The resource lambda calculus is short-sighted in its relational model

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The resource lambda calculus is short-sighted in its relational model

Flavien BREUVART

PPS, Paris Denis Diderot

26-28 June 2013

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Introduction

The resource calculus (∂Λ)

A resource sensitive extension of the pureλ-calculus with linear subterms thatcannot suffer any duplication nor erasingalong the reduction.

The relational model M

∞

M∞ is a reflexive object of MRel, which is theco-Kleisli category of the well known model of linear logicRel.

They have good properties:

• M_∞is a sensible modelof∂Λ (M⇑⇔[[M]] =∅).

• M_∞isfully abstractfor the pureλ-calculus ([Man09]).

• Both∂Λ andM_∞are well-behaved w.r.t. Taylor expansion.

Question of [Buc&Al 11]

IsM_∞ fully abstractfor∂Λ?

My answer [TLCA13]

NO

Flavien BREUVART PPS The resource lambda calculus is short-sighted in its relational model 2 / 1

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Introduction

The resource calculus (∂Λ)

A resource sensitive extension of the pureλ-calculus with linear subterms thatcannot suffer any duplication nor erasingalong the reduction.

The relational model M

∞

M∞ is a reflexive object of MRel, which is theco-Kleisli category of the well known model of linear logicRel.

They have good properties:

• M_∞ is a sensible modelof∂Λ (M⇑⇔[[M]] =∅).

• M_∞ isfully abstractfor the pureλ-calculus ([Man09]).

• Both∂Λ andM_∞ are well-behaved w.r.t. Taylor expansion.

Question of [Buc&Al 11]

IsM_∞ fully abstractfor∂Λ?

My answer [TLCA13]

NO

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Definition of ∂Λ

Grammar

(terms) Λ M,N::= x | λx.M | M P

(bags) Λ^b P,Q ::= [M^!] | [M] | [] | P·Q (sums) Λ^s L,M∈ Mf(Λ)

[N^!] is a usualλ-calculus argument, used as many time as wanted.

[N] is a linear argument, used exactly once.

[] the absence of arguments.

P·Q is the union of bags, i.e. parallel composition of arguments.

Λ^s possible outcomes of a computation are finite sums of terms.

Reduction (by examples)

(λx.x [x]) [u^!]→u[u] (λx.x [x]) [u]→0 (λx.x [x]) [u,v]→(u[v])+(v [u]) (λx.x [x^!]) [u]→u []

(λx.x [x^!]) [u,v,w]→(u [v,w])+(v [u,w])+(w [u,v])

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Observational semantic of ∂Λ

Reduction Strategy

outer-reduction: Reduction under anynon bangedcontext.

outer-normal forms: terms with no possible outer-reductions:

λx1...xk.y P1· · ·Pp

where

Pi = [N1, ...,Nn,M₁^!, ...,M_m^!] withN₁, ...,N_n in outer-normal form.

may-outer-convergence: converges to a sum containing a normal form:

M⇓ iff M→^∗Σ_iN_i such that ∃N_i∈ONF

Example

(λx.(x [M^!]))+Nis amonfform but notλx.(x [(λy.y)x]).

Operational order (v

_o

)

M v_oN iff ∀C(|.|), C(|M|)⇓ ⇒ C(|N|)⇓

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Definition of the CCC MRel

A categorical model of linear logic: Rel

Objects: Sets Morphisms: Relations Exponential: !A=M_f(A)

Remark: this is the free exponential.

The co-kleisli category: MRel

Objects: Sets Morphisms: relations fromM_f(A) andB Identities: 1A ={([α], α)|α∈A}

Composition: f;g :={(Σ_β∈baβ, α)|(b, α)∈g,(aβ, β)∈f} Cartesian product: ˘

i∈IAi:={(i, α)|i∈I, α∈Ai} Object of morphisms: A⇒B=Mf(A)×B All ccc-diagrams are given by the co-Kleisli construction from the exponential comonad.

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Definition of M

_∞

A reflexive object of MRel: M

∞

M0=∅ Mn+1=Mf(Mn)^(ω) M_∞=[ Mn

app andabs such that M∞ app

abs(M∞⇒M∞) are trivial.

Mf(Mn)^(ω): infinite quazi-everywhere empty lists of multisets overMn. This is the smallest solution of the equationD:= (!˘

ωD)^⊥ whose solutions are solutions ofD:= (!D)^⊥`D (em i.e. D:=D⇒D) but does not contain∅.

Grammar of M

∞

(M_∞) α, β ::= ∗ | a→α

M_f(M_∞) a,b ::= [α₁, ..., α_k] With the equation: ∗:= []→∗

so that∗represent theinfinite list of empty multisets.

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Interpretation of ∂λ into M

_∞

[[M]]^~^x ⊆(M_f(M_∞)^~^x×M_∞) [[P]]^~^x ⊆(M_f(M_∞)^~^x× M_f(M_∞))

Interpretation (by examples)

[[x]]^yx ={([][α], α)} [[λx.x]] ={[α]→α}

[[[x^!]]]^x ={(a,a)} [[[x, λy.x]]]^x ={([α, β],[α,[]→β]}

[[λy.y [x^!]]]^x ={(a,[a→α]→α} [[λy.y [x]]]^x ={([β],[[β]→α]→α)}

[[M+N]]^x = [[M]]^x ∪[[N]]^x [[[M^!,N^!]]] ={a·a⁰|a∈[[M]],a⁰∈[[N]]}

[[Ω]] =∅ [[Θ]] ={([Πα∈ab_α→β]·a)→β |(b_α→α)∈[[Θ]]}

Remark: One can also see the interpretation as typing judgment into the non-idempotent type-system that isM_∞.

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Full abstraction of M

_∞

for ∂λ ?

Sensibility

M_∞is sensible for ∂λ:

M⇓ ⇔ [[M]]6=∅

Inequational adequation

M_∞ is inequationally adequate for∂λ

[[M]]^~^x ⊆[[N]]^~^x ⇒ MvoN Those are obtained easily using Taylor expansion and its properties [TLCA13].

Failure of

Inequational completeness

MvoN ⇒ [[M]]^~^x ⊆[[N]]^~^x

Failure of completeness

M≡oN ⇒ [[M]]^~^x = [[N]]^~^x

These assertions are equivalents:

M voN⇔M+N≡oN [[M]]⊆[[N]]⇔[[M+N]] = [[N]]

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Full abstraction of M

_∞

for ∂λ ?

Sensibility

M_∞is sensible for ∂λ:

M⇓ ⇔ [[M]]6=∅

Inequational adequation

M_∞ is inequationally adequate for∂λ

[[M]]^~^x ⊆[[N]]^~^x ⇒ MvoN Those are obtained easily using Taylor expansion and its properties [TLCA13].

Failure of

Inequational completeness

Mv_oN ⇒ [[M]]^~^x ⊆[[N]]^~^x

Failure of completeness

M≡_oN ⇒ [[M]]^~^x = [[N]]^~^x

These assertions are equivalents:

M voN⇔M+N≡oN [[M]]⊆[[N]]⇔[[M+N]] = [[N]]

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A such that λx .x v

_o

A and [[λx.x ]] 6⊆ [[A]]

A := Θ [G, F

^!

]

G:=λuvw.w [(λx.x) [v^!]] F:=λuv1v2.u[(λx.x) [v₁^!] [v₂^!]]

A can be seen as a non deterministic While

Gis the stopping condition, andFthe continuation condition: A ≡β G[A^!] +F[A^!] ≡β G[A^!] +F[G[A^!]] +F²[A^!] ≡β · · ·

A as an infinite sum

A'Σ_n≥1Bnwith forn≥1:

B_n=λv₁. . .v_nw.w [(λx.x) [v₁^!] [v₂^!] · · · [v_n^!]]

G[A^!] →^∗_o B₁ For alli, F[B^!_i] →^∗_o B_i+1 A≡βB1+F[A] ≡β B1+B2+F²[A] ≡β (Σ^k_n=1Bn)+F^k[A]

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A such that λx .x v

_o

A and [[λx.x ]] 6⊆ [[A]]

A := Θ [G, F

^!

]

A can be seen as a non deterministic While

Gis the stopping condition, andFthe continuation condition:

A ≡β G[A^!] +F[A^!] ≡β G[A^!] +F[G[A^!]] +F²[A^!] ≡β · · ·

A as an infinite sum

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A such that λx .x v

_o

A and [[λx.x ]] 6⊆ [[A]]

A := Θ [G, F

^!

]

A can be seen as a non deterministic While

Gis the stopping condition, andFthe continuation condition:

A ≡β G[A^!] +F[A^!] ≡β G[A^!] +F[G[A^!]] +F²[A^!] ≡β · · ·

A as an infinite sum

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λx.x and Σ

_i

B

_i

A'Σ_n≥1Bn with forn≥1 : Bn=λv1. . .vnw.w [(λx.x) [v₁^!] [v₂^!] · · · [v_n^!]]

λx .x v

_o

Σ

_i

B

_i

If (λx.x)P1· · ·Pk converges then:

Bk P1· · ·Pk →^∗λw.w [(λx.x)P1· · · Pk]

Remark: We are using a context lemma for∂λ[TLCA13]

[[λx.x ]]6 ⊆ Σ

i

B

i

[∗]→∗∈[[λx.x]], but for everyn, [∗]→∗6∈[[Bn]]:

[∗]→∗∈[[Bn]]⇔([∗][]· · ·[][],∗)∈[[w [(λx.x)[v₁^!] [v₂^!]· · ·[v_n^]]]]^v¹^...vⁿ^w What isimpossible.

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Clues for the full abstraction

• M_∞ is fully abstract for theλ-calculus ([Manzonetto09])

Nomay non determinism

• M∞ is fully abstract for the purely linear fragment ([Buc &Al 12])

Nofixpoint combinator

• M_∞ is fully abstract for an extension of∂λwith tests ([Buc &Al 11])

presence of contexts of infinite range

• No intuitive counter-example found

intuition is mainly based on typed system

Our result, A := Θ [G, F

^!

]

We construct a counter-example as combination of anuntyped fixpoint and amay non deterministic argumentto create a term ofinfinite range whose behavior will not respect the semantics.

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Clues for the full abstraction

• M_∞ is fully abstract for theλ-calculus ([Manzonetto09]) Nomay non determinism

• M∞ is fully abstract for the purely linear fragment ([Buc &Al 12]) Nofixpoint combinator

• M_∞ is fully abstract for an extension of∂λwith tests ([Buc &Al 11]) presence of contexts of infinite range

Our result, A := Θ [G, F

^!

]

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Clues for the full abstraction

• M_∞ is fully abstract for theλ-calculus ([Manzonetto09]) Nomay non determinism

• M∞ is fully abstract for the purely linear fragment ([Buc &Al 12]) Nofixpoint combinator

• M_∞ is fully abstract for an extension of∂λwith tests ([Buc &Al 11]) presence of contexts of infinite range

Our result, A := Θ [G, F

^!

]

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The short-sightedness

Observations in ∂λ

They are performed by finite applicative contexts, thustheir range is finite.

Observations in M

∞

They are performed by elements ofM_∞ that are infinite lists, thus their range are infinite.

Range of ∂λ

∂λcontainsterms of infinite rangethat can outrange the operational observations but not the denotational one.

Over-approximation?

Over-approximations in denotational semantics aretoo strong.

Must/probabilistic?

Are must and probabilistic convergences more natural? In any case the problem of finding a fully abstract model of∂λ is reopen.

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The short-sightedness

Observations in ∂λ

They are performed by finite applicative contexts, thustheir range is finite.

Observations in M

∞

They are performed by elements ofM_∞ that are infinite lists, thus their range are infinite.

Range of ∂λ

∂λcontainsterms of infinite rangethat can outrange the operational observations but not the denotational one.

Over-approximation?

Over-approximations in denotational semantics aretoo strong.

Must/probabilistic?

Are must and probabilistic convergences more natural?

In any case the problem of finding a fully abstract model of∂λ is reopen.

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Conclusion

A term of infinite range

A=Θ[G,F^!]: A non deterministic while that canchooseits head variable

M

∞

is not inequationally fully abstract for ∂λ

(λx.x)voA but [[λx.x]]6⊆[[A]]

M

∞

is not fully abstract for ∂λ

(A+λx.x)≡oA but [[A+λx.x]]6= [[A]]