results

More Results

Studying forgetful/desugaring maps λlet → λ one gets

strong normalization of λlet, hence of λ˜ µ termination of the let rule, hence of ˜ µ rule of λ˜ µ the latter gives a focalization result: every t ∈ λ˜ µ has a unique ˜ µ-nf (which is a LJT proof, if t is typable)

Agnosticism theorem

take the s.c. characterization of CBN and CBV...

btain the isomorphic characterization in n.d...

and see how CBN and CBV are superimposed in λlet

“Co-control theorem”

given that the let rule is isomorphic to the ˜ µ rule...

ne understands why co-control is almost invisible in n.d.

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Introduction Sequent calculus Natural deduction Results Results

Duality

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Introduction Sequent calculus Natural deduction Results Results

Duality

Are control and co-control related by classical duality?

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Introduction Sequent calculus Natural deduction Results Results

Duality

Are control and co-control related by classical duality? Not within classical sequent calculus!

SLIDE 101

Introduction Sequent calculus Natural deduction Results Results

Duality

Are control and co-control related by classical duality? Not within classical sequent calculus! E.g. not in λµ˜ µ, because there

variables stand for terms (not co-continuations) ˜ µ is an ordinary term substitution former

SLIDE 102

Introduction Sequent calculus Natural deduction Results Results

Duality

Are control and co-control related by classical duality? Not within classical sequent calculus! E.g. not in λµ˜ µ, because there

variables stand for terms (not co-continuations) ˜ µ is an ordinary term substitution former

Yes in a system that unifies λ˜ µ and λlet - see the paper

SLIDE 103

Introduction Sequent calculus Natural deduction Results Results