HigherOrder Distributions for Differential Linear Logic
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Abstract
Linear Logic was introduced as the computational counterpart of the algebraic notion of linearity. Differential Linear Logic refines Linear Logic with a prooftheoretical interpretation of the geometrical process of differentiation. In this article, we construct a polarized model of Differential Linear Logic satisfying computational constraints such as an interpretation for higherorder functions, as well as constraints inherited from physics such as a continuous interpretation for spaces. This extends what was done previously by Kerjean for first order Differential Linear Logic without promotion. Concretely, we follow the previous idea of interpreting the exponential of Differential Linear Logic as a space of higherorder distributions with compactsupport, which is constructed as an inductive limit of spaces of distributions on Euclidean spaces. We prove that this exponential is endowed with a comonadic like structure, with the notable exception that it is functorial only on isomorphisms. Interestingly, as previously argued by Ehrhard, this still allows the interpretation of differential linear logic without promotion.
Keywords
Differential Linear Logic Categorical semantics Topological vector spaces1 Introduction
Denotational semantics interprets programs as functions which focuses not on how data from these programs are computed, but rather focusing on the input/output of programs and on data computed from other data [19]. Through the CurryHowardLambek correspondence, this approach refines into the categorical semantics of type systems. In particular, a study of the denotational model of the \(\lambda \)calculus for coherent spaces led Girard to Linear Logic [9] and the understanding of the use of resources as the computational counterpart of linearity in algebra. Differential Linear Logic (\(\mathrm {DiLL}\)) [7] is a refinement of Linear Logic which allows for a notion of linear approximation of nonlinear proofs. As a proofnet calculus, \(\mathrm {DiLL}\) originated from studying vectorial models of Linear Logic which in general are based on spaces of sequences, such as Köthe spaces and finiteness spaces [5].
Recently the first author argued in [14] that as a sequent calculus \(\mathrm {DiLL}\) has a “smooth" semantical interpretation where the exponential Open image in new window (the central object of Linear Logic) is interpreted as a space of distributions with compact support [18]. This semantical interpretation of \(\mathrm {DiLL}\) (along with the Linear Logic typed phenomena of duality and interaction) provides a strong argument that \(\mathrm {DiLL}\) should be considered as a foundation for a type theory of differential equations, whose semantics would be based on structures developed for mathematical physics. However one of the many divergences between the theoretical study of physical systems and the theoretical study of programming languages lies in the treatment of input data. In the study of differential equations, one generally only accepts a finite number of parameters: typically time and space [1]. While one of the fundamental aspects of the semantics of functional programming languages is the concept of higherorder types [4], which in particular allows programs to take other programs as inputs. Linking these two concepts together requires that when mathematical physics studies functions with finite dimensional domains, the denotational semantical counterpart will be studying functions whose codomains are spaces of functions (which are in general far from being finite dimensional).
This article gives a higherorder notion of distributions with compact support, following the model without higher order constructed by the first author in [14]. Indeed, only functions whose domains are finite dimensional were defined in [14], while no interpretation was given for functions whose domains are spaces of smooth functions. This latter notion relies on the basic intuition that even with a continuous and infinite set of input data, a program will at each computation use only a finite amount of data.
Content and Related Work. In this paper, we interpret the exponential as an inductive limit of spaces of distributions with compact support (Definition 7). Nonlinear proofs are thus interpreted as elements of a projective limit of spaces of smooth functions. In [3], Blute, Cockett, and Seely construct a general interpretation of an exponential as a projective limit of more basic spaces. In [13], Kriegl and Michor construct the free \(\mathcal {C}^\infty \)ring over a set X (thus a space of smooth functions) as a projective limit of spaces of smooth functions between Euclidean spaces. Our work thus differs on the fact that we reverse the use of projective and inductive limits for defining the exponential and that we use a finer indexation than the indexation used in [3, 13]. The reverse use of limits compared to the literature is motivated by the fact that we are cautious about polarities [16], while the finer indexing is for topological considerations. Indeed, we need to carefully consider the functoriality of the exponential and the topology on the objects.
Context. Differential Linear Logic (\(\mathrm {DiLL}\)) is a sequent calculus enriching Linear Logic (LL) with the possibility of linearizing proofs. This linearization is semantically understood as the differentiation at 0. Motivated by the need to explore the similarities between the differential structures inherited from logic and those inherited from physics, one would like to interpret formulas of \(\mathrm {DiLL}\) by general topological vector spaces and nonlinear proofs by smooth functions. The interpretation of the involutive linear negation of \(\mathrm {DiLL}\) leads to the requirement of reflexive topological vector spaces, that is, topological vector spaces E such that \(\mathcal {L}(\mathcal {L}( E, \mathbb {R}), \mathbb {R}) \simeq E\), otherwise expressed as \(E'' \simeq E\). In [14], the first author argued that in a classical smoothlinear setting, the exponential Open image in new window should be interpreted as a space of distributions with compact support [18], that is, Open image in new window . The first author also showed that this defines a strong monoidal functor Open image in new window from the category of Euclidean vector spaces to the category of reflexive locally convex and Hausdorff vector spaces. As reflexive spaces typically do not form a \(*\)autonomous category (or even a monoidal closed category), in [14] the first author constructs a polarized model of \(\mathrm {DiLL}\) structured as chirality [17]. This polarized structure is also necessary here. In Sect. 5, formulas of \(\mathrm {DiLL}_0\) are interpreted in two different categories, depending on whether they interpret a positive or a negative formula.
Organization of the Paper. Section 2 gives an overview of the development in \(\mathrm {DiLL}\) which led to this paper and gives some background in functional analysis. In Sect. 3 we discuss higherorder functions and distributions, and prove strong monoidality. Section 4 provides the interpretation of the dereliction and codereliction and the bialgebraic structure of the exponential. Finally in Sect. 5 we discuss the polarized interpretation of formulas.
Notation. In this article, we borrow notation from Linear Logic. In particular, we use \(\multimap \) to distinguish between linear functions and nonlinear ones, for example, \(f: E \multimap F\) would be linear continuous while \(g: E \rightarrow F\) would only be smooth. We also denote elements of Open image in new window and Open image in new window , which are index by linear continuous injective indexes \(f: \mathbb {R}^n \hookrightarrow E\), in bold with their indexing in subscript: Open image in new window or Open image in new window .
2 Preliminaries
2.1 Differential Linear Logic and Its Semantics
Following Fiore’s definition in [8], a categorical model of \(\mathrm {DiLL}\) is an extension of Seely’s axiomatization of categorical models of Linear Logic [20]. Explicitly a model of \(\mathrm {DiLL}\) consists of a \(*\)autonomous category \((\mathcal {L}, \otimes , 1, (\_)^*)\) with a finite biproduct structure \(\times \) with zero object 0, a strong monoidal comonad Open image in new window , and a natural transformation Open image in new window , called the codereliction operator, which interprets differentiation at zero. A particular important coherence for the codereliction is that composing the counit of the comonad Open image in new window with \(\bar{d}\) results in the identity (the top left triangle of Definition 1). Intuitively, this means that differentiating a linear map results in the same linear map.
Models of \(\mathrm {DiLL}_0\) in which promotion is not necessarily interpreted were studied by Ehrhard in his survey on Differential Linear Logic [6]. He introduces exponential structures which provides a categorical setting which differs from the traditional axiomatization of Seely’s models.
Definition 1
That we have a model of \(\mathrm {DiLL}_0\) and not of \(\mathrm {DiLL}\) fits well with our motivation, as we are looking for the computational counterpart of type theories modeled by analysis. \(\mathrm {DiLL}_0\) is indeed the sequent calculus which is refined into an understanding of Linear Partial Differential Equations in [14] and the meaning of promotion with respect to differential equations remains unclear. However, we are still able to construct a natural promotionlike morphism for our exponential (Definition 13).
2.2 Reflexive Spaces and Distributions
In this paper, we study and use the theory of locally convex topological vector spaces [12] to give concrete models of \(\mathrm {DiLL}\). Topological vector spaces are a generalization of normed spaces or metric spaces, in which continuity is only characterized by a collection of open sets (which may not necessarily come from a metric or a norm). In this section, we highlight some key concepts which hopefully will give the reader a better understanding of the difficulties of constructing models of \(\mathrm {DiLL}\) using smooth spaces. We refer respectively to [12] or [18] for details on topological vector spaces or distribution theory.
By a locally convex topological vector space (lcs), we mean a locally convex and Hausdorff topological vector space on \(\mathbb {R}\). Briefly, these are vector space endowed with a topology generated by convex open subsets such that the scalar multiplication and the addition are both continuous. For the rest of the section, we consider E and F two lcs.
Definition 2
Denote \(E \sim F\) for a linear isomorphism between E and F as \(\mathbb {R}\)vector spaces, and \(E \simeq F\) for a linear homeomorphism between E and F as topological vector spaces.
Definition 3
Denote \(\mathcal {L}_b(E,F)\) as the lcs of all linear continuous functions between E and F, which is endowed with the topology of uniform convergence on bounded subsets [12] of E. When \(F = \mathbb {R}\), we denote \(E' = \mathcal {L}_b(E, \mathbb {R})\) and is called the strong dual of E.
Definition 4
The following proposition is crucial to the constructions of this paper. In terms of polarization, it shows how semireflexivity is a negative construction, while reflexivity mixes positives and negative requirements.
Proposition 1

Semireflexivity is preserved by projective limits, that is, the projective limit of semireflexive lcs is a semireflexive lcs.

A lcs E is reflexive if and only if it is semireflexive and barrelled, meaning that every convex, balanced, absorbing and closed subspace of E is a 0neighbourhood.

Barrelled spaces are preserved by inductive limits, that is, the inductive limit of barrelled spaces is a barrelled space.
Next we briefly recall a few facts about distributions.
Definition 5
For each \(n \in \mathbb {N}\), a function \(f: \mathbb {R}^n \rightarrow \mathbb {R}\) is said to be smooth if it is infinitely differentiable. Let \(\mathcal {E}(\mathbb {R}^n)= \mathcal {C}^\infty (\mathbb {R}^n, \mathbb {R})\) denote the space of all smooth functions \(f: \mathbb {R}^n \rightarrow \mathbb {R}\), and which is endowed with the topology of uniform convergence of all differentials on all compact subsets of \(\mathbb {R}^n\) [12]. The strong dual of \(\mathcal {E}(\mathbb {R}^n)\), \(\mathcal {E}'(\mathbb {R}^n)\), is called the space of distributions with compact support.
We now recall the famous Schwartz kernel theorem, which states that the construction of a kernel of \(f \otimes g \in \mathscr {E}(\mathbb {R}^n) \otimes \mathscr {E}(\mathbb {R}^m) \mapsto f \cdot g \in \mathscr {E}(\mathbb {R}^{n+m})\) is in fact an isomorphism on the completed tensor product \(\mathscr {E}(\mathbb {R}^n) \hat{\otimes }\mathscr {E}(\mathbb {R}^m)\):
Theorem 1
Theorem 2
([14]). There is a firstorder polarized denotational model of \(\mathrm {DiLL}_0\) in which the exponential is interpreted as a space of distributions: Open image in new window .
This interpretation did not generalize to higherorder as we were unable to define Open image in new window for an infinite dimensional space E, even for those sharing the topological properties of spaces of smooth functions^{1}. For example, the definition of Open image in new window is in no way obvious. This is the problem we tackle in the following sections.
3 HigherOrder Distributions and Kernel
In this section we define spaces of higherorder functions and distributions, we prove that they are reflexive (Proposition 2) and verify a kernel theorem (Theorem 3).
Definition 6
Let E be a lcs and \({f: \mathbb {R}^n \hookrightarrow E}\) and \(g: \mathbb {R}^m \hookrightarrow E\) be two linear continuous injective functions. We say that \( f \leqslant g\) when \(n \leqslant m\) and \( f = g_{\mathbb {R}^n}\), that is, \(f = g \circ \iota _{n,m}\) where \(\iota _{n,m}: \mathbb {R}^n \rightarrow \mathbb {R}^m\) is the canonical injection.
The ordering \(\leqslant \) in the above definition provides an order on the set of dependent pairs (n, f) where \( n \in \mathbb {N}\) and \( f: \mathbb {R}^{\mathbb {N}} \hookrightarrow E\) is linear injective. This will allow us to construct an inductive limit (a categorical colimit) of lcs.
Definition 7
Let E any lcs.
 1.For every linear continuous injective function \(f: \mathbb {R}^n \multimap E\), define the \(lcs \) \(\mathscr {E}'_f(\mathbb {R}^n)\) as follows:$$\mathscr {E}'_f(\mathbb {R}^n):= \mathcal {C}^\infty (\mathbb {R}^n)'$$
 2.Define \(\mathscr {E}'(E)\), the space of distributions on E, as follows:that is, the inductive limit [12, Chapter4.5] (or colimit) in the category \(\textsc {TopVec} \) of the family of \(lcs \) \(\{ \mathscr {E}'_f(\mathbb {R}^n) \vert f: \mathbb {R}^n \multimap E~linear~continuous~injective\}\) directed under the inclusion maps defined as$$ \mathscr {E}'(E):= \varinjlim _{f: \mathbb {R}^n \multimap E} \mathscr {E}'_f(\mathbb {R}^n) $$when \(f \leqslant g\).$$S_{f,g}: \mathscr {E}'_g(\mathbb {R}^n) \rightarrow \mathscr {E}'_f(\mathbb {R}^m), \phi \mapsto ( h \mapsto \phi ( h \circ \iota _{n,m})) $$
Intuitively this definition of \(\mathscr {E}'(E)\) says that distributions with compact support on E are the distributions with a finite dimensional compact support \(K \subset \mathbb {R}^n\).
Proposition 2
For any lcs E, \(\mathscr {E}'(E)\) is a reflexive lcs.
The following proposition justifies the notation of \(\mathscr {E}'(\mathbb {R}^n)\) from Definition 5.
Proposition 3
If \(E \simeq \mathbb {R}^n\) for some \(n \in \mathbb {N}\), then \( \mathscr {E}'(E) \simeq \mathcal {C}^\infty (\mathbb {R}^n)'\).
As \(\mathscr {E}'(E)\) is reflexive, we give a special (yet obvious) notation for the strong dual of \(\mathscr {E}'(E)\).
Definition 8
For a reflexive lcs E, let \(\mathscr {E}(E)\) denote the strong dual of \(\mathscr {E}'(E)\).
Since the strong dual of a reflexive lcs is again reflexive [12], it follows by Proposition 3 that for any reflexive lcs E, \(\mathscr {E}(E)\) is also reflexive.
The strong dual of a projective limit is linearly isomorphic to the inductive limit of the duals, however as noted in [12, Chapter 8.8.12], the topologies may not coincide. When E is endowed with its Mackey topology (which is the case in particular when E is reflexive), then the topologies do coincide.
Proposition 4
The Kernel Theorem. We now provide the Kernel theorem for spaces \(\mathscr {E}(E)\). Indeed, the spaces of functions are the one which can be described as projective limits, and projective limits are the ones which commute with the completed projective tensor product \(\hat{\otimes }_{\pi }\). While we do not provide a proof here, we would like to highlight that the proof of this theorem depends heavily on the fact that the considered spaces of functions are nuclear spaces [12].
Theorem 3
We now give the definitions of functors Open image in new window and Open image in new window , both of which agree with the previous characterization described by the first author in [14] on Euclidean spaces \(\mathbb {R}^n\). However, as discussed in the introduction, while these functors can be defined properly on all objects, they will only be defined on isomorphisms. So let \(\textsc {Refl} _{iso}\) denote the category of reflexive lcs and linear homeomorphism between them.
Definition 9
Note that Open image in new window is defined by the universal property of the projective limit, that is, Open image in new window is uniquely defined by postcomposing by the projections \(\pi _g: \mathscr {E}(F') \rightarrow \mathscr {E}(\mathbb {R}^n)\) for each linear continuous injective function \(g: \multimap F'\). We also note that \(\mathbf f _{\ell ' \circ g}\) is welldefined since \(\ell '\) is injective and therefore so is \(\ell ' \circ g\). The universality of the projective limit also insures that Open image in new window is an isomorphism and that Open image in new window is functorial.
Definition 10
As before, Open image in new window is defined by the couniversal property of the inductive limit, that is, Open image in new window is defined by precomposition with the injections \(\iota _{f}: \mathscr {E}'_f(\mathbb {R}^n) \hookrightarrow \mathscr {E}'(E)\) for every linear continuous injective function \(f: \mathbb {R}\multimap E\). Functoriality of Open image in new window is ensured by functoriality of Open image in new window and reflexivity of the objects.
4 Structural Morphisms on the Exponential
We consider the exponential from the \(\mathrm {DiLL}\) model of convenient vector spaces in [2] as a guideline for defining the structural morphisms on Open image in new window . In that setting, structural operations can be defined on Dirac operations. For example, the codereliction \(d_{conv}\) maps \(\delta _x\) to x. Here the mapping \(\delta _x\) must be understood as the linear continuous function which maps \(x \in E\) to \(\left( (\mathbf {f}_f)_f \in \mathscr {E}(E') \mapsto \mathbf {f}( f^{1}(x) \right) \in \mathscr {E}'(E)\), which we show is well defined below.
4.1 Dereliction and Codereliction
Definition 11
Lemma 1
The morphisms \(d_E\) are natural with respect to linear homeomorphisms, that is, maps of \(\textsc {Refl} _{iso}\). Explicitly, if \(\ell : E \rightarrow F \in \textsc {Refl} _{iso}\) then Open image in new window .
Definition 12
Lemma 2
The morphisms \(\bar{d}_E\) are natural with respect to linear homeomorphisms, that is, maps of \(\textsc {Refl} _{iso}\). Explicitly, if \(\ell : E \rightarrow F \in \textsc {Refl} _{iso}\) then Open image in new window .
Finally, we observe that \(d_E\) and \(\bar{d}_E\) satisfy the allimportant coherence condition between derelictions and coderelictions.
Proposition 5
For a reflexive lcs E, \(d_E \circ \bar{d}_E = Id_E\).
4.2 (Co)contraction and (Co)weakening
Theorem 4
The morphisms \((w,\bar{w},c,\bar{c},d,\bar{d})\) satisfy the coherences of exponential structure on Open image in new window , as detailed in Definition 1.
We note that this does not give an exponential structure per say since \(\textsc {Refl} \) is not a monoidal category, as we will explain in Sect. 5. That said, in Sect. 5 we are still able to construct a polarized model of \(\mathrm {DiLL}_0\).
4.3 Comultiplication
The categorical interpretation of the exponential rules of linear logic requires a comonad Open image in new window . However in the case of this paper, the exponential Open image in new window is functorial only on isomorphisms. As such, one cannot interpret the promotion rule of Linear Logic, as this requires functoriality of Open image in new window on the interpretation of any proof (and typically on linear continuous maps which are not isomorphisms). That said, functoriality is the only missing ingredient, and one can still define natural transformations of the same type as the comultiplication and counit of the comonad. This section details this point, leaving the exploration of a functorial Open image in new window for future work.
Definition 13
This is well defined, as we can show as for the codereliction (5) that the term \(\mathbf {g}_g(g^{1} (\phi ))\) is unique when Open image in new window linear and \(\mathbf {g}_g \in \mathcal {C}^\infty _g(\mathbb {R}^m)\) varies. Moreover there is at least one linear function Open image in new window which has \(\phi \) in its image.
Lemma 3
The morphisms \(\mu _E\) are natural with respect to linear homeomorphisms, that is, maps of \(\textsc {Refl} _{iso}\). Explicitly, if \(\ell : E \rightarrow F \in \textsc {Refl} _{iso}\) then Open image in new window .
Proposition 6
For any reflexive lcs E, Open image in new window
The identity of Proposition 6 is one of the identities of a comonad. The other comonad identities require applying Open image in new window to \(\mu \) and d, which we cannot do in our context as Open image in new window is only defined on isomorphisms.
5 A Model of \(\mathrm {DiLL}_0\)
In Sect. 4 we defined the structural morphisms on the exponential and proved the equations allowing to interpret proofs of \(\mathrm {DiLL}_0\) by morphisms in \(\textsc {Refl} \), independent of cutelimination. We now detail which categories allow to interpret formulas of \(\mathrm {MALL}\). This will be done in a polarized setting generalizing the one of [14].
Polarization. So far we have constructed an exponential Open image in new window which is strong monoidal. However, the category of reflexive spaces is too big to give us a model of \(\mathrm {DiLL}_0\). Interpreting the multiplicative connective requires a monoidal setting, and reflexive spaces are not stable by topological tensor products. If we study more closely the definition of spaces of higherorder smooth functions, we see that their reflexivity follows from a more restrictive class of spaces. These spaces are however not stable by duality, thus resulting in a polarized model of \(\mathrm {DiLL}_0\).
Models of polarized linear logic are axiomatized categorically as an adjunction between a category of positives and a category of negative, where two interpretations for negation play the role of adjoint functors. These categories obey the axiomatic of chiralities [17].
Additives. Interpreting the additive connectives of linear logic is straightforward. The product \(\times \) and coproduct \(\oplus \) of lcs are linearly homeomorphic on finite indexes and therefore give biproducts, which leads to the usual commutative monoid enrichment as described in [8].
Multiplicatives. When sticking to finite dimensional spaces or normed spaces, duality is pretty straightforward in the sense that the dual of a normed space is still normed. This, however, is no longer the case when one generalizes to metric spaces. Indeed, the dual of a metric space may not be endowed with a metric. A Fréchet space, or (F)space, is a complete and metrizable lcs. The duals of these spaces are not metrizable in general, but they are (DF)spaces (see [10] for the definition):
Proposition 7

If E is metrizable, then its strong dual \(E'\) is a (DF)space.

If E is a (DF)space, then \(E'\) is an (F)space.
Typical examples of nuclear (F)spaces are the spaces of smooth functions \(\mathscr {E}(\mathbb {R}^n)\), while typical examples of nuclear (DF)spaces are the spaces of distributions with compact support \(\mathscr {E}'(\mathbb {R}^n)\). In particular, all these spaces are reflexive. In [14], the first author interpreted positive formulas as Nuclear (DF)spaces, while negative formulas were interpreted as (F)spaces. Following the construction of Sect. 3, we will consider respectively inductive limits and projective limits.
Definition 14
A lcs is said to be a Lnfspace if it is a regular projective limit of nuclear Fréchet spaces. The category of Lnfspaces and linear continuous injective maps is denoted \(\textsc {LNF}\). A lcs E is said to be a \(\textsc {Lndf} \)space if it is an inductive limit of nuclear complete (DF)spaces.
Proposition 8
 1.
A Lnfspace E is reflexive.
 2.
The dual of a Lnfspace is a \(\textsc {Lndf} \)space.
The above proposition can be proven using the same techniques as computing the dual of \(\mathscr {E}(E)\).
The difficulty of constructing a model of \(\textsc {MLL}\) in topological vector spaces is choosing the topology which will make the tensor product associative and commutative on the already chosen category of lcs. Contrary to what happens in a purely algebraic setting, the definition of a topological tensor product is not straightforward and several topologies can be defined, with each corresponding to a different notion of continuity for bilinear maps [10]. On nuclear spaces, such as \(\mathscr {E}(\mathbb {R}^n)\) and \(\mathscr {E}'(\mathbb {R}^n)\), most of these tensor product coincide with one another. In [14], both multiplicative connectors (\(\otimes \) and Open image in new window ) were interpreted as the completed projective (equivalently injective) tensor product \(\hat{\otimes }_{\pi }\) (see [12, 15.1 and 21.2]) This property is lost when working with limits. However, there is still a good interpretation of Open image in new window for Lnf spaces (which are thus the interpretation of negatives formulas). Indeed, the completed injective tensor product \(\hat{\otimes }_{\varepsilon }\) of a projective limit of lcs is the projective limit of the completed injective tensor products [12, 16.3.2]. Taking the duals of Theorem 3 applied to \(E'\) and \(F'\) gives the following:
Proposition 9
and shows that Open image in new window is interpreted by \(\hat{\otimes }_{\varepsilon }\). The multiplicative conjunction \(\otimes \) is interpreted as the dual of \(\hat{\otimes }_{\varepsilon }\), which may not be necessarily linearly homeomorphic to \(\hat{\otimes }_{\pi }\).
6 Conclusion
In this paper, we extended the polarized model of \(\mathrm {DiLL}\) without higher order constructed in [14] to a higherorder polarized model of \(\mathrm {DiLL}_0\). The motivating idea was that computation on spaces of functions used only a finite number of arguments. This lead to constructing an exponential on a reflexive lcs as an inductive limit of exponentials of finite dimensional vector spaces. While this exponential is only functorial for linear homeomorphisms we were still able to provide structural morphisms interpreting (co)weakening, (co)contraction, and (co)dereliction, and hints of a comonad.
The next step would be to extend the definition of the exponential in this paper to an interpretation of the promotion rule and thus of \(\mathrm {LL}\) – this could be done through epimono decomposition of arrows in \(\textsc {Refl} \). Another task is to properly work out which tensor product of reflexive space will provide a model of \(\mathrm {DiLL}\). Such a model should use chiralities [17], and underline the similarities between shifts and (co)dereliction.
More generally, this works highlights again that the interpretation of the exponential in lcs relies on a computing principle. Indeed, it always requires finding a higherorder extension of distributions. While what we have constructed here relies on a finitary principle, the construction of a free exponential in [3] relies on the principle that higherorder operations are computed on Dirac distributions \(\delta _x\). That is, the exponential is constructed following a discretization scheme. The appearance of such numerical methods in a semantic study of \(\mathrm {DiLL}\) provides another link between theoretical computer science and mathematical physics. This opens the door to studying relating numerical schemes of numerical analysis and the theoretical study of programming language.
Footnotes
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