A Type Theory for Linear Partial Differential Equations

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(1)A Type Theory For Partial Differential Equations Marie Kerjean [email protected] IRIF, Université Paris Diderot, Paris. The Curry-Howard-Lambek correspondance Cartesian closed categories Functions f : E → F. Scientific context :. Minimal Logic Implication E ⇒ F Classical Logic ((E ⇒ ⊥) ⇒ ⊥) ' E. + ∗-autonomous categories Linear functions f ∈ L(E, F ) Reflexive vector spaces E 00 ' E. Classical Linear Logic Linear Implication E ( F [2] ((E ( ⊥) ( ⊥) ' E. + differentiation Smooth functions f ∈ C ∞(E, F ). Differential Linear Logic Linear approximation to non-linear proofs [3]. Simply Typed λ-calculus Programs acting on contexts : µαE⇒⊥. < tE |αE⇒⊥ > [1]. Differential λ-calculus. D-DiLL Resolution of Linear Partial Differential Equations [4] (LPDE). Nuclear spaces, Distributions, and Differential operators. This work :. Simply Typed λ-calculus Programs λxE .tF. Conjecture : A theorical resolution of LPDE via context-program equivalence. Denotational models of Differential Linear Logic. Differential Linear Logic. Formulas must be interpreted by reflexive vector spaces,. Linear Logic encodes non-linearity into an exponential :. E ' E 00. !E ( F ' E ⇒ F. and functions by smooth functions. And deduces curryfication for non-linear programs from the curryfication for linear one : f : C ∞(E, F ). !E⊗!F '!(E × F ).. We use the theory of topological vector spaces : Nuclear spaces 0 ( ) Fréchet DF E0 `. Rn. Differential Linear Logic allows to compute a Linear function from a non linear one : its differentiation at 0.. f ∈ C ∞(R, R). E ⊗π. D(E ⇒ F )(0) = E ( F D(f )(0). ( )0 ⊗π = ⊗ε. Fig. 2: A polarized model of MLL Fig. 1: A classification. D-DiLL The exponential rules of DiLL are :. Smooth exponentials : Distributions A typical Nuclear Frćhet space is the space of smooth functions on Rn : ∞. n. C (R , R).. ` Γ, A ` Γ, ?A d. `Γ w ` Γ, ?A. ` Γ, !A, !A c̄ ` Γ, !A. ` Γ w̄ ` Γ, !A. ` Γ, ?A, ?A c ` Γ, ?A ` Γ, A ¯ ` Γ, !A d. A typical Nuclear DF spaces is the space of distributions with compact support : C ∞(Rn, R)0 := {φ : f ∈ C ∞(Rn, R) 7→ φ(f ) ∈ R}.. The exponentials rules of D-DiLL are :. Schwartz’ Kernel Theorem : ˆ ∞(F, R)0 ' C ∞(E × F, R)0 C ∞(E, R)0⊗C !Rn = C ∞(Rn, R)0.. ` Γ wD ` Γ, ?D A ` w̄D ` !D A. ` Γ, ?A, ?D A c ` Γ, ?D A D ` Γ, !A ` ∆, !D A c̄ D ` Γ, ∆, !D A. ` Γ, ?D A ` Γ, ?A dD ` Γ, !D A ¯ ` Γ, !A d. Differential Operator on Distributions D(g)(x) =. ∂ αg |α|≤n aα (x) ∂xα .. P. !D E := (D(C ∞(E))0. Short term goals :. Theorems :. • An indexed version of D-Dill with subtyping, with an understanding at the same time of all LPDO with constant coefficients, such that c̄(!∂ i )(!∂ j ) =!∂ i+∂ j .. • If D(g) is the differentiation at 0, !D E = E 00 ' E and DiLL = sums of D − DiLL.. • It will follows from the previous system that, syntactically, !D Rn = ⊥ if and only if, semantically, the LPDE does not have a solution.. • When the coefficients aα are constant, we have a model of D-DiLL.. References 1. A formulae-as-types notion of control. Timothy G. Griffin. POPL 1990 2. Linear Logic. Jean-Yves Girard, Theoretical Computer Science , 1987.. 3. Differential interaction nets. Thomas Ehrhard, Laurent Regnier, Th. Comp. Sci., 2006. 4. A Logical Account for Linear Partial Differential Equations, K. , 2018 preprint LATEX TikZposter.

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