The exponential logic of sequentialization

Linear logic has provided new perspectives on proof-theory, denotational semantics and the study of programming languages. One of its main successes are proof-nets, canonical representations of proofs that lie at the intersection between logic and graph theory. In the case of the minimalist proof-system of multiplicative linear logic without units (MLL), these two aspects are completely fused: proof-nets for this system are graphs satisfying a correctness criterion that can be fully expressed in the language of graphs. For more expressive logical systems (containing logical constants, quantifiers and exponential modalities), this is not completely the case. The purely graphical approach of proof-nets deprives them of any sequential structure that is crucial to represent the order in which arguments are presented, which is necessary for these extensions. Rebuilding this order of presentation - sequentializing the graph - is thus a requirement for a graph to be logical. Presentations and study of the artifacts ensuring that sequentialization can be done, such as boxes or jumps, are an integral part of researches on linear logic. Jumps, extensively studied by Faggian and di Giamberardino, can express intermediate degrees of sequentialization between a sequent calculus proof and a fully desequentialized proof-net. We propose to analyze the logical strength of jumps by internalizing them in an extention of MLL where axioms on a specific formula, the jumping formula, introduce constrains on the possible sequentializations. The jumping formula needs to be treated non-linearly, which we do either axiomatically, or by embedding it in a very controlled fragment of multiplicative-exponential linear logic, uncovering the exponential logic of sequentialization.


Introduction
Proof theory is concerned with proof systems, specifying how proofs are structured, and in turn, which formulae are provable.Different proof systems have different properties, such as soundness (the fact that all provable formulae are also valid -for a given semantics of formulae), completeness (the fact that all valid formulae are also provable -again, for a given semantics of formulae), canonicity (two proofs differ if and only if they have different interpretations -for a given semantics of proofs), ... Broadly speaking, Linear logic [11] has two main kinds of proof systems: Sequent calculus Proofs are trees whose nodes are labelled by inference rules transforming sequents of formulae.This representation -even though it is both historically decisive in the development of proof-theory since the work of Gentzen by emphasizing the role of the cut inference as well as for reductive logics (aka.proof-construction) by suggesting a natural notion of proof-goal -contains lot of unnecessary information (such as the specific order of application of some unrelated rules): it is far from being canonical.The loss of confluence in sequent calculus witnesses this loss of canonicity.Proof-nets are graphs whose nodes relate formulae to one another.A proof-net contains much less information than a sequent calculus proof with respect to the order of application of inference rules.This allows them to be canonical, for some fragments of linear logic at least.In particular, they are canonical for multiplicative linear logic without units (simply referred to as MLL in the following).As a by-product, confluence is recovered in MLL proof-nets.
Apart from the contrast on canonicity issue, a major difference between sequent proofs and proof-nets follows from the tree vs. graph structure and concerns logical correctness of the proof-object: sequent proofs being inductively defined proof objects, their logical correctness is naturally defined in a local way: soundness of the conclusion follows from the soundness of each premise and one only considers derivation trees that is built applying inferences in a correct way.On the other hand, proof-nets being graphs, there is no such thing as a root, in general, and logical correctness therefore becomes a global property of the graph.One of this property is saturation, the fact that the graph can be saturated with more edges to produce a proof tree, called a sequentialization of the net.It would be ad hoc to restrict solely to those graphs which are logically correct and one therefore consider two levels of proof objects: proof structures, which are graphs with some typing constraints on the edges and vertices; and proof-nets, which are logically correct proof structures.
For instance, consider the following rules of linear logic sequent calculus: They introduce the so-called multiplicative constants 1 and ⊥.The way they are introduced differ on their context: while 1 can be deduced thanks to an axiom, in an empty content, ⊥ needs to be juxtaposed to an already derived context.So, in a proof-net system, where operations are not on sequents of formulae but on formulae, this contextuality is missing.Indeed, both introductions are presented as a lone node with no input and an output: Context has thus to be recovered.One way to do so is via correctness criterion, an extra property verified by the graph, ensuring that there exists a coherent way of assigning its context to each formula.Another way is to add extra structure on top of the graph.The most common such structure is the box, introduced in Girard's seminal paper [11].Consider the sequent calculus rule introducing the exponential modality !: For a formula to receive a !, all the other formulae in the context have to be under the ?modality.A way to translate that into proof-nets is by requiring that a special region of the graph is enclosed so that every formula going out of it has a ?, except one corresponding to the !-cell.This is a direct translation of the sequent calculus (where rules are applied to sequents and not formulae) into proof-nets, in a graphtheoretically awkward fashion: Another more graph-theoretical way to introduce contexts are jumps, a recurrent feature in the litera- Fig. 1.A graphical jump ture on proof nets [13].They are purely geometrical (i.e.untyped) edges adding extra connection between links of a proof structure.Jumps were first introduced by Girard in [13] to add more sequentiality in proof nets with quantifiers, typically preventing sequentialization that would not match the variable-witness dependencies between ∀ and ∃ quantifier.Such jumps can also be found in unification nets, a more recent framework of proof nets with quantifiers that bears implicit witnesses [15].Jumps are also used as lighter feature than boxes to handle the sequentiality induced by sequent rules outside of MLL, typically the unit rules [4,14,17], additive rules [14,16] and exponentials for λ-nets [1].
On another line of work, Faggian and her collaborators developped a setting in which jumps are taken seriously as edges in a more general graph than just the mere syntactic tree of a formula.First in the context of L-nets [5,6,10], a parallel syntax for Ludics designs, then in the context of J-proof nets for HS + (a polarized and hypersequentialized sequent calculus) and MLL [8,9], their setting allows in particular to see sequent calculus proofs as proof-nets saturated with sequential edges, embedding both proof-nets and sequent calculus proofs in a same universe of partially sequentialized proof-structures, each of them living at one extremity on a de•sequentialization spectrum.
The figure above depicts a proof net with a jump (the red dashed arrow) from (the formula occurrence) (A `B) ⊗ C ⊥ to (the formula occurrence) C. As a consequence, when sequentializing such a net, the ⊗ rule introducing (A `B) ⊗ C ⊥ will always be placed above the ⊗ rule using C to introduce C ⊗ D ⊥ .Jumps are thus a way to constrain the set of sequentializations that are derivable from a proof net.
Proof structures with jumps offer a larger setting than usual proof structures, it is thus natural to search how the behave under cut elimination and how the usual correctness criteria on proof nets can be adapted to proof nets with jumps.

Contributions and organization of the paper.
In this paper, we show that graphical jumps as presented above can be internalised in the logic under study at the only cost of adding a special atom J that enjoys, as well as its dual, the property of a monoid and behaves as an exponential under cut elimination.This atom can itself be encoded in a fragment of linear logic, thus giving a way to find correctness criteria for proof structure with jumps through the ones of usual proof structures.
To keep this paper light and readable, we will focus our internalisation of jumps to MLL.In Section 2, we will recast the basic definitions on MLL sequent proofs and proof-nets.In Section 3 we start introducing our encoding of jumps as axioms in a cut-free setting.In particular, we will define MLL J a logic which is basically MLL plus a special atom J that enjoys left and right contractions rules.We will prove that our encoding is correct and that our jumps correspond exactly to the ones described in [9], in the sense that their correctness criterion matches with the usual Danos-Regnier criterion on our encoding.In Section 4 we will discuss the encoding of jumps in the dynamic setting of proofs with cuts and cut elimination, enriching MLL J with the appropriate exponential dynamics for J.This will make explicit the dynamics of jumps sketched in [9] and treated more exhaustively in [7].Finally, we will conclude in Section 5 by showing that the jump atom J can be implemented effectively in the exponential fragment of MELL + Mix.MLL is a minimal proof-system.Its formulae are given by the following grammar (with A ranging in a set of atomic formulae): Linear negation is inductively defined by Sequent calculus for MLL MLL proofs, in the sequent calculus, are given by the applications of the rules: Its dynamic of reduction, also called cut elimination is recalled in figure 2. The resulting calculus is (non deterministic and) not confluent.Remark 2.1 We have not made precise what sequents are, in the previous definition.There are many possible choices which have various impacts.We leave this as implicit and lightweight as possible in the remaining of the paper but need to discuss this here.We crucially need a notion of sequents which allows for two properties: (i) to trace formula occurrences along the proof, that is admit a well-defined notion of formula ancestor (ii) which validates the exchange rule (as an admissible rule at least).
Among the standard options at hand, there are sequents as ordered lists of formulas, sequents as sets of locative formula occurrences or finally sequents as indexed sets of formulas.On the other hand, sequents as multisets of formulas are not an option as it does not allow for a well-defined notion of formula occurrence, nor a well-defined notion of desequentialization (see below).The first option requires to consider some form of an exchange rule: sequents are lists of formulas, written ⊢ F 1 , . . ., F n , and the following exchange rule is necessary: . In such a situation, one can treat, as often, the exchange rule implicitly by considering that the set of inferences is complemented with the derived rules obtained by pre-and postcomposing every other rule with a series of exchanges.(See Example 2.2 for an illustration, if sequents are viewed as ordered lists there.)Each such derived rule is equipped with a notion of ancestor derived from the basic inference rules given above (as usually done in the proof theory literature [3]).When sequents are treated in a locative way, their formulas are equipped with a unique address (pairwise incomparable) and the notion of sub-address provides an explicit notion of formula ancestor, the exchange rule in unnecessary in this case.(See [2,12] for instance.)Last, with sequents as indexed sets of formulas, each inference rules is equipped with a relation between the indexing set of the conclusion of the rule and that of its premisses, constituting the ancestor relation.(Note that the above two presentations are concrete instances of this last presentation: with lists the indexing sets are the positions in the list, with locations, the indexing sets are finite sets of pairwise incomparable addresses.) In the following, we shall forget about the above considerations and simply assume that we have a well-defined notion of formula occurrence and formula ancestor.

Proof structures for MLL
Among sequent calculus proofs of a given sequent, some are essentially different, while some are essentially the same: a difference in the order of introduction of two connectives is typically a non-essential difference.A canonical representation of proofs can be found when moving to the highly parallel setting of proof-nets, a graphical representation of proofs.A MLL proof-structure is a directed graph whose nodes (links) and edges abide to the following typing constraints: Note that the type of the edges of a proof-structure can be recovered from the type of the edges going out its axiom links and the type of its links, so that they could be made implicit although they are often kept for readability reasons.Besides, we say that such a graph is a proof-structure for the sequent Γ if we obtain Γ by collecting all the formulas of its pending edges up to reordering.
Reflecting the dynamics of sequent calculus proofs, proof structures also define a (non-deterministic) graph rewriting calculus; the corresponding reductions are recalled in Figure 3. Cut elimination in proofstructures is non-deterministic but confluent and terminating.We write ⇓ R for the (unique) normal form of a proof-structure R; ⇓ ρ R if we want to make more precise the sequence of reduction steps leading to it 4 .

Sequent calculus proofs as (constrained) proof structures
Let us consider an MLL proof π.A formula F is a formula occurrence in π if it is the active conclusion of a rule in π (the formula having its topmost connective introduced by the rule or the two formulas introduced by an axiom rule).For F a formula occurrence in π, we define its life-time as sub-branch from its creation to its use.
For two rule occurrences r and r ′ in a proof π, we say that r is above (or occurs before) r ′ if r and r ′ belong to the same branch in the proof, and r ′ appears before r starting from the root.If r produces a formula F this is equivalent to saying that there exists F ′ , an active formula of r ′ , such that F is created before F ′ is used; in that case, we write F ≺ F ′ .
Every sequent proof π can be desequentialized into a proof structure R π , defined inductively.For a proof structure R, we note seq(R) its set of sequentializations, that is, the set of sequent proofs π such that R π = R. Desequentialization is preserved by cut elimination and, conversely, if π ∈ seq(R) and R → R ′ , then there exists π ′ ∈ seq(R ′ ) such that π → + π ′ .Note, however, that in general proofs in seq(R ′ ) do not correspond to the reduct of some proof in seq(R).
If π is a sequentialization of R then there is a one-to-one correspondence between the rules in π and the links in R (this correspondence is defined during the inductive procedure of desequentialization).By extension every edge in R can be associated with a formula occurrence in π throughout its life-time.In the following we will often make use of this correspondence, keeping the conversion from proof to net implicit.
Given a proof structure R, its set of sequentializations may be empty: in that case we say that R is incorrect.On the contrary, if seq(R) is non empty, we say that R is correct or that it is a proof net, meaning that it actually corresponds to a canonical representation of proofs.
A proof net can have several sequentializations, for example: Example 2.2 Consider the (correct) proof-structure R: It has two different sequentializations: On the first one, A is above D, not in the second.
Graphical jumps as depicted in Figure 1 are used to restrict the possible set of sequentializations of a net: if there is a jump from F to F ′ in R, then we expect F to appear above F ′ in any sequentialization of R. Writing jumps as a set of pairs of formula occurrences in R and writing L for a set of jumps, we define In the example above, the net with jump from (A `B) ⊗ C ⊥ to C has only one sequentialization left: The aim of this paper is to provide a logical encoding of jumps in proof nets that validates the definition of seq L (R).More precisely, given a proof net R and a set of jumps L, we want to define of proof net J L (R) such that seq(R) = seq(J L (R)).

Jumps as axioms
In this section, we show how the graphical jumps presented above can be encoded in cut-free MLL proof structures using the axiom link of fresh atomic formulas J to create synchronisation points.We first introduce the idea for a single jump, then generalize it to multiple jumps.

Case of a single jump
Definition 3.1 Let F and F ′ be two formulae.A jump from F to F ′ is the gadget: R be a proof-structure, and F , F ′ be two formula occurrences in R. We define J F →F ′ (R), the proof structure with a jump from F to F ′ , by grafting in R the jump from F to F ′ where F and F ′ are, and making all the types coherent by propagating downward the formula labels.
Let us illustrate the previous definition.Consider the (correct) proof-structure R in Example 2.2, we can add a jump from F = (A `B) ⊗ C ⊥ to C, resulting in the proof-structure J F →C (R), where the conclusion formulas have been updated to take into account the impact of adding the jump: This proof-structure is much more rigid than R. It also has only two sequentializations: and the one where the `rule shown in green is commuted below the ⊗ rule.They differ in a much lighter way than the two sequentializations of R: indeed, forgetting every occurrence of J in both of these two sequent calculus proofs yields only one of the two sequentializations of the original proof-net.The presence of the jump from F = (A `B) ⊗ C ⊥ to C forces F to be above C in every sequentialization of J F →C (R).Let us turn this example into a general theorem.

MLL J
From now on, we consider J as a special atom that does not appear in regular MLL formulae and proofs.We write MLL J (respectively PS J ) for the cut-free MLL proof system (resp.proof-structures) that does allow for the presence of J but with strong restrictions on its use described below.
MLL J has the same rules as MLL except for J-formulae that are kept separated from other formulae until being used: Proof structures and proof nets for MLL J are essentially the same as in MLL except for the typing restriction put on links ⊗ and `when dealing with J-formulae.This motivates to define MLL J -formulas by the following grammar: The following proposition is trivial by induction on the structure of MLL J proofs.
Proposition 3.2 If ⊢ Γ is derivable in MLL J , then Γ can be partitioned in ∆, Θ J such that every formula in ∆ is a MLL J -formula and every formula in Θ J is J ⊥ .
In particular, when removing J, MLL J is equivalent to MLL.More precisely: Definition 3.3 Let F be a MLL J -formula, the erasure F | J of J in F is inductively defined by: Let π be an η-expanded sequent calculus proof in MLL J (that is, all axiom rules are applied to atomic formulas only), the erasure erase J (π) is inductively defined by: is defined by applying a `rule on A| J and B| J to erase J (π ′ ); • if π is an application of a `J⊥ after π ′ then erase J (π) is defined as: erase J (π ′ ); • if π is an application of a ⊗ on A ⊗ B, combining subproofs π A and π B : • if B = J, then erase J (π) is defined by combining erase J (π A ) and erase J (π B ) with a ⊗ rule on A| J and B| J ; • if π is an application of a ⊗ J combining π ′ and ax J then erase J (π) is defined as erase J (π ′ ).
Let R be an MLL J -proof-structure the erasure, erase J (R), is inductively defined in the same way as erase J (π), that is by erasing every occurrence of J, J ⊥ and the `and ⊗ link immediately connected to those atoms as well as the axioms links between J and J ⊥ .Note that if ⊢ Γ is the conclusion of π (resp.R), then ⊢ Γ| J is the conclusion of erase J (π) (resp.erase J (R)).Erasures cannot be empty by construction of MLL J .Moreover, if π ∈ seq(R) then erase J (π) ∈ seq(erase J (R)).
MLL is included in MLL J .Moreover: In fact, for π ⊢ Γ, Θ J , erase J (π) is a MLL proof and there is an embedding erase J (π) ֒→ π of the rules of erase J (π) onto the rules of π, preserving their kind and the order between them.
Proof.By structural induction on π.Every case is straightforward, except maybe when ⊗ is the last rule of π: if F and G are the active formulae, one shall remark that by construction F | J or G| J = ∅.✷ Theorem 3.5 Let R be a proof-structure, and F and F ′ two formula occurrences, then erase J (seq (J In other words, the following diagram commutes: In particular, correctness of the proof-net enriched with a jump is euivalent to existence of particular sequentializations: Corollary 3.6 (correctness) Let R be a proof-structure, and F and F ′ two formula occurrences, then J F →F ′ (R) is correct if and only if there exists a sequentialization of R such that F is above F ′ .
Let us prove Theorem 3.5.
Proof.⊆.From Def. 3.3 it is immediate that erase J (seq(R)) ⊆ seq(erase J (R)).Now, let π be any sequentialization of R F →F ′ .π contains an axiom ⊢ J ⊥ , J, to which is applied a ⊗-rule tensoring J with F .So F has been created in the second branch leaving this ⊗-rule.At one point below, there is a `-rule between F ′ and J ⊥ .So the use of F ′ `J⊥ is below the creation of F in π, hence so is the use of F ′ in π| J .
⊇. Let π ∈ seq (F,F ′ ) (R).By hypothesis, the rule r introducing F is above the rule r ′ using F ′ in π, we can thus perform the following transformation adding three more rules in π (and propagating the changes in the resulting sequent accordingly): These 3 rules directly map to the jump gadget: the resulting proof is a sequentialization of J F →F ′ (R).✷ We can now model a single jump in a proof net.This situation is far too restricted: we now move to the modelling of multiple jumps.

Multiple jumps
The definition of jump given above suffers from an obvious drawback: there is no canonical way to consider two jumps having the same source or destination.To solve this issue, we need the ability to (i) parallelize: jumps with the same source or destination have no order, hence should not be represented as successive applications of `or ⊗ rules; (ii) contract: each jump needs two contractions: one on the source, one on the target.
Following these two requirements we ask J to have the following contraction properties: Hence, we disallow axioms on J-formulae and extend MLL J with the rules, for n ≥ 0, m > 0: and, in the same way, we add the following cells to PS J : In the ⊗ J rule, we also replace Ax J by J(n) for any n ≥ 0.
Note that having a weakening on J (i.e. the rule J(0)) is not required when dealing with cut-free proofs but we simply anticipate on the next section.
We generalise the encoding of a single jump to a list of jumps by making space to connect any edge with another one and then effectively realising the connections described in the list.Definition 3.7 Given a proof-structure R, we define the proof structure with synchronisation points J(R) as the proof structure R in which every edge F of R is replaced by the following gadget J(F ) (relabelling its successors to make all the types coherent): J ⊥ (F ⊗ J) `J⊥ The ⊗-link bounded to the J(1) axiom is the out-synchronisation point of F , while the `-link bounded to the J ⊥ (1) contraction is its in-synchronisation point.
Given two formula occurrences F and F ′ in R, a jump from F to F ′ is then encoded in J(R) by connecting the out-synchronisation point of F to the in-synchronisation point of F ′ , that is, by adding a J ⊥ edge from the J axiom link of J(F ) to the J ⊥ contraction link of J(F ′ ).
Finally, given L a set of pairs of formula occurrences in R, we define J L (R) the proof structure corresponding to J(R) with jumps between each pair of formula occurrences provided by L.
Figure 4 illustrates the encoding of a jump from F to F ′ where F also has n outgoing jumps and F ′ also has m incoming jumps.In the sequel we will sometimes hide these jumps-as-axioms encoding and schematize them with dashed arrow, which in the case of Figure 4 would correspond to:

Remark 3.8
It is striking to note how close our encoding of jumps can be from double negation translations used to add sequentializations in the translation of classical logic into intuitionistic logic.Indeed, in linear logic, double negation translation correspond to (A ⊸ ⊥) ⊸ ⊥ that is (A ⊗ 1) `⊥.This can be understood as the production of a token (1) once A is created, and the wait for this token to return before A can be consumed.In the above encoding of jumps, the J atom replaced 1, which allows many tokens to be produced and given to the environment after the creation of A (these tokens furthermore carry the information of coming from A thanks to the axiom link), while on the other side A can now wait for many specific tokens (coming from other formulae) to be put in the environment before it can actually be used.
The definition of erase J (() ) (see 3.3) does not change and following the same reasoning as for Theorem 3.5 we have: J ⊥ (F ′ ⊗ J) `J⊥ Theorem 3.9 Let R be a cut-free proof-structure, and L be a set of pairs of formula occurrences, then erase J (seq (J L (R))) = seq L (R).In other words, PS J P(MLL J ) PS P(MLL) seq P(erase J ( )) J L ( ) seq L Corollary 3.10 (correctness) Let R be a proof-structure, and L be a set of pairs of formula occurrences, then J L (R) is correct if and only if there exists a sequentialization of R such that F is above F ′ for every (F, F ′ ) ∈ L.
Remark 3.11 In [9], jumps are defined as edges from one link to one of the entry port of another link: a parr link having one entry port to which each of its premises are plugged; a tensor link having two entry ports, one for each of its premises.The correctness criterion for MLL proof structures with such graphical jumps is then a recast of the usual Danos-Regnier criterion: a proof structure is correct if and only if every of its switching paths is connected and acyclic, a switching path being here defined as a path which does not use any two edges entering trough the same port of a `link.
In the Danos-Regnier criterion for MLL proof structures, switching paths are defined as paths which never follow both premises of a `link.Viewing our in-synchronisation point as their entry port, the usual Danos-Regnier criterion on our encoding of jumps correspond exactly to their recast of the same criterion.Remark 3.12 In [15] Hughes introduces a MLL-encoding of his unification nets in order to prove their correction.Although we were not aware of it at the time we wrote this paper it must be acknowledged that this encoding is morally the same as our gadget from Definition 3.1 Being only interested in an encoding that preserves splitting tensors and switching cycles (i.e.correction), Hughes does not develop on the sequentialisation aspect of his encoding: in particular he shows no connection between his encoding and the set of sequentialisations of its nets, multiple cuts are not treated canonically and cuts are treated as tensors, evacuating any questions about their dynamics.

Cuts
We now investigate the internalisation of jumps in the presence of cuts.In particular, we need to define the dynamics of J and J ⊥ .
As intuited by [8], we expect jumps to propagate through cut elimination: suppose indeed that the left-hand side of a cut (on a formula F ) has jumps coming into it, while the right-hand side (F ⊥ ) has jumps coming out of it.It forces, in any sequentialization, the sub-formulae of the cut-formula to be above what is below the cut.This information has to be kept in some way during cut-elimination, even when the considered cut vanishes.Moreover, if n jumps enter into F and m jumps go out of F ⊥ , the reduction of the cut will produce m × n jumps: one for each composition of jumps.This reveals the exponential nature of jumps: each jump arriving onto one of the two cut formulae must be propagated towards every formula occurrences reached by a jump leaving one of the two cut formulae, which cannot be realized by the multiplicative combinatorics.Unsurprisingly we define a commutation rule between J ⊥ (m) and J(n) as: which corresponds to the following map from the monoidal structure of any monoidal category inter-L F ⊥ → ) (as jumps in L F → are duplicated when their target is the premise of a link, the resulting jumps are duplicated in the same way) and delete the one in L →F ∪ L →F ⊥ (J ⊥ edges corresponding to I F ∪ I F ⊥ are also deleted but they were only replications of remaining jumps in L).
Writing L ↓ {F, F ⊥ } for (L\{F, F ⊥ }) ∪ C F,F ⊥ , we have: Theorem 4.3 (bisimulation) Let R be a proof-structure, L be a list of pairs of formula occurrences, and F, F ⊥ be the premises of a cut in R, then Let R be a proof-structure and L be a list of pairs of formula occurrences, then Proof.Theorem 4.3 implies J ⇓ρL (⇓ ρ R) =⇓ ρ (J L R) for any reduction sequence ρ, and by confluence of cut elimination in proof-structure the reduction path does not matter.
Back in a cut-free context, one can apply Theorem 3.9 to get the result.✷

Correction
As done for tensor in Section 3, we restrict the use of cuts on J-formulae in order to have a well-defined notion of erasure of J from MLL J proofs to MLL proofs: MLL J is closed under cut elimination and following the same reasoning as before we still have Theorem 4.5 Let R be a proof-structure, and L be a set of pairs of formula occurrences, then erase J (seq (J L (R))) = seq L (R).
So as before we have an encoding (J L ) of graphical jumps in MLL J (and its corresponding decoding erase J ) such that: and, furthermore, by Theorem 4.3, there exists a bisimulation procedure so that the above diagram is stable under cut-elimination.

Encoding in MELL + Mix
In this section, in order to finish our logical internalization of jumps, we eventually show how jumps, as defined in Figure 3 and 4, can be implemented in the exponential fragment of MELL + Mix.
As explained in Section 3, we need J and J ⊥ to be able to contract.This can be achieved using the contraction rule of the ?-exponential in MELL, so our encoding of J and J ⊥ must involve this exponential modality.J only needs the ability to contract at the level of axioms while J ⊥ is free to contract everywhere, furthermore, J and J ⊥ must be dual from each other, as such, we set: where A J is a special atom that does not appear in regular MLL formulae.
In Figure 9 we show how the static properties of jumps (that are J(n ≥ 0) and J ⊥ (m > 0)) can be derived via this encoding.The dynamics properties of jumps given described in Section 2 can then be derived from the usual dynamics of exponential in MELL.

Conclusion and future work
We have presented a logical framework, MLL J , in which MLL-proof-structures with jumps can be encoded.This framework can be seen both as an extension of MLL with a special atom and as a fragment of MELL + Mix.As such, correctness criteria for proof-structures in these logics induce correctness criteria for MLL J -proof-structures.In particular, the Danos-Regnier correctness criterion for MLL induces correctness criterion for MLL J , which corresponds to the correctness criterion given in [8], in a setting where jumps in proof structures are treated from a purely graph-theoretical point of view.
Our encoding is compatible with the dynamics of proof-nets: cut elimination in our framework achieves the graphical dynamics of jumps considered in [8,7].On top of being the most natural one (it makes two jumps to compose if one targets the premise of a link involved in a cut and the other leaves one of these links), we believe that this dynamics is justified by the following conjecture: Conjecture 6.1 Let R be a MLL proof-net, and L be a set of pairs of formula occurrences such that sec L (R) = {π}, i.e.L adds enough constraints on R for its sequentialization set to be a singleton, then sec(⇓ J L (R)) = {π ′ | ∃ρ, π ′ =⇓ ρ π} In other words, the freedom of sequentialization induced by the cut elimination procedure in Jumped proof structures corresponds exactly to the choices made during cut elimination in the sequent calculus.⊇ is a consequence of Theorem 4.3, ⊆ however is left as future work.
Our framework draws a clear line between formulae encoding jumps and regular MLL formulae.This advocates for easy extensions to other logics such as MALL or MELL with or without the Mix rule.In particular we plan, in expanding this work to MELL to study in detail the correspondence between the dynamics that is induced by our modelling of jumps and the cut-elimination considered in [7] for the exponentials as well as how the notion of cones introduced by di Giamberardino as a replacement and generalization of exponential boxes translates in our setting.
This work fits in a general tendency to try to make sense logically of the graph-theoretic foundations of proof-nets.Here, we reinterpreted jumps as a purely logical structure.

Fig. 9 .
Fig.9.Encoding of the combinators on J and J ⊥