Choco - November 2019 Higher-Order Distributions for Linear Logic Marie Kerjean (Inria Rennes -LS2N) Based joints works with Jean-Simon Lemay (Oxford) 1 / 49
Choco - November 2019 Higher-Order Distributions for Linear Logic Differentiation and Duality (in denotational semantics) Marie Kerjean (Inria Rennes -LS2N) Based joints works with Jean-Simon Lemay (Oxford), Christine Tasson (IRIF), Yoann Dabrowski (Lyon 1) 2 / 49
Curry-HOward for COmputing differentials ◮ As pure mathematicians study differentiation as a local and linear approximation of functions. ◮ As applied mathematicians we study and approximate infinite objects in numerical analysis. ◮ As logicians, what do we have to say about the computation of differentials ? 3 / 49
Curry-Howard-Lambek for Computing differentials As logicians, what do we say to death the computation of differentials ? 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 − DiLL Functional Analysis 4 / 49
The logic is : ◮ (linear) Classical : A ⊥ := A ⊸ ⊥ and A ⊥⊥ ≃ A . ◮ Higher-Order : λf.λg.f ( g ). The models should be: ◮ Reflexive. A ′ := L ( A, R ) and A ≃ A ′′ ◮ Higher-Order. f : R n → R but f : C ∞ ( R n , R ) → R 5 / 49
The logic is : ◮ (linear) Classical : A ⊥ := A ⊸ ⊥ and A ⊥⊥ ≃ A . ◮ Higher-Order : λf.λg.f ( g ). The models should be: ◮ Reflexive. A ′ := L ( A, R ) and A ≃ A ′′ ◮ Higher-Order. f : R n → R but f : C ∞ ( R n , R ) → R I will present these result not necessarily in chronological order. ◮ Part I: Classical Smooth models of Differential Linear Logic. ◮ Part II: Higher-Order Smooth models of Differential Linear Logic. 6 / 49
Linear logic, once and for all 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 ′′ 7 / 49
Linear logic, once and for all 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 ′′ 8 / 49
Linear logic, once and for all 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 ′′ 9 / 49
Linear logic, once and for all 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 ′′ 10 / 49
Just a glimpse at Differential Linear Logic Differential Linear Logic f : ! A ⊢ B ℓ : A ⊢ B ¯ d d ℓ : ! A ⊢ B D 0 ( f ) : A ⊢ B From a non-linear proof we can ex- A linear proof is in particular non- tract a linear proof linear. f ∈ C ∞ ( R , R ) d ( f )(0) 11 / 49
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 ⊢ Γ , A w d ⊢ Γ , ? A ⊢ Γ , ? A ⊢ Γ , ? A ⊢ Γ , ! A, ⊢ ∆ , ! A ¯ ⊢ Γ , A ⊢ w ¯ ¯ c d ⊢ ! A ⊢ Γ , ∆ , ! A ⊢ Γ , ! A � A particular point of view on differentiation induced by duality. Normal functors, power series and λ -calculus. Girard, APAL(1988) Differential interaction nets , Ehrhard and Regnier, TCS (2006) 12 / 49
Ok, just a little bit more A, B := A ⊗ B | 1 | A ` B |⊥| A ⊕ B | 0 | A × B |⊤| ! A | ! A � ? A � := C ∞ ( � A � ′ , R )’ � ! A � := C ∞ ( � A � , R ) ′ functions distributions Exponential rules of DiLL 0 ⊢ Γ , f : ? A, g : ? A c ⊢ Γ , ℓ : A ⊢ Γ w d ⊢ Γ , cst 0 : ? A ⊢ Γ , f.g : ? A ⊢ Γ , ℓ : ? A ⊢ Γ , φ : ! A, ⊢ ∆ , ψ : ! A ¯ ⊢ Γ , v : A w ¯ ¯ c ⊢ ! A d ⊢ Γ , ∆ , φ ∗ ψ : ! A ⊢ Γ , ( f �→ D 0 ( f )) : ! A 13 / 49
Classical Models of Differential Linear Logic in Functional Analysis. A bit of context about linear logic and duality 14 / 49
Smoothness and Duality Objectives Spaces : E is a locally convex and Haussdorf topological vector space. Functions : f ∈ C ∞ ( R n , R ) is infinitely and everywhere differentiable. The two requirements works as opposite forces . � A cartesian closed category with smooth functions. � Completeness, and a dual topology fine enough. � Interpreting ( E ⊥ ) ⊥ ≃ E without an orthogonality: � Reflexivity : E ≃ E ′′ , and a dual topology coarse enough. . 15 / 49
What’s not working A space of (non necessarily linear) functions between finite dimensional spaces is not finite dimensional. dim C 0 ( R n , R m ) = ∞ . 16 / 49
What’s not working A space of (non necessarily linear) functions between finite dimensional spaces is not finite dimensional. dim C 0 ( R n , R m ) = ∞ . We can’t restrict ourselves to finite dimensional spaces. The tentative to have a normed space of analytic functions fails (Girard’s Coherent Banach spaces). ◮ We want to use power series. ◮ For polarity reasons, we want the supremum norm on spaces of power series. ◮ But a power series can’t be bounded on an unbounded space (Liouville’s Theorem). ◮ Thus functions must depart from an open ball, but arrive in a closed ball. Thus they do not compose. ◮ This is why Coherent Banach spaces don’t work. 17 / 49
What’s not working A space of (non necessarily linear) functions between finite dimensional spaces is not finite dimensional. dim C 0 ( R n , R m ) = ∞ . We can’t restrict ourselves to finite dimensional spaces. The tentative to have a normed space of analytic functions fails (Girard’s Coherent Banach spaces). ◮ We want to use power series. ◮ For polarity reasons, we want the supremum norm on spaces of power series. ◮ But a power series can’t be bounded on an unbounded space (Liouville’s Theorem). ◮ Thus functions must depart from an open ball, but arrive in a closed ball. Thus they do not compose. ◮ This is why Coherent Banach spaces don’t work. We can’t restrict ourselves to normed spaces. 18 / 49
MLL in TopVect It’s a mess. Duality is not an orthogonality in general : ◮ It depends of the topology E ′ β , E ′ c , E ′ w , E ′ µ on the dual. ◮ It is typically not preserved by ⊗ . ◮ It is in the canonical case not an orthogonality : E ′ β is not reflexive. Monoidal closedness does not extends easily to the topological case : ◮ Many possible topologies on ⊗ : ⊗ β , ⊗ π , ⊗ ε . ◮ L B ( E ⊗ B F, G ) ≃ L B ( E, L B ( F, G )) ⇔ ”Grothendieck probl` eme des topologies”. 19 / 49
Which interpretation for formulas L ? [Ehr02] [Ehr05] [DE08] C ∞ ( R n , R ) is not finite dimensional countable bases of vector spaces Reflexive anc complete : e.g. C ∞ ( R n , R ) Coherent Banach spaces [Gir99] a norm is too restrictive 20 / 49
Smoothness and Duality Smoothness Spaces : � A � is a locally convex and Haussdorf topological vector space. Functions : f ∈ C ∞ ( R n , R ) is infinitely and everywhere differentiable. A coinductive definition : f is smooth iff it is differentiable and its differentials everywhere are smooth. E ′ := L ( E, R ) Reflexivity E ′′ ≃ ? E Vec Topvec E E ′ E ′ B E ′′ Semi-Reflexivity E ′′ ∼ ? E In general, reflexive spaces enjoy poor stability properties . 21 / 49
Smoothness and Duality Smoothness Spaces : � A � is a locally convex and Haussdorf topological vector space. Functions : f ∈ C ∞ ( R n , R ) is infinitely and everywhere differentiable. In general, reflexive spaces enjoy poor stability properties . ◮ No closure by E �→ E ′′ . ◮ No stability by linear connectives ⊗ , ` , − ⊸ − . Keep calm and polarize 22 / 49
Chiralities: a categorical model for polarized MLL Syntax Negative Formulas: N, M := a | ? P | N ` M | ⊥ | N & M | ⊤ | Positive Formulas: P, Q := a ⊥ | ! N | P ⊗ Q | 0 | P ⊕ Q | 1 ( ) ⊥ L ˆ ( P op , ⊗ , 1) ( N , ` , ⊥ ) ⊥ ⊥ P N ´ ( ) ⊥ R N ⊥ R ⊥ L ≃ N N ( ˆ p, m ` n ) ≃ N ( ˆ ( p ⊗ m ⊥ ) , n ) 23 / 49
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