Linear Logic and Intersection Types

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Introduction Non-Idempotent Types Inhabitation

Linear Logic and Intersection Types

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Agenda for Today

1 Introduction

2 Non-Idempotent Types

3 Inhabitation

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Agenda

1 Introduction

3 Inhabitation

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Simply versus Intersection Types

Simply typedterms⊆strongly-normalizingterms.

Simply typedterms+strongly-normalizingterms.

(e.g.λx.xx).

Finite polymorphism

Simply types No

Intersection Types Yes x:(σ→σ)∧σ`xx:σ

Intersection typedterms=strongly-normalizingterms.

Intersection type systems areundecidable.

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Intersection Types

SystemC(Coppo-Dezani).

Contains aconstant typeωto typeALLterms.

SystemP(Pottinger).

No constant type.

Types in both systems enjoy ACI Axioms.

Associativity (σ∧ρ)∧τ ∼ σ∧(ρ∧τ) Commutativity σ∧ρ ∼ ρ∧σ

Idempotence σ∧σ ∼ σ

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Some Typical Results in λ -Calculus by means of Intersection Types

Definition

LetRbe any reduction sequence.

A termtis anR-normal formiff there is nopsuch thatt→Rp.

A termtisR-weakly normalizingiff there is some reduction sequencet→^∗_Rp, wherepis a normal form.

A termtisR-strongly normalizingiff there is no infinite reduction sequence t→R→R. . ..

The previous three notions will be considered for different reduction relations, notably for β-reduction.

A termtisβ-head normalizingifft→^∗_βλx1...xn.yt1...tm(n,m≥0). Ahead contextis a context of the shape(λx₁...x_n.)t1...t_m(n,m≥0) A termtissolvableiff there is a head contextCs.t.C[t]→^∗_βλx.x

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Some Typical Results by means of Intersection Types

Theorem

A termtissolvableiff it isβ-head normalizingiffΓ`t:σin systemCfor someΓ^and σsuch thatσ/ω.

Theorem

A termtisβ-weakly normalizingiffΓ`t:σin systemCfor someΓandσsuch that ω<pos⁺([Γ, σ]).

Theorem

A termtisstrongly-normalizingiffΓ`t:σin systemPfor someΓandσ.

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Comments

Only qualitative (and not quantitative) information is provided in the previous theorems.

No relation between types and consumption of resources.

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Towards Non-Idempotent Intersection Types

The idempotence axiom is ruled out, i.e.σ∧σ/σ

The non-idempotent intersection operator∧can be seen as the multiplicative linear logic connective⊗.

Quantitave models forλ-calculi (De Carvalho, Ehrhard).

Head-Normalization, Solvability, Weak-Normalization and Strong-Normalization can be proved bycombinatorialarguments (weightedsubject reduction properties).

Implicit Complexity⇒use of ressources (e.g. substitution) can be measured.

Partial substitution (c.f. Proof-Nets) can be seen as a measured operation of substitution.

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Agenda

1 Introduction

3 Inhabitation

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Towards non-idempotent types

Pioneers: P. Gardner, A. Kfoury, D. de Carvalho.

Several typing systems sharing the same principles.

Intuitionistic calculi, classical calculi, etc.

Called alsoquantitative types.

We choose here to present the main principles of non-idempotent types on a language with partial substitution.

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The Linear Substitution Calculus

Inspired from Milner’s calculus with partial substitution and the structural lambda calculus.

Terms:λ-terms withexplicit substitutions.

Contexts are terms with one whole^.

Reduction Relation→_LSCis at adistanceand performspartialsubstitution:

(λx.t)[y1/v1]. . .[yn/vn]u 7→_B t[x/u][y1/v1]. . .[yn/vn] C[[x]][x/u] 7→_c C[[u]][x/u]

t[x/u] 7→_Gc t if|t|_x=0

Example

(xx)[x/I]→_c(xI)[x/I]→_c(II)[x/u]→_GcII→_Bx[x/I]→_cI[x/I]→_Gc I

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Linear-Head Reduction

Inspired from Milner’s calculus.

With Linear-Logic Proof-Nets flavour.

Conexion with Krivine’s Abstract Machine.

Linear-Head Reduction is astandard strategy.

Linear-Head Reduction (lineary) performs reduction only onhead-positions.

Linear-Head Reductiondoes not eraseterms.

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Linear-Head Reduction Formally

Linear-Head Contexts:LH::=|λx.LH|LHt|LH[x/t].

Linear-Head Reduction (written→lhr): rules{B,hc}closed byhead-linear contexts.

(λx.t)[y1/v1]. . .[y_n/vn]u 7→_B t[x/u][y1/v1]. . .[yn/vn]

LH[[x]][x/u] 7→hc LH[[u]][x/u]

Remark:→lhr⊆→LSC.

Example

(xx)[x/I]→hc(I x)[x/I]→By[y/x][x/I]→hcx[y/x][x/I]→hcI[y/x][x/I]

is a linear-head reduction

(xx)[x/I]→c(xI)[x/I]→c(II)[x/I]→Gc II→By[y/I]→cI is not

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Types for Milner’s calculus

Types:

Strict Types

Non-idempotent intersection

Intersection types are represented by multisets (σ∧σis{{σ, σ}}_).

σ ::= α|A→σ (linear types) A ::= {{ }} |B (multiset types)

B ::= {{σk}}_k∈K(I,∅) (non-empty multiset types) Typing Rules:

Relevance (no weakening) Multiplicative rules

Syntax-directed typing rules

Typed termsmay containuntyped subterms(I=∅)

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The Type System H W

x:{{σ}} `x:σ (var) Γ`t:σ

Γ\x`λx.t:Γ(x)→σ (→I) Γ`t:{{σk}}_k∈K→σ (∆k`u:σk)k∈K

Γ +k∈K∆k`tu:σ (→E) x:{{σk}}_k∈K;Γ`t:σ (∆k`u:σk)k∈K

Γ +k∈K∆k`t[x/u] :σ (Cut)

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Example

(var) x:{{{{ }} →σ}} `x:{{ }} →σ

(→E) x:{{{{ }} →σ}} `xv:σ

(Cut) x:{{{{ }} →σ}} `(xv)[y/u] :σ

Typed terms may contain (pink) untyped subterms.

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Measuring Typing Derivations

Definition

Given a type derivationΦ, we define themeasureM(Φ)as the number of occurrences of typing rulesvar,→I,→EandCutinΦ.

Example

(var) x:{{{{ }} →σ}} `x:{{ }} →σ

(→E)

x:{{{{ }} →σ}} `xv:σ

(Cut) x:{{{{ }} →σ}} `(xv)[y/u] :σ

The measure of this type derivation is3.

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Relevance

Lemma (Weak Relevance)

IfΓ`H W t:σ, thendom(Γ)⊆fv(t).

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Technical Tools for Characterizing Head-Linear Normalization

Positionof terms are finite words on the alphabet{0,1}.

A positionp∈pos(t)is atyped occurrenceofΦ^{if either}p=, orp=ip⁰(i=0,1) andp⁰∈pos(t|i)is a typed occurrence ofsomeof their corresponding

subderivations.

A redex occurrence oftwhich is also a typed occurrence ofΦ^{is a}typed redex occurrenceoftinΦ^.

x:{{{{τ, τ}} →τ}} `x:{{τ, τ}} →τ

y:{{{{ }} →τ}} `y:{{ }} →τ y:{{{{ }} →τ}} `yz:τ

y:{{{{τ}} →τ}} `y:{{τ}} →τ z:{{τ}} `z:τ y:{{{{τ}} →τ}},z:{{τ}} `yz:τ x:{{{{τ, τ}} →τ}},y:{{{{ }} →τ,{{τ}} →τ}},z:{{τ}} `x(yz):τ

x:{{{{τ, τ}} →τ}} `x:{{τ, τ}} →τ

y:{{{{ }} →τ}} `y:{{ }} →τ y:{{{{ }} →τ}} `yz:τ

y:{{{{ }} →τ}} `y:{{ }} →τ y:{{{{ }} →τ}} `yz:τ x:{{{{τ, τ}} →τ}},y:{{{{ }} →τ,{{ }} →τ}} `x(yz):τ

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Results: towards characterization of head-linear normalization

Lemma (Weighted Subject Reduction for Typed Redex Occurrences)

WheneverΠ:Γ`H Wt:σandt→LSCt⁰reduces atyped redex occurrence, then Π⁰:Γ`H Wt⁰:σandM(Π)>M(Π⁰).

Corollary (Weighted Subject Reduction for Linear-Head Reduction)

WheneverΠ:Γ`_{H W}t:σandt→_lhrt⁰, thenΠ⁰:Γ`_{H W}t⁰:σandM(Π)>M(Π⁰).

Lemma (Subject Expansion for the Linear Substitution Calculus)

IfΠ:Γ`H Wt:σandt⁰→LSCt, thenΠ⁰:Γ`H W t⁰:σ.

Corollary (Subject Reduction for Linear-Head Reduction)

IfΠ:Γ`H Wt:σandt⁰→_lhrt, thenΠ⁰:Γ`H W t⁰:σ.

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Results: characterization of head-linear normalization

Theorem

The following statements are equivalent:

tislhr-weakly normalizing.

tisH W-typable.

Proof.uses, between others,

WeightedSubject Reduction (combinatorial argument for non-idempotent types).

Subject Expansion.

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Results: towards characterization of weak normalization

PositiveandNegativeoccurrences of types are defined as follows.

A∈pos⁺(A).

A∈pos⁺({{σk}}k∈K)if∃kA∈pos⁺(σk);A∈pos⁻({{σk}}k∈K)if∃kA∈pos⁻(σk). A∈pos⁺(E→τ)ifA∈pos⁻(E)orA∈pos⁺(τ);A∈pos⁻(E→τ)ifA∈pos⁺(E)or A∈pos⁻(τ).

A∈pos⁺(Γ)if∃y∈dom(Γ)s.t.A∈pos⁻(Γ(y));A∈pos⁻(Γ)if∃y∈dom(Γ)s.t.

A∈pos⁺(Γ(y)).

A∈pos⁺([Γ, τ])ifA∈pos⁺(Γ)orA∈pos⁺(τ);A∈pos⁻([Γ, τ])ifA∈pos⁻(Γ)or A∈pos⁻(τ).

As an example,{{ }} ∈pos⁺({{ }}), so that{{ }} ∈pos⁻({{ }} →σ),{{ }} ∈pos⁺(x:{{{{ }} →σ}})and {{ }} ∈pos⁺([x:{{{{ }} →σ}}, σ]).

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Results: characterization of weak normalization

Theorem (Weak-Normalization)

A termtisLSC-weakly normalizingiffΓ`t:σin systemH Wfor someΓandσsuch that{{ }}<pos⁺([Γ, σ]).

The following term isLSC-weakly normalizing:

z:α`(λx.z)(∆∆) :α The following term is notLSC-weakly normalizing:

z:{{{{ }} →α}} `z(∆∆) :α

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The Type System S

Γ\x`λx.t:Γ(x)→σ (→I) Γ`t:{{σk}}_k∈K→σ (∆k`u:σk)k∈K I,∅

Γ +k∈K∆k`tu:σ (→E1)

x:{{σk}}_k∈K;Γ`t:σ (∆k`u:σk)k∈K I,∅ Γ +k∈K∆k`t[x/u] :σ (Cut1) Γ`t:{{ }} →σ ∆`u:τ

Γ + ∆`tu:σ (→E2) Γ`t:σ ∆`u:τ x<dom(Γ) Γ + ∆`t[x/u] :σ (Cut2)

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Relevance

Lemma (Strong Relevance)

IfΓ`_St:σ, thendom(Γ)=fv(t).

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Results: characterization of strong normalization

Theorem (Strong-Normalization)

A termtisLSC-strongly normalizingifftisS-typable.

Proof.uses

Postponementoferasingsteps (which does not hold inλ-calculus).

WeightedSubject Reduction for systemS. Subject Expansion for systemS_.

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Agenda

1 Introduction

3 Inhabitation

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Turning the inhabitation problem into a decidable

question

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Typing and Inhabitation Problems

Typing Inhabitation

?`t: ? Γ`?:A

Simple Types: Decidable Decidable

Idempotent Types: Undecidable Undecidable (Urzyczyn) Non-idempotent Types: Undecidable Decidable

(Bucciarelli-K.-Ronchi Della Rocca)

Decidability of restricted classes of idempotent types (Kurata & Takahashi), (Urzyczyn), (Bunder), (Ku´smierek), (Dudenhefner & Rehof), . . .

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The Typing System for the Lambda-Calculus

Γ\x`λx.t:Γ(x)→σ (→I) Γ`t:{{σi}}_i∈I→σ (∆i`u:σi)i∈I

Γ +i∈I∆i`tu:σ (→E)

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Approximants

How to restrict the search space of the algorithm?

We use the normalization property to restrict the search space toapproximants:

a,b,c ::= Ω| N N ::= λx.N | L L ::= x| La

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The Algorithm

a∈T(Γ +(x:A), τ) x<dom(Γ)

λx.a∈T(Γ,A→τ) (Abs)

Γ = +i=1...nΓi (bi∈TI(Γi, σi))i=1...n

xb₁. . .b_n∈T(Γ +(x:{{A₁→. . .→A_n→τ}}), τ) (Head)

(ai∈T(Γi, σi))i∈I ↑_i∈Ia_i _

i∈I

a_i∈TI(+i∈IΓi,{{σi}}_i∈I) (Union)

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Soundness and Completeness

Theorem

The algorithm terminates, is sound and complete.

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Some Final Remarks

Non-idempotentintersection types are particularly pertinent for calculi with ressources.

Arithmeticalarguments for terminating proofs.

New logical characterization oflinear-headnormalization.

The inhabitation problem for intersection types has been proved to be undecidable, but breaking the idempotency of intersection types makesinhabitation decidable.