Fossacs 2019 Higher-Order Distributions for Linear Logic Marie Kerjean & Jean-Simon Lemay Inria Bretagne - LS2N - Nantes & University of Oxford logo_oxford.png 1 / 34
Differentiating Programs - Differentiating Functions ◮ Differentiation in Theoretical Computer Science : Automatic Differentiation, Incremental Computing, Differential Linear Logic ... [Discrete] ◮ Differentiation in Mathematics : Differential Geometry, Numerical Analysis, Functional analysis ... [continuous] Differentiation in Computer Science the same as Differentiation in Mathematics ? 2 / 34
Differentiating Programs - Differentiating Functions ◮ Differentiation in Theoretical Computer Science : Automatic Differentiation, Incremental Computing, Differential Linear Logic ... [Discrete] ◮ Differentiation in Mathematics : Differential Geometry, Numerical Analysis, Functional analysis ... [continuous] Is differentiation in Logic the same as Differentiation in Functional Analysis ? ◮ [K18] : Models of Differential Linear Logic with Distributions and Differential Equations, without Higher-Order. C ∞ ( R n , R ) ◮ Today : going to Higher Order. C ∞ ( E , R ) ? 3 / 34
Differentiating Programs - Differentiating Functions ◮ Differentiation in Theoretical Computer Science : Automatic Differentiation, Incremental Computing, Differential Linear Logic ... [Discrete] ◮ Differentiation in Mathematics : Differential Geometry, Numerical Analysis, Functional analysis ... [continuous] Is differentiation in Logic the same as Differentiation in Functional Analysis ? ◮ [K18] : Models of Differential Linear Logic with Distributions and Differential Equations, without Higher-Order. C ∞ ( R n , R ) ◮ Today : going to Higher Order. C ∞ ( E , R ) ? 4 / 34
Differentiating Programs - Differentiating Functions ◮ Differentiation in Theoretical Computer Science : Automatic Differentiation, Incremental Computing, Differential Linear Logic ... [Discrete] ◮ Differentiation in Mathematics : Differential Geometry, Numerical Analysis, Functional analysis ... [continuous] Is differentiation in Logic the same as Differentiation in Functional Analysis ? ◮ [K18] : Models of Differential Linear Logic with Distributions and Differential Equations, without Higher-Order. C ∞ ( R n , R ) Today : going to Higher Order. C ∞ ( E , R ) ? ◮ From mathematics to computer science. ⇐ Higher-Order 5 / 34
Differentiating Programs - Differentiating Functions ◮ Differentiation in Theoretical Computer Science : Automatic Differentiation, Incremental Computing, Differential Linear Logic ... [Discrete] ◮ Differentiation in Mathematics : Differential Geometry, Numerical Analysis, Functional analysis ... [continuous] Is differentiation in Logic the same as Differentiation in Functional Analysis ? ◮ [K18] : Models of Differential Linear Logic with Distributions and Differential Equations, without Higher-Order. C ∞ ( R n , R ) Today : going to Higher Order. C ∞ ( E , R ) ? ◮ From models for physics to models for computing. ⇐ Higher-Order 6 / 34
Curry-Howard-Lambek The syntax mirrors the semantics. Programs Logic Semantics Proof of A ⊢ B f : A → B . fun (x:A)-> (t:B) Types Formulas Objects Execution Cut-elimination Equality 7 / 34
Curry-Howard-Lambek The syntax mirrors the semantics. Programs Logic Semantics Proof of A ⊢ B f : A → B . fun (x:A)-> (t:B) Types Formulas Objects Execution Cut-elimination Equality Coherence spaces [Girard87] λ -calculus Linear maps f : A ⊸ B Non-linear maps f : ! A ⊸ B
Curry-Howard-Lambek The syntax mirrors the semantics. Programs Logic Semantics Proof of A ⊢ B f : A → B . fun (x:A)-> (t:B) Types Formulas Objects Execution Cut-elimination Equality Coherence spaces [Girard87] λ -calculus Linear maps f : A ⊸ B Non-linear maps f : ! A ⊸ B Linear Logic [Gir87] Linear proofs f : A ⊢ B ! A ⊸ B ≃ A ⇒ B Non-linear proofs f : ! A ⊢ B
Curry-Howard-Lambek The syntax mirrors the semantics. Programs Logic Semantics Proof of A ⊢ B f : A → B . fun (x:A)-> (t:B) Types Formulas Objects Execution Cut-elimination Equality Coherence spaces [Girard87] λ -calculus Linear maps f : A ⊸ B Non-linear maps f : ! A ⊸ B Linear Logic [Gir87] Linear proofs f : A ⊢ B ! A ⊸ B ≃ A ⇒ B Non-linear proofs f : ! A ⊢ B Vectorial Models [Ehrhard02/05] Power series f = � n f n Differentiation D 0 : f �→ f 1 Differential Linear Logic [Ehrhard&Regnier06] 10 / 34
Linear logic Usual Implication A linear implication A ⇒ B = ! A ⊸ B C ∞ ( A , B ) ≃ L (! A , B ) A proof is linear when it uses only once its hypothesis A. 11 / 34
Linear logic Usual implication Linear Implication A linear implication A ⇒ B = ! A ⊸ B C ∞ ( A , B ) ≃ L (! A , B ) A proof is linear when it uses only once its hypothesis A. 12 / 34
Linear logic Usual implication Linear implication A linear implication A ⇒ B = ! A ⊸ B C ∞ ( A , B ) ≃ L (! A , B ) Exponential A proof is linear when it uses only once its hypothesis A. 13 / 34
Linear logic A linear implication A ⇒ B = ! A ⊸ B C ∞ ( A , B ) ≃ L (! A , B ) A focus on linearity ◮ 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 ˆ ⊗ ! B ◮ Classicality is about a linear involutive negation : A ⊥ := A ⊸ ⊥ A ′ := L ( A , R ) A ⊥⊥ ≃ A A ≃ A ′′ 14 / 34
Just a glimpse at Differential Linear Logic Differential Linear Logic f : ! A ⊢ B ℓ : A ⊢ B ¯ d d D 0 ( f ) : A ⊢ B ℓ : ! A ⊢ B From a non-linear proof we can A linear proof is in particular extract a linear proof non-linear. f ∈ C ∞ ( R , R ) d ( f )(0) Normal functors, power series and λ -calculus. Girard, APAL(1988) 15 / 34
Getting a smooth model of classical Differential Linear Logic ? Smoothness Spaces : E is a locally convex and Haussdorf topological vector space. Functions : f ∈ C ∞ ( R n , R ) is infinitely and everywhere differentiable. These two requirements work as opposite forces. � Handling smooth functions : some completeness. � Interpreting the involutive linear negation ( E ⊥ ) ⊥ ≃ E : Reflexive spaces. 16 / 34
Getting a smooth model of classical Differential Linear Logic ? Smoothness Spaces : E is a locally convex and Haussdorf topological vector space. Functions : f ∈ C ∞ ( R n , R ) is infinitely and everywhere differentiable. These two requirements work as opposite forces. � Handling smooth functions : some completeness. × 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. 17 / 34
Getting a smooth model of classical Differential Linear Logic ? Smoothness Spaces : E is a locally convex and Haussdorf topological vector space. Functions : f ∈ C ∞ ( R n , R ) is infinitely and everywhere differentiable. These two requirements work as opposite forces. × Handling smooth functions : some completeness. � Interpreting the involutive linear negation ( E ⊥ ) ⊥ ≃ E : Reflexive spaces. Weak topologies for Linear Logic , K. LMCS 2015. 18 / 34
Getting a smooth model of classical Differential Linear Logic ? Smoothness Spaces : E is a locally convex and Haussdorf topological vector space. Functions : f ∈ C ∞ ( R n , R ) is infinitely and everywhere differentiable. These two requirements work as opposite forces. � Handling smooth functions : some completeness. � Interpreting the involutive linear negation ( E ⊥ ) ⊥ ≃ E : Reflexive spaces. 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 19 / 34
Exponential : from ressources to distributions ◮ Linear Logic has long been interpreted in terms of discrete models and resource consumption. quantitative semantics : ! A := � n A ⊗ n 20 / 34
Exponential : from ressources to distributions ◮ Linear Logic has long been interpreted in terms of discrete models and resource consumption. quantitative semantics : ! A := � n A ⊗ n ◮ 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 ) ′ 21 / 34
Exponential : from ressources to distributions ◮ Linear Logic has long been interpreted in terms of discrete models and resource consumption. quantitative semantics : ! A := � n A ⊗ n ◮ 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 ) ′ ◮ The space of distributions with compact support E ′ ( R n ) := C ∞ ( R n , R ) ′ , whose elements are for example : � φ f : g ∈ C ∞ ( R n , R ) �→ fg . δ x : g �→ g ( x ) 22 / 34
Recommend
More recommend
