LICS 2018, Oxford A Logical Account for Linear Partial Differential Equations Marie Kerjean IRIF, Universit´ e Paris Diderot kerjean@irif.fr 1/ 19
Linear Logic is about joining Logic and Algebra. Differential Linear Logic is about joining Logic and Differentiation. In this talk, we join Logic and Mathematical Physics, Via Linear Partial Differential Equations and a generalization of Differential Linear Logic. 2/ 19
Linear Logic is about joining Logic and Algebra. Differential Linear Logic is about joining Logic and Differentiation. In this talk, we join Logic and Mathematical Physics, Via Linear Partial Differential Equations and a generalization of Differential Linear Logic. This takes place in a more general setting: Computer Science is drifting from Discrete Mathematics to Analysis. 2/ 19
Smoothness Differentiation Differentiating a function f : R n → R at x is finding a linear approximation D ( f )( x ) : v �→ D x ( f )( v ) of f near x . 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. 3/ 19
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 4/ 19
Linear Logic A decomposition of the implication A ⇒ B ≃ ! A ⊸ B Usual non-linear implication A linear proof is in particular non-linear. A ⊢ B is linear. ! A ⊢ B is non-linear. A ⊢ Γ dereliction ! A ⊢ Γ 4/ 19
Linear Logic A decomposition of the implication A ⇒ B ≃ ! A ⊸ B Linear implication A linear proof is in particular non-linear. A ⊢ B is linear. ! A ⊢ B is non-linear. A ⊢ Γ dereliction ! A ⊢ Γ 4/ 19
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 linear proof is in particular non-linear. A ⊢ B is linear. ! A ⊢ B is non-linear. A ⊢ Γ dereliction ! A ⊢ Γ 4/ 19
Differential Linear Logic ⊢ Γ , A ⊥ ⊢ ∆ , A ¯ d d ⊢ Γ , ? A ⊥ ⊢ ∆ , ! A From a non-linear proof we can A linear proof is in particular extract a linear proof non-linear. 5/ 19
Differential Linear Logic ⊢ Γ , ℓ : A ⊥ ⊢ ∆ , v : A ¯ d d ⊢ Γ , ℓ : ? A ⊥ ⊢ ∆ , ( f �→ D 0 ( f )( v )) : ! A From a non-linear proof we can A linear proof is in particular extract a linear proof non-linear. 5/ 19
Differential Linear Logic ⊢ Γ , ℓ : A ⊥ ⊢ ∆ , v : A ¯ d d ⊢ Γ , ℓ : ? A ⊥ ⊢ ∆ , ( f �→ D 0 ( f )( v )) : ! A From a non-linear proof we can A linear proof is in particular extract a linear proof non-linear. Cut-elimination: ⊢ ∆ , A ⊥ ⊢ Γ , v : ! A ¯ d d ⊢ ∆ , ? A ⊥ ⊢ Γ , ! A cut ⊢ Γ , ∆ � ⊢ ∆ , A ⊥ ⊢ Γ , A cut Γ , ∆ 5/ 19
Differential Linear Logic ⊢ Γ , ℓ : A ⊥ ⊢ ∆ , v : A ¯ d d ⊢ Γ , ℓ : ? A ⊥ ⊢ ∆ , ( f �→ D 0 ( f )( v )) : ! A From a non-linear proof we can A linear proof is in particular extract a linear proof non-linear. Cut-elimination: ⊢ ∆ , ℓ : A ⊥ ⊢ Γ , v : A ¯ d d ⊢ ∆ , ℓ : ? A ⊥ ⊢ Γ , D 0 ( )( v ) : ! A cut Γ , ∆ ⊢ ∆ , ℓ : A ⊥ ⊢ Γ , x : A � cut Γ , ∆ , D 0 ( ℓ )( x ) = ℓ ( x ) : R = ⊥ 5/ 19
Just a glimpse at Differential Linear Logic A , B := A ⊗ B | 1 | A ` B |⊥| A ⊕ B | 0 | A × B |⊤| ! A | ! A Exponential rules of DiLL 0 ⊢ Γ , ? A , ? A c ⊢ Γ , A ⊢ Γ w d ⊢ Γ , ? A ⊢ Γ , ? A ⊢ Γ , ? A ⊢ Γ , ! A , ⊢ ∆ , ! A ¯ ⊢ Γ , A ⊢ ¯ ¯ w c d ⊢ ! A ⊢ Γ , ∆ , ! A ⊢ Γ , ! A Normal functors, power series and λ -calculus. Girard, APAL(1988) Differential interaction nets , Ehrhard and Regnier, TCS (2006) 6/ 19
Linear Partial Differential Equations with constant coefficient Consider D a LPDO with constant coefficients: ∂ α � D = a α ∂ x α . α, | α |≤ n The heat equation in R 2 ∂ 2 u ∂ x 2 − ∂ u ∂ t = 0 u ( x , y , 0) = f ( x , y ) Theorem (Malgrange 1956) c ( R n , R ) ′ such that For any D LPDOcc, there is E D ∈ C ∞ DE D = δ 0 , and thus for any φ ∈ C ∞ ( R n , R ): D ( E D ∗ φ ) = φ 7/ 19
Linear Partial Differential Equations with constant coefficient Consider D a LPDO with constant coefficients: ∂ α � D = a α ∂ x α . α, | α |≤ n The heat equation in R 2 ∂ 2 u ∂ x 2 − ∂ u ∂ t = 0 u ( x , y , 0) = f ( x , y ) Theorem (Malgrange 1956) c ( R n , R ) ′ such that For any D LPDOcc, there is E D ∈ C ∞ DE D = δ 0 , and thus for any φ ∈ C ∞ ( R n , R ): output D ( E D ∗ φ ) = φ input 7/ 19
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 d ⊢ Γ , ? A ⊢ Γ , ! A D − DiLL ⊢ Γ , ? D A d D ⊢ Γ , ! D A ¯ d D ⊢ Γ , ? A ⊢ Γ , ! A 8/ 19
What this work is about: A new exponential ! D . D is a Linear partial Differential Operator with constant coefficients: DiLL = D 0 − DiLL Because of A ≡ A ⊥⊥ ⊢ Γ , ? D 0 A ⊢ Γ , ! D 0 A ¯ d d ⊢ Γ , ? A ⊢ Γ , ! A D − DiLL ⊢ Γ , ? D A d D ⊢ Γ , ! D A ¯ d D ⊢ Γ , ? A ⊢ Γ , ! A 8/ 19
What this work is about: the same cut-elimination Cut-elimination models resolution of the Linear Partial Differential Equations on Distributions D ψ = φ . ⊢ ∆ , ? D A ⊥ ⊢ Γ , ! D A ¯ d D d D ⊢ ∆ , ? A ⊥ ⊢ Γ , ! A cut ⊢ Γ , ∆ � ⊢ Γ , ! D A ⊢ ∆ , ? D A cut ⊢ Γ , ∆ 9/ 19
It’s all about semantics And getting a smooth model of Differential Linear Logic with involutive linear negation. 10/ 19
It’s all about semantics And getting a smooth model of Differential Linear Logic with involutive linear negation. � A � = � A � ′′ , spaces are reflexive � A ⇒ B � = C ∞ ( � A � , � B � ) 10/ 19
Challenges We encounter several difficulties in the context of topological vector spaces: � Finding a category of tvs and smooth functions which is Cartesian closed. Requires some completeness, and a dual topology fine enough. � Interpreting the involutive linear negation ( E ⊥ ) ⊥ ≃ E : Reflexive spaces, and a dual topology coarse enough. 11/ 19
Challenges We encounter several difficulties in the context of topological vector spaces: � Finding a category of tvs and smooth functions which is Cartesian closed. Requires some completeness, and a dual topology fine enough. × Interpreting the involutive linear negation ( E ⊥ ) ⊥ ≃ E . Convenient differential category Blute, Ehrhard Tasson Cah. Geom. Diff. (2010) Mackey-complete spaces and Power series , K. and Tasson, MSCS 2016. 11/ 19
Challenges We encounter several difficulties in the context of topological vector spaces: × Finding a category of tvs and smooth functions which is Cartesian closed. Requires some completeness, and a dual topology fine enough. � Interpreting the involutive linear negation ( E ⊥ ) ⊥ ≃ E : Reflexive spaces, and a dual topology coarse enough. Weak topologies for Linear Logic , K. LMCS 2015. 11/ 19
Challenges We encounter several difficulties in the context of topological vector spaces: � Finding a category of tvs and smooth functions which is Cartesian closed. Requires some completeness, and a dual topology fine enough. � Interpreting the involutive linear negation ( E ⊥ ) ⊥ ≃ E : Reflexive spaces, and a dual topology coarse enough. We construct in this paper a polarized solution to this problem. 11/ 19
Distributions are everywhere ◮ Distributions with compact support are elements of C ∞ ( R n , R ) ′ , seen as generalisations of functions with compact support: � φ f : g ∈ C ∞ ( R n , R ) �→ fg . ◮ In a classical and Smooth model of Differential Linear Logic, the exponential is a space of Distributions. ! A ⊸ ⊥ = A ⇒ ⊥ 12/ 19
Distributions are everywhere ◮ Distributions with compact support are elements of C ∞ ( R n , R ) ′ , seen as generalisations of functions with compact support: � φ f : g ∈ C ∞ ( R n , R ) �→ fg . ◮ 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 ) 12/ 19
Distributions are everywhere ◮ Distributions with compact support are elements of C ∞ ( R n , R ) ′ , seen as generalisations of functions with compact support: � φ f : g ∈ C ∞ ( R n , R ) �→ fg . ◮ 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 ) ′′ ≃ C ∞ ( E , R ) ′ 12/ 19
Distributions are everywhere ◮ Distributions with compact support are elements of C ∞ ( R n , R ) ′ , seen as generalisations of functions with compact support: � φ f : g ∈ C ∞ ( R n , R ) �→ fg . ◮ 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 ) ′′ ≃ C ∞ ( E , R ) ′ ! E ≃ C ∞ ( E , R ) ′ 12/ 19
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