Gluing resource proof-structures: inhabitation and inverting the Taylor expansion

A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be expanded into a set of resource proof-structures: its Taylor expansion. We introduce a new criterion characterizing (and deciding in the finite case) those sets of resource proof-structures that are part of the Taylor expansion of some MELL proof-structure, through a rewriting system acting both on resource and MELL proof-structures. We also prove semi-decidability of the type inhabitation problem for cut-free MELL proof-structures.


Introduction
The Taylor expansion. Girard's linear logic (LL, [Gir87]) is a refinement of intuitionistic and classical logic that isolates the infinitary parts of reasoning in two dual modalities: the exponentials ! and ?. They give a logical status to operations of memory management such as copying and erasing: a linear (i.e. exponential-free) proof corresponds-via the Curry-Howard isomorphism-to a program that uses its argument linearly, i.e. exactly once, while an exponential proof corresponds to a program that can use its argument at will.
The intuition that linear programs are analogous to linear functions (as studied in linear algebra) while exponential programs mirror a more general class of analytic functions got a technical incarnation in Ehrhard's work [Ehr02,Ehr05] on LL-based denotational semantics for the λ-calculus. This investigation has been then internalized in the syntax, yielding the resource λ-calculus [ER03,ER08], inspired by [Bou93]: there, copying and erasing are forbidden and replaced by the possibility to apply a function to a bag of resource λ-terms which specifies how many times (in a finite number) an argument can be linearly passed to the function, so as to represent only bounded computations.
The Taylor expansion [ER08] (more precisely, its support; but here and throughout the paper we do not consider the rational coefficients in the Taylor expansion) associates with every ordinary λ-term a-generally infinite-set of resource λ-terms, recursively approximating Quasi-proof-structures and mix. Our rewrite rules consume proof-structures from their conclusions. The rule corresponding to boxes in MELL opens a box by deleting its principal door (a !-cell) and its border, while for a resource proof-structure it deletes a !-cell and separates the different copies of the content of the box (possibly) represented by such a !-cell. This operation is problematic in a twofold way. In a resource proof-structure, where the border of boxes is not marked, it is not clear how to identify such copies. On the other side, in a MELL proof-structure the content of a box is not to be treated as if it were at the same level as what is outside of the box: it can be copied many times or erased, while what is outside boxes cannot, and treating the content in the same way as the outside suppresses this distinction, which is crucial in LL. So, we need to remember that the content of a box, even if it is at depth 0 (i.e. not contained in any other box) after erasing the box wrapping it by means of our rewrite rules, is not to be mixed with the rest of the structure at depth 0. π · · · In order for our proof-structures to provide this information, we need to generalize them and consider that a proof-structure can have a forest of boxes, instead of just a tree: this yields the notion of quasi-proof-structure (Definition 4.5). In this way, according to our rewrite rules, opening a box by deleting its principal door amounts to taking a box in the tree and disconnecting it from its root, creating a new tree. We draw this in a quasi This is not merely a technical remark, as this generalization gives a status to the mix rule of LL: indeed, mixing two proofs amounts to taking two proofs and considering them as one, without any other modifications. Here, it amounts to taking two proof-structures, each with its box-tree, and considering them as one by merging the roots of their trees (see the mix step in Definition 6.3). We embed this design decision up to the level of formulas, which are segregated in different zones that have to be mixed before interacting. Indeed, our rewrite rules actually act on conclusions arranged as a list of lists of MELL formulas.
Geometric invariance and emptiness: the filled Taylor expansion. The use of forests instead of trees for the nesting structure of boxes, where the different roots are thought of as the contents of long-gone boxes, has an interesting consequence in the Taylor expansion: indeed, an element of the Taylor expansion of a proof-structure contains an arbitrary number of copies of the contents of the boxes, in particular zero. If we think of the part at depth 0 of a MELL proof-structure as inside an invisible box, its content can be deleted in some elements of the Taylor expansion just as any other box. 4 As erasing completely the conclusions would cause the Taylor expansion not preserve the conclusions (which would lead to technical complications), we introduce the filled Taylor expansion (Definition 5.8), which contains not only the elements of the usual Taylor expansion, but also elements of the Taylor expansion where one component has been erased and replaced by a -cell (daimon), representing lack of information, apart from the number and types of the conclusions. Roughly, a -cell is a placeholder for any DiLL 0 proof-structure of given conclusions.
Gluability and cut-free gluability. Our gluability criterion is based on local rewritings, which yields a geometric and modular approach to the inverse Taylor expansion problem. From the geometric point of view, there is nothing special in the elementary rewrite step corresponding to the cut rule, and it is natural to prove our gluability criterion in presence of cuts (Theorem 8.3.1). But from the proof-theoretical point of view, the gulf separating cut-free proof-structures from the others shows up: for every MELL formula A, the set of MELL proof-structures with conclusion of type A is never empty, while this might very well be the case if we restrict to cut-free MELL proof-structures (see Example 8.5 and Remark 8.8). The modularity of our proofs allows to straightforwardly adapt the criterion to the cut-free case (Theorem 8. 3.2) and thus to state correctly (and easily solve) the type inhabitation problem for (cut-free) MELL proof-structures (Theorem 8.10).
Outline. Section 3 recalls some preliminary notions on graph theory. In Section 4 we define (MELL and DiLL 0 ) proof-structures and quasi-proof-structures with atomic axioms. Section 5 defines the notion of Taylor expansion. Section 6 introduces the rewriting rules on lists of lists of formulas and lifts them to MELL quasi-proof-structures via the functor qMELL. In Section 7 we lift the rewriting rules to DiLL 0 quasi-proof-structures via the functor PqDiLL 0 and we show our first main result: the Taylor expansion induces a natural transformation between the two functors. Section 8 proves two other main results: the solution of the inverse Taylor expansion problem (via a gluability criterion) and the solution of the type inhabitation problem in MELL. Section 9 proves that the inverse Taylor expansion problem is decidable in the finite case. In Section 10 we show how to adapt our method when axioms are not necessarily atomic. Section 11 concludes with some final remarks.

Preliminaries on graphs
Graphs with half-edges. There are many formalizations of the familiar notion of graph. Here we adopt the one due to [BM07]: 5 a graph is still a set of edges and a set of vertices, but edges are now split in halves, allowing some of them to be hanging. Splitting every edge in two has at least four features of particular interest to represent LL proof-structures: • two half-edges are connected by an involution, thus defining an edge linking two vertices (possibly the same vertex). The fixed points of this involution are "hanging" edges, linked to a vertex only on one endpoint: they are well suited for representing the conclusions of a proof-structure. In this way it is also easy to define some intuitive but formally tricky operations such as grafting/substituting a graph into/for another graph (see Example 3.4); • given any vertex v in a graph τ , it is natural to define the corolla of v, that is v itself with the half-edges linked to it; τ is the union of its corollas, glued together by the involution; • while studying proof-structures, it is often necessary to treat them both as directed and undirected graphs. With this definition of graph, an orientation, a labeling and a coloring, are structures on top of the structure of the undirected graph (see Definition 3.3); • this definition of graph allows a uniform syntax to represent both proof-structures and other structures of interest (e.g., the box-tree of a proof-structure). In this way, we avoid appealing to ad hoc conditions in the definitions of proof-structure and Taylor expansion, which are then more compact and rely only on notions from graph theory.
Given two graphs τ and τ , it is always possible to consider their disjoint union τ τ defined as the disjoint union of the underlying sets and functions.
A one-vertex graph with set of flags F and involution the identity function id F on F is called a corolla (the endpoint of each flag is the only vertex); it is usually denoted by * F . Given a graph τ = (F τ , V τ , ∂ τ , j τ ), a vertex v ∈ V τ defines a corolla . A graph τ can be seen as the disjoint union of the corollas of its vertices, and the involution gluing some flags (the ones that are not tails in τ ) in edges.
Definition 3.2 (graph morphism and isomorphism). Let τ, σ be two graphs. A graph morphism h : τ → σ from τ to σ is a couple of functions (h F : A graph morphism is empty (resp. an identity) if its components are empty (resp. identity) functions. A graph isomorphism is a graph morphism whose components are bijective.
Intuitively, a graph morphism preserves tails and edges. The category Graph has graphs as objects and graph morphisms as arrows: indeed, graph morphisms compose in a associative way (by composing their components) and identity graph morphisms are neutral for such a composition. Graph is a monoidal category, with disjoint union as a monoidal product.
Graphs with structure. Some structures can be put on top of a graph.
• A labeled graph (τ, τ ) with labels in L is a graph τ and a function τ : V τ → L.
• A colored graph (τ, c τ ) with colors in C is a graph τ and a function c τ : ). • An ordered graph (τ, < τ ) is a graph τ together with an order < τ on its flags. Different structures on a graph τ can combine modularly, for instance τ can be endowed with a labeling τ and an orientation o τ , so as to form a directed labeled graph (τ, o τ , τ ); and conversely, a directed labeled graph (τ, o τ , τ ) can be seen as a directed graph (τ, o τ ), forgetting the labeling τ . Roughly, a graph is labeled (resp. colored) when labels are associated with its vertices (resp. edges and "hanging" edges). In a directed graph, an input (resp. output) of a vertex v is a-half or hanging-edge incoming in (resp. outgoing from) v.
Graphs can be depicted in diagrammatic form, vertices are represented by small rounded boxes and flags by wires touching their endpoint. As a graph is just a disjoint union of corollas glued by the involution, we only need to depict corollas ( Figure 1, on the left) and place the two halves of an edge next to each other (Figure 1, on the right). In directed graphs, inputs of a corolla are depicted above the corolla, outputs below; arrows also show their orientation. The color of a flag f (if any) is written next to f . The label of a vertex v (if any) is written inside v. If ordered, flags of a corolla are depicted increasing from left to right. A dotted gray wire denote an arbitrary number (possibly 0) of flags (see Figures 8 and 9).
Each enrichment of the graph structure introduced in Definition 3.3 induces a notion of structure-preserving morphism (and isomorphism) that extends the notion of graph morphism (and graph isomorphism) seen in Definition 3.2, so as to form an associated category. Moreover, different kinds of structure preservation can be combined in a modular way. For instance, given two directed labeled graphs (τ, o τ , τ ) and (σ, o σ , σ ) both with labels in L, (in particular, they can be seen as two directed graphs (τ, o τ ) and (σ, o σ ), forgetting labels), a directed graph morphism h : this means that h F maps input (resp. output) flags of τ to input (resp. output) flags of σ. And a directed labeled isomorphism h : Trees and paths. An undirected path on a graph τ is a finite sequence of flags ϕ = (f 1 , . . . , f 2n ) for some n ∈ N such that, for all , otherwise it is the empty (undirected ) path, which is between any vertex and itself; the length of ϕ is n. Note that ϕ cannot cross any tail of τ . Two vertices in a graph are connected if there is an undirected path between them.
Let τ be a graph: τ is connected if any vertices v, v ∈ V τ are connected; a connected component of τ is a maximal (with respect to the inclusion of flags and vertices) connected subgraph of τ ; τ is acyclic (or a forest) if it has no cycles; τ is a tree if it is a connected forest. Note that a forest can be seen as a non-empty finite sequence of trees, each tree is a connected component; so, an empty forest can be seen as a non-empty finite sequence of empty trees.
A rooted tree τ is a non-empty directed tree such that each vertex has exactly one output. So, by finiteness, τ has exactly one output tail f : the endpoint of f is called the root of τ .
Remark 3.5. Let τ and τ be two rooted trees, and h : τ → τ be a directed graph morphism. As h F preserves tails and orientation, h V maps the root of τ to the root of τ . Rooted trees and directed graph morphisms over them form a category RoTree.
A directed path on a directed graph τ is an undirected path ϕ = (f 1 , . . . , f 2n ) for some n ∈ N where f 2i−1 is output and f 2i is input for all 1 i n. We say that ϕ is from ∂ τ (f 1 ) to ∂ τ (f 2n ) if n > 0; otherwise it is the empty (directed) path, from any vertex to itself.
The set of directed paths on a directed tree τ is finite. As such, we define the reflexivetransitive closure, or free category, τ of τ as the directed graph with same vertices and same tails as τ , and with an edge from v to v for any directed path from v to v in τ . The operator (·) lifts to a functor from the category RoTree to the category of directed graphs, so for every morphism f in RoTree, f is a directed graph morphism between directed graphs.
Pullback in the category of graphs. The category of graphs has all pullbacks, a fact that we use extensively. We recall here all the definitions and facts involved in that affirmation. Definition 3.6 (pullback). Let C be a category. Let X, Y , Z be objects of C, and f : X → Z and g : Y → Z be arrows of C. A pullback of f and g is the triple (P, ! X , ! Y ) where P is an object of C and ! X : P → X and ! Y : P → Y are arrows of C such that diagram (3.1) commutes and, for any (Q, h : Q → X, k : Q → Y ) making the same diagram commute, there is a unique arrow u : Q → P factorizing h and k, i.e. such that diagram (3.2) commutes.
A pullback (P, ! X , ! Y ) of f : X → Z and g : Y → Z is unique (up to unique isomorphism); P is usually denoted by X × Z Y (leaving f, g implicit) and depicted like in diagram (3.3).
Roughly, the pullback σ × τ ρ is obtained as a sort of "lax intersection" of the "similar" vertices of σ and ρ (they are "similar" when they are sent to the same vertex in τ , by g V and h V respectively), and keeping whatever flags that are in the "intersection". So, σ × τ ρ is the maximal graph that is "compatible" with both σ and ρ.
The pullback construction lifts to directed, labeled, colored and ordered graphs. Figure 2. Cells, with their labels and their typed inputs and outputs.

MELL proof-structures and quasi-proof-structures
We present here a purely graphical definition of MELL proof-structures, following the noninductive approach of [GPT19]: a MELL proof-structure R is essentially a labeled directed graph |R| together with some additional information to identify its boxes. Our definition is completely based on standard notions (recalled in Section 3) from graph theory: it is formal (with an eye towards complete computer formalization) but avoids ad hoc technicalities to identify boxes. Indeed, the inductive and ordered structure of the boxes of R is recovered by means of a tree A R and a graph morphism box R from |R| to A R which allows us to recognize the content and the border of all boxes in R. We use n-ary vertices of type ? collapsing weakening, dereliction and contraction (like in [DR95]). In this way, we get a canonical representation of MELL proof-structures with respect to operations like associativity and commutativity of contractions, or neutrality of weakening with respect to contraction. One of the benefits of our purely graphical definition of proof-structure is that notions involving proof-structures such as isomorphism (see below), Taylor expansion (see Section 5) and correctness graph (see [GPT19]) can be defined rigorously by means of standard (and not ad hoc) mathematical tools from graph theory, making them more "manageable". 4.1. MELL formulas and lists. Given a countably infinite set of propositional variables X, Y, Z, . . . , MELL formulas are defined by the following inductive grammar: Linear negation (·) ⊥ is defined via De Morgan laws 1 ⊥ = ⊥, (A ⊗ B) ⊥ = A ⊥`B⊥ and (!A) ⊥ = ?A ⊥ , so as to be involutive, i.e. A ⊥⊥ = A for any MELL formula A (we say that A and A ⊥ are dual ). Variables and their negations are atomic formulas; ⊗ and`(resp. ! and ?) are multiplicative (resp. exponential ) connectives; 1 and ⊥ are (multiplicative) units.
We call any finite sequence a list. The empty list is denoted by . A list (x 1 , . . . , x n ) with n > 0 and x i = for all 1 i n is denoted by n or . Note that a list is not empty.

4.2.
Proof-structures. We define here proof-structures corresponding to some subsystems and extensions of LL: MELL, DiLL and DiLL 0 . Full differential linear logic (DiLL) is an extension of MELL (with the same language as MELL) provided with both promotion rule (i.e., boxes) and co-structural rules for the !-modality (the duals of the structural rules handling the ?-modality): DiLL 0 and MELL are particular subsystems of DiLL, respectively the promotion-free one (i.e., without boxes) and the one without co-structural rules. We reuse the syntax of proof-structures given in [GPT19], based on the graph notions introduced in Section 3. Note that, unlike [PT09], we allow the presence of cuts (vertices of type cut), even though we are not interested in cut-elimination and its study is left to future work. In Figure 2 we depicted the corollas associated with all types of vertices.
A (DiLL) proof-structure is a triple R = (|R|, A, box) where: A (DiLL) proof-structure R = (|R|, A, box) is said to be: (1) MELL if its structured graph |R| has no vertices of type and • all vertices in |R| of type ! have exactly one input, 9 We deal with atomic axioms only to simplify the presentation in the next sections. The general case with non-atomic axioms (i.e. c(o1) and c(o2) are still dual but not necessarily atomic) is discussed in Section 10. 10 This implies that c(ij) = c(i k ) for all 1 j, k n. There are no conditions on the number n of inputs. 11 There are no conditions on the number of outputs o1, . . . , op (besides p > 0) or on their types. 12 Intuitively, A represents the tree-structure of the nested boxes of R, see Remark 4.2 below. 13 The structured graph |R| of R is more structured than a directed graph such as A : |R| is directed but also labeled, colored, ordered. When we talk of a morphism between two structured graphs where one of the two, say σ, is less structured than the other, say τ , we mean that τ must be only considered with the same structure as σ. Thus, in this case, box is a morphism from ( R , oR)-discarding R, cR, <R-to A . 14 It means that for any input flag f in A there is exactly one input f of some vertex of type ! in |R| such that boxF (f ) = f ; but, for any input f of some vertex of type ! in |R|, boxF (f ) need not be an input flag in A (by definition of directed graph morphism, boxF (f ) is necessarily an input flag in A ). In particular, if |R| has no vertices of type !, A is a root with its output and no inputs; and if R is the empty graph, box is the empty graph morphism from |R| to such a A. Intuitively, given a vertex v of type ! in |R|, a box is associated with (and only with) each input f of v such that boxF (f ) is an input flag in A and not only in A : f represents the principal door in the border of such a box, indeed by definition of directed graph morphism, boxV ((∂ R • j R )(f )) = boxV (∂ R (f )), and for all f ∈ F R , if f = f then boxF (f ) = boxF (f ). 15 Roughly, it says that the border of a box is made of inputs of vertices of type ! or ? in |R|.
• the bijection induced by box F is defined on the whole set v∈V R , (v)=! in |R| (v); 16 (2) DiLL 0 (or resource) if A consists only of the root with its output, and either |R| has no vertices of type or |R| is a daimon, i.e. |R| has only one vertex and it is of type ; (3) empty (which is both MELL and DiLL 0 ) if its graph R is empty; it is denoted by ε; (4) cut-free if its structured graph |R| has no vertices of type cut, otherwise it is with cuts.
Our MELL proof-structures correspond to usual MELL proof-structures (as in [dCT12]). Our DiLL 0 proof-structures correspond to usual DiLL 0 proof-structures (as in [ER06b]) except that we allow also (proof-structures that are) daimons. Daimons will be used in the Taylor expansion to deal with the content of a box taken 0 times (see Section 5). Our interest for DiLL proof-structures is just to have a unitary syntax subsuming both MELL and DiLL 0 without considering cut-elimination: for this reason, unlike [Pag09,Tra09,Tra11], our DiLL proof-structures are not allowed to contain a sum of DiLL proof-structure inside a box. 17 Given a proof-structure R = (|R|, A, box), the output tails of |R| are the conclusions of R. The type of R is the list of the types of these conclusions, ordered according to < |R| .
Borrowing the terminology from [ER06b,Laf90], in proof-structures, we speak of cells instead of vertices, and of -cell for a cell of type . An hypothesis is a cell without inputs. Roughly, any non-root vertex v in A induces a subgraph of A made up of all vertices "above v" with their inputs and outputs: the preimage of this subgraph through box is the box of v in |R|. The preimage of the root of A through box is the part of |R| outside any box.
More precisely, with every flag f of |R| such that box F (f ) is an input flag of A 18 is associated a box B f , that is the subgraph of |R| (which is actually a proof-structure) made up of all the cells v (with their inputs and outputs) such that there is a directed path on A from is not in B f . The tree-structure of A expresses the usual nesting condition of boxes: two boxes in |R| are either disjoint or contained one in the other.
The depth of a cell v of R is the length of the directed path in A from box V (v) to the root of A. The depth of R is the maximal depth of the cells of R. Figure 3 a MELL proof-structure S = (|S|, A S , box S ) is depicted. The box-function box S is kept implicit by means of colors: the colored areas in |S| represent boxes (the preimages of non-root vertices of A S through box S ), and the same color is used on A S to show where each box is mapped by box S . 16 It means that for any input flag f in A there is exactly one vertex of type ! in |R| whose unique input f is such that boxF (f ) = f ; and, if f is the (only) input some vertex of type ! in |R|, boxF (f ) is an input flag in A. Intuitively, a box is associated with (and only with) the unique input of each vertex of type ! in |R|.
17 This restriction is without loss of generality, because a box containing a sum of proof-structures is equivalent (see [Tra09]) to a co-contraction of the boxes of each proof-structure in the sum. Also, the vertex of type corresponds to the empty sum inside a box: roughly, it is the content of an "empty box". 18 By the constraints on box, this condition can be fulfilled only by inputs of !-cells in |R|, and an input of a !-cell need not fulfill it; in particular, if R is a MELL proof-structure, then this condition is fulfilled by all and only the inputs of !-cells (and such an input is unique for any !-cell) in |R|; but if R is a DiLL0 proof-structure, this condition is not fulfilled by any flag in |R| (since A has no inputs) and so box is a directed graph morphism associating the root of A with any cell of R. Thus, in a DiLL0 proof-structure ρ = (|ρ|, Aρ, boxρ), there are no boxes, Aρ and boxρ do not induce any structure on |ρ|: ρ can be identified with |ρ|.
The box-tree A S (without dotted lines) and the reflexive-transitive closure A S of A S (with also dotted lines).
The proof-structures we have defined depend on the carrier sets for vertices and flags. It is natural to abstract away from them, and identify two proof-structures that differ only in the name of their vertices and flags, through a handy notion of isomorphism (i.e. structurepreserving bijection) inherited from the one of graph isomorphism defined in Section 3.
Note that if R is a MELL or DiLL 0 proof-structure isomorphic to a proof-structure R , then R is respectively MELL or DiLL 0 . Often we implicitly consider proof-structures up to isomorphism (as we do, for instance, in every figure representing proof-structures).

4.3.
Quasi-proof-structures. We need to consider more general structures in order to accommodate our rewrite rules, as discussed in Section 2, p. 5 (the rewrite rules are defined in Sections 6 and 7). This is why we extend all the definitions to tuples of proof-structures.
For convenience, given a proof- A conclusion of R is any conclusion of any component of R.
A quasi-proof-structure R is MELL (resp. DiLL 0 ) if all components of R are MELL (resp. DiLL 0 ) proof-structures. 19 The 0-ary tuple of proof-structures is not a quasi-proof-structure. Indeed, every quasi-proof-structure has at least one (possibly empty) component. Note that the components of a quasi-proof-structure are independent of each other, in particular they may differ in the number of their conclusions or in their type.
The box-forest F R (without dotted lines) and the reflexive-transitive closure F R of F R (with also dotted lines).
The short notation R = (|R|, F R , box R ) for quasi-proof-structures makes sense because the structured graph |R| = (|R 1 |, . . . , |R n |) can be seen as the disjoint union of its components |R 1 |, . . . , |R n |, and similarly for F R and box R . In particular, the box-forest F R is the disjoint union of the box-trees A 1 , . . . , A n of R 1 , . . . , R n . The box-function box R locates not only the boxes on |R|, but also the different components of R on |R| (see also Remark 4.7).
The only delicate point is the definition of the order < |R| for the conclusions of the quasiproof-structure R = (R 1 , . . . , R n ), given the order We often identify the conclusions of R and their order < |R| with a finite initial segment of N \ {0} and its natural order. Note that ε has no conclusions, akin to ε, as ε is empty.
The type of a quasi-proof-structure R = (R 1 , . . . , R n ) is a list Γ = (Γ 1 , . . . , Γ n ) of lists of MELL formulas such that Γ i is the type of R i for all 1 i n. When n = 1, we often identify R and R 1 (a quasi-proof-structure with only one component and a proof-structure), or Γ and Γ 1 (the type of a quasi-proof-structure with only one component and the type of a proofstructure). So, the type of a quasi-proof-structure R determines if R is a proof-structure.
The type of an empty quasi-proof-structure ε is a non-empty list of empty lists of MELL formulas, where each occurrence of the empty list of MELL formulas is the type of a component ε of ε. More precisely, is the type of ε, and n is the type of nε for every n > 0.  4.7 (components). A difference arises between a DiLL 0 proof-structure and a DiLL 0 quasi-proof-structure. Both have no boxes. In the former, its box-tree and box-function do not induce any structure on its structured graph, so we can identify a DiLL 0 proof-structure with its structured graph. In the latter, its box-forest and box-function do induce a structure on its structured graph: indeed, its box-forest is made up only of roots with their output and its box-function separates the components of its structured graph, by mapping the vertices to the roots. So, we cannot identify a DiLL 0 quasi-proof-structure with its structured graph.

The Taylor expansion
The Taylor expansion T (R) [ER08] of a MELL (or more generally DiLL) proof-structure R is a possibly infinite set of DiLL 0 proof-structures: roughly, each element of T (R) is obtained from R by approximating-i.e. replacing-each box B in R with n B copies of its content (for some n B ∈ N), recursively on the depth of R. Note that n B depends not only on B but also on which "copy" of all boxes containing B we are considering. Usually, the Taylor expansion T (R) of a MELL proof-structure R is defined globally and inductively [MP07,PT09,CV18]: with R is directly associated the whole set T (R) by induction on the depth of R. A drawback of this approach is that, for each element of T (R), the way the copies of the content of a box are merged is defined "by hand", which is syntactically heavy.
Following [GPT19], we adopt an alternative non-inductive approach, which strongly refines [GPT16]: the Taylor expansion is defined pointwise (see Example 5.2 and Figure 5). Indeed, proof-structures have a tree structure made explicit by their box-function. The definition of the Taylor expansion uses this tree structure: first, we define how to "expand " a tree via the notion of thick subtree [Bou09] (Definition 5.1; roughly, it states the number of copies of each box to be taken, recursively), we then take all these expansions of the box-tree of a proof-structure and we pull them back to the underlying graphs (Definition 5.4), finally we forget the tree structures associated with them (Definition 5.5). Thus, pullbacks give an abstract and elegant way to define the merging of copies of the content of a box in an element of T (R). The use of pullbacks is made possible because all the ingredients in our definition of a proof-structure live in the category of directed graphs (as defined in Section 3).
An advantage of our approach is that it smoothly generalizes to quasi-proof-structures.
Example 5.2. The following is a graphical presentation of a thick subforest (τ, h) of the box-forest F R of the quasi-proof-structure in Figure 4, where the directed graph morphism Intuitively, it means that τ is obtained from F R by taking 3 copies of the blue box, 1 copy of the red box and 4 copies of the orange box; in the first (resp. second; third) copy of the blue box, 1 copy (resp. 0 copies; 2 copies) of the purple box has been taken.
Remark 5.3 (roots). If (τ, h) is a thick subforest of a forest σ of rooted trees, h sets up a bijection between the roots of τ and σ, by definition of directed graph morphism; so, we can identify the roots of τ with the ones of σ. In particular, if σ is made of roots with their output and no inputs, the only thick subforest of σ is σ with its identity graph morphism.  Figure 5. An element ρ of the Taylor expansion of the MELL quasi-proofstructure R in Figure 4, obtained from the element of T pr (R) in Example 5.2.
The crucial point is to pull back the expansion of a forest to quasi-proof-structures.
, computed in the category of directed graphs and directed graph morphisms. 20 Given a quasi-proof-structure R and t = (τ t , h t ) ∈ T pr (R), the directed graph R t inherits the types of its vertices and flags by pre-composition of |R| and c |R| with the graph morphism p R : R t → |R|. The order on the flags of R t is induced by the one on |R| via p R .
Let [τ t ] be the forest made up of the roots of τ t and ι : τ t → [τ t ] be the graph morphism sending each vertex of τ t to the root below it; ι induces by post-composition a morphism is a DiLL 0 quasi-proof-structure, and it is a DiLL 0 proof-structure if R is a proof-structure. We can now define the Taylor expansion T (R) of a quasi-proof-structure R (an example of an element of a Taylor expansion is in Figure 5).
An element (R t , [τ t ], h t ) of the Taylor expansion of a quasi-proof-structure R has much less structure than the pullback (R t , p t , p R ): the latter indeed is a DiLL 0 quasi-proof-structure R t coming with its projections |R| which establish a precise correspondence between cells and flags of R t and cells and flags of R: a cell in R t is labeled (via the projections) by both the cell of |R| and the branch of the box-forest of R it arose from. But where R t is without its projections p t and p R loses the correspondence with R.
Remark 5.6 (conclusions). By definition and Remark 5.3, the Taylor expansion preserves conclusions and type: each element of the Taylor expansion of a quasi-proof-structure R has the same conclusions and the same type as R. More precisely, there is a bijection ϕ from the conclusions of a quasi-proof-structure R to the ones in each element ρ of T (R) such that 20 So, |R| is considered as directed graph (see Footnote 13), forgetting that it is colored, labeled, ordered.  Figure 6. An element of the filled Taylor expansion of the MELL quasiproof-structure R in Figure 4, obtained as an emptying of ρ in Figure 5.
i and ϕ(i) have the same type and the same root (i.e. box R F (i) = box ρ F (ϕ(i)) up to the bijection of Remark 5.3). So, the types of R and ρ are the same (as a list of lists).
Example 5.7 (Taylor of the empty). By Definitions 4.1 and 4.5, the box-tree of ε is just a root with its output and no inputs (we call it a root-tree), and the box-forest of nε is a list of n root-trees, for all n > 0; the box-function of ε is the empty graph morphism and the box-function of nε is a list of n such morphisms. By Remark 5.
where τ is a root-tree (resp. a forest of n root-trees) and h is the identity graph morphism on τ . By Definition 5.5, T (ε) = {ε} and T (nε) = {nε} for all n > 0.
5.1. The filled Taylor expansion. As discussed in Section 2 (p. 6), the rewriting rules we will introduce in Section 6 need to "represent" the emptiness introduced by the Taylor expansion (taking 0 copies of a box) so as to preserve the conclusions. Thus, an element of the filled Taylor expansion T (R) of a quasi-proof-structure R (an example is in Figure 6) is obtained from an element of T (R) where some components can be erased and replaced by daimons with the same conclusions (hence T (R) ⊆ T (R)). The filled Taylor expansion (and not the plain one) will play a crucial role in Section 7 to define a natural transformation.
Definition 5.8 (emptying, filled Taylor expansion). The emptying of a DiLL 0 proof-structure ρ with at least one conclusion is a daimon whose conclusions and type are the same as ρ.
An emptying of a DiLL 0 quasi-proof-structure ρ is a DiLL 0 quasi-proof-structure obtained from ρ by replacing some (possibly none) components of ρ having at least one conclusion with their emptying.
The filled Taylor expansion T (R) of a quasi-proof-structure R is the set of all the emptyings of all the elements of its Taylor expansion T (R).
, and r 1 , . . . , r n are some roots of [τ t ], the emptying of ρ on r 1 , . . . , r n , denoted by ρ r 1 ...rn , is the element of T (R) obtained by substituting a daimon for each component of ρ with at least one conclusion corresponding to the root r i (i.e., if such a conclusion is an output of a vertex v, then h tV (v) = r i ), for all 1 i n.
Remark 5.9 (conclusions of emptying). By construction, the filled Taylor expansion preserves conclusions and type, as the Taylor expansion does (Remark 5.6): conclusions and type of a quasi-proof-structure R and of any emptying of any element of T (R) are the same.
Remark 5.11 (connection). The Taylor expansion does not "create connection"; the filled Taylor expansion does, but only in a component that has been filled by a daimon. More precisely, let i and j be conclusions of a quasi-proof-structure R = (|R|, F R , box R ), let ρ = (|ρ|, F ρ , box ρ ) ∈ T (R) and let ρ r 1 ...rn ∈ T (R) be the emptying of ρ on the roots r 1 , . . . , r n . If Our aim is to deconstruct (MELL or DiLL 0 ) proof-structures from their conclusions. To do that, we introduce the category of schedulings. The arrows of this category are sequences of deconstructing rules, acting on lists of lists of MELL formulas. These arrows act through functors on (MELL or DiLL 0 ) quasi-proof-structures, exhibiting their sequential structure.
Definition 6.1 (the category Sched). Let Sched be the category of schedulings whose • objects are lists Γ = (Γ 1 ; · · · ; Γ n ) of lists of MELL formulas, with n 0; • arrows are freely generated by composition of the elementary schedulings in Figure 7; identities are the empty sequence of elementary schedulings, called empty scheduling. A scheduling is any arrow ν : Γ → Γ in Sched. We write the composition of schedulings by juxtaposition in the diagrammatic order; so, if ν : Γ → Γ and ν : Γ → Γ , then νν : Γ → Γ . If ν = a 1 · · · a k where k 0 and the a i 's are elementary schedulings, the length of ν is k. Example 6.2.`1`2`3 ⊗ 1 ⊗ 3 exc 1 exc 2 mix 2 ax 2 exc 2 mix 2 ax 2 ax 2 is a scheduling from (X ⊗ Y ⊥ )`((Y ⊗ Z ⊥ )`(X ⊥`Z )) to 3 , the list of 3 empty lists of MELL formulas.
The category Sched acts on MELL quasi-proof-structures, exhibiting a sequential structure in their construction, by means of the rewrite rules in Figure 8 (Definition 6.3). To ease their reading, we make explicit the name of the active conclusions (to be intended as positive   Except mix i , exc i and ? c i , the action of an elementary scheduling is a rewrite rule on MELL quasi-proof-structures that destroys either a cut-cell of depth 0 or a cell whose output is a conclusion. The types of the conclusions are updated according to Figure 7. The action of a scheduling is just the composition of a number (possibly none) of these rewrite rules.
The action of schedulings cannot decrease the number of components of a MELL quasiproof-structure, and the action of mix i is the only one that increases this number.
Among the actions of elementary schedulings in Figure 8, the cases mix and box are the only ones that change the box-forest of a quasi-proof-structure: mix splits a tree in two distinct trees by splitting its root, while box merges a root of a tree with a non-root vertex just above it. For instance, the action mix i rewrites the MELL proof-structure S in Figure 3 to the MELL quasi-proof-structure R in Figure 4, when i is the conclusion of S of type ?!1.
The way the action of an elementary scheduling is defined (Definition 6.3) automatically rules out the possibility that a MELL quasi-proof-structure rewrites to a module that is not a MELL quasi-proof-structure. For instance, the elementary scheduling ? d 1 cannot act on the MELL proof-structure Q below, because it would yield a module Q that is not a MELL quasi-proof-structure, as in Q it would be impossible to define a box-function that fulfills the constraints of Definition 4.1 for MELL (see Footnote 15); Q is just a DiLL 0 proof-structure.
Actions of schedulings can been seen as arrows in the category Rel of sets and relations.
Lemmas 6.5, 6.6, 6.7 and Proposition 6.8 express some properties of our rewrite rules.
Let R, R be MELL quasi-proof-structures, a be an elementary scheduling. We say that: • a applies to R if R exc i 1 ···exc in a R for some elementary schedulings exc i 1 , . . . , exc in (n 0); • R a R is nullary ? c -splitting if a = ? c i and its action from R to R splits the ?-cell of R with output i and h 0 inputs into two ?-cells, one with h inputs and one with 0 inputs.
Lemma 6.6 (applicability of elementary scheduling). Let R be a non-empty MELL quasiproof-structure. Then an elementary scheduling a ∈ {mix i , ax i , No elementary scheduling applies to an empty quasi-proof-structure.
To deconstruct a MELL quasi-proof-structure R, nullary ? c -splitting actions are problematic because they increase any reasonable size for R. Luckily, the lemma below says roughly that, in MELL, nullary ? c -splitting actions are superfluous as means of destruction. Remark 9.6 will show that they are needed to achieve the naturality result (Theorem 7.5).
Lemma 6.7 (splitting). Let R be a MELL quasi-proof-structure with at least a conclusion i. If a = ? c i and R a R is nullary ? c -splitting, then for some MELL quasi-proof-structure S: (1) either i is the output of a ?-cell of R with k 2 inputs and R a S is not nullary ? c -splitting; (2) or i is the output of a ?-cell of R with 0/1 inputs and in the same component as i there are conclusions i 0 , . . . , i n with n 0 (possibly i = i j or i j = i j for some 0 j, j n; i 0 may also be an input of a cut-cell of depth 0) such that R exc i 1 ···exc in S a S with a ∈ {mix i 0 , Lemmas 6.5, 6.6 and 6.7 are proven by simple inspection of the rewrite rules in Figure 8. The proof of Lemma 6.5 has a subtle case: given R ∈ qMELL(Γ ), if ν = ? c i there is a (unique) R ∈ qMELL(Γ) such that R ν R because any conclusion of R of type ?A is the output of a ?-cell (thanks to atomic axioms, Footnote 9; for non-atomic axioms, see Section 10).
Given a scheduling ν, we say R ν R has nullary ? c -splittings if there is an elementary scheduling a such that R ν R = R ν 1 R 1 a R 2 ν 2 R and R 1 a R 2 is nullary ? c -splitting. Every MELL quasi-proof-structure can be unwound from its conclusions to an empty one by the action of some scheduling, and nullary ? c -splittings are not needed for that.
Proof. Let the size of a MELL quasi-proof-structure R be the triple (p, q, r) where: Multisets are well-ordered as usual, triples are well-ordered lexicographically.
By Lemmas 6.6 and 6.7, it suffices to show that such a size is left unchanged by exc i and decreases with any action of ? c i that is not nullary ? c -splitting (p decreases) and with any action in Figure 8 other than the ones of exc i and ? c i (the action of mix i leaves p and q unchanged and decreases r; the remaining actions leave p unchanged and decrease q).
Proposition 6.8 and the last statement in Lemma 6.6 imply that actions of elementary schedulings define a weakly normalizing rewrite system on MELL quasi-proof-structures: the normal forms are exactly the empty quasi-proof-structures nε, for all n > 0. This rewrite system is not strongly normalizing because of the actions of exc i and, more significantly, ? c i when are nullary ? c -splitting. An example of an indefinitely extendable rewriting is below. The action of a scheduling ν such that R ν ε, read backward, can be seen as a sequence of elementary steps to build the MELL quasi-proof-structure R from an empty one ε. An example of a rewriting from some R to ε is in Figure 11 (Section 8). Intuitively, a scheduling ν : (Γ) → "encodes" a way to build a MELL proof-structure of type Γ from scratch.

Naturality of unwinding DiLL 0 quasi-proof-structures
We show here how the actions of schedulings act on DiLL 0 quasi-proof-structures, mimicking the behavior of the actions of schedulings on MELL quasi-proof-structures seen in Section 6. Schedulings are the same, the novelty is that DiLL 0 quasi-proof-structures have no boxes and might have daimons. The bridge between the MELL and DiLL 0 frameworks is given by the filled Taylor expansion, which actually defines a natural transformation (Theorem 7.5).
For Γ a list of lists of MELL formulas, qDiLL 0 (Γ) is the set of DiLL 0 quasi-proof-structures of type Γ (we assume they have the same conclusions). For any set X, its powerset is P(X).
Roughly, any rewrite rule in Figure 9-except Figures 9i and 9j-mimics the behavior of the corresponding rule in Figure 8 using a -cell. Since every elementary scheduling has at least two versions of its action, one in Figure 8 and one (or two for box i ) in Figure 9, when Π a Π , some elements of Π may rewrite according to one version of the action of a, others elements according to another version. In any case, all the elements of Π have the same conclusions, as the elements of Π do. Types are updated according to Figure 7. 23 More precisely, if a ⊆ qDiLL0(Γ) × P(qDiLL0(Γ )), in the rewrite rule a in Figure 8 the quasi-proofstructure on the right-hand-side must be replaced by the singleton set containing it.
Lists Γ k and Γ k may be empty, except in Figure 9d since a -cell has at least an output. The action in Figure 9i acts on a component made of one co-weakening (a !-cell without inputs) and some weakenings (?-cells without inputs), it erases them and creates a -cell, whose outputs are all the conclusions of the component. If we just erased the co-weakening with output of type !A, we would lose the information about the type A; this is one of the reasons why a -cell is needed. Intuitively, a co-weakening represents a box taken 0 times, so there is no information about the content of the box, apart from its type. The daimon is a "universal" DiLL 0 proof-structure representing this information plainly missing.
The action in Figure 9j separates the copies of a candidate for a box. It requires that, in the graph of the component on the left of box i , ρ j is not connected to ρ j for j = j , except for the !-cell and the ?-cells in the conclusions. Read in reverse, the rule associates with a non-empty finite set of DiLL 0 quasi-proof-structures {ρ 1 , . . . , ρ m } the merging of ρ 1 , . . . , ρ m , that is the DiLL 0 quasi-proof-structure depicted on the left of box i . Intuitively, ρ 1 , . . . , ρ m represent m > 0 possible copies of a box, whose "compatibility" must be analyzed "in parallel": this is why the action rewrites a single DiLL 0 quasi-proof-structure to a (finite) set of them.  Figure 10. Examples of actions of schedulings for DiLL 0 .
Note that a -cell can be created only by the rewrite rule in Figure 9i, and erased only by the action of elementary scheduling called hypothesis in Figure 9c: Example 7.2. Consider the DiLL 0 proof-structures below (X is a propositional variable).
Consider the DiLL 0 proof-structures π and π below.
and therefore, for every n > 1, the scheduling ν 1 n acts on {π} in the way below The rewriting on DiLL 0 quasi-proof-structures behaves differently than on MELL quasiproof-structures. In DiLL 0 , Lemma 6.6 and Proposition 6.8 are false. Indeed, in Figure 10 no elementary scheduling applies to the singleton Π of a non-empty DiLL 0 proof-structure.
We can now compare the functors qMELL and PqDiLL 0 from Sched to Rel. In other words, diagram (7.1) below commutes for every scheduling ν : Γ → Γ .
That is, given Π ν Π with Π ⊆ T (R ), we can simulate backward the rewriting to R (here the co-functionality of actions on qMELL expressed by Lemma 6.5 comes in handy) so that R ν R and Π ⊆ T (R) (square (7.2)); and conversely, given R ν R , we can simulate forward the rewriting for any Π ⊆ T (R), so that Π ν Π for some Π ⊆ T (R ) (square (7.3)). In both squares, the same kind of action is performed on DiLL 0 and on MELL.
Proof. T is a family of arrows of Rel indexed by the objects in Sched. We have to show that squares (7.2)-(7.3) commute for any scheduling ν : Γ → Γ , and actually, it is enough to show commutation for elementary schedulings. Let a : Γ → Γ be an elementary scheduling.
Let (t, h) be a thick subforest of F, let r 1 , . . . , r n be some roots in F (and in t, by Remark 5.3), and let ρ r 1 ...rn ∈ Π be the element of the filled Taylor expansion T (R) of R associated with (t, h) and r 1 , . . . , r n (i.e. ρ r 1 ...rn is the emptying on r 1 , . . . , r n of the element of T (R) obtained via the thick subforest (t, h) of F, see Definition 5.8). By Remark 5.9, we can identify the conclusions of R and of ρ r 1 ...rn . The case a = exc i is trivial since it just swaps the order of two consecutive conclusions. Other cases for a: • If a = mix i , let r be the root in F corresponding to the conclusion i (i.e. box F (i) is the output of r) and let p, . . . , p + k be the conclusions of the component S of R containing i. In the undirected graph S , none of the conclusions p, . . . , i is connected to any conclusion i + 1, . . . , p + k. By Definitions 5.5 and 5.8, as ρ r 1 ...rn ∈ Π ⊆ T (R), r is a root in the box-forest of ρ r 1 ...rn , and p, . . . , p + k are the conclusions of ρ r 1 ...rn corresponding to r. There are two cases, according to Remark 5.11: the connected components of i and i+1 are disjoint in the undirected graph ρ r 1 ...rn ; i and i+1 belong to the same connected component of ρ r 1 ...rn , and their component in ρ r 1 ...rn is a daimon with conclusions p, . . . , i, i+1, . . . , p+k. In both cases the rule mix i is also applicable to ρ r 1 ...rn , yielding a DiLL 0 quasi-proofstructure ρ . The box-forest F of R is obtained from the box-forest F of R by root-splitting 25 the tree with root r in two trees with roots r 1 , r 2 . Let t be a forest obtained from t in a similar way. Let h : t → F be the directed graph morphism let k be such that the rule a acts on the conclusions i − k, . . . , i of a component of R, and let be the type of the cell in R whose output is the conclusion i (or whose outputs are the conclusions i − 1 and i, if = ax). In ρ r 1 ...rn either there is a -cell with = ax whose output is the conclusion i, or there is a ax-cell whose outputs are the conclusions i − 1 and i, or the component containing the conclusion i is a daimon with conclusions i − k, . . . , i. Clearly a is applicable to ρ r 1 ...rn , which yields a DiLL 0 quasi-proof-structure ρ . The box-forest F of R is F, the one of R. We verify that ρ is the element of the filled Taylor expansion T (R ) of R associated with the thick subforest (t, h) of F . • If a = cut i , let c be the cut-cell of depth 0 in R to which a is applied. As in the case above, the cut-cell c has either one image in ρ r 1 ...rn or is represented by a -cell (with at least one conclusion). In both cases, cut i is applicable to ρ r 1 ...rn , yielding ρ . The box-forest F of R is F, the one of R. We verify that ρ is the element of the filled Taylor expansion T (R ) of R associated with the thick subforest (t, h) of F . • If a = box i , consider the component of R whose conclusions are i−k, . . . , i with k 0.
In the component π of ρ r 1 ...rn with conclusions i−k, . . . , i, there are three cases: (a) π is a daimon whose conclusions are i−k, . . . , i; so, the rule in Figure 9h applies; (b) π consists of a !-cell with output tail i and no inputs; and of k ?-cells without inputs; therefore, the rewrite rule in Figure 9i applies; (c) π has a !-cell c with output tail i and at least an input, k ?-cells c 1 , . . . , c k immediately above each of the other k conclusions, and other cells that can be split in m pairwise disjoint sub-graphs of π , where m 1 and π is π without c, c 1 , . . . , c k and their output; therefore, the rewrite rule in Figure 9j applies. In any case, box i applies to ρ r 1 ...rn , yielding a family ρ 1 , . . . , ρ m of DiLL 0 proofstructures, where m = 1 in cases (a) and (b), while m 1 in case (c). The box-forest F of R is obtained from the box-forest F of R by erasing the root b corresponding to the conclusions i−k, . . . , i (i.e., box F (i−k) = · · · = box F (i) is the output of b): the new root b of this tree of F is the unique vertex immediately above b in F. We have h −1 (b ) = {b 1 , . . . , b m }, and m trees t 1 , . . . , t m , where b j is the root of t j . The directed graph morphisms h j : t j → F are defined accordingly. We verify that ρ j is the element of T (R ) associated with the thick subforest (t j , h j ) of F . F , box ). Let Π be a witness of composition, i.e. a set in PqDiLL 0 (Γ ) such that Π a Π and (Π , R ) ∈ T Γ .
We want to exhibit a MELL quasi-proof-structure R such that R a R and Π is a part of the filled Taylor expansion T (R) of R. By co-functionality of qMELL(a) (Lemma 6.5), we have a candidate for such an R: the preimage of R by this co-functional relation a . In other terms, if a : Γ → Γ and R ∈ qMELL(Γ ), then there exists a unique R = (|R|, F, box) ∈ qMELL(Γ) such that R a R (Lemma 6.5). We only have to check that Π ⊆ T (R). If Π = ∅ then trivially Π ⊆ T (R). So, we suppose that Π = ∅. 25 It means that F = F except that in F the root r with its output is replaced by the roots r 1 and r 2 with their output, and that the inputs of r are split into the inputs of r 1 and the inputs of r 2 . Let ρ ∈ Π and {ρ 1 , . . . , ρ m } ⊆ Π such that ρ a {ρ 1 , . . . , ρ m }. In all cases except a = box i , we have m = 1 i.e. {ρ 1 , . . . , ρ m } = {ρ 1 }. For a = box i , let (t , h ) be a thick subforest of F , let r 1 , . . . , r k be some roots in F (and in t , by Remark 5.3), and let ρ 1 = ρ r 1 ...r k ∈ Π be the element of the filled Taylor expansion T (R ) of R associated with (t , h ) and r 1 , . . . , r k (i.e. ρ r 1 ...r k is the emptying on r 1 , . . . , r k of the element of T (R ) obtained via the thick subforest (t , h ) of F , see Definition 5.8). By Remark 5.9, we can identify the conclusions of R and of ρ r 1 ...r k . The case a = exc i is trivial since it just swaps the order of two consecutive conclusions. Other cases for a: ρ is the element of T (R) associated with r 1 , . . . , r k and the thick subforest (t, h) of F. • If a = mix i , let s 1 , s 2 be the roots in F (and in t , by Remark 5.3) corresponding to the is the output of s). We define the thick subforest (t, h) of F: t is the forest obtained from t by merging its roots s 1 and s 2 into the root s, and h : t → F is such that We verify that ρ is the element of T (R) associated with r 1 , . . . , r k and the thick subforest (t, h) of F. • If a = box i , we describe the rule for non-empty box (Figure 9j); the easier rules for daimoned and empty boxes (Figures 9h and 9i) are left to the reader. Let (t 1 , h 1 ), . . . , (t m , h m ) be thick subforests of F , let r 1 , . . . , r k be some roots in F (and in t 1 , . . . , t m , by Remark 5.3). For all 1 j m, let ρ j be the element of T (R ) associated with (t j , h j ) and r 1 , . . . , r k (i.e. ρ j is the emptying on r 1 , . . . , r k of the element of T (R ) obtained via the thick subforest (t j , h j ) of F , see Definition 5.8). The forests t j 's differ from each other only in the rooted tree corresponding to the conclusion i of the ρ j 's. Let t be the forest made of all the rooted trees on which the forests t 1 , . . . , t m do not differ, and of the union of the rooted trees on which the forests differ, joined by a new root; define h from h 1 , . . . , h m accordingly. We verify that ρ is the element of T (R) associated with r 1 , . . . , r k and the thick subforest (t, h) of F.
The rewriting system, the filled Taylor expansion (which generalizes the usual Taylor expansion) and the notion of quasi-proof-structure (which generalizes usual proof-structures) are designed to obtain the naturality result (Theorem 7.5), as shown in the example below.
Example 7.6 (the need for quasi). Extending proof-structures to quasi-proof-structures, with dashed boxes at the root level to separate their components, is crucial to achieve naturality (Theorem 7.5) when dealing with a box in a MELL proof-structure R copied zero times in an element of its Taylor expansion. Take for instance ρ 1 , ρ 2 and R below. Figure 8 to separate their components-we would have R box 2 R . To make square (7.3) commute (as required by naturality), we observe the following facts: (1) the action of box 2 on ρ 2 cannot simply erase the !-cell, otherwise we would lose the match with the type of R ; so, the action of box 2 on ρ 2 must create a -cell of type 1; (2) thus, when {ρ 1 , ρ 2 } box 2 Π , the relation between Π and R is not Π ⊆ T (R ), because of the -cell; this is why we introduced the filled Taylor expansion, so that Π ⊆ T (R ); (3) the two 1-cells in R must be handled in two different ways; indeed, the second one comes from the content of the box in R, and can be erased by the Taylor expansion of R by taking 0 copies of the box (as done in ρ 2 ), while the first 1-cell cannot; to mark this difference, and to make explicit the scope of the -cell introduced by the action of box 2 on ρ 2 , we split the two conclusions of R into two distinct components via R mix 1 R 0 .
According to our rewrite rules, R mix 1 R 0 box 2 R 0 and this rewriting commutes with the filled Taylor expansion on quasi-proof-structures, as required by square (7.3) for naturality.

Gluability of DiLL 0 quasi-proof-structures and inhabitation
The properties of actions of schedulings allow us to prove two of our main original results: (1) a characterization of the sets of DiLL 0 proof-structures that are in the Taylor expansion of some MELL proof-structure (inverse Taylor expansion problem, Theorem 8.3); (2) a characterization of the lists of MELL formulas inhabited by some cut-free MELL proofstructure (type inhabitation problem for cut-free MELL proof-structures, Theorem 8.10). Both characterizations are constructive: when we claim the existence of an object (a MELL proof-structure with a certain property), we can actually construct a witness of it, through our rewrite rules. Thus, our characterizations can be seen as nondeterministic procedures, because the rewritings to build witnesses are not unique. Our procedures also decide the problems above in special cases of interest (Theorem 9.9, Proposition 8.12). 8.1. Gluability. We introduce the gluability criterion to solve the inverse Taylor expansion problem. We actually solve it in two different domains. We look for a solution: (1) in the set of MELL proof-structures (Theorem 8.3.1); (2) in the set of cut-free MELL proof-structures (Theorem 8.3.2).
We say that a scheduling is cut-free if does not contain the elementary scheduling cut i .
Lemma 8.1 (cut-free scheduling). Let R be a MELL quasi-proof-structure and ν be a scheduling such that R ν ε. We have that R is cut-free if and only if ν is cut-free.
Proof. Let R be MELL quasi-proof-structure. If R is cut-free and R a R , then a = cut i because the action R cut i R requires the presence of a cut-cell in R; moreover, R is cut-free because no rewrite rule in Figure 8-when read left to right-introduces a cut-cell.
Conversely, if R is with cuts and R a R where R is cut-free, then a = cut i , because among the rewrite rules in Figure 8 only cut i erases a cut-cell.
A straightforward induction on the length of the scheduling ν allows us to conclude.
(2) Π is cut-free gluable if and only if Π ⊆ T (R) for some cut-free MELL proof-structure R.
(2) Just repeat the same proof as above, paying attention that: • in the "if" part, if R is cut-free, then the scheduling ν is cut-free by Lemma 8.1; • in the "only if" part, if Π ν { ε} for some cut-free scheduling ν : (Γ) → , then by Lemma 8.1 R is cut-free, since R ν ε according to naturality (Theorem 7.5).
Theorem 8.3 means that the action of a scheduling rewriting a set Π of DiLL 0 proofstructures to the empty proof-structure can be seen, read in reverse, as a sequence of elementary steps to build a MELL proof structure that includes Π in its Taylor expansion.
Example 8.4. An example of the action of a scheduling starting from the singleton of a DiLL 0 proof-structure ρ and ending in { ε} is in Figure 11 (it is by no means the shortest possible rewriting). When replayed backward, by naturality (Theorem 7.5) it induces a MELL proof-structure R such that ρ ∈ T (R). In Figure 11 we represent simultaneously the rewriting from Π to { ε} and the rewriting from R to ε.
We introduced two different gluability criteria (Theorems 8.3.1 and 8.3.2) because, given a non-empty set Π of cut-free DiLL 0 proof-structures without -cells, the existence of a solution to the inverse Taylor expansion problem depends on whether we look for it among MELL proof-structures or among cut-free MELL proof-structures (Example 8.5).
Example 8.5. Let Σ = {σ 0 } as in Figure 12. This singleton of a cut-free DiLL 0 proofstructure without -cells is gluable (as shown by the action of the scheduling in Figure 12) but not cut-free gluable (as {σ 0 } box 1 {σ} and, apart from cut i , no elementary schedulings apply to {σ} because of its type). Indeed, σ 0 "masks" any information about the content B of the box represented by its !-cell, apart from its type !X which does not allow B to be a cut-free MELL proof-structure (see also Section 8.2). Gluability (with cuts) of {σ} is related to the fact that there is a MELL proof-structure of type X with cuts, for every atomic formula X (see Remark 8.8). Note that σ 0 ∈ T (S 0 ), and σ ∈ T (S) and σ ∈ T (S ), where S 0 , S and S are the MELL proof-structures with cuts represented in Figure 12.
Example 8.6. The three DiLL 0 proof-structures ρ 1 , ρ 2 , ρ 3 below are not gluable as a whole, but are gluable two by two (this is a slight variant of the example in [Tas09,). In fact, there is no MELL proof-structure whose Taylor expansion contains ρ 1 , ρ 2 , ρ 3 , but any pair of them is in the Taylor expansion of some MELL proof-structure.
Indeed, for any i, j ∈ {1, 2, 3} with i = j, ρ i and ρ j are in the Taylor expansion of R ij below.
Example 8.6 can be generalized to finite sets of any cardinality: for any n > 1, there is a finite set Π of n + 1 DiLL 0 proof-structures that is not gluable as a whole, but all subsets of Π of at most n elements are gluable. This means that, for every n > 1, gluability of a set Π of DiLL 0 proof-structures is not reducible to test that all subsets of Π with at most n elements satisfy some n-ary coherence relation, unless (obviously) Π is a finite set with at most n elements. As explained in Section 1, this difficulty in the inverse Taylor expansion problem is typical of MELL and does not arise in the λ-calculus.
Remark 8.7 (infinite sets). The gluability criterion (Theorem 8.3) is not limited to finite sets of DiLL 0 proof-structures, it holds for infinite sets too. This raises the question of the relationship between gluability and finite gluability, namely: when every finite subset of a given infinite set of DiLL 0 proof-structures is gluable, is the infinite set itself gluable too? The answer is negative. Let Π be an infinite set of DiLL 0 proof-structures such that every finite subset of Π is gluable. There are two cases for Π. • Π itself is gluable. It is typically the case when Π is exactly the Taylor expansion of a MELL proof-structure R with at least one box, such as in the following example. We might say that Π is infinite in width: infinity is related to the number of copies chosen for the boxes of R, a number that has to be finite but can be arbitrarily large.
• Π is not gluable. This is the case in the following set (we draw three elements only).
Every finite subset of Π is gluable, but the whole infinite set Π is not. We might say that Π is infinite in depth: in Π, for every n > 0 there is a DiLL 0 proof-structure with n !-cells. Any finite subset Π n of Π (n is the maximal number of !-cells in the elements of Π n ) is included in the Taylor expansion of a MELL proof-structure with depth n, and indeed the action of the scheduling ν 1 n defined in Example 7.3 rewrites Π n to { ε}. 26 However, the whole Π can only been seen as the Taylor expansion of an "infinite" proof-structure with infinitely deep nested boxes. Such infinite proof-structures are not in the syntax of MELL. An ongoing research relates them to non-well-founded proofs for fixed-point logics [DS19,DPS21].
In the inverse Taylor expansion problem, both kinds of infinity-in width and in deptharise already in the λ-calculus, they are not specific to MELL. An instance of a set of resource λ-terms that is infinite in width is the Taylor expansion of the λ-term xy. The set ], . . . } of resource λ-terms is an example of infinity in depth: every finite subset of Π is included in the Taylor expansion of some λ-term (a Church numeral), but Π is not included in the Taylor expansion of any λ-term.
8.2. Inhabitation. We can characterize type inhabitation in MELL proof-structures. We first put the question of type inhabitation for MELL proof-structures in a proper way.
Remark 8.8 (inhabitation with cuts is trivial). For every atomic formula X there is a MELL proof-structure of type X with cuts, the one called S in Figure 12. More generally, for every MELL formula A (resp. list Γ of MELL formulas), there is a MELL proof-structure of type A (resp. Γ) possibly with cuts. Indeed, for A just "η-expand" S in Figure 12, by replacing each ax-cell in S with the "canonical" MELL proof-structure of type A, A ⊥ . For Γ = (A 1 , . . . , A n ), repeat the same procedure for each A i separately, and juxtapose the MELL proof-structures to yield a unique MELL proof-structure R of type Γ (take R = ε if Γ = ).
Therefore, type inhabitation for MELL proof-structures is trivial, if cut-cells are allowed.
Type inhabitation for MELL proof-structures is a non-trivial problem only if we restrict to cut-free MELL proof-structures. Indeed, given two propositional variables X = Y , there is no cut-free MELL proof-structure of type X or Y, X. So, the problem is the following: Data: a list Γ of MELL formulas; Question: does there exist a cut-free MELL proof-structure of type Γ? 26 In fact, if π h is the element of such a Π with exactly h !-cells, {π h }  We study the type inhabitation problem for cut-free MELL proof-structures via schedulings, following the idea that a scheduling ν : (Γ) → "encodes" a MELL proof-structure of type Γ.
Let us see the behavior of cut-free schedulings. For some list Γ of MELL formulas there is no cut-free scheduling ν : (Γ) → . For instance, if X = Y are propositional variables and we exclude cut i , then there are no elementary schedulings from (X), while the only other elementary schedulings from (Y, X) are mix 1 and exc 1 . The non-existence of a cut-freescheduling from (X) or (Y, X) to goes hand in hand with the non-existence of a cut-free MELL proof-structure of type X or Y, X, by Proposition 6.8 and Lemma 8.1.
Conversely, for any list Γ of MELL formulas, a MELL proof-structure R(ν) of type Γ is associated with every scheduling ν : (Γ) → : such a R(ν) is built by reading ν backward, it exists and is uniquely determined because the action of ν is co-functional (Lemma 6.5).
Definition 8.9 (building from scheduling). Let Γ = be a list of lists of MELL formulas, and ν : Γ → n be a scheduling with n > 0. The MELL quasi-proof-structure R(ν) is defined by induction on the length of ν: if ν is empty, R(ν) = nε; otherwise ν = aµ for some elementary scheduling a, and R(ν) is the only MELL quasi-proof-structure R such that R a R(µ).
Clearly, for any scheduling ν : Γ → , R(ν) is of type Γ and R(ν) ν ε, and R(ν) is cut-free if and only if so is ν (Lemma 8.1). Actually, cut-free schedulings are a way to characterize the (lists of) MELL formulas that are the type of some cut-free MELL proof-structure.
Theorem 8.10 (type inhabitation for cut-free MELL proof-structures). Let Γ be a list of MELL formulas. There is a cut-free MELL proof-structure of type Γ if and only if there is a cut-free scheduling ν : (Γ) → .
Proof. If ν : (Γ) → is a cut-free scheduling, R = R(ν) is a MELL quasi-proof-structure of type Γ. As Γ is a list of MELL formulas, R is a proof-structure. By Lemma 8.1, R is cut-free.
Conversely, if R is a cut-free MELL proof-structure of type Γ, then R ν ε for some cut-free scheduling ν : (Γ) → , by Proposition 6.8 and Lemma 8.1.
Lemma 6.5, Definition 8.9 and the proof of Theorem 8.10 say that, given a list Γ of MELL formulas, any cut-free scheduling from (Γ) to can be seen, when read backward, as a sequence of elementary steps to build a cut-free MELL proof-structure of type Γ from scratch.
Consider now the procedure P 1 below, given a list Γ of MELL formulas as input: (1) for all n 0, take all cut-free schedulings from (Γ) of length n (there are finitely many); (2) as soon as a list of the form is reached by a cut-free scheduling, accept Γ.
Theorem 8.10 guarantees that such a nondeterministic procedure accepts Γ if and only if there is a cut-free MELL proof-structure of type Γ, and any cut-free scheduling ν that allows the procedure to accept Γ also constructs a witness R(ν). Therefore, we can conclude: otherwise, for a daimon ρ there is no ρ such that {ρ} cut i {ρ } (see Figure 9d).
Therefore, by Lemma 9.2, to prove that a set Π of DiLL 0 proof-structures is gluable, it suffices to show that Π rewrites to a -full set Π . A more subtle question is to prove that if Π cannot rewrite to a -full set Π , then Π is not gluable (Lemma 9.5 below). Definition 9.3 ( -/¬ -actions, canonicity). Let Π be a set of DiLL 0 quasi-proof-structures with the same conclusions. Suppose that Π a Π where a = cut i (resp. a = cut i ) is an elementary scheduling that acts on the conclusion i of all elements of Π (resp. Π ). If for every ρ ∈ Π (resp. ρ ∈ Π ) the conclusion i of ρ (resp. ρ ) is the output of a -cell, then we say that Π a Π is a -action. If Π ν Π and for every elementary scheduling a such that So, given a set Π 0 of DiLL 0 quasi-proof-structures and a -action Π 0 a 1 . . . an Π n , for all 1 j n the elementary scheduling a j acts on a conclusion of a daimon in all elements of Π j−1 . On the other hand, in a ¬ -action Π 0 a 1 . . . an Π n , for all 1 j n the elementary scheduling a j acts on the output of a -cell with = in some element of Π j−1 .
Remark 9.4 (fullness). Let Π be a set of DiLL 0 quasi-proof-structures, and ν be a scheduling. By straightforward induction on the length of ν, we can prove the facts below.
• If Π is -full and Π ν Π for some set Π , then Π ν Π is a -action and Π is -full.
The following lemma relies on the fact that -actions can always be delayed.
Proof. Immediate consequence of the following fact: if Π 1 , Π 2 , Π 3 are sets of DiLL 0 quasiproof-structures such that Π 1 a 1 Π 2 is a -action and Π 2 a 2 Π 3 is not a -action for some elementary schedulings a 1 , a 2 , then there are elementary schedulings c 1 , c 2 and a set Σ of DiLL 0 quasi-proof-structures such that Π 1 c 1 Σ is not a -action and Σ c 2 Π 3 is a -action.
Akin to MELL, nullary ? c -splitting actions in DiLL 0 are superfluous as means of destruction. They may be needed for some elements in a set Π of DiLL 0 quasi-proof-structures, but not for all of them, thus nullary ? c -splitting actions are useless for the set Π as a whole.
Proof. By Remark 9.4, it suffices to prove that for every canonically gluable set Π of DiLL 0 quasi-proof-structures, there is a scheduling ν such that Π ν Π is a ¬ -action without ? c -splittings, for some -full Π . Indeed, by Lemma 9.2, there is a scheduling µ such that Π µ { ε}, it is a -action by Remark 9.4 and is clearly without nullary ? c -splittings.
Given a ¬ -action Π ν Π where Π is -full, we prove by induction on ? c 0 (Π ν Π ) ∈ N that it is possible to build a ¬ -action Π ν Π without nullary ? c 0 -splittings for some -full Π . Clearly, it is enough to show that if ? c 0 (Π ν Π ) > 0 then we can find a -full Π and a where Π is -full, a = ? c k and Π 1 a Π 2 is nullary ? c -splitting. Since the ?-cells with 0 inputs of Π 2 created by Π 1 a Π 2 must disappear in Π (which contains only -cells), there are only two cases for Π 2 ν 2 Π : ? wi Π 4 erases the ?-cells without inputs and their unique output (a conclusion of each element of Π 3 ) created by Π 1 a Π 2 ; (2) or Π 2 into a -cell, according to the rewrite rule in Figure 9i applied to all the elements of Π 3 .
In Case 1, just erase the actions a and µ 21 Π 4 is (essentially) the same as Π 2 ν 21 Π 3 except that the name of the conclusions on which the elementary actions apply have changed (since the action Π 1 a Π 2 is no more present). Clearly, Π ν Π is a ¬ -action and ? c 0 (Π ν Π ) > ? c 0 (Π ν Π ).
In Case 2, erase the action a and slightly modify the action box i : indeed, let Π ν Π = • Π 1 µ 21 Π 3 is the same as Π 2 ν 21 Π 3 except that the name of the conclusions on which the elementary actions apply have changed (since the action Π 1 a Π 2 is no more present); • the set Π 3 is Π 3 with one conclusion and one ?-cell with 0 inputs less in each of its elements; • the action Π 3 box i−1 Π 4 applies the rewrite rule in Figure 9i to the conclusion i − 1 of every element in Π 3 , which corresponds to the conclusion i of every element in Π 3 ; • the -cell of each element in Π 4 created by Π 3 box i−1 Π 4 has one output less than the -cell of each element in Π 4 created by Π 3 box i Π 4 ; the action Π 4 µ 22 Π and the set Π are obtained from Π 4 ν 22 Π and Π accordingly (in particular Π is -full).
By Lemmas 9.5 and 9.7, a set Π of DiLL 0 quasi-proof-structures is gluable if and only if it is canonically gluable without nullary ? c -splittings. This restricts the search space for rewritings to decide if Π is gluable, making gluability a decidable problem in the finite case.
Proposition 9.8 (decidability of finite gluability). The problem below is decidable.
Proof. We first define some notions that will be used in the proof. Akin to the size of a MELL quasi-proof-structure used in the proof of Proposition 6.8, the size s(Π) of a finite set Π of DiLL 0 quasi-proof-structures is the quadruple (p, q, r, s) where: • p is the finite multiset of the number of inputs of each ?-cell with at least one input in each element of Π (hence, p is a finite multiset of positive integers); • q is the number of cells in Π different from ?-cells without inputs and from -cells; • r is the finite multiset [k 1 , . . . , k n ] where n is the number of conclusions in each element of Π and, for all 1 i n, k i is the number of ρ ∈ Π whose conclusion i is the output of a ?-cell with at least one input; • s is the finite multiset of the number of conclusions of each component of each element of Π.
Finite multisets are well-ordered as usual, quadruples are well-ordered lexicographically.
We say that a non--full set Π of DiLL 0 quasi-proof-structures safely rewrites to Π if Π ν Σ a Π for some set Σ, some scheduling ν only made of an arbitrary number (possibly 0) of exc i such that Π ν Σ is a ¬ -action, and some elementary scheduling a = exc i such that Σ a Π is neither a -action nor nullary ? c -splitting.
For every finite set Π of DiLL 0 quasi-proof-structures, exactly one of the following holds: (1) either Π is -full (this includes the case Π = { ε}) and then it is gluable by Lemma 9.2; (2) or Π is not -full, and every scheduling ν such that Π ν Π is a ¬ -action for some Π is only made of an arbitrary number -possibly 0 -of exc i (this includes the case when no elementary scheduling applies to Π, in particular when two elements of Π have different types); then, by definition, Π is gluable if Π = ∅, and it is not gluable otherwise; (3) or Π is not -full, and there are h > 0 sets Π 1 , . . . , Π h such that Π safely rewrites to Π j for all 1 j h, and if Π safely rewrites to Π then Π = Π j for some 1 j h; then, Π is canonically gluable without nullary ? c -splittings (which is equivalent to be gluable, by Lemmas 9.5 and 9.7) if and only if so is at least one among Π 1 , . . . , Π h .
Let us prove that when neither Case 1 nor Case 2 holds, we are in Case 3. If neither Case 1 nor Case 2 holds, Π is not -full and Π ν Σ a Π for some sets Σ, Π , some scheduling ν only made of exc i such that Π ν Σ is a ¬ -action, and some elementary scheduling a = exc i such that Σ a Π is not a -action. If a = ? c i and Σ a Π is nullary ? c -splitting, then i is the output of a ?-cell in every element of Σ, and so there is a set Π such that Σ c Π is neither a -action nor nullary ? c -splitting, with c ∈ {? c i , ? wi , ? d i } (if c = ? c i , Σ c Π just splits the inputs of the ?-cell with output i in some element of Σ in a way different from Σ a Π ). Therefore, Π safely rewrites to a finite set of DiLL 0 quasi-proof-structures. Clearly, it is impossible that Π safely rewrites to infinitely many sets, since each element of Π has a finite number of conclusions. Hence, Case 3 holds.
The nondeterministic procedure to decide if a finite set Π of DiLL 0 quasi-proof-structures is gluable consists of repeating the step described in Case 3, until all the sets Π 1 , . . . , Π n which Π rewrites to are of the form described in Case 1 or 2: Π is gluable if and only if so is at least one among Π 1 , . . . , Π n . The procedure terminates because (recall the size above): • if Π exc i Π then s(Π) = s(Π ) but infinite chains of exc i are irrelevant (see Footnote 27); • if Π safely rewrites to Π then s(Π) > s(Π ) (according to the lexicographic order).
The size used in the proof of Proposition 9.8 is well defined only if the set Π of DiLL 0 quasi-proof-structures is finite. The finiteness hypothesis in deciding gluability of Π is necessary, as we have seen in Remark 8.7: the infinite in depth Π shown there is not gluable and there are infinite rewritings from Π that are canonical and without nullary ? c -splittings.
Theorem 9.9 (decidability). The finite inverse Taylor expansion problem is decidable.
Proof. Let Π be a finite set of DiLL 0 proof-structures without -cells. By Proposition 9.8, we can decide if Π is gluable. By Theorem 8.3, since there is no -cell in Π, this is equivalent to deciding if Π ⊆ T (R) for some MELL proof-structure R.
The nondeterministic procedure to decide the finite inverse Taylor expansion problem is given by the proof of decidability of finite gluability (Proposition 9.8) and is constructive: if a finite set of DiLL 0 proof-structures without -cells is included in the Taylor expansion T (R) of some MELL proof-structure R, our procedure allows us to construct one of these R.

Non-atomic axioms
Our (quasi-)proof-structures (Definitions 4.1 and 4.5, Figure 2) deal with atomic axioms only: the two outputs of any ax-cell are dual atomic formulas. We can relax the definitions and allow also the presence of non-atomic axioms: the outputs of any ax-cell are typed by dual MELL formulas, not necessarily atomic. We can extend our results to this more general setting, with some technical complications. The only trouble is with exponential axioms, where the outputs of an ax-cell are typed by MELL formulas of the form !A ⊥ , ?A. Indeed, consider the daimons σ and σ and the MELL proof-structure S below.
We have that {σ} ? c 3 {σ } ⊆ T (S ). But no elementary scheduling ? c 3 can be applied backwards to S (because of the lack of a ?-cell), which breaks naturality (Theorem 7.5). This is actually due to the fact that, with exponential axioms, co-functionality (Lemma 6.5) fails for the rewrite rule ? c i : read from right to left, ? c i is a functional but not total relation.
The solution we propose here to deal with non-atomic axioms asks for a technical refinement of only few notions, the rest is unchanged and goes smoothly. The definitions of module, proof-structure and quasi-proof-structure (Definitions 4.1 and 4.5) change as follows.
(1) In modules, non-atomic axioms are allowed, except the exponential ones. We add a new type of cells, the !ax-cells, with no inputs and two outputs of type !A ⊥ and A. (2) In proof-structures (and hence in quasi-proof-structures), we require that, for every !ax-cell whose outputs have type !A ⊥ and A, the output o of type A is an input of a ?-cell (more precisely, o is an half of an edge whose other half is the input of a ?-cell). The requirement "atomic" in the elementary scheduling ax i in Figure 7 is replaced by "nonexponential", and we add the new elementary scheduling !ax i below to the list in Figure 7, (Γ 1 ; · · · ; Γ k−1 ; c(i), c(i+1); Γ k+1 ; · · · ; Γ n ) !axi − − → (Γ 1 ; · · · ; Γ k−1 ; ; Γ k+1 ; · · · ; Γ n ) if c(i) = ?A = c(i+1) ⊥ whose action-also in its daimoned version-is like the hypothesis in Figures 8c and 9c: The way the action of an elementary scheduling is defined (Definitions 6.3 and 7.1) automatically rules out the possibility that a quasi-proof-structure rewrites to a module that is not a quasi-proof-structure. For instance, the elementary scheduling ? d 1 does not apply to the (MELL and DiLL 0 ) proof-structure Q below, because it would yield a module Q that is not a quasi-proof-structure (see Item 2 above).
Apart from the changes we mentioned, all definitions and claims in Sections 4 to 9 remain unaffected and valid (just add !ax i to the list of elementary schedulings that can apply to R in Lemmas 6.6 and 6.7.2). The counterexample to naturality shown at the beginning of this section does not apply here, as S is not a MELL quasi-proof-structure (because of its exponential axioms). In our framework with non-atomic axioms, the only MELL proofstructures containing σ in their filled Taylor expansion are the ones below (up to the order of their conclusions), and the elementary scheduling ? c 3 applies backward to each of them.
The way we propose here to deal with non-atomic axioms is more elegant and less ad hoc than the one we presented in [GPT20, Section 7]. In the latter, we allowed the outputs of a -cell to be inputs of a ?-cell, and we changed the action of the elementary scheduling ? c i for DiLL 0 quasi-proof-structures in the daimoned case (Figure 9f), by explicitly requiring the presence of a ?-cells above the conclusion i. Consequently, in [GPT20] we had to redefine the filled Taylor expansion in the non-atomic case, and thus to recheck that naturality holds for all the rewrite rules. Here instead we do not have to redefine the filled Taylor expansion and we have to check naturality (Theorem 7.5) in the non-atomic case only for the action of !ax i .
As another possible solution to deal with non-atomic axioms, instead of using generalized ?-cells in Definition 4.1 (with an arbitrary number of inputs of type A and one output of type ?A, as in [DR95]), we could have used different kinds of ?-cells for dereliction (one input of type A, one output of type ?A), contraction (two inputs of type ?A, one output of type ?A), and weakening (no inputs, one output of type ?A), as in [Gir87]. Such a choice should be supported by a rework of the elegant definition of the Taylor expansion of a MELL proof-structure via the notion of pullback (Definitions 5.4 and 5.5), since it collapses dereliction, contraction and weakening in a generalized ?-cell.

Conclusions and perspectives
Daimons for empty boxes. Our gluability criterion (Theorem 8.3) solves the inverse Taylor expansion problem in an "asymmetric" way: we characterize the sets of DiLL 0 proof-structures without -cells that are included in the Taylor expansion of some MELL proof-structure, but in general DiLL 0 proof-structures might contain -cells (while MELL proof-structures cannot, see Definition 4.1). Daimons and emptyings are needed to get a natural transformation (via the filled Taylor expansion, Theorem 7.5), which is the main ingredient to prove our gluability criterion. But we are interested in frameworks without -cells. This asymmetry is Vol. 18:2 GLUING RESOURCE PROOF-STRUCTURES: INHABITATION AND INVERTING TAYLOR 4:43 technically inevitable, due to the fact that a gluable set of DiLL 0 proof-structures might not contain any information on the content of some box, when they take 0 copies of it.
Comparison with Pagani and Tasson [PT09]. Pagani and Tasson's solution [PT09] to the inverse Taylor expansion problem is less general than ours, because it characterizes finite sets of DiLL 0 proof-structures that are included in the Taylor expansion of some cut-free MELL proof-structure with atomic axioms. We do not have these limitations (finite, cut-free, atomic axioms), our characterization yields a decision procedure (Theorem 9.9) in the finite case as in [PT09], and we can smoothly adapt our criterion to the cut-free case (Theorem 8.3.2, which is not trivial as explained in Example 8.5). In [PT09], the restrictions to cut-free and to atomic axioms aim to simplify their presentation and might be overcome. But the restriction to finite sets of DiLL 0 proof-structures is somehow essential in their approach. Roughly, their machinery takes a finite set Π of DiLL 0 proof-structures as input, juxtaposes its elements in a unique graph π and then runs a rewriting by means of some tokens that go through the whole π and try to merge its components in a MELL proof-structure whose Taylor expansion includes Π, if any. If Π were an infinite set, π would be an infinite graph and, apart from the technical intricacies of dealing with infinite objects, it would be impossible to provide a characterization in that case. In particular, their approach could not distinguish whether Π is infinite in width (which can be in the Taylor expansion of some MELL proof-structure) or infinite in depth (which cannot, see Remark 8.7). Our approach, instead, defines a rewrite relation on a single-finite-element of Π and extends it to the whole (possibly infinite) Π by requiring that the same rewrite rule applies to each element of Π (Definition 7.1). Thus, we can accommodate the case where Π is infinite.
We believe that our rewriting rules are also simpler than the ones in [PT09] and rely on a more abstract and less ad hoc property (naturality), that allows us to prove also semi-decidability of another problem: type inhabitation for cut-free MELL proof-structures.
Finally, Pagani and Tasson's solution of the inverse Taylor expansion problem is affected by another limitation, even though not particularly emphasized in [PT09]: their Theorem 2 (analogous to the "only if" part of our Theorem 8.3) assumes not only that their rewriting starting from a set of DiLL 0 proof-structures terminates but also that it ends on a MELL proof-structure, according to their definition of MELL proof-structure. This is limiting when in the set of DiLL 0 proof-structures there is no information about the content of a box. For instance, consider the singletons Π and Π of DiLL 0 proof-structures below. Pagani and Tasson's rewrite rules do not distinguish the two singletons, each one is included in the Taylor expansion of cut-free MELL proof-structures with an "empty box", due to the lack of information. So their notion of MELL proof-structure is wider and non-standard (because it allows the presence of empty boxes). On the contrary, our rewrite rules distinguish Π and Π : via the action of cut-free schedulings, the former can rewrite to {ε} and is included in the Taylor expansion of the MELL proof-structure R above, the latter cannot rewrite to {ε} and is not part of the Taylor expansion of any cut-free MELL proof-structure. Summing up, our characterization follows the standard notion of MELL proof-structures (unlike [PT09]) and is more fine-grained and informative than the one in [PT09]. The λ-calculus, connectedness and coherence. Our rewriting system and gluability criterion might help to prove that a binary coherence relation can solve the inverse Taylor expansion problem for MELL proof-structures fulfilling some geometric property related to connectedness, despite the impossibility for the full MELL fragment. Such a coherence would extend the coherence criterion for resource λ-terms. Note that our gluability criterion is actually an extension of the criterion for resource λ-terms. Indeed, in the case of the λ-calculus, there are three rewrite steps, corresponding to abstraction, application and variable (which can be encoded in our rewrite steps), and coherence is defined inductively: if a set of resource λ-terms is coherent, then any set of resource λ-term that rewrites to it is also coherent. Presented in this way, the main difference between the λ-calculus and "connected" MELL (concerning the inverse Taylor expansion problem) would not be because of the rewriting system, but because the structure of any resource λ-term uniquely determines the rewriting path, while for DiLL 0 proof-structures we have to quantify existentially over all possible paths. This is an unavoidable consequence of the fact that proof-structures do not have a tree-structure, contrary to λ-terms.
Moreover, it is possible to match and mix different rewritings. Indeed, consider three DiLL 0 proof-structures pairwise gluable: proving that they are gluable as a whole amounts to computing a rewriting from the three rewritings witnessing their pairwise gluability. Our rewriting system has been designed with that mixing-and-matching operation in mind, in the particular case where the boxes are connected. This is reminiscent of [GPT16], where we also showed that a certain property enjoyed by the λ-calculus can be extended to proof-structures, provided they are connected inside boxes. We leave it as future work.
Functoriality and naturality. Our functorial point of view on proof-structures might unify many results. Let us cite two of them.
• A sequent calculus proof of Γ can be translated to a path from the empty sequent to Γ.
This could be the starting point for the formulation of a new correctness criterion. • The category Sched can be extended with a higher structure-transforming it from a category into a 2-category-which allows cut-elimination to be represented as a 2-arrow. The functors qMELL and PqDiLL 0 can also be extended to 2-functors, so as to prove via naturality that cut-elimination and the Taylor expansion commute.