A concrete semantics without higher order

8.4 D-DiLL

8.4.3 A concrete semantics without higher order

Let us define formally the semantics ofD´DiLLwhich was used informally throughout the last sections. This section is peculiar, as it uses the notations used in mathematical physics to interpret the rules ofD´DiLL, see theorem8.4.11.

In this section we show that the categories EUCL, NDFand NF, together with distributions of compact support and a LPDODwith contant coefficients, form a first-order model ofD´DiLL. As for Smooth DiLL, we define

!Das a strong monoidal functor between EUCLand NDF. Extending this to a higher-order model could be possible using the same techniques as in section7.5.

ConsiderD:D¹pRⁿq //D¹pRⁿqa LPDOcc:

Dpf, xq “ ÿ

αPNⁿ

aαB^αfpxq.

In section8.3.2.2we saw that we could makeDact on any euclidean space:

Definition 8.4.8. If m ă n and f P DpR^mq, one defines an extension of f to R^m as f : px1, ...xnq ÞÑ fpx1, ..., xmq, and thus define D onDpR^mq. It m ą nand f P DpR^mq, one define Dpfq as D applied to the restriction offtoRⁿ:Dpfq:“Dppx1, . . . , xnq ÞÑfpx1, . . . , xn,0, . . . ,0qq.

Remark8.4.9. In particular, the fundamental solutionEDP!DRⁿcan be restricted or extended toED PR^m, for anym.

$N wD Figure 8.3:Cut-elimination for the exponential rules ofD´DiLL

We interpret finitary formulasA, Bas euclidean spaces. One has indeed1 » K “RandJ » 0“ t0u. The connectives of LL are interpreted in EUCL, NFand NDFas in section7.4.

Definition 8.4.10. We recall the definition of the functor!_Dwhich was made explicit in section8.3.2.2:

!D:

The introduction of!_Dbreaks the symmetry of the Kernel theorem:

Theorem 8.4.11. ConsiderD : D¹pRⁿq //D¹pRⁿqa LPDOcc. Then for anym ě0, we have natural isomor-phisms

mD,n: !DpR^n`nq »!DRⁿbˆπ!Rⁿ.

Proof. This theorem encodes the definition8.3.5, which allows to extendsD defined on EpRⁿq toEpR^n`mq.

This theorem is then directly deduced from the Kernel theorem7.3.5: asEpR^n`mq » EpRⁿqbEpRˆ ^mqwe have DpEpR^n`mqq »DEpRⁿqbEpˆ R^mq. Taking the dual gives us the desired result.

From this strong monoidal isomorphism and the biproduct structure on TOPVEC, one deduces the interpretation of w_D,w¯_D,c_D andc¯_D, following the categorical procedure introduced by Fiore (see section2.4.2.1). These interpretations are to be compared to the ones for the model ofDiLLwith distributions of Chapter7, which are detailed in Section7.4.3.

Definition 8.4.12. We give the following interpretation forwD,w¯D,cDand¯cD:

• In Section2.4.2), we interpreted the coweakening as the mapw¯A:! //!Aresulting of the application of the functor!on the initial mapuA : I //A. Concretely, we haveI “ t0uu_R^mp0 P t0uq ÞÑ0 P R^m, and

• The cocontractionc¯: !_Db! //!_Dis interpreted by the convolution product (see prop.7.4.12)

¯ c_D:

#!_DR^mb!ˆ R^m //!_DR^m

φbψÞÑφ˚ψ (8.11)

and is well defined thanks to corollary8.1.24.

• The interpretation of weakeningwD: !D //1is defined as:

w_D:φP!_DR^mÞÑ ż

Dφ.

It is well defined as by definitionDφhas compact support. As explained in Section7.4.3, integration on distributions with compact supportψPE¹pEqis defined asψapplied to a function constant at1:

ψ“ψpconst1q.

Whenψis the generalization of a functionf with compact support, then we have indeed ψpconst1q “

Remark8.4.13. Notice that the interpretation ofcDfollows the intuitions of theorem8.4.11:

f PDpˇ EpR^mˆR^mqq

is in fact inDpˇ EpR^mqqbˆEpR^mq, as differentiation occurs only on the firstmvariables. In particular, ifmą0, then the differentiation onR^mˆRⁿis defined as:

DˇbId_R^m :C⁸pRⁿˆR^mq //C⁸pRⁿˆR^mq

Thus the fundamental solutionEDP!DpR^m`nqcorresponds toEDbδ0P!DRⁿb!R^m.

Dereliction and the codereliction are defined as previously forD₀´DiLL(see Section8.3.2.1).

Definition 8.4.14. We interpret the derelictiondD: ! //!Das

dD,EpφPE¹pRⁿqq ÞÑ pED˚φq and coderelictiond¯_D: ! //!_Das

d¯D,E :pφP!DRⁿq ÞÑ pDφq PE¹pRⁿq.

Proposition 8.4.15. This denotational semantics of the rules ofD´DiLLis preserved by cut-elimination (see figure8.3).

In figure8.4.3, we provide the cut-elimination rules annotated with the interpretation of the proofs, to support the proof of proposition8.4.15.

Proof. • Cut elimination betweend_Dandd¯_D. By equation8.3one has for everyφPE¹pRⁿqandf PEpRⁿq:

dD,E˝d¯D,Epφqpfq “ED˚ pφpDpfqq

“φpE_D˚Dpfqq

“φpfq

• Cut elimination betweenwDandw¯D:

wD˝w¯Dp1PRq “ ż

DED“ ż

δ0“1

• Cut elimination betweencDandw¯D: by definition of the interpretation ofcD, one has:

cD˝w¯Dp1q “ pfD, gq PDpˇ EpRⁿqq bEpRⁿq ÞÑw¯DpxÞÑfpxqgpxqq.

Thus asw¯_Dp1q “E_D, and by making use of remark8.4.13, we have:

cD˝w¯Dp1q “EDbδˆ 0ppxÞÑfpxqgpxqqq which corresponds to the interpretation of the reduced cut-rule.

• Cut elimination between¯cDandwD: considerφP!DR^mandψP!R^m. We are used to see the cocontrac-tion as the convolucocontrac-tion between distribucocontrac-tion, but remember that we can also understand it as the resulting distribution which sums in the codomain of its function (see Proposition7.4.12). Then:

wD˝c¯Dpφ, ψq “ ż

Drf PEDpR^mq ÞÑφpxÞÑψpx¹ÞÑfpx`x¹qqqs

“ ż

rf ÞÑφpxÞÑDψpx¹ ÞÑfpx`x¹qqqs

“φpxÞÑDψpx¹ÞÑconst₁px`x¹qqq

“φpconstDψpconst₁qq

Again by definition8.3.5,Dapplies only toφ. Thus the interpretation of the cut-rule and its reduced form corresponds.

• Cut elimination between¯cDandcD: considerφP!DR^mandψP!R^m. Then up to the kernel isomorphism we have:

cD˝c¯D“!DpAÂqÝ^!ÝÝÝ^D⁴ÑÂ !DAÝÝÝÑ^!^DÔÂ !DpAÂq Thus

cD˝¯cDpφP!DpAˆAqq “f PDˇEpAˆAq ÞÑ p¯cDφqpxÞÑfpx, xqq

“f ÞÑφppx, yq ÞÑfpx, xqfpy, yqq.

Through the Kernel isomorphism we can suppose by density thatφ“φ¹bψ, withφ¹ P!DAandψP!A, andf “f¹bgwithf¹ P?DA^KandgP?A^K. Thus

pcD˝¯cDqpφP!DpAˆAqq “f¹bgˆ ÞÑφ¹bψppx, yq ÞÑf¹pxqgpyqq

“f¹bgˆ ÞÑφ¹pf¹qφpgq,

and the last proposition corresponds to the interpretation of the reduced cut-rule betweencDand¯cD.

$N w_D

Figure 8.4:Cut-elimination for the exponential rules ofD´DiLL, annotated with the semantics

Chapter 9

Conclusion

In this thesis, we conducted a study of the semantics of Differential Linear Logic: our goal was to find a model of DiLLin which functions were smooth (making coincide the intuitions of analysis concerning differentiation and the requirements of logic), and spaces continuous and reflexive (asDiLLfeatures an involutive linear negation).

We met this goal by constructing several model for it. During this study, it appeared that apolarizedwas better fit to interpretDiLL, as we may want to interpret differently the linear negation on the positives or and the negative ( Chapter6), or as a space and its dual may not belong to a same monoidal closed category (Chapter7). Throughout this study, we highlighted the fact spaces distributions with compact support are the canonical interpretation for the exponential. In this spaces of distributions, one can resolve Linear Partial Differential Equations, and we provide a sequent calculus supporting the idea that the exponential is the space of solutions for a Linear Partial Differential Equation.

Directly following this thesis: Among other points, let us mention what could appear in this thesis and is not:

1. Categorical models ofDiLL_0,polpresented in section2.5.2.3are not optimal. Indeed, they always provide for an interpretation of the promotion rule, even when this one does not figure in DiLL₀. Moreover, the symmetry between the dereliction and codereliction rules is broken in the categorical axiomatization: dere-liction comes with the strong monoidal adjunction between!andU, while codereliction is ad-hoc. In view of results in Chapter8, we should have a symmetric axiomatization.

2. In Chapter6, one would want an interpretation for the exponential which is intrinsically bornological, thus emphasizing on the fact that the adjunction between convenient spaces and complete Mackey spaces is the good linear classical refinement of convenient spaces.

3. In Chapter7, we detailed a model ofDiLLusing the theory of Nuclear Spaces. One should explore the link between this model and the model of Köthe spaces, using in particular Fourier transformation. In particular, as any nuclear fréchet space is a subset of a denumerable product of the Kothe spaces of rapidly decreasing sequences [44, 21.7.1], introducing subtyping may lead to a complete semantics.

4. In Chapter8, the proof that!DEis reflexive is cruelly missing. We also miss a good notion of categorical model forD´DiLL.

5. Chapter 8 gives a logical account for the notion of fundamental solution for Linear Partial Differential Operator. Let us note that the Cauchy Problem for Linear Partial Differential Equation behaves well with respect to solutions defined on cones, and that a link with the recent models of probabilistic programming by Ehrhard, Pagani and Tasson may exists [21].

6. This thesis emphasized the importance of reflexivity when constructing models of Differential Linear Logic.

One should now investigate the differentialλ-calculus in terms of linear continuations, and linear exceptions, in the spirit of [16] and [35].

What’s more important: In Chapter8, we build a deterministic sequent calculusD´DiLLwith a concrete first-order denotational model in which applying the dereliction rule corresponds to solving a linear partial differ-ential equation, with the basic intuition that several expondiffer-ential exists, and each one is associated with a Linear Differential Operator

1. The priority is to work towards completing the Curry-Howard-Lambek correspondence for Linear Partial Differential Equations. This means finding the good categorical axiomatization for models ofD´DiLL, and most importantly refining the differentialλ-calculus [22] into a deterministic calculus for Linear Partial Differential Equations. This should be done by understanding computationally the proof of the Existence of ED.

2. In Chapter7, we gave a sequent calculusD-DiLL, which corresponds to afixedLinear Partial Differential OperatorD. We would like to generalize this procedure, by constructed a graduated sequent calculus à la BLL[34], which would describe any Linear Partial Differential operator with constant coefficient.

3. Of course, the global objective is to work towards a computational understanding of non-linear Partial Dif-ferential Equations.

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Appendices

Appendix A

Index of symbols

We indicate symbols, and then give the name of the mathematical object they denote, and the page or section in which they are introduced.

Chapter 2

• LpE, Fq,LinpE, Fq,LpE, Fq: the vector space of all (resp. of all bounded, resp of all continuous) linear functions betwenn the lcsEandF.

• w,c,d: the weakening, contraction and dereliction rules ofLL, page17.

• pT, d, µq: a co-monad with co-unitdand co-multiplicationµ., page20.

• L!Co-Kleisli category of the co-monad!, page20.

• ˆ,´: the shifts ofLLpol, page24.

• N ,P: categories involved in a model ofLLpol, page28.

• w,¯ ¯c,d¯: the co-weakening, co-contraction, co-dereliction rules ofDiLL, page35.

• ˛,O,u,4,n: a biproduct structure on a category , page40.

Chapter 3

• E_B : whenBis a absolutely convex and weakly-closed subset ofE, it is the vector space generated byB, normed by the distance toB, page57

• E˜the completion of a lcsE, page59.

• lim ÝÑ,lim

ÐÝ: the lcs injective (resp. projective) limit of |cs, page58.

• C_c⁸pRⁿq “DpRⁿq: the vector space of all smooth functionsf :Rⁿ //Rwith compact support,endowed with the topology of uniform convergence of compact subsets of every derivatives of finite order, page60

• C⁸pRⁿq “EpRⁿq: the vector space ofallsmooth functionsf :Rⁿ //R, endowed with the topology of uniform convergence of compact subsets of every derivatives of finite order, page60.

• Ew,E¹_w˚: the weak and weak* topologies onEandE¹respectively, page63.

• Eµ˚,E_µ¹ : the Mackey* and Mackey topologies onEandE¹respectively, page68.

• E_β¹ : the strong dual, page68

• bπ,bˆπ: the projective tensor product, the completed projective tensor product, page74.

• bε,bˆε: the injective tensor product, the completed injective tensor product, page74.

Chapter 5

• bi: the inductive tensor product, page86.

• HⁿpE, Fq: The space of n-linear symmetric separately continuous functions fromEtoF, page89.

• !: the exponential for which arrows in the co-kleisli category are formal power series.

• !1: the exponential for which arrows in the co-kleisli category are formal power series without constant coefficients, whose composition correspond to the Faa di Bruno Formula.cde"

Chapter 6

• p_q¯ ^born: the bornologification of a lcs , page108.

• p_qˆ ^M : the Mackey-completion of a lcs, page110.

• ¯_^conv: The bornologification of the Mackey-completion, making a spaceconvenient, page111.

• Eµ: The Mackey topology on the lcsE, considered as the dual (equivalently pre-dual) ofE_µ¹, page69.

• ε: theεproduct, page78.

Chapter 7

• D¹pRⁿq “C_c⁸pRⁿq¹_β, the space of distributions„ page132.

• E¹pRⁿq “C⁸pRⁿq¹_β, the space of distributionswith compact support, page132.

• f, f¹, g, g¹, h, h¹...: smooth functions, sometimes with compact support. If it is not explicitly mentioned,f¹ will never designate the derivative off, asfis not in general a function defined onR.

• φ, φ¹, ψ, ψ¹. . . : Distributions, sometimes with compact support. Beware that Hormander [41,42] uses reverse notations for functions and distributions.

Chapter 8

• D0: the operator mapping a smooth function to its differential at0(which is linear continuous).

• D: a Linear Partial Differential Operator with constant coefficients.

D“ ÿ

αPNⁿ

aα

B^αf Bx^α Dˇ :“ ÿ

αPNⁿ

p´1q^αaα

B^αf Bx^α

• D₀´DiLL: a non-deterministic sequent calculus for which bothDandD₀results in a model.

• D´DiLL: a deterministic sequent calculus for whichDresults in a model.

Appendix B

Models of Linear Logic based on the Schwartz ε-product.

This chapter consist in a submitted paper in collaboration with Y. Dabrowksi. In contrary to chap-ters 6, 7, we develop here an unpolarized, and therefore more difficult, approach to smooth and reflexive model of DiLL. From the interpretation of Linear Logic multiplicative disjunction as the ε-product defined by Laurent Schwartz, we construct several models of Differential Linear Logic based on usual mathematical notions of smooth maps. We isolate a completeness condition, called k-quasi-completeness, and an associated notion stable by duality calledk-reflexivity, allowing for a

˚-autonomous category ofk-reflexive spaces in which the dual of the tensor product is the reflexive version of theεproduct. We adapt Meise’s definition of Smooth maps into a first model of Differential Linear Logic, made ofk-reflexive spaces. We also build two new models of Linear Logic with con-veniently smooth maps, on categories made respectively of Mackey-complete Schwartz spaces and Mackey-complete Nuclear Spaces (with extra reflexivity conditions). Varying slightly the notion of smoothness, one also recovers models ofDiLLon the same˚-autonomous categories. Throughout the article, we work within the setting of Dialogue categories where the tensor product is exactly the ε-product (without reflexivization).

Dans le document manuscript (Page 171-189)