A logical account for LPDEs

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LICS 2018, Oxford

A Logical Account

for Linear Partial Differential Equations

Marie Kerjean

IRIF, Universit´e Paris Diderot [email protected]

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Linear Logic is about joining Logic and Algebra.

Differential Linear Logic is about joining Logic and Differentiation.

In this talk, we join Logic andMathematical Physics, ViaLinear Partial Differential Equationsand a generalization of Differential

Linear Logic.

This takes place in a more general setting: Computer Science is drifting from Discrete Mathematics to Analysis.

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Linear Logic is about joining Logic and Algebra.

Differential Linear Logic is about joining Logic and Differentiation.

In this talk, we join Logic andMathematical Physics, ViaLinear Partial Differential Equationsand a generalization of Differential

Linear Logic.

This takes place in a more general setting: Computer Science is drifting from Discrete Mathematics to Analysis.

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Smoothness

Differentiation

Differentiating a functionf :Rⁿ→R atx is finding a linear approximationD(f)(x) :v 7→D_x(f)(v) of f nearx.

f ∈ C^∞(R,R)

d(f)(0)

A coinductive definition

Smooth functions are functions which can be differentiated everywhere in their domain and whose differentials are smooth.

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Linear Logic

A decomposition of the implication A⇒B '!A(B

A linear proof is in particular non-linear.

A`B is linear. !A`B is non-linear.

A`Γ

dereliction

!A`Γ

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Linear Logic

A decomposition of the implication A⇒B '!A(B Usualnon-linear implication

A`Γ

dereliction

!A`Γ

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Linear Logic

A decomposition of the implication A⇒B '!A(B Linear implication

A`Γ

dereliction

!A`Γ

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Linear Logic

A decomposition of the implication A⇒B '!A(B

Exponential: Usually, the duplicable copies of A.

Here the exponential is a space of Solution to a Differential Equation.

A`Γ

dereliction

!A`Γ

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Differential Linear Logic

`Γ,A^⊥

`Γ,?A^⊥ d

`∆,A d¯

`∆,!A A linear proof is in particular

non-linear.

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

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Differential Linear Logic

`Γ, `:A^⊥

`Γ, `: ?A^⊥ d

`∆,v :A

d¯

`∆,(f 7→D0(f)(v)) : !A A linear proof is in particular

non-linear.

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Differential Linear Logic

`Γ, `:A^⊥

`Γ, `: ?A^⊥ d

`∆,v :A

d¯

`∆,(f 7→D0(f)(v)) : !A A linear proof is in particular

non-linear.

Cut-elimination:

`Γ,v : !A d¯

`Γ,!A

`∆,A^⊥

`∆,?A^⊥ d

`Γ,∆ cut

`Γ,A `∆,A^⊥ Γ,∆ cut

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Differential Linear Logic

`Γ, `:A^⊥

`Γ, `: ?A^⊥ d

`∆,v :A

d¯

`∆,(f 7→D₀(f)(v)) : !A A linear proof is in particular

non-linear.

Cut-elimination:

`Γ,v :A d¯

`Γ,D₀( )(v) : !A

`∆, `:A^⊥

`∆, `: ?A^⊥ d Γ,∆ cut

`Γ,x:A `∆, `:A^⊥ Γ,∆,D₀(`)(x) =`(x) :R=⊥ cut

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Just a glimpse at Differential Linear Logic

A,B :=A⊗B|1|A`B|⊥|A⊕B|0|A×B|>|!A|!A

Exponential rules ofDiLL0

`Γ,?A,?A

`Γ,?A c

`Γ w

`Γ,?A

`Γ,A

`Γ,?A d

`Γ,!A, `∆,!A

¯

`Γ,∆,!A c

` w¯

`!A

`Γ,A d¯

`Γ,!A

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

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Linear Partial Differential Equations with constant coefficient

ConsiderD a LPDO with constant coefficients:

D = X

α,|α|≤n

aα

∂^α

∂x^α.

The heat equation inR²

∂²u

∂x² −^∂u_∂t = 0 u(x,y,0) =f(x,y)

Theorem (Malgrange 1956)

For anyD LPDOcc, there isE_D ∈ C_c^∞(Rⁿ,R)⁰ such that DED =δ0, and thus for any φ∈ C^∞(Rⁿ,R):

D(E_D∗φ) =φ

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Linear Partial Differential Equations with constant coefficient

ConsiderD a LPDO with constant coefficients:

D = X

α,|α|≤n

aα

∂^α

∂x^α.

The heat equation inR²

∂²u

∂x² −^∂u_∂t = 0 u(x,y,0) =f(x,y)

Theorem (Malgrange 1956)

For anyD LPDOcc, there isE_D ∈ C_c^∞(Rⁿ,R)⁰ such that DED =δ0, and thus for any φ∈ C^∞(Rⁿ,R):

output D(E_D∗φ) =φinput

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What this work is about: A new exponential !

_D

.

D is a Linear partial Differential Operator with constant coefficients:

DiLL

`Γ,A

`Γ,?A d

`Γ,A d¯

`Γ,!A

D−DiLL

`Γ,?_DA d_D

`Γ,?A

`Γ,!_DA d¯D

`Γ,!A

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What this work is about: A new exponential !

_D

.

D is a Linear partial Differential Operator with constant coefficients:

DiLL=D₀−DiLLBecause of A≡A^⊥⊥

`Γ,?_D₀A

`Γ,?A d

`Γ,!_D₀A d¯

`Γ,!A

D−DiLL

`Γ,?_DA d_D

`Γ,?A

``Γ,Γ,!_D!AA d¯_D

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What this work is about: the same cut-elimination

Cut-elimination models resolution of the Linear Partial Differential Equations on DistributionsDψ=φ.

``Γ,Γ,!_D!AA d¯_D

`∆,?_DA^⊥ d_D

`∆,?A^⊥

`Γ,∆ cut

`Γ,!DA `∆,?DA

`Γ,∆ cut

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It’s all about semantics

And getting a smooth model of Differential Linear Logic with involutive linear negation.

JA⇒BK=C^∞(JAK,JBK) JAK=JAK

00,spaces arereflexive

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It’s all about semantics

And getting a smooth model of Differential Linear Logic with involutive linear negation.

JA⇒BK=C^∞(JAK,JBK) JAK=JAK

00,spaces arereflexive

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Challenges

We encounter several difficulties in the context of topological vector spaces:

X Finding a category of tvs and smooth functions which is Cartesian closed. Requires somecompleteness, and a dual topology fine enough.

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

Reflexive spaces, and a dual topology coarse enough.

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Challenges

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

Convenient differential categoryBlute, Ehrhard Tasson Cah. Geom. Diff.

(2010)

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

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Challenges

× Finding a category of tvs and smooth functions which is Cartesian closed. Requires some completeness, and a dual topology fine enough.

Weak topologies for Linear Logic, K. LMCS 2015.

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Challenges

We construct in this paper a polarized solution to this problem.

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Distributions are everywhere

I Distributions with compact support are elements of

C^∞(Rⁿ,R)⁰, seen as generalisations of functions with compact support:

φ_f :g ∈ C^∞(Rⁿ,R)7→

Z fg.

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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Distributions are everywhere

φ_f :g ∈ C^∞(Rⁿ,R)7→

Z fg.

!A(⊥=A⇒ ⊥ L(!E,R)' C^∞(E,R)

(!E)⁰⁰' C^∞(E,R)⁰

!E ' C^∞(E,R)⁰

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Distributions are everywhere

φ_f :g ∈ C^∞(Rⁿ,R)7→

Z fg.

!A(⊥=A⇒ ⊥ L(!E,R)' C^∞(E,R)

(!E)⁰⁰' C^∞(E,R)⁰

!E ' C^∞(E,R)⁰

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Distributions are everywhere

φ_f :g ∈ C^∞(Rⁿ,R)7→

Z fg.

!A(⊥=A⇒ ⊥ L(!E,R)' C^∞(E,R)

(!E)⁰⁰' C^∞(E,R)⁰

!E ' C^∞(E,R)⁰

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Topological models of DiLL

Coherent Banach spaces, Girard 2004, anormis too restrictive

Nuclear Fr´echet spaces are reflexive and complete C^∞(Rⁿ,R) is not finite dimensional

Let us take the other way around, through Nuclear Fr´echet spaces.

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A Smooth classical Differential Linear Logic with Distributions

Fr´echet spaces DF-spaces Nuclear spaces

⊗_π =⊗

Rⁿ E⁰ E

⊗_π

`

C^∞(Rⁿ,R) !Rⁿ=C^∞(Rⁿ,R)⁰ ( )⁰

( )⁰

Seely’s isomorphism corresponds to Schwartz Kernel Theorem.

Getting a model with Higher-order was done in a recent collaboration with JS Lemay.

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Another exponential is possible

D a Linear Partial Differential operator with constant coefficients:

!_DE = (D(C^∞(E,R))⁰ that is !_DRⁿ={φ∈(C_c^∞(Rⁿ))⁰,Dφ∈!Rⁿ}.

d¯_D :

(!_DE →Dφ

φ7→(f 7→φ(D(f)))

d_D :

(!E →!DE ψ7→ψ∗ED

E_D is the fundamental solution ofD.

Getting back to LL whenD =D₀

!_D₀A' L(A,R)⁰ 'A⁰⁰'A by reflexivity.

WhenD =Id, !DA= !A.

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What this work is about: the same cut-elimination

Cut-elimination models resolution of the Differential Equations on DistributionsDψ=φ.

`Γ, φ:!_DA d¯D

`Γ,Dφ: !A

`∆,g : ?_DA^⊥ d_D

`∆,E_D ∗g : ?A^⊥

`Γ,∆,D(φ)(E_D ∗g) :R=⊥ cut

`Γ, φ: !_DA `∆,g : ?_D(A)

`Γ,∆, φ(g) :R=⊥ cut

D(E_D∗φ)(g) =D(φ)(E_D∗g) =φ(g) ψ=E_D∗φ

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What this work is about: the same cut-elimination

`Γ, φ:!_DA d¯D

`Γ,Dφ: !A

`∆,g : ?_DA^⊥ d_D

`∆,E_D ∗g : ?A^⊥

`Γ,∆,D(φ)(E_D ∗g) :R=⊥ cut

`Γ, φ: !_DA `∆,g : ?_D(A)

`Γ,∆, φ(g) :R=⊥ cut

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What this work is about: the same cut-elimination

`Γ, φ:!_DA d¯D

`Γ,Dφ: !A

`∆,g : ?_DA^⊥ d_D

`∆,E_D ∗g : ?A^⊥

`Γ,∆,D(φ)(E_D ∗g) =φ(g) :R=⊥ cut

`Γ, φ: !_DA `∆,g : ?_D(A)

`Γ,∆, φ(g) :R=⊥ cut

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Intermediates rules for D

DiLL

`Γ w

`Γ,?A

`Γ,?A,?A

`Γ,?A c

`Γ,A

`Γ,?A d

`Γ

¯

`Γ,!A w

`Γ,!A `∆,!A

¯

`Γ,∆,!A c

`Γ,x :A d¯

`Γ,D0( )(x)!A

D−DiLL

`Γ w

`Γ,?_DA

`Γ,?A,?DA

`Γ,?DA c

`Γ,?DA d_D

`Γ,?A

` w¯_D

`ED: !DA

`Γ, φ: !A `∆, ψ:!_DA

¯ cD

`Γ,∆, φ∗ψ:!_DA

`Γ, ψ:!_DA d¯

`Γ,Dψ: !A

A deterministiccut-elimination.

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Logic in Computer Science: Curry-Howard-Lambek

This Talk: Linear Partial Differential Equations are the Semantics forD−DiLL

Mathematical Physics Theorical computer science

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Conclusion

Take away

Linear Logic andDiLLextends to Linear Partial Differential Operators, in which !A is interpreted by a space of distributions, and a space of solutions to a Differential Equation, and

cut-elimination computes the solution.

Now that we’ve build a bridge with functional analysis, there’s A LOT of exciting possibilities.

Two priorities

I Curry-Howard: a deterministic LPDE calculus.

I Most importantly: towards non-linear PDEs.