Higher-Order Distributions for Linear Logic

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Fossacs 2019

Higher-Order Distributions for Linear Logic

Marie Kerjean & Jean-Simon Lemay

Inria Bretagne - LS2N - Nantes & University of Oxford

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Differentiating Programs - Differentiating Functions

I Differentiation in Theoretical Computer Science : Automatic Differentiation, Incremental Computing, Differential Linear Logic... [Discrete]

I Differentiation in Mathematics : Differential Geometry, Numerical Analysis, Functional analysis ... [continuous]

Differentiation in Computer Science the same as Differentiation in Mathematics ?

I [K18] : Models of Differential Linear Logic with Distributions and Differential Equations, without Higher-Order. C^∞(Rⁿ,R) I Today : going to Higher Order. C^∞(E,R) ?

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Differentiating Programs - Differentiating Functions

I Differentiation in Mathematics : Differential Geometry, Numerical Analysis,Functional analysis ... [continuous]

Is differentiation in Logic the same as Differentiation in Functional Analysis ?

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Differentiating Programs - Differentiating Functions

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Differentiating Programs - Differentiating Functions

From mathematics to computer science.

⇐ Higher-Order

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Differentiating Programs - Differentiating Functions

From models for physics to models for computing.

⇐

Higher-Order

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Curry-Howard-Lambek

The syntax mirrors the semantics.

Programs Logic Semantics

fun (x:A)-> (t:B) Proof ofA`B f :A→B.

Types Formulas Objects

Execution Cut-elimination Equality

λ-calculus

Coherence spaces [Girard87] Linear mapsf :A(B Non-linear maps f :!A(B Linear Logic [Gir87]

Linear proofsf :A`B Non-linear proofs f :!A`B

!A(B'A⇒B

Vectorial Models [Ehrhard02/05] Power seriesf =P

nfn

Differentiation D₀ :f 7→f₁ Differential Linear Logic [Ehrhard&Regnier06]

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Curry-Howard-Lambek

λ-calculus

Coherence spaces [Girard87]

Linear mapsf :A(B Non-linear maps f :!A(B

Linear Logic [Gir87] Linear proofsf :A`B Non-linear proofs f :!A`B

!A(B'A⇒B

nfn

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Curry-Howard-Lambek

λ-calculus

Linear mapsf :A(B Non-linear maps f :!A(B Linear Logic [Gir87]

!A(B'A⇒B

nfn

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Curry-Howard-Lambek

λ-calculus

Linear mapsf :A(B Non-linear maps f :!A(B Linear Logic [Gir87]

!A(B'A⇒B

Vectorial Models [Ehrhard02/05]

Power seriesf =P

nfn

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Linear logic

A linear implication

A⇒ B = ! A ( B C^∞(A,B)' L(!A,B) Usual Implication

A proof is linear when it uses only once its hypothesis A.

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Linear logic

A linear implication

A ⇒ B = ! A( B C^∞(A,B)' L(!A,B) Usual implication

Linear Implication

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Linear logic

A linear implication

A ⇒ B = ! A ( B C^∞(A,B)' L(!A,B) Usual implication

Linear implication

Exponential

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Linear logic

A linear implication

A ⇒ B = ! A ( B C^∞(A,B)' L(!A,B)

A focus on linearity

I Higher-Order is about Seely’s isomoprhism.

C^∞(A×B,C)' C^∞(A,C^∞(B,C)) L(!(A×B),C)' L(!A,L(!B,C))

!(A×B)'!A⊗!Bˆ

I Classicality is about a linear involutive negation :

A^⊥ :=A(⊥ A⁰ :=L(A,R)

A^⊥⊥'A A'A⁰⁰

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Just a glimpse at Differential Linear Logic

Differential Linear Logic

` :A`B

`: !A`B d

f : !A`B d¯ D₀(f) :A`B A linear proof is in particular

non-linear.

From a non-linear proof we can extract a linear proof

f ∈ C^∞(R,R)

d(f)(0)

Normal functors, power series andλ-calculus.Girard, APAL(1988) Differential interaction nets, Ehrhard and Regnier, TCS (2006)

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Getting a smooth model of classical Differential Linear Logic ?

Smoothness

Spaces : E is a locally convex and Haussdorf topological vector space.

Functions: f ∈ C^∞(Rⁿ,R) is infinitely and everywhere differentiable.

These two requirements work as opposite forces.

X Handling smooth functions : some completeness.

X Interpreting the involutive linear negation (E^⊥)^⊥ 'E: Reflexive spaces.

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Getting a smooth model of classical Differential Linear Logic ?

Smoothness

× Interpreting the involutive linear negation (E^⊥)^⊥ 'E. Reflexive spaces

Convenient differential category Blute, Ehrhard Tasson Cah. Geom. Diff.

(2010)

Mackey-complete spaces and Power series, K. and Tasson, MSCS 2016.

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Getting a smooth model of classical Differential Linear Logic ?

Smoothness

× Handling smooth functions : some completeness.

Weak topologies for Linear Logic, K. LMCS 2015.

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Getting a smooth model of classical Differential Linear Logic ?

Smoothness

A model of LL with Schwartz’ epsilon product, Dabrowski and K., 2018.

A logical account for PDEs, K.,LICS18 [A polarized solution, no higher-order]

Higher-Order Distributions, Lemay and K.,Fossacs19

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Exponential : from ressources to distributions

I Linear Logic has long been interpreted in terms of discrete models and resource consumption.

quantitative semantics: !A:=P

nA^⊗ⁿ

I In a classical and Smooth model of Differential Linear Logic, the exponential is a space of Distributions.

!A(⊥=A⇒ ⊥ L(!E,R)' C^∞(E,R)

(!E)⁰⁰ ' C^∞(E,R)⁰

!E ' C^∞(E,R)⁰

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Exponential : from ressources to distributions

nA^⊗ⁿ

!A(⊥=A⇒ ⊥ L(!E,R)' C^∞(E,R)

(!E)⁰⁰ ' C^∞(E,R)⁰

!E ' C^∞(E,R)⁰

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Exponential : from ressources to distributions

nA^⊗ⁿ

!A(⊥=A⇒ ⊥ L(!E,R)' C^∞(E,R)

(!E)⁰⁰ ' C^∞(E,R)⁰

!E ' C^∞(E,R)⁰

I The space of distributions with compact support E⁰(Rⁿ) := C^∞(Rⁿ,R)⁰, whose elements are for example :

φ_f :g ∈ C^∞(Rⁿ,R)7→

Z fg. δ_x :g 7→g(x)

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Exponential : from ressources to distributions

nA^⊗ⁿ

!A(⊥=A⇒ ⊥ L(!E,R)' C^∞(E,R)

(!E)⁰⁰ ' C^∞(E,R)⁰

!E ' C^∞(E,R)⁰

I LL and Distribution Theory enjoy the same computing principle same computing principles : Seely’s isomorphisms are Kernel theorems.

!A⊗!B '!(A×B) C^∞(E,R)⁰⊗Cˆ ^∞(F,R)⁰ ' C^∞(E ×F,R)⁰ .

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Which category of tvs should interpret formulas ?

Reflexive spaces enjoy poor stability properties.

I It is typically not preserved by ⊗.

I Nor by L( , ).

Reflexivity takes many forms :

I It depends of the topologyE_β⁰, E_c⁰, E_w⁰ , E_µ⁰ on the dual.

I The dual is not reflexive : one cannot close by bidual as with biorthogonals.

Monoidal closedness does not extends easily to the topological case:

I Many possible topologies on⊗: ⊗_β, ⊗_π, ⊗_ε. I L_B(E ⊗_BF,G)' L_B(E,L_B(F,G))

⇔”Grothendieck probl`eme des topologies”.

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Topological models of DiLL

[Ehr02] [Ehr05] [DE08]

countable bases of vector spaces

Coherent Banach spaces [Girard99]

a normis too restrictive Reflexive anc complete :

e.g. C^∞(Rⁿ,R)

C^∞(Rⁿ,R) is not finite dimensional

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Polarized model of Smooth differential Linear Logic [K.18]

Typical Nuclear Fr´echet spaces are spaces of [smooth, holomorphic, rapidly decreasing ...] functions.

Fr´echet spaces C^∞(Rⁿ,R) DF-spaces

!Rⁿ=C^∞(Rⁿ,R)⁰

Nuclear spaces

⊗_ε' ⊗_π

Rⁿ ( )⁰

( )⁰

What about C^∞(!Rⁿ,R) or !!Rⁿ ?

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Constructing some notion of smoothness which leaves stable the class of reflexive topological vector space.

We tackle this issue through the space of distribution ConsiderE a topological vector space.

I Define an order on linear injections f :Rⁿ,→E by f ≤g :=∃ι:Rⁿ,→R^m,f =g ◦ι.

I Define the action of a distribution on E with respect to these linear injections:

E⁰(E) := lim−→

f:Rⁿ(E

E_f⁰(Rⁿ)

directed under the inclusion maps defined as

S_f_,g :E_g⁰(Rⁿ)→ E_f⁰(R^m), φ7→(h7→φ(h◦ι_n,m)) This is similar to work onC^∞-algebras [KainKrieglMichor87], which we need to refine to obtain reflexivity. ^{27 / 34}

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A good inductive limit

Because the distributions spaces with which we build the inductive limit are extremenly regular, we have

I E⁰(E) is always reflexive.

weakly quasi-complete : E =E⁰⁰ algebraically.

barrelled E 'E⁰⁰ topologically.

I E⁰(E) is the dual of a projective limit of spaces of functions : E(E) := lim←−

f:Rⁿ(E

E_f(Rⁿ)

φ∈ E⁰(E) acts onf= (f_f)_f_:_Rⁿ_,→E.

where f_f ∈ C^∞(Rⁿ,R).

The Kernel Theorem lifts to Higher-Order : E(E) ˆ⊗E(F)' E(E ⊕F)

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Reflexivity is enough for the structural morphisms

Because we worked with reflexive spaces at the beginning, we can built natural transformations :

dE :











!(E)→E⁰⁰ 'E φ 7→( `

|{z}

E(R

∈E⁰ 7→φ[(

Rⁿ→R

z}|{`◦f )_f_:_Rⁿ_,→E ∈ E(E)]

| {z }

R

)

d¯_E :







E →!E '(E(E))⁰

x 7→((f_f)_f_:_Rⁿ_(E⁰)7→D₀f_f(f⁻¹(x))

wheref is injective such thatx ∈Im(f) . And interpretations for (co)-weakening and (co)-contraction follow from the Kernel Theorem.

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We have obtain polarized model of Differential Linear Logic :

CoLim NDF, ˆ⊗, ⊕ E⁰(F)

Lim NF,`, × E(E)

F E

( )⁰ ( )⁰

... without promotion

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We have obtain polarized model of Differential Linear Logic :

CoLim NDF, ˆ⊗, ⊕ E⁰(F)

Lim NF,`, × E(E)

F E

( )⁰ ( )⁰

... without promotion

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We don’t have a Cartesian Closed Category

This definition gives us functoriality only on isomorphisms :

! :







Refliso →Refliso

E 7→ E⁰(E)

`:E (F 7→!` ∈ E(F⁰) where

(!`)(φ)(

g

) =φ((

g ` ◦ f

|{z}

Rn,→F

)_f_:Rⁿ_,→E).

No category with smooth functions as maps.

We have however a good candidate to make a co-monad of our functor.

µ_E :











!E →!!E

φ7→ (g_g)_g ∈ E(!E)'lim−→

g

C_g^∞(R^m)

!

7→g_g(g⁻¹(φ)) when φ∈Im(g) and g is injective

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Conclusion

What we have :

A Higher-Order exponential extending the notion of distributions, which interpret classical Differential Linear Logic without promotion.

E⁰(E) := lim−→

f:Rⁿ(E

E_f⁰(Rⁿ)

Perspectives :

I Linearity / Non-linearity , Solution /Parameter, Positive / Negative :

give a categorical structure to the several interactions at stakes.

I Lifting this exponential to a co-monad:

finer handling of indexations.

I Constructing exponentials via methods from Numerical Analysis :

!E =< δ_x,x ∈E > [BET12]

Cut-elimination through Numerical Schemes.

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Computing in Higher-Dimension - Computing Solutions

I If we wanted only smoothness and no reflexivty, we could have used :

!E = < δ

_x

, x ∈ E >

By Fr¨olicher and Kriegl, as used by Blute, Ehrhard and Tasson.

That’s a discretisation scheme.

I In [K18] we showed thatcut-elimination is the resolution of certain class of differential equations for which we have an explicit one-step resolution .

generalize to partial differential equations with no explicit solution.

Let’s embed numerical schemes into cut-elimination.

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