Curry-Howard isomorphism for sequent calculus at lastcs.ioc.ee/logic-rakvere/espiritosanto-slides.pdf · Introduction Sequent calculus Natural deduction Results Goal In the same sense

Introduction Sequent calculus Natural deduction Results

Curry-Howard isomorphism for sequent calculusat last

Jose Carlos Espırito Santo

Centro de MatematicaUniversidade do Minho

[email protected]

Estonian-Finnish Logic MeetingRakvere, Estonia

14 November 2015


Goal

In the same sense as the simply-typed λ-calculus (resp.combinators) corresponds to natural deduction (Hilbertsystems), what variant of the simply-typed lambda-calculusdoes correspond to the sequent calculus for intuitionisticimplicational 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, ameaningful, perhaps familiar, computational behavior isrecognized


Goal

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

what variant of the simply-typed lambda-calculusdoes correspond to the sequent calculus for intuitionisticimplicational logic?






Goal







Goal







Goal







Goal







An open question

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

Sørensen-Urzykzyn, 2006: sequent calculus corresponds toexplicit substitutions, but “this is just the beginning of thestory”

Wikipedia: “The structure of sequent calculus relates to acalculus whose structure is close to the one of some abstractmachines”


An open question





An open question





An open question





Basic unanswered questions

What does a variable stand for in a sequent calculus proofexpression?

What is the related substitution operation?

What is co-control?





What is co-control?





What is co-control?


Plan

Sequent calculus λµ

Internal interpretation

Natural deduction λlet

External interpretationExplains co-control, justifies design of λµ

Results


SEQUENT CALCULUS


Preliminaries

Overview of λµ

System λµ: 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 λThis copy is a formal vector notation

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

Concept of co-continuationµx .t (known from λµµ) becomes a co-control operator

Variables in proof expressions stand for co-continuations

λµ is developed hand-in-hand with isomorphic naturaldeduction system λlet


Preliminaries

Overview of λµ










Preliminaries

Overview of λµ










Preliminaries

Overview of λµ










Preliminaries

Overview of λµ










Preliminaries

Overview of λµ










Preliminaries

Overview of λµ










Preliminaries

Herbelin’s λ-calculus

Permutation-free, focused fragment of sequent calculus

Two-sorted syntax

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

Sketch of two computational interpretationsInternal interpretation

Lists are stacksCuts tl are states in a rudimentary abstract machine

External interpretation

Lists l = u1 :: · · · :: um :: [] are “evaluationcontexts”/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 :: []) · · · )


Preliminaries



Two-sorted syntax










Preliminaries



Two-sorted syntax










Preliminaries



Two-sorted syntax










Preliminaries



Two-sorted syntax










Preliminaries

Vector notation

RecallM,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)


Preliminaries

Vector notation


| x ~N (2nd form)




M[] = M

(x ~Q)N ~N = x( ~QN ~N)

(P ~Q)N ~N = P( ~QN ~N)


Preliminaries

Vector notation


| x ~N (2nd form)




M[] = M

(x ~Q)N ~N = x( ~QN ~N)

(P ~Q)N ~N = P( ~QN ~N)


Preliminaries

Vector notation


| x ~N (2nd form)




M[] = M

(x ~Q)N ~N = x( ~QN ~N)

(P ~Q)N ~N = P( ~QN ~N)


Preliminaries

A variant of Parigot’s λµ-calculus...

... called λµ-calculus

... whose management of sequents’ RHS we want to “dualize”

Control points M a, where a, b, · · · are co-variables

Control operator µa.M

Control operation: if C = ([·]N1 · · ·Nm) b

C[µa.M]→ [C/a]M

Context substitution (rather than “structural substitution”)

[C/a](M a) = C[M ′] where M ′ = [C/a]M


Preliminaries







C[µa.M]→ [C/a]M




Preliminaries







C[µa.M]→ [C/a]M




Preliminaries







C[µa.M]→ [C/a]M




Preliminaries







C[µa.M]→ [C/a]M




Preliminaries







C[µa.M]→ [C/a]M




The λµ-calculus

Syntax of proof expressions

(Terms) t, u, v ::= λx .t (1st form)| x k (2nd form, co-control point)| tk (3rd form)

(Generalized vectors) k ::= [] (empty vector)| µx .v (co-control operator)| u :: k (vector constructor)


The λµ-calculus

Syntax of proof expressions

(Terms) t, u, v ::= λx .t (1st form)| x k (2nd form, co-control point)| tk (3rd form)

(Generalized vectors) k ::= [] (empty vector)| µx .v (co-control operator)| u :: k (vector constructor)


The λµ-calculus

Typing

Recall

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

Sequents Γ ` t : A and Γ|A ` k : A

Typing/inference rules

AxΓ|A ` [] : A

CutΓ ` t : A Γ|A ` k : B

Γ ` tk : B

PassΓ, x : A|A ` k : BΓ, x : A ` x k : B Act Γ, x : A ` v : B

Γ|A ` µx .v : B

⊃LΓ ` u : A Γ|B ` k : C

Γ|A ⊃ B ` u :: k : C⊃R

Γ, x : A ` t : BΓ ` λx .t : A ⊃ B


The λµ-calculus

Typing

Recall




AxΓ|A ` [] : A


Γ ` tk : B


Γ|A ` µx .v : B

⊃LΓ ` u : A Γ|B ` k : C

Γ|A ⊃ B ` u :: k : C⊃R

Γ, x : A ` t : BΓ ` λx .t : A ⊃ B


The λµ-calculus

Typing

Recall




AxΓ|A ` [] : A


Γ ` tk : B


Γ|A ` µx .v : B

⊃LΓ ` u : A Γ|B ` k : C

Γ|A ⊃ B ` u :: k : C⊃R

Γ, x : A ` t : BΓ ` λx .t : A ⊃ B


The λµ-calculus

Derived syntax

Concatenation of two generalized vectors k@k ′

[]@k ′ = k ′

(u :: k)@k ′ = u :: (k@k ′)(µx .t)@k ′ = µx .tk ′


The λµ-calculus

Derived syntax

Concatenation of two generalized vectors k@k ′

[]@k ′ = k ′

(u :: k)@k ′ = u :: (k@k ′)(µx .t)@k ′ = µx .tk ′


The λµ-calculus

Derived syntax: co-continuations

Recall


Co-continuations

H ::= x [·] | t([·]) |H[u :: [·]]

Hence co-continuations have the forms

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

Co-continuation substitution [H/x ]t and [H/x ]k

[H/x ](x k) = H[k ′] with k ′ = [H/x ]k


The λµ-calculus


Recall


Co-continuations

H ::= x [·] | t([·]) |H[u :: [·]]




[H/x ](x k) = H[k ′] with k ′ = [H/x ]k


The λµ-calculus


Recall


Co-continuations

H ::= x [·] | t([·]) |H[u :: [·]]




[H/x ](x k) = H[k ′] with k ′ = [H/x ]k


The λµ-calculus


Recall


Co-continuations

H ::= x [·] | t([·]) |H[u :: [·]]




[H/x ](x k) = H[k ′] with k ′ = [H/x ]k


The λµ-calculus

Reduction

Recall


Reduction rules

(β) (λx .t)(u :: k) → (u(µx .t))k(µ) H[µx .t] → [H/x ]t(ε) t[] → t

(π1) (x k)k ′ → x (k@k ′)(π2) (tk)k ′ → t(k@k ′)

Interp: formal vector notation with first-class co-controlβ: function callµ: co-control operationε and πi : vector bookkeeping in relaxed vector notation


The λµ-calculus

Reduction

Recall


Reduction rules


(π1) (x k)k ′ → x (k@k ′)(π2) (tk)k ′ → t(k@k ′)



The λµ-calculus

Reduction

Recall


Reduction rules


(π1) (x k)k ′ → x (k@k ′)(π2) (tk)k ′ → t(k@k ′)



The λµ-calculus

Reduction

Recall the µ-rule: H[µx .t]→ [H/x ]t

Partition of µ

(ρ) y (µx .t) → [y ([·])/x ]t(σ) t(µx .t) → [t([·])/x ]t(τ) H[u :: µx .t] → [H[u :: [·]]/x ]t

Logical procedurescut-elimination β, σ, ε, π

µ-elimination = co-control ρ, τ, σpermutation τ, σ, ε, πfocalization ρ, τ


The λµ-calculus

Reduction


Partition of µ





The λµ-calculus

Reduction


Partition of µ





NATURAL DEDUCTION


Preliminaries

Developing natural deduction

Bidirectional syntax

Let-expressions (“Lets” vs cuts)

Lets vs explicit substitutions

Isomorphic to sequent calculus


Preliminaries







Preliminaries







Preliminaries







The system

Syntax of proof terms of λlet

(Terms) M,N,P ::= λx .M (abstraction)| app(H) (applicative term)| let x := H inP (let-expression)

(Heads) H ::= x (variable)| hd(M) (head term)| HN (application)


The system

Syntax of proof terms of λlet

(Terms) M,N,P ::= λx .M (abstraction)| app(H) (applicative term)| let x := H inP (let-expression)

(Heads) H ::= x (variable)| hd(M) (head term)| HN (application)


The system

Typing in λlet

Recall

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

Sequents Γ ` M : A and Γ B H : A

Rules

Γ, x : AB x : AHyp

Γ B H : A Γ, x : A ` P : B

Γ ` let x := H inP : BLet

Γ B H : AΓ ` app(H) : A

WCoercionΓ ` M : A

Γ B hd(M) : ASCoercion

Γ, x : A ` M : B

Γ ` λx .M : A ⊃ BIntro

Γ B H : A ⊃ B Γ ` N : AΓ B HN : B

Elim


The system

Typing in λlet

Recall



Rules

Γ, x : AB x : AHyp

Γ B H : A Γ, x : A ` P : B



WCoercionΓ ` M : A


Γ, x : A ` M : B


Γ B H : A ⊃ B Γ ` N : AΓ B HN : B

Elim


The system

Typing in λlet

Recall



Rules

Γ, x : AB x : AHyp

Γ B H : A Γ, x : A ` P : B



WCoercionΓ ` M : A


Γ, x : A ` M : B


Γ B H : A ⊃ B Γ ` N : AΓ B HN : B

Elim


The system

Derived syntax

Recall


Continuations

K ::= app([·]) | let x := [·] inP | K[[·]N]


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


The system

Derived syntax

Recall


Continuations

K ::= app([·]) | let x := [·] inP | K[[·]N]


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


The system

Reduction in λlet

Recall


Rules

(beta) hd(λx .M)N → hd(let x := hd(N) inM)(let) let x := H inP → [H/x ]P

(triv) app(hd(M)) → M(head1) hd(app(H)) → H(head2) K[hd(let x := H inP)] → let x := H inK[hd(P)]

Comments: bidirectional, agnostic, computational λ-calculus


The system

Reduction in λlet

Recall


Rules





The system

Reduction in λlet

Recall


Rules





Mapping sequent calculus

Map Θ : λµ→ λlet

x (u1 :: · · · um :: []) 7→ app(xN1 · · ·Nm)x (u1 :: · · · um :: µy .v) 7→ let y := (xN1 · · ·Nm) inP

t(u1 :: · · · um :: []) 7→ app(hd(M)N1 · · ·Nm)t(u1 :: · · · um :: µy .v) 7→ let y := (hd(M)N1 · · ·Nm) inP

Replace each left-introduction by an elimination

Inversion of the associativity of non-abstractions




















Proof expressions of λµ as instructions to build λlet proofs

tk 7→ K[hd(M)] where t 7→ M and k 7→ Kx k 7→ K[x ] where k 7→ K

u :: k 7→ K[[·]N] where u 7→ N and k 7→ Kµx .v 7→ let x := [·] inP where v 7→ P

[] 7→ app([·])






Proof expressions of λµ as instructions to build λlet proofs

tk 7→ K[hd(M)] where t 7→ M and k 7→ Kx k 7→ K[x ] where k 7→ K

u :: k 7→ K[[·]N] where u 7→ N and k 7→ Kµx .v 7→ let x := [·] inP where v 7→ P

[] 7→ app([·])


RESULTS


Results

Isomorphism

Isomorphism Θ : λµ ∼= λlet

Bijections of syntactic classes

terms t M termsgen. vectors k K continuations

co-continuations H H heads

Remark: exchange of primitive/derived concepts

Isomorphism of reduction relations

β betaµ letε trivπ head


Results

Isomorphism









Results

Isomorphism









Results

Isomorphism









Results

Isomorphism









Results

Curry Howard at last

The two faces of the coin:

Internal interpretation: formal vector notation with first-classco-controlExternal interpretation: bidirectional, agnostic, computationalλ-calculus

Example: expression x k

Internal interpretation: co-control point, fill k in the hole ofthe co-continuation H that will substitute xExternal interpretation: fill head x in the hole of continuationK, where K = Θk - an instruction for the construction of aλlet 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 notationExternal interpretation: rule beta of λlet written in thenotation of instructions that λµ is


Results









Results





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

External interpretation: fill head x in the hole of continuationK, where K = Θk - an instruction for the construction of aλlet derivation (fill x in k better notation for this reading)




Results









Results







Internal interpretation: β-rule of a relaxed vector notation

External interpretation: rule beta of λlet written in thenotation of instructions that λµ is


Results









Results

More Results

Studying forgetful/desugaring maps λlet→ λ one gets

strong normalization of λlet, hence of λµtermination of the let rule, hence of µ rule of λµthe latter gives a focalization result: every t ∈ λµ has a uniqueµ-nf (which is a LJT proof, if t is typable)

Agnosticism theorem

take the s.c. characterization of CBN and CBV...obtain 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...one understands why co-control is almost invisible in n.d.


Results

More Results


strong normalization of λlet, hence of λµ

termination of the let rule, hence of µ rule of λµthe latter gives a focalization result: every t ∈ λµ has a uniqueµ-nf (which is a LJT proof, if t is typable)

Agnosticism theorem





Results

More Results


strong normalization of λlet, hence of λµtermination of the let rule, hence of µ rule of λµ

the latter gives a focalization result: every t ∈ λµ has a uniqueµ-nf (which is a LJT proof, if t is typable)

Agnosticism theorem





Results

More Results



Agnosticism theorem





Results

More Results



Agnosticism theorem





Results

More Results



Agnosticism theorem

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

obtain the isomorphic characterization in n.d...and see how CBN and CBV are superimposed in λlet




Results

More Results



Agnosticism theorem

take the s.c. characterization of CBN and CBV...obtain the isomorphic characterization in n.d...

and see how CBN and CBV are superimposed in λlet




Results

More Results



Agnosticism theorem





Results

More Results



Agnosticism theorem



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

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


Results

More Results



Agnosticism theorem





Results

Duality

Are control and co-control related by classical duality?

Not within classical sequent calculus!

E.g. not in λµµ, because there

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

Yes in a system that unifies λµ and λlet - see the paper


Results

Duality







Results

Duality







Results

Duality







Results

Duality







Results

THANKS

Curry-Howard isomorphism for sequent calculus at lastcs.ioc.ee/logic-rakvere/espiritosanto-slides.pdf · Introduction Sequent calculus Natural deduction Results Goal In the same sense

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