Computing Analysis through Logic
50
0
0
Texte intégral
(2) DiLL. Smooth. Tensor products. Polarized setting. Curry-Howard for semantics Programs λ xA .tB. Types Execution. Resources λ-calculus. Logic Proof of A ` B Formulas Cut-elimination. Semantics f :A→B Objects Equality. Differential Linear Logic Vectorial Models Linear Logic. λ-calculus. Min. Logic. Normal functors.
(3) DiLL. Smooth. Tensor products. Polarized setting. Curry-Howard for semantics Programs λ xA .tB. Types Execution. Logic Proof of A ` B Formulas Cut-elimination. Resources λ-calculus. Categories Morphisms Objects Equality. Concrete models functions Space Equality. Differential Linear Logic Seely categories. Vectorial Models. Linear Logic. λ-calculus. Min. Logic. Normal functors.
(4) DiLL. Smooth. Tensor products. Polarized setting. Curry-Howard for semantics Programs λ xA .tB. Types Execution. Logic Proof of A ` B Formulas Cut-elimination. Resources λ-calculus. Categories Morphisms Objects Equality. Functional analysis Smooth functions Topological vector spaces. Equality Differential categories. Topological vector spaces. Seely categories. Vectorial Models. Differential Linear Logic. Linear Logic. λ-calculus. Min. Logic. Normal functors.
(5) DiLL. Smooth. Tensor products. Polarized setting. Outline I Categorical Models of Differential Linear Logic What’s our shopping list ?.
(6) DiLL. Smooth. Tensor products. Polarized setting. Outline I Categorical Models of Differential Linear Logic What’s our shopping list ? I Non-linear proofs: convenient vector spaces What’s a good notion of smooth function ?.
(7) DiLL. Smooth. Tensor products. Polarized setting. Outline I Categorical Models of Differential Linear Logic What’s our shopping list ? I Non-linear proofs: convenient vector spaces What’s a good notion of smooth function ? I Linear proofs: Grothendieck ”problème des topologies” What’s a good topological tensor product ?.
(8) DiLL. Smooth. Tensor products. Polarized setting. Outline I Categorical Models of Differential Linear Logic What’s our shopping list ? I Non-linear proofs: convenient vector spaces What’s a good notion of smooth function ? I Linear proofs: Grothendieck ”problème des topologies” What’s a good topological tensor product ? I Duality. Can we even have an involutive linear negation ?.
(9) DiLL. Smooth. Tensor products. Polarized setting. Categorical Models of Differential Linear Logic.
(10) DiLL. Smooth. Tensor products. Polarized setting. Differential Linear Logic features linear proofs, non-linear proofs, and is a classical logic. As such, its models must be: I Cartesian closed with smooth functions. I Endowed with a biproduct I Monoidal closed with linear functions. I ∗-autonomous. Equivalently, we are looking for a ∗-autonomous differential category. ! has a meaning in functional analysis and the differential setting is natural. Is differentiation in DILL the same one as in continuous mathematics? We are looking for continous objects and smooth functions..
(11) DiLL. Smooth. Tensor products. Cartesian closed and smooth A space of (non necessarily linear) functions between finite dimensional spaces is not finite dimensional. dim C 0 (Rn , Rm ) = ∞. We can’t restrict ourselves to finite dimensional spaces.. Polarized setting.
(12) DiLL. Smooth. Tensor products. Cartesian closed and smooth A space of (non necessarily linear) functions between finite dimensional spaces is not finite dimensional. dim C 0 (Rn , Rm ) = ∞. We can’t restrict ourselves to finite dimensional spaces. Girard’s Coherent Banach spaces fail. We can’t restrict ourselves to normed spaces.. Polarized setting.
(13) DiLL. Smooth. Tensor products. Cartesian closed and smooth A space of (non necessarily linear) functions between finite dimensional spaces is not finite dimensional. dim C 0 (Rn , Rm ) = ∞. We can’t restrict ourselves to finite dimensional spaces. Girard’s Coherent Banach spaces fail. We can’t restrict ourselves to normed spaces. Metrics instead of norms ? Metrisable spaces are not stable under duality.. Polarized setting.
(14) DiLL. Smooth. Tensor products. Polarized setting. Cartesian closed and smooth A space of (non necessarily linear) functions between finite dimensional spaces is not finite dimensional. dim C 0 (Rn , Rm ) = ∞. We can’t restrict ourselves to finite dimensional spaces. Girard’s Coherent Banach spaces fail. We can’t restrict ourselves to normed spaces. Metrics instead of norms ? Metrisable spaces are not stable under duality.. Definition A topological vector space is a vector space endowed with a convex topology making sum and scalar multiplication continuous..
(15) DiLL. Smooth. Tensor products. Polarized setting. A few intuitions on topological vector spaces.
(16) DiLL. Smooth. Tensor products. Polarized setting. Smooth functions and convenient spaces.
(17) DiLL. Smooth. Tensor products. Polarized setting. Smooth maps à la Frölicher,Kriegl and Michor A smooth curve c : R → E is a curve infinitely many times differentiable. c f (c) f. A smooth function f : E → F is a function sending a smooth curve on a smooth curve. In Banach spaces, the definition coincides with the usual one (all iterated derivatives exists and are continuous).. The Convenient Setting of Global Analysis Kriegl & Michor, (1997) Linear spaces and differentiation theory, Frölicher & Kriegl, (1988)..
(18) DiLL. Smooth. Tensor products. Polarized setting. A dream come true ... Cartesian closedeness C ∞ (E, C ∞ (F, G)) ' C ∞ (E × F, G). Differentiation Any smooth map is Gateau-differentiable and the differentiation operator ∞ ∞ C (E, F ) → C (E, L(E, F )) d¯ : f (x + ty) − f (x) f 7→ x 7→ y 7→ lim t→0 t is well-defined, linear and bounded. ... when topological vector spaces are Mackey-complete.
(19) DiLL. Smooth. Tensor products. Polarized setting. Bounded sets and smooth functions Linear bounded function := sending a bounded set on a bounded set.. `. B. U. `−1. B. U. bounded set := absorbed by any 0-neighborhood. continuity is sufficient but not necessary. Monoidal closedeness Linear bounded maps over tvs and bornogical tensor product form a monoidal closed category..
(20) DiLL. Smooth. Tensor products. Polarized setting. Mackey-completeness. A complete locally convex topological vector space is a locally-convex topological vector space in which every Cauchy net converges..
(21) DiLL. Smooth. Tensor products. Polarized setting. Mackey-completeness. A Mackey-complete locally-convex topological vector space is a locally convex topological vector space in which every Mackey-Cauchy sequence converges..
(22) DiLL. Smooth. Tensor products. Polarized setting. Mackey-completeness A Mackey-complete locally-convex topological vector space is a locally convex topological vector space in which every Mackey-Cauchy sequence converges.. Mackey-Cauchy net A net (xγ )γ∈Γ such that there is a net of scalars λγ,γ 0 decreasing towards 0 and a bounded set B of E such that: ∀γ, γ 0 ∈ Γ, xγ − xγ 0 ∈ λγ,γ 0 B.. x2 x5 x4 x3. B x1. A very weak condition and works well with bounded sets..
(23) DiLL. Smooth. Tensor products. Polarized setting. A convenient differential category A convenient differential category Blute, Ehrhard, Tasson, (2012). Mackey-complete spaces, linear bounded maps and smooth functions form a differential category, and as such a model of intuitionistic differential linear logic.. Warning bounded maps badly accomodate duality Then one needs to switch to linear continuous functions, topological duals and topological tensor products..
(24) DiLL. Smooth. Tensor products. Polarized setting. ⊗B `=ε Linear maps and topological tensor products A = A⊥⊥ ≡ JAK ' JAK00.
(25) DiLL. Smooth. Tensor products. Polarized setting. MLL in TopVect. JA⊥ K := LB (JAK, K) = JAK0B . I Topologies of uniform convergence on elements of B ⊂ P(E). I The topology on E determines E 0 . I The topology on E 0 determines whether E ' E 00 . It’s a mess..
(26) DiLL. Smooth. Tensor products. Polarized setting. A variety of topological duals : 0 I Many topological duals Eβ0 , Ec0 , Ew , Eµ0 I Reflexivity is typically not preserved by ⊗. I Not an orthogonality : Eβ0 is not reflexive. I Many possible topologies on ⊗: ⊗β , ⊗π , ⊗ε . I Monoidal closedness ⇔ ”Grothendieck problème des topologies”..
(27) DiLL. Smooth. Tensor products. Polarized setting. Weak spaces, A negative interpretation of DiLL 0 I Ew : simple convergence on points of E. I Ew∗ : simple convergence on points of E 0 .. = ˆ = ( )w. Vec E0. TopVec Weak 0 Ew. E. Ls (E, Fw ) Ew. 0 0 (Ew ). Fw Ew ⊗ Fw.
(28) DiLL. Smooth. Tensor products. Polarized setting. Weak spaces, A negative interpretation of DiLL Weak spaces are reflexive, when the dual is the weak dual. = ˆ = ( )w. Vec E0. TopVec Weak 0 Ew. E. Ls (E, Fw ) Ew ' E. 0 0 (Ew ). 00. Fw Ew ⊗ Fw.
(29) DiLL. Smooth. Tensor products. Polarized setting. Weak spaces, A negative interpretation of DiLL The tensor product needs to undergo a shift to be a weak space. = ˆ = ( )w. Vec E0. TopVec Weak 0 Ew. E. Ls (E, Fw ) Ew. 0 0 (Ew ). (Ew ⊗ Fw )w Fw Ew ⊗ Fw.
(30) DiLL. Smooth. Tensor products. Polarized setting. Weak spaces, A negative interpretation of DiLL The ` is internally endowed with its weak topology. = ˆ = ( )w. Vec E0. TopVec Weak 0 Ew. E. Ls (E, Fw ) Ew. 0 0 (Ew ). Fw Ew ⊗ Fw Ls (E, Fw )0 ∼ E 0 ⊗ F.
(31) DiLL. Smooth. Tensor products. Polarized setting. Weak spaces, a negative quantitative interpretation of DiLL Without completeness, no smooth exponential is possible. ! Weak∞. ⊥. f =(f1 , f2 , . . . , fn ). L(!E, K) '. Weak. U d¯ d Q. n. Hn (E, K). Weak topologies for Linear Logic, K. LMCS 2015..
(32) DiLL. Smooth. Tensor products. Polarized setting. Topological Tensor products E ⊗ F is a vector space which misses a topology. E×F h may be continuous separately continuous B-hypocontinuous .... h. f. G. f˜. E⊗F I hypocontinuity : for all bounded set B, {h(B, ·)} and {h(·, B)} are equicontinuous. I Any bornology B defines a notion of hypocontinuity, and thus a topological tensor product ⊗B . I ⊗B may or may not be associative according to the condition on spaces and the bornology considered. Théorie des Distributions à valeurs vectorielles, II Schwartz, (1958).
(33) DiLL. Smooth. Tensor products. Polarized setting. The ε product Only one good ` EεF := Lε (Ec0 , F ) where Ec0 is E 0 with the topology compact-open, and the whole space is endowed with the topology of uniform convergence on equicontinuous sets of Ec0 .. ⇒ C ∞ (E, F ) ' C ∞ (E, R)εF when E and F are complete.. A monoidal category by Schwartz The ε is associative and commutative on quasi-complete spaces. Théorie des Distributions à valeurs vectorielles, I Schwartz, (1957).
(34) DiLL. Smooth. Tensor products. Polarized setting. Models based on `=ε Models of Linear Logic based on the Schwartz’ ε product Dabrowski & K.. A first smooth model of DiLL : k-Ref I Associativity of ε through a minimal completeness condition. I ((Ec0 )0c )0c ' Ec0 . I A new model of MALL.. Smooth functions with parameters in C ⊂ k-Ref CC∞ (E, F ) := ∞ ∞ {f : E → F, ∀X ∈ C, ∀c ∈ Cco (X, E) ⇒ f ◦ c ∈ Cco (X, F )}. but the differential is linear bounded ... A new induced topology Dereliction : E ,→ CC∞ (Eµ0 , R), induces a new topology SC (E) on E..
(35) DiLL. Smooth. Tensor products. Polarized setting. Models based on `=ε Then when E is Mackey-complete : C Fin Ban {0}. SC (E) The Schwartzification of E The Nuclearification of E The weak topology on E. Smooth and classical models of LL I k-complete spaces with Arens reflexivity. I Schwartz Mackey-complete spaces with Mackey reflexivity. I Nuclear Mackey-complete spaces with Mackey reflexivity.. Differentiation: one drawback Differentials are bounded in general, and we need an ad-hoc definition to have continuous differentials..
(36) DiLL. Smooth. Tensor products. Polarization. Polarized setting.
(37) DiLL. Smooth. Tensor products. Polarized setting. Not Not ... Who’s there ?. DILL = LL + Diff. `. ⊗. ⊕. smoothness. & MALL. ! ?.
(38) DiLL. Smooth. Tensor products. Polarized setting. Divide and Conquer DILL = LL + Diff.. ⊕. &. ⊗. `. P. N. smoothness ! ?. I Polarized models are behind the MLL interpretation of several smooth models of DiLL. I It offers a new point of view on reflexivity as a mathematical notion..
(39) DiLL. Smooth. Tensor products. Polarized setting. 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 Semantics (negative chirality): ( )⊥L (P op , ⊗, 1). ⊥. ˆ (N , `, ⊥). P. ( )⊥R N ⊥R ⊥L ' N N (ˆp, m ` n) ' N (ˆ(p ⊗ m⊥ ), n) Dialogue categories and chiralities Melliès, (2016). ⊥ ´. N.
(40) DiLL. Smooth. Tensor products. Polarized setting. With the Weak Dual, a negative interpretation. (−)0w. (Weakop , `w , R). TopVec. a. a. (TopVec, ⊗w , R). (−)w. ι (−)0w. in which ι denotes the inclusion functor.. Weak.
(41) DiLL. Smooth. Tensor products. Polarized setting. Involutive duals. Coarse. 0 Ew. Eµ0. Eβ0. Fine. I Weak reflexivity and Mackey reflexivity is immediate. I Strong reflexivity is the traditional one and is much harder to attain. It decompose as: I the algebraic equality between E and (Eβ0 )0 , equivalent to some weak completeness condition. I the topological correspondence E ,→ (Eβ0 )0β , called barrelledness..
(42) DiLL. Smooth. Tensor products. Polarized setting. With the strong dual, a dialogue chirality Barrelled spaces were introduced by Bourbaki as the good setting for Banach-Steinhauss theorem.. (−)0σ. (wqComplop , `w , R). Mackey. (−)0µ. Banach-Steinhauss ! Monoidal closedness Barrelled ! positive Complete ! negative. a. a. (Barr, ⊗β , R). (−)w. (−)µ. Weak.
(43) DiLL. Smooth. Tensor products. Polarized setting. A negative interpretation for the ε product. S ((−)0µ ) S (−). (ComplµSchop , ε, R) TopVec. a. a. ˆM (ubTopVec, ⊗ β , R). Sch. ι (−)0µ. Functions with domains E ∈ ubTopVec are linear continuous iff they are linear continuous continuous differentiation..
(44) DiLL. Smooth. Tensor products. Polarized setting. With bornological spaces, a positive interpretation A compatibilty condition between bounded and open sets : E bornological := ` : E → F continuous if and only if ` bounded. E convenient := bornological and Mackey-complete.. (−)0µ. (Mcoop , `b , R). bTopVec. (Born((−)0µ ))conv. a. a. M. ˆ β , R) (Conv, ⊗. Top. TopVec. Born. Bornological ! Positive [Tasson, PhD thesis, 2009] [Blute, Ehrhard, Tasson, A convenient differential Category, 2012][K.Tasson, Mackey-Complete spaces and Power series, 2018].
(45) DiLL. Smooth. Tensor products. Polarized setting. Conclusion DIY : make you own smooth model of DiLL I Smooth function are the conveniently smooth one, but will get you a linear bounded differentiation. I A weak completion criterion migh be enough: Mackey-complete, Quasi-complete ... I You want a dual topology coarse enough to be reflexive (e.g : weak dual) but fine enough to be complete (e.g. : Arens-Dual, Mackey-Dual) It is unlikely you will have it all ... Perspectives I Logically : computational content in distribution theory ( !E) ! I Categorically : could we put duality at the heart of the axiomatization ? I Tangent, Probabilistic ....
(46) DiLL. Smooth. Tensor products. Polarized setting. The Mackey-Arens Theorem, by Barr Eµ. E 7→ (E, E 0 ). Mackey. ⊥. (F, E) 7→ Fw(E). Fw. ⊥. Weak. Dual pairs. (E, F ) 7→ Eµ(F ). F 7→ (F, F 0 ). L(Eµ , F ) =L(E, Fw ). On ∗-autonomous categories of topological vector spaces, M. Barr Cahiers Topologie Géom. Différentielle Catég., 2000. On convex topological vector spaces, G. Mackey, Trans. Amer. Math. Soc., 1946..
(47) DiLL. Smooth. Tensor products. Polarized setting. With the Mackey Dual, almost a positive interpretation. (−)0µ ι. (TopVec, `µ , R). Mackey. (−)0µ. in which ι denotes the inclusion functor. → Stability properties for ⊗µ , but nothing for the `. a. a. (Mackey, ⊗µ , R). (−)µ. Weak.
(48) DiLL. Smooth. Tensor products. Polarized setting. Conclusion. DiLL = LL + Diff. `. ⊗. ⊕. smoothness. & MALL. ! ?.
(49) DiLL. Smooth. Tensor products. Polarized setting. Conclusion. DILL = LL + Diff.. ⊕. & smoothness. ⊗. `. P. N. ! ?.
(50) DiLL. Smooth. Tensor products. Conclusion DILL = LL + Diff.. ⊕. & smoothness. ⊗. `. P. N. Conjecture: Two chiralities for two interactions.. Polarized setting.
(51)
Documents relatifs
By constructing a differential characteristic of certain S-box difference pairs in each round, such that a differential involves plaintexts bits and data bits to the input of
We introduce in particular the Geneva 1 fragment, which consists of the set of NNF formulas of shape ψ ∧ G(φ), where ψ is an FOLTL formula featuring existential quantifiers only at
Based on this result, models based on deterministic graph- ings – or equivalently on partial measured dynamical systems – can be defined based on realisability techniques using
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des
Once you move to vector functions of several variables, an inevitable move for purposes of geometry and physics, derivatives of such functions are linear maps between vector
Similarly to how we proved it for transition matrices, we have the following result.
Similarly to how we proved it for transition matrices, we have the following result.
Such definition is based on the novel notion of symmetric tensor power of QCS’s, which, in turn, is an analogue in the QCS’s setting of the symmetric tensor product of Hilbert spaces,
