concrete fully complete models of mll- and mall-thomasc/lics.pdf · 2018-08-09 · to shift from a...

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Concrete fully complete models ofMLL− andMALL−

Thomas David CuvillierQueen Mary University of London

London, [email protected]

AbstractThis paper focusses on defining a concrete fully completemodel for linear logic, where previous models rely cruciallyon quotient or 2-categorical tools such as dinatural trans-formations. This research presents a new categorical con-struction mixing double-glueing and the Chu-construction.We build a relational semantics of proof structures based onpermutations. We then use the new categorical constructionto restrict the relation category and obtain full complete-ness for MLL without units. Moreover, our model can beenriched with a hypercoherence structure to yield a staticfully complete model for MALL without units. We make useof this result to prove the Pagani conjecture about hypercor-rectedness. Our proof notably relates hypercoherence andclosure-operators.

Keywords Linear Logic, Denotational Semantics, Full com-pleteness

1 Introduction1.1 General introductionLinear logic is one of the most studied logics in theoreticalcomputer science, especially from the denotational semanticspoint of view. Several important models, such as coherencespaces [13] or games [2], have been first introduced as deno-tational models for linear logic, before being reused success-fully for programming languages. Undoubtedly an importantcriterion to judge the relevance of a model is the propertyof full completeness, presented and described in [2] as “thetightest possible connection between syntax and semantics”.The main goal of the present research is to find a “concrete”fully complete model. Precisely, we want to avoid the useof 2-categorical tools to model type variables, together withnot relying on a quotient. We hope that such a model couldprovide important insights for the study of linear logic, butalso related and connected fields in semantics, such as se-mantics of programs[20] and processes [26]. In this paper, weconsider two logics:MLL− (MLL without units), andMALL−

(MALL without additive nor multiplicative units).As of today, most models of logics with type variables

are built using dinatural transformations [5, 7, 19], with theintuition that a proof should behave uniformly at variabletypes [6]. Consequently, the interpretations of the proofsare not concrete, regardless of how simple the morphisms

LICS’18, July 09–12, 2018, Oxford, UK2018.

of the initial category were. The first goal of this study isto shift from a 2-categorical setting to a first-order category.As MLL− proof structures are permutations, we first devisea semantic account of permutations. We develop a way tomodel them that takes inspiration from the concurrent gamemodel of Abramsky and Mellies [5]. The morphisms of thismodel were closure operators σ originating from a stableoutput function f : σ (x) = x ⊔ f (x). Our morphisms canalternatively be seen as lattices, permutations, linear out-put functions associated with closure operators, or a modelof data-flow. Notably, we can compose them relationally,through tracing of permutations, or through the Kahn se-mantics of data-flow. Similarly we can compose them bycomposing their associated closure operators.

We wish to discriminate those morphisms that come froma denotation of a proof. We refine them using ideas relatinggames and static models, especially those developed by Hy-land and Schalk [16] when developing the abstract games.We enrich the domains, on which the strategies act, witha new notion of Opponent and Player positions, definedthrough a correspondence with Chu-dualization. Further-more, we wish to equip each object of the category with aset of strategies and counter-strategies, as in the standarddouble-glueing construction. In order tomix the two togethersuccessfully, we introduce a new categorical constructionthat showcase how the double-glueing construction and theChu-dualization can be combined together by requiring thatthe strategies only act on P-positions, and counter-strategieson O-positions. We prove that the model thus obtained isfully-complete for MLL−.

This model admits a canonical extension that models theadditive connectives of MALL−. We conjecture that the re-sulting model is fully-complete and display some early re-sults in this direction.We demonstrate how the fully-complete ofMLL− can be

extended to provide full completeness for MALL− by enrich-ing it with hypercoherence. This follows the proof previ-ously established in [7] for double-glued hypercoherences.However, this one was designed with dinatural transfor-mations in mind, and we show here that it adapts well toour 1-categorical model. In contrast to the games model ofMALL− [5, 21] where morphisms should be considered “upto equivalence”, the model that we have developed is static;therefore each element in the function space model definesa morphism. The proof we provide notably establishes anew link between hypercoherences and closure operators,

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LICS’18, July 09–12, 2018, Oxford, UK Thomas David Cuvillier

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Axiom :⊢ A⊥,A

⊢ Γ,A ⊢ A,∆Cut :

Γ,∆⊢ Γ,A,B,

Parr :⊢ Γ,A` B,

⊢ Γ,A ⊢ ∆,BTensor :

⊢ Γ,∆ ⊢ A ⊗ B⊢ Γ,A

⊕ 1 :⊢ Γ,A ⊕ B

Γ ⊢ B⊕ 2 :

⊢ Γ,A ⊕ B

⊢ Γ,A ⊢ Γ,B& :

⊢ Γ,A & B

Figure 1. Sequent Calculus for MALL−

yielding a new insight in the computational time of an hy-percoherence [8, 12]. This model furthermore provides anindirect proof of the Pagani conjecture about proof structureand hypercoherences [22]: hypercorrectedness together withslice-connectedness entails sequentializability. In particular,as Tranquilli successfully characterized hypercorrectednessof proof structures [24], this leads to a new definition ofproof-net for MALL−.In Section 2, we briefly recall the Chu-dualization and

the double-glueing construction, before exposing how weblend them together. In Section 3 we present our semanticsof permutations, and how the previous construction can beapplied to it. We then proceed to show how the resultingmodel is fully-complete in Section 4. In Section 5, we showsome preliminary results about the canonical extension ofthe previous model with products. Finally, in Section 6, werecall the Pagani conjecture. We then present a new modelwith hypercoherence, such that the hypercoherence and Chu-condition are orthogonal: they do not interfere with eachother. We show how the resulting model is fully-complete.We conclude that the Pagani conjecture is true.

1.2 MALL−,MLL− and their proof structuresWe focus on theMALL− fragment of linear logic, that is, themultiplicative additive one without units. This is presentedin Figure 1. The restriction to the rules “Axiom, Cut, Parr,Tensor” forms the MLL− fragment. We will also refer to theMIX rule, defined as follows:

⊢ Γ ⊢ ∆MIX⊢ Γ,∆

We recall the definition of Hughes-Van Glabbeek proofstructures [15]. We see a sequent as a multi-set of parse trees,that we call a parse forest. A &-resolution is the result oferasing one argument sub-graph of each &-occurrence. Anadditive resolution is the result of deleting one argumentsub-graph of each ⊕,&-occurrence. We say that an additiveresolution is on a &-resolution, if it can be obtained throughdeleting argument sub-trees of the ⊕-occurrences of this&-resolution.An axiom-link is an edge between occurrences of com-

plimentary literals. A linking on Γ is a set λ of disjointaxiom links such that

⋃λ partitions the set of leaves of an

additive resolution of Γ, called Γ λ. That is, each literal ofthe additive resolution is under a unique axiom-link of λ. A

X⊥ ⊕ (Y ⊕ X ) , (X & X ) ⊗ (Z ⊕ Z ) , (Z⊥⊗ Z ) `Z⊥

⊕

⊕ &

⊗

⊕ ⊗

`Figure 2. AMALL− proof-structure

MALL− proof structure comprises a MALL− sequent, seenas a forest, together with a set of linkings Λ = λ1, ..., λn,such that for each &-resolution of Γ, there is a unique linkingλ ∈ Λ such that Γ λ is on this &-resolution. We refer tothis property as (P1), named “resolution condition” in theseminal paper [15]. Note that each linking λ canonically de-fines a bijection from negative to positive literals of Γ λ.To each MALL− proof can be canonically associated a proof-structure. The proof structures coming from denotation ofproofs are called proof-nets.

A proof structure for a sequent Γ ofMLL− is the same as aproof structure for it seen as aMALL− sequent. As there is aunique &-resolution, there is a unique linking, that partitionsall leaves of the formula. InMLL−, a linking can alternativelybe presented as a bijection from negative to positive literals.Given a proof structure P of MLL−, a switching S is a

choice, for each` occurrence in the parse forest, of the left orthe right premise, written S(`). We write S(`) for the non-choosen premise. The correction graph consists of, given aproof structure Ps and a switching S, the graph obtained byremoving from Ps all edges ` ↔ S(`). We say that a proofstructure is acyclic (resp connected) if, for all switchings Son it, the correction graphs are acyclic (resp connected). Aproof structure Ps ofMLL− is a proof net if and only if it isacyclic and connected [9].As each linking of a MALL− proof structure defines a

MLL− proof structure on the corresponding additive resolu-tion, one can speak about connectedness and acyclicity of alinking. A proof structure ofMALL− is a proof net only if, forall its switching λ, Γ λ is connected and acyclic. These prop-erties are referred to as slice connectedness, slice acyclicityrespectively, and slice correctedness for the conjunction ofthe two . Slice correctedness (referred to as (P2)) is howevernot enough for a proof structure to be sequentializable: oneneeds the additional toggling criterion [15].Proof nets for MALL− and MLL− capture precisely the

nature of proofs modulo equivalence. That is, two proofs areequivalent if and only if they are denoted by the same proofstructure. The equivalence relation between proofs will notbe presented here, and we refer to the literature for moredetails [25].

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2 Chu double-glueingWe recall the reader that sound categorical models ofMLLare star-autonomous categories, and star-autonomous cat-egories with product to model MALL. Star-autonomous cat-egories are monoidal closed categories, such that A ⊸ ⊥ ⊸⊥ ≃ A. Writing (A)⊥ for A ⊸ ⊥, this amounts to a invo-lutive negation. A denotational semantics is the data of acategorical model C, and a functor ⟦⟧ from the formulas andproofs (up to equivalence) of the logic into C. A model isfully-complete if every morphism in C is the denotation ofa proof: ⟦⟧ is full. We write I for the unit of the monoidaltensor, and ⊥ for I⊥. ⊥ is the unit of the ` monoidal tensor,defined as the negation of ⊗ : A` B = (A⊥ ⊗ B⊥)⊥.

2.1 Chu-dualizationWe introduce a new categorical construction that combinesChu-dualization and the simplest form of double-glueing,namely along hom-functors. We start by briefly recalling thetwo constructions, and then display how we blend the twotogether.

Definition 2.1. [17] Let C be a symmetric monoidal closedcategory with products (written & here, with unit 1). Thenits Chu-dualization Cd = C × Cop is a star-autonomouscategory, with the following tensor product:

(U ,X ) ⊗ (V ,Y ) = (U ⊗ V ,U ⊸ Y &V ⊸ X )

whose unit is (I , 1), and the negation (U ,X )⊥ = (X ,U ). Fur-thermore, if C has coproducts, denoted ⊕ with unit 0, thenCd has products and coproducts defined as follows:

• (U ,X ) &Cd (V ,Y ) = (U &V ,X ⊕ Y ).• (U ,X ) ⊕Cd (V ,Y ) = (U ⊕ V ,X & Y ).• 1Cd = (1, 0).• 0Cd = (0, 1).

Hyland and Schalk used this construction for establishinga functor from the category of games into a star-autonomouscategory case Reld. The functor sent an arena to its set of Pand O-positions respectively [16].

2.2 Double-glueing constructionWe now introduce the double glueing construction.

Definition 2.2. [17] Given a star-autonomous category C,we define the double-glued category DG as having:

• objects triples (A, S, S⊥) where A is an object of C,S ⊆ C(I ,A) and S⊥ ⊆ C(A,⊥). S is the set of values,while S⊥ defines the co-values.

• arrows f : (A, S, S⊥) → (B,T ,T⊥) morphismsf : A → B ∈ C(A,B) such that:

∀s ∈ S .s; f ∈ T and ∀t ∈ T⊥. f ; t ∈ S⊥.

DG inherits the star-autonomous categorical structurefrom C. The monoidal product in DG is defined as follows :

(A, S, S⊥) ⊗ (B,T ,T⊥) = (A ⊗ B, S ⊗ T ,Z )

where S ⊗ T = I ≃ I ⊗ Is⊗t−−−→ A ⊗ B, s ∈ S, t ∈ T . On

the other hand, Z is slightly more delicate to define. Givenz ∈ C(A ⊗ B,⊥), and u : C(I ,A) we define the interaction ofu against z, denoted u;A z : B → ⊥ as the morphism:

u;A z = B ≃ I ⊗ Bu⊗idB−−−−−→ A ⊗ B

z−→ ⊥.

Similarly, we can define the interaction of z against a mor-phism x : I → B by :

x ;B z = A ≃ A ⊗ IidA⊗x−−−−−→ A ⊗ B

z−→ ⊥.

We expose the structure of Z .

Z = z ∈ C(A ⊗ B,⊥) | ∀s ∈ S .s;A z ∈ S⊥ ∧ ∀t ∈ T .t ;B z ∈ T⊥

The morphisms of Z send values to co-values. The unit forthe monoidal tensor is:

IDG = IC, idI ,C(I ,⊥)

The negation simply swaps values and covalues, while negat-ing the original object.

(A, S, S⊥)⊥ = (A⊥,¬S⊥,¬S)

where ¬S = f ⊥ : (⊥⊥ = I ) → A⊥ | f : A → ⊥ ∈ S,and similarly for S⊥. Furthermore, if C has products andcoproducts, then so does DG.

(A, S, S⊥) & (B,T ,T⊥) = (A & B, S &T , S⊥ ⊕ T⊥)

where C(I ,A & B) ⊇ S & T = f : I → A & B | f ;πA ∈

S∧ f ;πB ∈ T where πA : A&B → A and πB : A&B → B arethe two products projections. Similarly, S⊥⊕T⊥ ⊆ C(A&B →

⊥) = πA; f : A & B → ⊥ | f ∈ S⊥ ⊎ πB ; f : A & B → ⊥ |

f ∈ T⊥. Finally, the additive units are as follows:• 0DG = (0,C(I , 0), ∅)• 1DG = (1, ∅,C(1, I ))

2.3 The Chu double-glueing constructionIn this section, we demonstrate how the two constructionscan be naturally combined together. Given an object (U ,X )

of the Chu-dualization Cd of a category, we interpretU asthe object of Player (P ) positions, while X is the object ofOpponent (O) positions. We require that the values fromthe double glueing construction happen to be morphisms toP-positions, while the co-values depart from O-positions.

Definition 2.3. Given a monoidal closed category withproductsC, we define the categoryCDG, calledChudouble-glueing construction, whose objects are 4-tuples (U ,X , S, S⊥),where U ,X ∈ C, S ⊆ C(I ,U ) is the set of strategies, andS⊥ ⊆ Cop(X , I ) = C(I ,X ) is the set of counter-strategies.The morphisms (U ,X , S, S⊥) → (V ,Y ,T ,T⊥) are pairs ofmorphisms (f ,д), f ∈ C(U ,V ), д ∈ Cop(X ,Y ) such that∀x ∈ S .x ;C f ∈ T and ∀y ∈ T⊥.д;Cop y ∈ S⊥.

CDG is a monoidal category, with monoidal tensor:(U ,X , S, S⊥) ⊗ (V ,Y ,T ,T⊥) = (U ⊗ V ,U ⊸ Y & X ⊸ V , S ⊗ T ,Z )

(f1,д1) ⊗ (f2,д2) = (f1 ⊗ f2, f1 ⊸ д1 & f2 ⊸ д2)

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where Z = (f ,д), f ∈ C(I ,U ⊸ Y ),д ∈ C(I ,X ⊸ V ) |

∀s ∈ S .eval(f , s) ∈ T⊥ ∧ ∀t ∈ T .eval(д, t) ∈ S⊥. The unit ofthe monoidal product is:

I = (I , 1, idI ,C(I , 1)).It is star-autonomous, with negation as follows:

(U ,X ,T ,T⊥)⊥ = (X ,U ,T⊥,T )

Furthermore, if C has coproducts, then CDG got productsand coproducts:

• (U ,X , S, S⊥) ⊕ (V ,Y ,T ,T⊥) = (U ⊕ V ,X & Y , S ⊕ T , S⊥ &T⊥).• (U ,X , S, S⊥) & (V ,Y ,T ,T⊥) = (U &V ,X ⊕ Y , S &T , S⊥ ⊕ T⊥).• 0CDG = (0, 1, ∅,C(I , 1)).• 1CDG = (1, 0,C(I , 1), ∅).

Theorem 2.4. If C is a monoidal closed category with prod-ucts and coproducts, then CDG is star-autonomous with prod-ucts and coproducts.

CDG is more precise than DG in the sense that there isa clear notion of polarities that appears for both objects andmorphisms.

3 A relational account of permutationsAn MLL proof structure is, fundamentally, simply a type-respecting bijection between literals of opposite polarities. Adecisive step towards the construction of our model consistsin building a suitable semantics for such functions. Namely,we target a model that can be given a dual dynamic-staticaccount: the morphisms rest in Rel and compose accordingly,while being possibly interpreted as strategies, and composedas such. Our model takes heavy inspiration from the concur-rent games model of Abramsky and Mélliès[5], and we willtherefore work within the realm of domain theory, althoughthis structure does not appear to be crucial for our model.To start, we briefly recall some standard definitions fromdomain theory.

All domains/sets we will be working with are finite, so wewill not be concerned with continuity and limits of chains. Adomain (D,⊑,⊥) is a poset (D,⊑)with aminimal element⊥.A set of elements S is compatible, written ↑ S if it is bounded.A domain is bounded complete if every compatible S has aleast upper bound, denoted

⊔S . A bounded complete domain

automatically has meets, written.. A bounded complete

domain is a (complete) lattice if every subset ofD is compat-ible. A prime of a bounded complete domain is an elementp ∈ D such that ∀S ⊆ D st ↑ S .p ⊑

⊔S ⇒ ∃x ∈ S .p ⊑ x .

Pr(D) stands for the subset of primes of D. Furthermore, Dis prime algebraic, also called di-domain if it is boundedcomplete and any element can be written as a finite joinof primes. A domain (D,⊑,⊥) is qualitative if it is primealgebraic and no distinct primes are related by ⊑. A qualita-tive domain is a coherence domain if, given two elementsx ,y, written as sets of primes x =

⊔i pi , y =

⊔j qj , then

(∀i, j .pi ↑ qi ) ⇔ x ↑ y. That is, a coherence domain isentirely defined by its set Pr(D) together with the binary

coherence relation between primes. A polarized domain(D,⊑,⊥, λ) is a di-domain (D,⊑,⊥) together with a polar-ity function λ : Pr(D) → −1, 1. Given an element x ofa polarized domain, we write pos(x) for the join of all thepositive primes under x , and neg(x) for the one of negativeprimes. A coherence domain is polarized if it is polarized asa di-domain. As we will speak about those later, we recall thedefinition of closure operator on a prime algebraic domain,and introduce the concept of P-closure operators.

Definition 3.1. Given a di-domain D, we write D⊤ for itsupper bound completion, that consists in adding a maximalelement ⊤. D⊤ is a lattice. A closure operator σ on D, writ-ten σ : D, is a function σ : D⊤ → D⊤ that satisfies thefollowing properties:

• σ 2(x) = σ (x) (σ is idempotent)• σ (x) ⊒ x (σ is extensive)• x ⊒ y ⇒ σ (x) ⊒ σ (y) (σ is increasing)

Furthermore, if D is polarized, then σ is positive, or a P-closure operator, if:

• ∀x ∈ D.σ (neg(x)) , ⊤. (σ is civil)• ∀x ∈ D.σ (x) , ⊤ ⇒ (∀p ∈ Pr(D).(p ⊑ σ (x) ∧ ¬(p ⊑

x)) ⇒ λ(p) = 1)• ∀x ∈ D, σ (x) = σ (neg(x)) ⊔ pos(x).

Given σ ,τ : D, we write ⟨σ ,τ ⟩ ∈ D⊤ for the element:⟨σ ,τ ⟩ =

⊔(σ τ )n(⊥) | n ∈ N.

that corresponds to the resulting position of playingσ againstτ . Note that ⟨σ ,τ ⟩ = ⟨τ ,σ ⟩.

Proposition 3.2. σ positive, τ negative entails ⟨σ ,τ ⟩ , ⊤.

We introduce the denotation function ⟦⟧Qual that mapseach formula of MLL to a polarized qualitative lattice. Wemodel each literal by a polarized, labelled version of theSierpinski domain O, and the unit by a flat domain.

⟦X⟧Qual = (⊥,⊤X ,⊥ ⊑ ⊤,⊥, λ(⊤X ) = 1) = OX

⟦I⟧Qual = (⊥, (⊥,⊥),⊥, λ = ∅)

Our lattices can be negated by negating the polarity function:(D,⊑,⊥, λ)⊥ = (D,⊑,⊥,−λ).

The tensor product of two domains is given by their Carte-sian product. In particular, note that Pr(D1×D2) = (Pr(D1)×

⊥2) ⊎ (⊥1 × Pr(D2)) ≃ Pr(D1) ⊎ Pr(D2), where ⊎ is thecoproduct in the category of sets and functions. We can thenform the function λ1 ⊎λ2 : Pr(D1 ×D2) → −1, 1. Formally,this yields:

(D1,⊑1,⊥1, λ1) × (D2,⊑2,⊥2, λ2) =

(D1 × D2, (⊑1 × ⊑2), (⊥1,⊥2), λ1 ⊎ λ2).

In particular, this allows us to assign to each formula ofMLLits associated polarized qualitative lattice:

⟦A ⊗ B⟧Qual = ⟦A⟧Qual × ⟦B⟧Qual

⟦A` B⟧Qual = (⟦A⟧⊥Qual × ⟦B⟧⊥Qual)⊥ = ⟦A⟧Qual × ⟦B⟧Qual

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We now present the morphisms of our model. Since wewill later consider a slightly different model whose objectsare coherence domains, which are more general than quali-tative lattices, our definitions of morphisms are written forthem. The morphisms are monotone functions f that arealso continuous: f (⊥) = ⊥. Furthermore, we require themorphisms to be linear , meaning stable and additive.

• (Stability) ∀x ,y ∈ D,x ↑ y ⇒ f (x ⊓y) = f (x) ⊓ f (y)• (Additivity) x ↑ y ⇒ f (x ⊔ y) = f (x) ⊔ f (y)

Since f is stable, it is entirely defined by its trace:tr(f ) = (x , p) | p ∈ Pr(D),x ∈ D minimal s.t p ⊑ f (x).

The reverse operation being:f (x) =

⊔p | (y, p) ∈ tr(f ),y ≤ x.

The fact that a function f is additive entails that the firstcomponent of each pair in its trace is a prime.We enforce f tobe polarized, that is, in each pair of tr(f ) the first componentis anO-prime and the second a P-prime. Furthermore, since fshould denote a proof structure, we also impose the followingproperties, that we call typed-permutation conditions.

• ∀(p1, p′1), (p2, p′2) ∈ tr(f ).p′1 , p′2 ⇒ p1 , p2 that is,tr(f ) is an injective partial function from negative topositive primes.

• ∀p ∈ Pr(D).Neg(p) ⇒ ∃p′.(p, p′) ∈ tr(f ) : tr(f ) istotal.

• ∀p ∈ Pr(D).Pos(p) ⇒ ∃p′.(p′, p) ∈ tr(f ) : tr(f ) issurjective.

• (p, q) ∈ tr(f ), p ∈ ⟦X⟧ ∧ p ∈ ⟦Y⟧ ⇒ X = YThis above essentially states that the trace, and therefore f

establishes a bijection between negative and positive primes,and that this function corresponds to a proof-structure. Toeach proof structure can be associated such a morphism, asstated in the following proposition.

Proposition 3.3. Let π be a bijection between negative andpositive primes of a coherent polarized domain. Then f , definedas follows:

f (x) =⊔

p | π−1(p) ⊑ x

is a linear monotone function, whose trace is indeed (p,π (p).

We introduce the category PermRel.

Definition 3.4. We define the category PermRel as having:• polarized qualitative lattices arising from denotationsof formulas of MLL as objects.

• As morphisms A → B linear functions f : A⊥ × B →

A⊥ × B satisfying the typed-permutation conditionsas morphisms.

Given a morphism f , we write f • for the set of positionsof interactions, that is defined to be the closure under finitecompatible union of the following set:

f ⋔ = a ⊔ f (a) | (a, f (a)) ∈ tr(f ).

That is:f • = x | ∃y1, ...,yn ∈ f ⋔ . ↑ y1, ...,yn ∧ x =

⊔y1, ...,yn.

Note that ⊥ =⊔

∅ ∈ f •. To understand the dynamicalcontent of f •, one needs to consider σ (x) = x ⊔ f (x). σ isa P-closure operator, and hence a strategy in the sense ofconcurrent games. Furthermore, f • ⊆ σ • (where σ • is theset of fix-points of σ ). f • selects the positions of σ • that canonly be reached from ⊥ through an interaction against anOpponent-strategy that plays O-primes. In other terms, f •selects those positions of σ • that are relevant.

Lemma 3.5. We can equivalently define f ;д in the four fol-lowing ways:

• composing f ,д through the Kahn semantics of data-flow[18].

• composing relationally their sets of positions: (f ;д)• =f •;Rel д•

• by trace composition of their underling permutation [1].• by composing their associated closure-operators, andextracting the "output function" of the resulted closure-operator.

Therefore, the positions of interactions give us a relationalaccount of permutations. Furthermore, it refines the set ofpositions of the associated closure operator.

Proposition 3.6. • Given amorphism f : D of PermRel,f • forms a sub-lattice of D.

• PermRel is star-autonomous.

4 Dualization and full-completeness4.1 The categoryOf course the category PermRel is not fully-complete, sinceto each morphism corresponds a proof-structure, but thisone does not necessarily satisfy the correctedness conditions.We show in this section howwe can apply ideas coming fromSection 2 to our model. Note that we also take inspirationfrom the graph games of Hyland and Schalk [16]. The objectsof our new category CDGPermRel are of the form:

(D, P ,O,T ,T⊥)

with D being an object of PermRel, P ,O ⊆ D and T ⊆ P(P)and T⊥ ⊆ P(O). We denote:⟦X⟧CDGPermRel = (⟦X⟧PermRel, ⊤X , ⊥, ⊤X , ⊥)

The product is defined as in section 2.3. Namely:A ⊗ B = DA × DB , PA × PB ,OA × PB ⊎ PA ×OB ,TA ×TB ,T

⊥A⊗B

T⊥A⊗B = R ∈ P(OA⊗B ) |

∀Q ∈ TA.Q;A R ∈ T⊥B ∧ ∀Q ∈ TB .R ;B Q ∈ T⊥

A

The unit I and the negation are defined by:ICDGPermRel = (⟦I⟧PermRel, ⊥, ∅, ⊥, ∅)

(D, PD,OD,TD,T⊥D )

⊥ = (D⊥,OD, PD,T⊥D ,TD).

We define QA = OA ⊎ PA. Note that, unfortunately, ourmodel does not cope well with units. For instance, PI`I = ∅,

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and therefore T(⊥⊗⊥)`(I`I ) = ∅, despite the fact that thereare two proofs ⊢ ⊥ ⊗ ⊥, I ` I . However, it works well whenit comes to the restriction of MLL without units MLL−. Thecategorical models of MLL− are no longer star-autonomouscategories but semi-star autonomous categories, whose defi-nition can be found in [10].

Definition 4.1. CDGPermRel is the category with objectsthe 5-tuples A = (DA, PA,OA,TA,T

⊥A ), where DA is an object

of PermRel, PA ⊆ DA, OA ⊆ DA, TA ⊆ P(PA), and T⊥A ⊆

P(OA) that are freely generated from ⟦X⟧ by tensor andnegation. ThemorphismsA → B aremaps f : PermRel(DA,DB )such that:

• ∃R ∈ TA⊸B satisfying R ⊆ f • ∩QA⊸B• f • ∩QA⊸B ⊆ PA⊸B

Note that we have not used the condition f • ∩ QA⊸B ∈

TA⊸B but aweaker one in order to prove composition. Indeed,let us consider two morphisms of CDGPermRel as followsf : A ⊸ B and д : B ⊸ C such that f • ∩QA⊸B ∈ TA⊸B andsimilarly for д. Then, in order to establish composition, wemust prove that (f •;д•) ∩QA⊸C ∈ TA⊸C . By definition ofthe Chu double-glueing construction, (f • ∩QA⊸B ) ;Rel (д• ∩QB⊸C ) ∈ TA⊸C . The difficulty comes from the fact thatthere might be some b < QB such that (a,b) ∈ f •, (b, c) ∈д•, (a, c) ∈ QA⊸C . However, using the weakened conditionas in the definition allows us to circumvent the problem.

Proposition 4.2. CDGPermRel is a semi star-autonomouscategory.

Actually, it is a sub-semi-star-autonomous category ofPermRel. That is, the tensor, and negation operations liftfrom PermRel to CDGPermRel.

4.2 Full completenessMDNF stands for multiplicative disjunctive normal form. Itis the form for formulas of MLL− that results from applyingthe transformations (A`B) ⊗C → A` (B ⊗C) and its rightvariant in order to dispose of the` that are not at the bottomlevel in a formula.

Definition 4.3. A sequent Γ of MLL is MDNF if ; Γ is amultiset of formulas Fi and each Fi = ⊗jXi, j where Xi, j is aliteral.

A model is MDNF fully complete if it is fully complete forevery MDNF-sequent. That is, for every MDNF sequent Γ,every morphism R : I → ⟦Γ⟧ is the denotation of a proof.ProvingMLL− full-completeness relying on MDNF-formulasand sequents is an argument drawn from [23], itself takinginspiration from [3]. The proof structures, and proof nets,ofMLL−-MDNF formulas are much simplified compared tothe ones of MLL−. Indeed, as there is no `, there is no needto consider switchings anymore. We use the term “block” torefer to a formula Fi of Γ.

Proposition 4.4. CDGPermRel is MDNF fully-complete.

Given a morphism f of PermRel, and the proof structurePs canonically associated with it, we have to prove that Psis acyclic and connected. Note that we will make use of thefollowing lemma, whose proof is easily carried by induction.Lemma 4.5. ∀F formula ofMLL−:

• TF , ∅ ∧T⊥F , ∅.

• ∀R ∈ TF .R , ∅ and ∀Q ∈ T⊥F .Q , ∅.

• PF ∩OF = ∅.

Proof of 4.4. We start by showing that there is no cycle insidea block. Let us assume that there is one, and we name theblock F1. We call R the relation ofTΓ such that R ⊆ f •∩QΓ .Then let us pick a counter strategy Q ∈ T⊥

F2`....`Fn, and,

making it interact with R through R ;F2`...`Fn Q = S, weget that S ∈ TF1 . Let us assume that the block is of the shapeF1 = Y1⊗Y2⊗ ...⊗Yl ⊗X ⊗X⊥, and the strategy establishes anaxiom link between X and X⊥. Then let us pick a position xof S. As x X⊥ = x X , this entails that x X⊥ ∈ PX⊥ ⇔

x X ∈ OX and x X⊥ ∈ OX⊥ ⇔ x X ∈ PX . Therefore,x X ⊗ X⊥ ∈ OX ⊗X⊥ , and x < PF1 . This is a contradiction.Hence there is no cycle inside a block.

Next let us assume there is a cycle that goes through sev-eral blocks, and let us pick one of minimal length; therefore, itonly goes at most once through each block, passing throughtwo literals exactly. Given a block Fi on the cycle, we writeFi = Gi ⊗ l1

i ⊗ l2i , where l

1i , l

2i are the literals that belong

in the cycle. We then consider that the Fi have been rear-ranged in the order of the cycle, starting from l1

1 . This isdrawn in Figure 3. We write F1, .., Fm for the blocks throughwhich the cycle passes, and Fm+1, ..., Fn for the remainingones. Let us pick a counter-strategy Q ∈ T⊥

F2`..`Fn. Then,

writing S = R ;F2`...`Fn Q, we get S ∈ TF1 , In particu-lar, S is non-empty and therefore ∃x ∈ S, x ∈ PF1 ,∃y ∈

OF2`...`Fn .(x ,y) ∈ R and in particular (x ,y) ∈ f • ∩ QΓ .This implies ∀n ≥ i > 1.y Fi ∈ OFi . As x is a P-position,this implies that x l1

1 ∈ Pl 11and x l2

1 ∈ Pl 21. As y F2

is an O-position, it implies that there is exactly one literall of F2, such that y l ∈ Ol . This has to be l1

2 , as x l21 is a

P-position, and the polarity of y in l21 must be the opposite.

This implies that y l22 is positive. Following the same rea-

soning, it implies that y l13 is negative, and y l2

3 positive.Repeating this, we finally obtain that y l1

m is negative andy l2

m is positive. But the polarity ofy l2m is the opposite of

the one of x l11 , that is positive. Hencey l

2m is positive and

negative, bringing a contradiction. We deduct the acyclicityof the proof structure.We now prove connectedness. Note that every literal of

a block belongs to the same connected component. Sup-pose that there are two connected components (the resultgeneralises straightforwardly for more), that we split intotwo sets of blocks F1, ..., Fk and Fk+1, ..., Fm . We pick a ran-dom block in the first connected component, and nameit F1. We consider a counter-strategy Q ∈ T⊥

F2`F3`....`Fm.

Let us consider a position (x ,y) such that y ∈ Q and x ∈

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G1 ⊗ l11 ⊗ l2

1 G2 ⊗ l12 ⊗ l2

2 G3 ⊗ l13 ⊗ l2

3.... Gm⊗ l1

m⊗l2m

⊗

⊗

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⊗

⊗

⊗

⊗

⊗

Fm+1 .. Fn

Figure 3. A cycle inside a proof structure corresponding to a MDNF sequent

R ;F2⊗...⊗Fm y as above. Then by definition, x is a player po-sition, and all positions y Fi are O-positions. In particular,y Fk+1 ` ... ` Fm ∈ OFk+1`...`Fm . On the other hand, wealready know that the part of the relation in Fk+1 ` ...` Fmcorresponds to an acyclic proof structure. Furthermore, as itis connected, it corresponds to a proof. Therefore it is a deno-tation of a proof and belongs in PFk+1`...`Fm by soundness.Consequently, y ∈ PFk+1`...`Fm ∩OFk+1`...`Fm = ∅. This isa contradiction, and the structure is connected.

Finally, we simply note that MDNF full completeness en-tailsMLL− full completeness in any semi star-autonomouscategory such that each morphism corresponds a proof-structure. This has already been devised in [3, 23]. Thisallows us to conclude this section.

Theorem 4.6. CDGPermRel is fully-complete for MLL−.

As a byproduct, we can deduce that every morphism ofCDGPermRel can be formed by tensor and negation startingfrom axiom-links, and hence satisfy a stronger conditionthan the one expressed in the definition of CDGPermRel.

Corollary 4.7. CDGPermRel can be presented as the cate-gory having same objects as before and as arrows A → B mor-phisms f : PermRel(DA,DB ) such that f • ∩QA⊸B ∈ TA⊸B .

5 Towards the additivesThe model admits a natural extension to the additives. Giventwo polarized lattices A,B we define their sum as follows:

A ⊎ B = (DA ⊎ DB ,⊑A ⊎ ⊑B , λA ⊎ λB ).

A ⊕ B is not a domain anymore, as it does not have a uniqueminimal element.

Definition 5.1. A polarized (qualitative) sum-lattice (respsum-domain) is an object of the form ⊎iDi , where each Diis a polarized (qualitative) lattice (resp domain).

The tensor product lifts straightforwardly to polarizedsum-domains: given two polarized sum-domains

⊎Di and⊎

Dj , then (⊎

i Di ) ⊗ (⊎Dj ) = (

⊎i Di )×(

⊎Dj ) ≃

⊎i, j Di ×

Dj . Similarly, the negation is defined by negating all objects(⊎

i Di )⊥ =

⊎i D

⊥i . This lifting from domains to objects via

sums is analogous to the family construction [4]. Using this,

we can extend the model to cover the additives.

A ⊕ B =(inl(DA) ⊎ inr(DB ), inl(PA) ⊎ inr(PB ), inl(OA) ⊎ inl(OB ),

inl(TA) ⊎ inr(TB ), inl(T⊥A ) × inr(T⊥

B ))

A & B =(inl(DA) ⊎ inr(DB ), inl(PA) ⊎ inr(PB ), inl(OA) ⊎ inl(OB ),

inl(TA) × inr(TB ), inl(T⊥A ) ⊎ inr(T⊥

B ))

Definition 5.2. We define CDGPermRelA to be the cate-gory with denotations of formulas ofMALL− as objects, andas morphisms A → B relations R : ⟦A ⊸ B⟧ such that,∀Ψ additive resolutions, ∀x ∈ R .x ∈ R Ψ ⇒ (∃f ∈

CDGPermRel(A → B Ψ).x ∈ f • ∧ R Ψ = f •).

Proposition 5.3. Let R be a morphism of CDGPermRelA.Then for every &-resolution Ψ there exists a unique additiveresolution Φ on Ψ such that R Φ , ∅ and R Φ is adenotation of anMLL− proof.

In particular our morphisms define Hughes-Van-Glabbekproof structures [15], and those satisfy slice-correctedness.We have not, at this stage, been able to prove or refute afull-completeness result for this model.

6 Hypercoherence, full-completeness andthe Pagani Conjecture

6.1 Hypercoherence and the Pagani conjectureHypercoherences were proposed by Ehrarhrd in [11] asa refinement of the concept of coherence, originally de-signed by Girard [14]. Pagani conjectured in [22] that theywere powerful enough to characterize sequentializable proof-structures of MALL−, at the condition that theses ones wereslice-connected. Morphisms of the category of hypercoher-ence are called hyperclique.

Conjecture 6.1. (Pagani). If Θ is a proof structure, Θ is slice-connected and Θ is a hypercerclique for any interpretation ⟦⟧from proof structures to hypercoherence spaces then Θ is aproof net.

It is important at this point to recall that if any interpre-tation of an MLL− proof structure is a hyperclique, then itis sequentializable in MLL− +MIX, or, in other terms, it isacyclic but not necessarily connected [24]. In particular, ifthe proof structureΘ is an hyperclique for any interpretationand it is slice-connected, then Θ is slice-correct.

Here we will accomplish two steps at once. First, we willprovide a fully-complete model ofMALL− arising from our

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first fully complete model of section 4, by enriching it withhypercoherence. Secondly, we prove that this model providesa proof of the Pagani conjecture through semantical means.The main ingredient of the proof is a copycat of the proofof full-completeness for a category of double-glued hyper-coherence provided in [7], adapted to cope with our model.However, an important obstacle to overcome is to prove thatthe condition (P1) composes, and is tackled through a newlyestablished relation between hypercoherence and closureoperators.

6.2 Two categoriesWe slightly change the interpretation of the atomic variables,in order for it to have hypercoherent and hyper-incoherentsubsets. For each type variable X we introduce a count-ably infinite set of names AX . We set A = ⊎AX . We definePermNomRel similarly as PermRel, but replacing OX witha larger domain.

Definition 6.2. The objects of PermNomRel are polarizedcoherent domains arising from denotations of formulas ofMLL as object with:

⟦X⟧PermNomRel = ⊥ ⊎ AX ,⊥ ⊏ AX ,⊥.

⟦I⟧PermNomRel = ⟦I⟧PermRel

The morphisms are linear functions satisfying the typed per-mutations. We furthermore require that if (p, q) ∈ tr(f ), thenp, q corresponds to the same element of AX , and that tr(f )is consistent: writing inl for the straightforward inclusion⟦l⟧ → ⟦F⟧when l is an occurrence of a type variable in F , weimpose (inl (p), inl ′(q)), (inl (p′), inl ′′q′) ∈ tr(f ) ⇒ l ′ = l ′′.CDGPermNomRel is this category canonically decoratedwith Chu double-glued conditions as above with:

⟦X⟧CDGPermNomRel = (⟦X⟧PermNomRel,AX , ⊥,

a | a ∈ AX , ⊥)

CDGPermNomRel is fully complete for MLL−, and theproof is the same as above.WewritePfin for the finite subsetsoperator, and P∗

fin for the non-empty finite subsets operator.

Definition 6.3. A polarized coherence hypercoherence spaceis a pair (A, Γ(A)) where A is a polarized coherence objectand Γ(A) ⊆ P∗

fin(DA) is its set of hypercoherence. Γ(A) issuch that every singleton from A belongs in Γ(A).

We furthermore write Γ∗(A) for the subset of Γ(A) of el-ements that are not singletons. Γ(A) is the disjoint unionof Γ∗(A) with all singletons from A. We define Γ⊥(A) =P∗fin(DA)\Γ

∗(A), and we write Γ⊥∗(A) for the subset of Γ⊥(A)consisting of non-singleton sets. We will introduce the cate-gory HypPermRel, whose objects are qualitative polarized

coherence hypercoherence spaces as objects:⟦X⟧HypPermRel = (⟦X⟧PermNomRel,

Γ∗(X ) = x ⊎ ⊥ | x ⊆ Pfin,>1(AX )),

⟦I⟧HypPermRel = (⟦I⟧PermNomRel, Γ(I ) = ⊥)

where we denote Pfin,>1(AX ) the set of finite subsets of AXof cardinal more than 1. That is, Γ(X ) is the set of singletons,together with the sets a1, ...,an ,⊥. The way to think aboutit is that ⊥ switches the hypercoherence:, a ∈ Γ(X ) anda,⊥ ∈ Γ⊥(X ), a1, ...,an ∈ Γ⊥(X ), and a1, ...,an ,⊥ ∈

Γ(X ). For the remaining objects, it is defined by repeatedapplications of the required rules associated with the con-nectives. These were defined in [11], and we recall thembelow.Γ(A ⊗ B) = w ∈ P∗

fin(DA × DB ) |

w DA ∈ Γ(A) ∧w DB ∈ Γ(B)

Γ(A ⊕ B) = w ∈ Pfin(DA ⊎ DB ) | w ∈ Γ(A) ∨w ∈ Γ(B)

Γ(A⊥) = Γ⊥(A)

Definition 6.4. HypPermRel is the category containing:• as objects polarized coherence hypercoherence spaces(A, Γ(A)) generated from ⟦X⟧HypPermRel, ⟦I⟧HypPermRelusing ⊗, ⊕ and (.)⊥.

• as morphisms A → B relations R : DA⊸B such that:1. (P1) For each &-resolution Ω of A ⊸ B, there ex-ists a unique additive resolution Ψ on Ω such thatR Ψ , ∅ and R Ψ ∈ PermNomRel(A ⊸ B Ψ).

2. R is a hyperclique: ∀s ∈ P∗fin( R ), s ∈ Γ(A ⊸ B).

HypPermRel will allow us to study precisely the effect ofhypercoherence on the shape of the morphisms. In order tostudy the Pagani conjecture, we shall enforce that, at themultiplicative level, the relations are connected. Therefore,we design a second category HypCDGPermRel by addingthe Chu-conditions at the multiplicative level.

Definition 6.5. HypCDGPermRel is defined having:• the objects are built from ⟦X⟧HypPermRel, ⊗, ⊕ and (.)⊥.• the morphisms A → B are those relations R : DA⊸Bsuch that:1. (P1) for each &-resolution Ω of A ⊸ B, there is aunique additive resolution Ψ on Ω such that R Ψ , ∅ and R Ψ ∈ CDGPermNomRel(A ⊸ B Ω)

2. ∀s ∈ P∗fin( R ), s ∈ Γ(A ⊸ B).

Proposition 6.6. HypCDGPermRel is a sub-semi-star-autonomous category of HypPermRel.

Proposition 6.7. Let Θ a proof structure satisfying the hy-potheses of the Pagani conjecture. Then Θ can be denoted as amorphism of HypCDGPermRel.

Therefore, if one proves that HypCDGPermRel is fullycomplete forMALL−, then one can conclude that each proof-structure satisfying the hypotheses of the Pagani conjecture

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is actually sequentializable, that is, a proof-net, hence solvingthe conjecture.

6.3 Hypercoherence and closure operatorsIn this section we establish that each (P1) hypercoherence,that is, a morphism of HypPermRel, produces a concur-rent operator. Establishing this allows us to prove that mor-phisms of HypPermRel compose. Given two morphisms R :HypPermRel(A,B), andQ : HypPermRel(B,C), a ⊕-resolutionΨA of A, a &-resolution ΨC of C , the goal is to prove thatthere is a unique additive resolution ΦB of B such that ΨA R ΦB , ∅, and ΦB Q ΨC , ∅. That is, there exists aunique additive resolution on which R and Q meets.We start by defining a polarized di-domain for each for-

mula of MALL−, where each prime corresponds to the leftor right branch of an additive resolution.

• ⟦X⟧ = (⊥, (⊥,⊥),⊥) for each literal X .• ⟦A ⊗ B⟧ = ⟦A` B⟧ = ⟦A⟧ × ⟦B⟧• ⟦A ⊕ B⟧ is defined by a lifted sum: (⟦A⟧ ⊎ ⟦B⟧)⊥. Thetwo added initial primes are P-primes.

• ⟦A⊥⟧ = ⟦A⟧⊥, where the (.)⊥ operation consists inswapping the polarity of the primes. Consequently,⟦A & B⟧ = ⟦A⊥ ⊕ B⊥⟧⊥.

This interpretation is the same as the one of [5], withtrivial instantiation for each literal. Let us note that eachposition of the domain corresponds to a partial additive reso-lution of the formula, and the maximal positions correspondprecisely to total additive resolutions. We define a polarityfor each position of our domain, that ranges over a three-elements set: O, P ,N . The unique element ⊥ of ⟦X⟧, ⟦X⊥⟧has polarity N , whereas the root of ⟦A ⊕ B⟧ is O (seen asopponent winning), and the root element of ⟦A&B⟧ is P . Forthe multiplicative case, we devise the polarity as follows:

⊗ O P NO O O OP O P PN O P N

For the`, we setK`L = (K⊥⊗L⊥)⊥ whereO⊥ = P , P⊥ = O ,and N⊥ = N .Given R a relation of HypPermRel(A,B), we strip it by

forgetting all the elements of A in it: we focus on the ⊥

elements of the relation.

R = x ∈ R | ∃DA.x = ⊥A.

Of course, R ⊆ R , and R is a hyperclique. Note that ˜(.)maps each subset of R that is a relation of PermNomRel toa singleton. Respectively, each element in R corresponds toa PermNomRel relation subset of R .

Definition 6.8. We define SimpleHypRel as the full sub–category of HypPermRel restricted to those objects that arefreely built from ⟦I⟧HypPermRel using ⊗, ⊕, (.)⊥.

Given R ∈ HypPermRel(A,B), R ∈ SimpleHypRel(A, B),where the ˜(.)-operation sends a formula A to the same for-mula where each occurrence of atomic variable has been re-placed by an occurrence of I : ˜(.) acts as a functorHypPermRel →SimpleHypRel. Given R ∈ HypPermRel(A), then R projectsnaturally to a set of maximal positions of the domain ⟦A⟧.We define the corresponding closure operator σR on ⟦A⟧ asfollows:

σ R (x) =.y ∈ R | y ⊒ x

Similarly, given a &-resolution Ψ on A, one can associatea closure operator to it. Given the set of maximal positionsy of ⟦A⟧ that correspond to those additive resolutions thatare on Ψ, we define:

τΨ(x) =.y | y ⊒ x ,y on Ψ

Then let us consider a relation R ∈ SimpleHypRel(A,B)and Q ∈ SimpleHypRel(B,C), and a pair of &-resolutions(ΨA,ΨC ) of (A⊥,C). Then in order to prove composition, onemust prove that the following element:

⟨τΨA ;A σ R ,σQ ;C τΨC ⟩

is maximal. This element then corresponds to the uniqueadditive resolution such that both relations have elementsin it.

Lemma 6.9. Let τΨA ,σR ,σQ ,τΨC as above. Then:• τΨA ;A σR ∈ SimpleHypRel(B).• σQ ;τΨC ∈ SimpleHypRel(B⊥).

And finally, we prove the required property.

Proposition 6.10. Let R ∈ SimpleHypRel(A),Q ∈ SimpleHypRel(A⊥). Then ⟨σ R ,τQ⟩ is maximal, that isR ∩ Q , ∅

To establish this proposition given two simple hyperco-herences as above, we prove that the interaction betweentheir associated concurrent operators never reach a deadlock.This follows from the following lemma.

Proposition 6.11. Given R ∈ SimpleHypRel(A), and σ R

the associated closure operator, then σ R satisfies the following:• σ R is positive. This entails that given τ a negative clo-sure operator, ⟨σ ,τ ⟩ , ⊤.

• σ R is total:∀x ∈ σ •, eitherx has polarity P , orN . In par-ticular, this entails that given τ ∈ SimpleHypRel(A⊥),then ⟨σ ,τ ⟩ has polarity N , and hence is maximal.

The core of the proof is the following property:

Lemma 6.12. Let x be an element of the domain A, and x =y maximal | y ⊒ x. Then:

• x ∈ Γ∗(A) ⇔ x is P.• x ∈ Γ⊥∗(A) ⇔ x is O.• x ∈ Γ(A) ∩ Γ⊥(A) ⇔ x is N.

We conclude that morphisms of HypPermRel (andHypCDGPermRel) compose, and hence forms a category.

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αi,m−1 ⊗ α⊥i,m αi,m ⊗ (Bi1 &i Bi2) Mi [Ci,1[α⊥i,1] ⊕Ci,2[α⊥i,1]] αi,2 ⊗ α⊥i,2 ... αi,m−1 ⊗ α⊥i,m Ξ Fi+1

axi,1

axi,2

Figure 4. Local structure of the cycle

6.4 MALL− full completenessWeprove thatHypCDGPermRel is fully complete forMALL−

by checking that the proof provided in [7] by Blute, Hamanoand Scott for their category of double-glued hypercoherencesadapts well to HypCDGPermRel. Their use of the double-glueing allows them to obtain a category that rejects theMIX rule. However, this one did not provide a proof of thePagani conjecture since the condition that each linking λ isconnected is enforced by mixing hypercoherence and double-glueing.One of the key elements for the proof of [7] is first to

prove a full-completeness argument for PALL− +MIX usingHypPermRel. Indeed, this enables us to assign to each mor-phism of HypPermRel a GirardMALL− proof structure. Theproof makes full use of the structure of ⟦X⟧HypPermRel, thatis, it would not work if it were the Sierpinski domain.Assuming that there is a proof structure coming from a

morphism of HypCDGPermRel with a cycle, we can eventu-ally prove that there is a morphism ofHypPermRel encodinga proof structure with a cycle in it of the type ⊢ F1, ..., Fnwhere:

Fi = αi,m ⊗ (Bi1 & Bi2),Ni [α⊥i+1,1,α

⊥i+1,1],

αi+1,1 ⊗ α⊥i+1,2, ....,αi+1,m−1 ⊗ α⊥

i+1,m ,Ξi

with Ξi = Ei11 ⊕ Ei12, ...,Eim−1 ⊕ Eim , li1, ..., lir ,Ni = Mi [Ci,1[α

⊥i ] ⊕ Ci,2[α

⊥i ]] where Mi is a context made

out of ` and ⊕. Furthermore, the linkings are such that thechoice of B1 entails the choice of the left α⊥

i , and respectivelyfor B2 and the right one, while otherwise letting the otherlinks that appear in the cycle intact. Locally, the linking is asin figure 4. We prove that given a relation R ofHypPermRelencoding such a proof structure there exists S ⊆ R suchthat S ∈ Γ⊥,∗(F1 ` ... ` Fm). Consequently, R is not anhyperclique, and such a proof structure is rejected.

Theorem 6.13. HypCDGPermRel is a fully complete modelofMALL−. Furthermore, there is a one-to-one correspondencebetween the morphisms of HypCDGPermRel on MALL− andthe equivalence classes of MALL− proofs of linear logic.

This theorem relies on the characterization of equivalenceclasses of MALL− using Hughes and Van-Glabbeek proofstructures [25].

Corollary 6.14. The Pagani conjecture is true.

A direct consequence of that corollary is a new sequen-tialization criterion for the Hughes-Van-Glabbek proof struc-ture [15]. This one simply consists in mixing the hypercor-rectedness criterion devised by Tranquilli [24] with slice-connectedness, that is, the (P2 -MLL) condition of [15]. In

particular, if the morphisms of CDGPermRelA give rise tohypercorrect proof structures then the conjectured full com-pleteness result for MALL− holds.

References[1] S. Abramsky. Abstract Scalars, Loops, and Free Traced and Strongly

Compact Closed Categories. CALCO’05.[2] S. Abramsky and R. Jagadeesan. New foundations for the geometry of

interaction. In LICS, 1992.[3] S. Abramsky and R. Jagadeesan. Games and full completeness for

multiplicative linear logic. J. Symb. Log., 59(2):543–574, 1994.[4] S. Abramsky and G. McCusker. Call-by-Value Games. CSL ’97.[5] S. Abramsky and P-A. Melliès. Concurrent games and full complete-

ness. LICS’99.[6] E. S. Bainbridge, P. J. Freyd, A. Scedrov, and P. J. Scott. Functorial

polymorphism. Theor. Comput. Sci., 70, 1990.[7] R. Blute, M. Hamano, and P. J. Scott. Softness of hypercoherences and

MALL full completeness. Ann. Pure Appl. Logic, 131, 2005.[8] P. Boudes. Hypercoherences et Jeux. PhD thesis, Université de la

Mediterranée - Aix Marseille II, 2002.[9] V. Danos and L. Regnier. The structure of multiplicatives. Archive for

Mathematical Logic, 28(3):181–203, 1989.[10] Hughes D.J.D, R. Houston, and A. Schalk. Modelling Linear Logic

without Units (Preliminary Results).[11] T. Ehrhard. Hypercoherences: A Strongly Stable Model of Linear Logic.

In Mathematical Structures in Computer Science, 1993.[12] T. Ehrhard. Parallel and serial hypercoherences. Theoretical Computer

Science, 247(1):39–81, 2000.[13] J-Y. Girard. Linear logic. Theor. Comput. Sci., 50:1–102, 1987.[14] Jean-Yves Girard. The system f of variable types, fifteen years later.

Theoretical Computer Science, 1986.[15] D. J. D. Hughes and R. J. Van Glabbeek. Proof Nets for Unit-free

Multiplicative-Additive Linear Logic. In LICS 2003.[16] M. Hyland and A. Schalk. Abstract Games for Linear Logic Extended

Abstract. ENTCS, 1999.[17] M. Hyland and A. Schalk. Glueing and orthogonality for models of

linear logic. Theoretical Computer Science, 294(1):183–231, 2003.[18] G. Kahn and D. Macqueen. Coroutines and Networks of Parallel

Processes. Research report, 1976.[19] R. Loader. Linear logic, totality and full completeness. LICS’94.[20] G.McCusker. Games and Full Abstraction for a FunctionalMetalanguage

with Recursive Types. PhD thesis, University of Oxford, 1997.[21] P-A. Mellies. Asynchronous games 4: A fully complete model of

propositional linear logic. LICS ’05.[22] Michele Pagani. Proof Nets and Cliques: towards the Understanding of

Analytical Proofs. PhD thesis, 2006.[23] A. Schalk and H. Steele. Constructing Fully Complete Models for

Multiplicative Linear Logic. LICS ’12, 2012.[24] P. Tranquilli. A Characterization of Hypercoherent Semantic Correct-

ness in Multiplicative Additive Linear Logic. CSL ’08.[25] Rob J. van Glabbeek and Dominic J. D. Hughes. MALL proof nets

identify proofs modulo rule commutation. 2016.[26] P. Wadler. Propositions As Sessions. ICFP ’12, 2012.

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A Proofs of section IITheorem A.1. If C is a monoidal closed category with products and coproducts, then CDG is a star-autonomous with productsand coproducts.

The proof is a simple mix of the proof for Cd and DG. We repeat it here for completeness sake.

Proof. We first prove that this is a category. Given two morphisms (f1,д1) and (f2,д2) that are compatible for composition,their composition is defined by;

(f1,д1); (f2,д2) = (f1;C д1, f2 Copд2)

The composition is automatically strictly associative, and given an object (U ,X ,T ,T⊥), (idU , idX ) is automatically the identity.Given an object (U ,X ,T ,T⊥), then:

(U ,X ,T ,T⊥) ⊗ I = (U ⊗ I ,U ⊸ 1 × I ⊸ X ,T × idI , (f ,д) | DG conditions ,д ∈ C(I , 1))≃ (U ⊗ I ,X ,T × idI , f | DGconditions for f )

That is, given an object A, A ⊗ I ≃ I . Since a morphism is defined as a object of A ⊸ B, monoidal closure is straightforward.and hence star-autonomy as well. The products-coproducts obviously satisfy their universal property, and it is routine to checkthat A & 1 ≃ A and A ⊕ 0 ≃ A.

B Proofs of section IIILemma B.1. If σ is positive, τ is negative, then ⟨σ ,τ ⟩ , ⊤.

Proof. The proof is done by induction on the growing chain yn defined as follows:• y0 = ⊥.• yn+1 = σ (yn) if n is even.• yn+1 = τ (yn) if n is odd.

We prove by induction that for each n in the chain σ (yn) = σ (neg(yn)) and τ (yn) = τ (pos(yn)). This is true for the case casen = 0. We do the induction case, dealing first with σ . If n is even then σ (yn+1) = σ (σ (yn)) = σ (yn), making the inductionstraightforward. In the case where n is odd, σ (yn+1) = (σ (τ (yn)). As σ is positive, σ (yn+1) = pos(yn+1) ⊔ σ (neg(yn)). As τ isnegative, and τ (yn) , ⊤ by induction, pos(τ (yn)) = pos(yn). Therefore, σ (yn+1) = σ (neg(yn+1)) ⊔ pos(yn). On the other hand,σ (yn) = σ (neg(yn)). Therefore, pos(yn) ⊆ σ (yn) ⊆ σ (yn+1). Consequently, σ (yn+1) = σ (neg(yn+1)) as required. The proof for τis done on an equal footing. In particular, this leads to σ (yn) , ⊤, τ (yn) , ⊤, and hence yn , ⊤. Consequently, ⟨σ ,τ ⟩ , ⊤.

We recall the Kahn semantics of dataflow [1, 8]. Given f : A× B and д : B ×C , then f ;д(a, c) = (a′, c ′) ∈ A×C where (a′, c ′)is defined to be the least solution of the following equations:

• f (a,b) = (a′,b ′)• д(b ′, c) = (b, c ′)

Note that b ′ can be computed as the limit of the growing chain⊔

i ∈N((πB ; f (a, _)) (πB ;д(_, c)))i (⊥). As the domain onlyadmits finite chains, f ,д are automatically continuous, and hence b,b ′ always exists and are well defined.

Proposition B.2. • f • forms a lattice.• PermRel forms a category, and (f ;д)• = f • ;Rel д•.

Proof. To show that f • forms a lattice, we have to prove it is stable under finite union and intersection. The stability underfinite union follows from its definition. Now, given a finite family of xi in f •, xi =

⊔j (pj ⊔ f (pj )), then

.xi =

⊔pk ⊔ f (pk ) |

∀i .pk ≤ xi , (this follows from a simple computation using that tr(f ) is a bijection between negative and positive primes), andhence

.xi ∈ f •.

We now start to prove that linear functions compose, and that the additional criteria are respected.Let us consider two functions f : A × B and д : B ×C , together with (a, c) ∈ A ×C . We assume that f ;д(a, c) = (a′, c ′), that

is (a′, c ′) is the minimal pair solution of the equations:• f (a,b) = (a′,b ′)• д(b ′, c) = (b, c ′).

In that case, we say that (a,b), (b ′, c) satisfy the equational system for the composition.To tackle the proof, we will use an additional property of our functions, namely their stability under the complement

operator , denoted \. Given x ,y two elements of the qualitative domain, seen as sets of primes, we write x \ y for the element⊔p | p ∈ Pr(D), p ⊑ x ∧ p @ y. Then our functions satisfy the following property:

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f (x \ y) = f (x) \ f (y).

This is straightforward to prove, seeing tr(f ) as a bijection between negative and positive primes.The composite is stable: Let us compare f ;д(x ⊓ y) versus f ;д(x) ⊓ f ;д(y). Let x = (xA,xC ), and y = (yA,yC ) be compatible.

Let x1B , x

2B the minimal elements of interaction, that is, such that f (xA,x1

B ) = (x ′A,x

2B ) and д(x

2B ,xC ) = (x1

B ,x′C ). Similarly, we

introduce y1B ,y

2B . As f ,д are stable, we have:

f (xA ⊓ yA,x1B ⊓ y1

B ) = (x ′A ⊓ y ′

A,x2B ⊓ y2

B )

д(x2B ⊓ y2

B ,xC ⊓ yC ) = (x1B ⊓ x2

B ,x′C ⊓ y ′

C ).

Therefore, (xA ⊓yA,x1B ⊓y1

B ), (x2B ⊓y2

B ,xC ⊓yC ) satisfy the equational system. However, they might not be minimal. Therefore,we already know that:

f ;д(xA ⊓ yA,xC ⊓ yC ) ⊑ f ;д(xA,xC ) ⊓ f ;д(yA,yC ).

Let suppose there is some lesser v1,v2 such that f (xA ⊓ yA,vB ) = (wA,v2) and д(v2,xC ⊓ yC ) = (v1,wC ) and v1 ⊑ x1

B ⊓ y1B ,

v2 ⊑ x2B ⊓ y2

B (where at least one of the inequalities is strict). We write z1 for x1B ⊓ y1

B , and z2 for x2B ⊓ y2

B . The followingequations hold:

f (⊥, z1 \v1) = f (xA ⊓ yA, z1) \ f (xA ⊓ yA,v1) = (z ′A, z2 \v2)

and

f (z2 \v2,⊥) = f (z2,xC ⊓ yC ) \ f (v2,xC ⊓ yC ) = (z1 \v1, z ′′C ).

Therefore, (⊥, z1\v1) and (z2\v2,⊥) satisfy the equational system. But therefore, so does (xA,x1B \(z

1\v1)) and (x2B \(z

2\v2),xC ).Or, x1

B and x2B are minimal. This implies xB1 ⊓ (z1 \v1) = ∅ (and similarly for x2

B , z2,v2). Or, as z1 = x1

B ⊓ y1B , this entails that

z1 ⊑ v1. Or v1 was chosen to be minimal so z1 = v1. So x1B \ (z1 \v1) = x1

B ⇒ z1 = v1. A similar pattern leads us to concludethat z2 = v2. So, overall, z1, z2 are minimal. As a result, this entails:

f ;д(x1A ⊓ x2

B ,x1C ⊓ x2

C ) = f ;д(x1A,x

1C ) ⊓ f ;д(x2

A,x2C ),

that is, the composite function is stable.The composite is additive: Composition of additivity is proven as follows. Given (xA,x

1B ), (x

2B ,xC ) and (yA,y

1B ), (x

2B ,yC )

satisfying the equational system, we get (xA ⊔yA,x1B ⊔y1

B ), (x2B ⊔y2

B ,xC ⊔yC ) satisfying it as well, following the additivity of f .This proves that f ;д(xA⊔yA,xC ⊔yC ) ⊑ f ;д(xA,xC )⊔ f ;д(yA,yC ). Now as f ;д is monotone, then f ;д(x⊔y) ⊒ f ;д(x)⊔ f ;д(y)and by combining the two inequalities we get the result.

The separation conditions compose: We prove the proposition by analysing how f ;д acts on a negative prime. Suppose p ∈ Aand Neg(p). Then f (p,⊥) will trigger exactly one positive prime p′. Either this prime is in A, in which case f ;д(p) = p′, orthis prime is in B. Now, this prime p′, that is negative from the д point of view, will trigger, through д, a unique positive primep′′. This chain of p continues through f , д, playing new primes on B. As there is only a finite number of primes (compatiblewith each other) in B the process ends, that is, either f finishes the chain by playing in A, or д finishes it by playing in C .Therefore, each negative prime triggers one and only one positive prime through f ;д. Furthermore, this one is unique. Finally,by backtracking, one can see that every positive prime is triggered through f ;д by a single negative prime.

Polarised function do compose: We need to prove that for each couple (p, p′) ∈ tr(f ;д) then Neg(p),Pos(p′). This is straight-forward.

Equivalence sequential -relational composition: Let (xA,xC ) ∈ (f ;д)•. This implies that there existy1B ,y

2B , such that f (neg(xA),y

1B ) =

(pos(xA),y2B ), д(y

2B , neg(xC )) = (y1

B , pos(xC )). Furthermore, as f ,д is polarized then Neg(y1B ) and Pos(y

2B ). Therefore, (xA,y

1B ⊔

y2B ) in f •, (y2

B⊔y1B ,xC ) ∈ д•. Thus, (xA,xC ) ∈ f • ;Rel д•, and (f ;д)• ⊆ f • ;Rel д•. Now, let us consider (xA,xB ) ∈ f •, (xB ,xC ) ∈ д•.

Then, let us consider the set of those xi such that (xA,xi ) ∈ f •, (xi ,xC ) ∈ д•. This set is closed under compatible intersection.Therefore, we can take a least element. We rename this element xB . Now, f (neg(xA), neg(xB )) = (pos(xA), pos(xB )) andд(pos(xB ), neg(xC )) = (neg(xB ), pos(xC )). Furthermore, neg(xB ), pos(xB ) are minimal satisfying this set of equations. Hence(xA,xC ) ∈ (f ;д)•, and (f ;д)• = f •;д•.

Note that, as our the traces of our functions act as bijections, the composition through de Kahn semantics of data-flow isequivalent to the composition of permutations trough tracing. Furthermore, given σ = x ⊔ f (x) and τ = x ⊔д(x), then the f ;дis a output function for σ ;τ . That is, σ ;τ (x) = x ⊔ f ;д(x). Since for any f like this, σ (x) = f (x) when Neg(x), f is entirelydefined by σ . So we can compose f ,д by composing their corresponding closure operators, and extract the canonical outputfunction associated with the composite.

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PermRel is star-autonomous: the monoidal product being the cartesian product, and the negation consists in reversingpolarities on objects, together with taking f ⊥ as being the monotone linear function whose trace arise from the bijectiontr(f )−1 on morphisms. Finally, the unit of the monoidal product is the qualitative domain with no primes I .

C Proofs of section IVC.1 Proofs of section IV-ADefinition C.1. A symmetric semi-monoidal closed category is described by the following data:

• A category C.• Two functors ⊗ : C × C → C and⊸: Cop × C → C.• Three natural isomorphisms corresponding to the following properties:1. (symmetry) A ⊗ B ≃ B ⊗ A.2. (associativity) A ⊗ (B ⊗ C) ≃ (A ⊗ B) ⊗ C .3. (closure) C(A ⊗ B,C) ≃ C(A,B ⊸ C).

• A functor J : C → Set together with a natural isomorphisms C(A,B) ≃ J (A ⊸ B).A semi star-autonomous category is a symmetric semi-monoidal closed category with a full and faithful functor (.)⊥ : C →

Cop and a natural isomorphism:• C(A ⊗ B,C⊥) ≃ C(A ⊗ B⊥,C).

We now focus on proving that morphisms compose. In order to tackle the proof of this property we introduce the functionI that generalises the concept of O and P-position. The function I : D→ Z \ 0 (set that we refer to as Z∗ in the future )sends any position to a number, that is supposed to represent a generalised notion of polarity. It it sound in the sense that itsends P-positions to 1 and O-positions to −1. We will refer to I(x) as the payoff of x .The function I is defined by induction as follows:• For ⟦X⟧, I(⊥) = −1 and I(⊤X ) = 1.• Given any formula F , IF⊥ (x) = −IF (x).• Given two formulas A and B, I is computed according to the following table , where given an element (x ,y) ∈ ⟦A ⊗ B⟧,we write p,q for IA(x) and IB (y) respectively, and distinguish between the cases where p (respectively q) are positiveand negative:

⊗ p > 0 p < 0

q > 0 p + q − 1 if q > |p | then p + qif q ≤ |p | then p + q − 1

q < 0 if p > |q | then p + qif p ≤ |q | then p + q − 1 p + q

Another way of presenting this table is by introducing two new functions:

• η : Z→ Z∗ :

n 7→ n if n > 0n 7→ n − 1 if n ≤ 0

• ι : Z∗ → Z :

n 7→ n if n > 0n 7→ n + 1 if n < 0

This is not hard to see thatη, ι are inverse to one another. Then the above table could be synthesised intop⊗q = η(ι(p)+ι(q)−1).This allows us to conclude about the associativity of the ⊗:

p ⊗ (q ⊗ r ) = η(ι(p) + ι(η(ι(q) + ι(r ) − 1)) − 1) = η(ι(p) + ι(q) + ι(r ) − 2) = (p ⊗ q) ⊗ r .

By duality, we obtain the associativity for ` as well. We will also make use of the following property in a future proof.

Lemma C.2. • ∀m ∈ Z∗.m ` 1 =m ⊗ 2.• ∀m ∈ Z∗.m ⊗ −1 =m ` −2.

Proof. We only need to prove the first point by duality. That is, −(−m ⊗ −1) = (m ⊗ 2).• ifm > 0 thenm ⊗ 2 =m + 2 − 1 =m + 1. On the other and (−m ⊗ −1) = (−m − 1) as expected.• ifm = −1 thenm ⊗ 2 = 1, and (−m ⊗ −1) = 1 − 1 − 1 = −1 as expected.• ifm < −1 thenm ⊗ 2 = 2 +m − 1 =m + 1, and (−m ⊗ −1) = −m − 1.

We do the proof for CDGPermNomRel, but the proof is perfectly identical replacing CDGPermNomRel with CDGPermRel.13

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Lemma C.3. Let A ∈ Obj(CDGPermNomRel) :• Let x ∈ PA. Then I(x) = 1.• Let x ∈ OA. Then I(x) = −1.• Let x ∈ QA = PA ⊎OA. Then I(x) = 1 ⇒ x ∈ PA, I(x) = −1 ⇒ x ∈ OA.

Proof. The third point is a direct consequence of the two first ones. We prove the two first points together. The proof isby induction on the structure of the formula. If it is atomic, then it is by definition. In the case of A = A1 ⊗ A2, thenI(x1 ⊗ x2) = I(x1) ⊗ I(x2) = 1 ⊗ 1 = 1 since x1 ∈ PA1 ,x2 ∈ PA2 by definition of PA1⊗A2 . Finally, if A = A1 ` A2, thenI(x) = I(x1)`I(x2), and I(x1) = −1,I(x2) = 1, or the other way around I(x1) = 1,I(x2) = −1, by definition of PA`B . Then−1 ` 1 = 1, and we conclude that I(x) = 1. The proof for the OA`B and OA⊗B is dealt with similarly.

We will also need the following lemma.

Lemma C.4. • ∀F formula of MLL−.TF , ∅ ∧T⊥F , ∅,

• ∀F formula ofMLL−, ∀R ∈ TF .R , ∅ and ∀R ′ ∈ T⊥F .R

′ , ∅.

Proof. This is proven by mutual induction along the structure of the formula F . For the first point, the atomic case is bydefinition. In the case F = A⊗B, then we simply consider a relation R A× R B ∈ TA⊗B and a counter-strategy PA×τB ⊎τA×PB ,where τA ∈ TA⊥ , and τB ∈ TB⊥ . The case A` B is dealt with similarly.

For the second point the atomic case is by definition. For the elements of T , the inductive case ⊗ is automatic. For T⊥A⊗B ,

given a relation R ∈ T⊥A⊗B , then given Q ∈ TA, Q;A R ∈ T⊥

B . Hence Q;A R , ∅ by inductive hypothesis and hence R , ∅.The case for relations in A` B is proven along the same lines.

Proposition C.5. Let f ∈ PermRel(A,B) such that ∃R ∈ TA⊸B .R ⊆ f •. Then for any x ∈ f •,I(x) = 1.

Proof. Instead of working with A⊥ ` B, we work with a general formula of MLL−. So let f • : D where D is the qualitativepolarized domain ⟦F⟧Qual for a formula F ofMLL−. As there exists R inTF such that R ⊆ f • ∩QF , this implies, by the lemmaabove C.4 that there exists some x ∈ f • such that x ∈ PF . Consequently I(x) = 1.

Now we prove the following property: let us assume x such that I(x) = 1, and primes p, p′ such that p O-prime, p′ P-prime,p, p′ ↑ x , and both p, p ∈ x or p, p′ < x . Then I(x ⊎ p, p′) = 1 or I(x \ p, p′) = 1. That is, if we start from a position xof payoff 1, and do, or undo, a pair of OP primes, then we reach a new position of payoff 1. As every position of f • can beobtained from one another by adding or removing pairs ofOP-primes (this follows from the definition of f •) this will allow usto conclude that every position y of f • satisfies I(y) = 1.

In order to establish that, we simply need to prove that given a position x of payoff n, then if we add an opponent prime toit, or remove a player prime from it, we reach a position n ⊗ −1. By duality, if we add a player prime, or remove an opponentprime to a position of payoff n, then we reach a position of payoff n ` 1. Therefore, if we start from a position of payoff 1, andadd, or remove, a couple of OP primes then we will reach a position of payoff 1 ⊗ −1 ` 1 = 1.The proof is done by induction on the structure of the formula F . If x is a position of an atomic formula X , then if x is

a prime a, x has payoff 1. Then if we remove a prime from x we end up in the position ⊥, that has payoff −1 = 1 ⊗ −1.Similarly, if x has payoff −1 then if we add a P-prime to it, we reach a position of payoff 1 = −1 ` 1. The same reasoningworks for X⊥. We now focus on the induction case. Imagine that x is a position of a qualitative polarized domain denotationof a formula A = A1 ⊗ A2, and I(x) = 1. Then let us imagine we add a O-prime to x , and we consider without lossof generality that this O prime is in A1. Then setting n1 = I(x A1), n2 = I(x A2), we reach a position of payoff(n1 ⊗ −1)` n2 = (n1 ` −2)` n2 = (n1 ` n2)` −2 = (n1 ` n2) ⊗ −1. The other cases are dealt similarly.

So we can conclude that ∀x ∈ f •.I(x) = 1.

This was the last missing element needed to prove composition. As we proved composition only for morphisms of MLL−,we have to work in the category without units.

Proposition C.6. CDGPermNomRel is a semi star-autonomous category.

Actually, it is a sub semi star-autonomous category of PermRel. That is, the tensor, and negation operation lift from PermRelto CDGPermNomRel.

Proof. We start by proving that the morphisms compose. We consider two morphisms f : A → B and д : B → C ofCDGPermNomRel. Given R ∈ TA⊸B such that R ⊆ f • ∩ QA⊸B , and Q ∈ TB⊸C such that Q ⊆ д• ∩ QB⊸C , then R ;Q ⊆

(f ;д)• ∩ QA⊸C , and, by definition, R ;Q ∈ TA⊸C . To finish, we simply need to prove that (f •;д•) ∩ QA⊸C ⊆ PA⊸C . As14

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∃R ∈ TA⊸C such that R ⊆ f •, we know by proposition C.5 that ∀x ∈ f •.I(x) = 1. In particular, by lemma C.3, we canconclude that ∀x ∈ f • ∩QA⊸B .x ∈ PA⊸B as expected.

C.2 Proofs of section IV-BProposition C.7. In any semi star-autonomous category such that each morphism is the denotation of a proof structure,MDNFfull completeness entails full completeness.

Proof. Suppose that F is a formula such that there is a morphism f : I → F whose corresponding proof structure is cyclic(and, or, disconnected) for a given switching S . First, we handle acyclicity. Let us select an occurrence ` such that the cyclepasses through it. Then either the ` is at a lowest level, in which case we do nothing. Or there is a ⊗ that appears beforein the parse-tree. That is, it occurs in a context of the form (A ⊗ (B `C)). Then if the switching selects the formula B, wepost-compose f with the morphism that is identity almost everywhere but transforms (A ⊗ (B `C)) into ((A ⊗ B)`C). Thenthere is still a cycle in the newly obtained proof structure corresponding to the new morphism. By doing so repeatedly, weeventually obtain a MDNF formula that has a cycle in it. However, such a morphism is rejected by the model. Therefore, thereis no cycle in the original formula. The connectedness proof works on a similar basis.

Finally, we prove that CDGPermNomRel can be given a simpler presentation by replacing the condition:∃R ∈ TA⊸B .R ∈ f • ∩QA⊸B

f • ∩QA⊸B ⊆ P

in the definition, with:f • ∩QA⊸B ∈ TA⊸B .

We recall that the problem with the second condition was that we were not able to prove the composition. However, now,we can rely on the full-completeness result in order to establish that the second condition composes. Indeed, two morphismsbeing denotation of proofs of MLL−, they will compose just as their proof compose.Proposition C.8. CDGPermNomRel can be presented as the category having

• Polarised coherence objects equipped with Chu-structure as objects: A = (DA, PA,OA,TA,T⊥A ), where OA, PA ⊆ DA, TA ⊆

P(PA), T⊥A ⊆ P(OA), such that each object is the denotation of a MLL− formula.

• As morphisms A → B morphisms f : PermRel(DA,DB ) such that f • ∩QA⊸B ∈ TA⊸B .

Proof. To differentiate, we name CDGPermNomRel2 the latest construction, and CDGPermNomRel1 the former. The identitiesare morphisms of CDGPermNomRel2. Furthermore, given morphisms f ,д of CDGPermNomRel2, f ⊗ д is a morphism ofCDGPermNomRel2, just as is f ⊥. Therefore, any cut-free morphism of MLL− can be modelled within CDGPermNomRel2.Furthermore, (forgetting that the composition is not yet defined), CDGPermNomRel2 is a sub-semi-star-autonomous categoryof CDGPermNomRel1 (in the sense that every morphism of CDGPermNomRel2 is a morphism of CDGPermNomRel1 and theyshare the same tensor and negation). AsCDGPermNomRel1 is fully-complete, so isCDGPermNomRel2, that is, everymorphismof CDGPermNomRel2 is the denotation of a proof. Therefore, the composition of two morphisms of CDGPermNomRel2 resultsin a morphism that is the denotation of a proof. This one is then a morphism ofCDGPermNomRel2. That is,CDGPermNomRel2is a category.Therefore, CDGPermNomRel2 is a sub-semi-star-autonomous category of CDGPermNomRel1 able to faithfully model

axioms, and, as CDGPermNomRel1 is fully-complete for MLL−, CDGPermNomRel2 = CDGPermNomRel1.

D Proofs of section VLemma D.1. 1. Let Ψ be a ⊕ resolution of A, and R ∈ TAΨ. Then R injects naturally into TA.

2. Let Ψ be a &-resolution of A, and Q ∈ T⊥AΨ. Then Q injects naturally into T⊥

A .3. Let R ∈ TA, A formula ofMALL−, and let Ψ be a &-resolution on A. Then R Ψ ∈ TAΨ.4. Let Q ∈ T⊥

A , A formula ofMALL−, and let Ψ be a ⊕-resolution on A. Then R Ψ ∈ T⊥AΨ.

Proof. The proof is done by mutual induction on the formula A. As the conditions (1) and (2) are mirror of each other, andsimilarly for (3) and (4), one can focus on only one of them. We tackle (2) and (3) alongside.

• if A = B ` C . We start with (3) and consider Ψ an &-resolution. Then we split Ψ into ΨB ,ΨC . Let R ∈ TB`C , andlet Q ∈ T⊥

BΨB. Then by induction Q ∈ T⊥

B . Hence Q; ( R ΨB ) ∈ TC . By induction, Q; ( R ΨBΨB ) ∈ TCΨC . Thus,R Ψ ∈ TB`C . We tackle (2). Given Q ∈ T⊥

B`CΨ, then Q = QB ` QC , and QB ∈ T⊥BΨB

, QC ∈ T⊥CΨC

, hence we canconclude by induction.

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• If A = B ⊗ C , to tackle (3) we need to consider R ∈ TB⊗C . Then R = R 1 ⊗ R 2 and we can apply the induction. Onthe other hand to prove (2) we consider Q ∈ T⊥

B⊗CΨ, Ψ being a &-resolution of B ⊗ C , that we split into ΨB ,ΨC . Thenlet us consider R ∈ TB . Then by induction, R ΨB ∈ TBΨB . In particular, R ΨB ;Q ∈ T⊥

CΨC. Then by induction,

ΨB ;Q ∈ T⊥C .

• The ⊕, & cases are straightforward.

Lemma D.2. Let R ∈ CDGPermRelA(F ), F formula of M ⊕ LL−. Then there exists a unique additive resolution Ψ such thatR ∈ TAΨ.

Proof. The proof is done by induction on the structure of the formula. If A = B ⊗ C and A = B ⊕ C then the induction isstraightforward. IfA = B`C , then suppose for contradiction that R ∩C lies in two additive resolutions. then ∀Q ∈ T⊥

B .Q; R ∈

TC , and hence TC . Therefore, by induction hypothesis, Q; R is on a single additive resolution. We claim that on aM ⊕ LL−

formula F , there is a maximal counter-strategy MF defined by induction.• Atomic case X , Mx = OX .• In the case F = F1 ` F2,MF = MF1 ×MF2 .• In the case F = F1 ⊗ F2, MF = PF1 ×MF2 ⊎ PF2 ×MF1 .• Finally, in the case A ⊕ B, QA⊕B = |MA ×MB .

One can check by induction that∀Q ∈ T⊥F .Q ⊆ MF . In particular, M; R is on a single additive resolution, and hence ∀Q.Q; R

are on the same additive resolution. Since every element of OB is hit by M, we conclude that R PC is on a unique additiveresolution. Since R ∈ CDGPermRelA, if R OB would hit two different additive resolutions, so would R PB , as the elementscome from relations of PermRel. Hence R B is on a single additive resolution. By symmetry, R is on a unique additiveresolution.

PropositionD.3. Let R be a morphism ofCDGPermRelA. Then for every &-resolutionΨ, there exists a unique additive resolutionΦ on Ψ such that R Φ , ∅ and R Φ is a denotation of aMLL− proof.

Proof. Let F a formulla of MALL−, R ∈ TF , and Ψ a &-resolution. Then by D.1, R Ψ ∈ TFΨ. Furthermore, R Ψ ∈

CDGPermRelA(F Ψ). As F Ψ is a formula M ⊕ LL−, we can use D.2 to conclude that there exists a unique ⊕ resolution Φon it such that R Ψ injects into R ΦΨ, and R ΨΦ ∈ TFΦΨ. As this one is a formula ofMLL, we get that R Ψ.Φ is adenotation of a MLL− proof by full-completeness.

E Proofs of Section VIE.1 Proof of section CLemma E.1. Let τΨA ,σR ,σQ ,τΨC as above. Then:

• τΨA ;A σR ∈ SimpleHypRel(B).• σQ ;τΨC ∈ SimpleHypRel(B⊥).

Proof. We only prove the first point, the second being symmetric. Let R ∈ SimpleHypRel(A,B), ΨA a ⊕-resolution of A. Thenfor all &-resolutions ΦB of B, there is a unique element x ∈ R such that x on ΨAΦB . So for every &-resolution of ΦB , thereis an element x ∈ (ΨA R ΦB ), and this element is unique. So R ΨA B is (P1). Furthermore, one needs to provethat for every subset Q ⊆ (ΨA R ) B then Q ∈ Γ(B). Either there is only one element in ΨA Q A, and thereforethis element is both incoherent and coherent. Then as ΨA Q ⊆ Γ(A⊥ ` B), this entails (ΨA Q) B ∈ Γ(B). Either thereare two or more elements. In this case, they are in the same &-resolution of A⊥, but on different ⊕-ones. This would entail(ΨA Q) A⊥ ∈ Γ⊥∗(A⊥), and therefore (ΨA Q) B ∈ Γ∗(B).

One can note that the only possible elements of (σ •R A) ∩ τ •ΨA are the maximal elements that correspond to the ΨA R .

Therefore, τΨA ;A σ R = σ(ΨA R )B = σQ , where Q = ΨA R B ∈ SimpleHypRel(I ,B).

Proposition E.2. Let R ∈ SimpleHypRel(A),Q ∈ SimpleHypRel(A⊥). Then ⟨σ R ,τQ⟩ is maximal, that is R ∩ Q , ∅

To establish this proposition given two simple hypercoherences as above, we prove that the interaction between theirassociated concurrent operators never reach a deadlock. This follows from the following lemma.

Lemma E.3. Given R ∈ SimpleHypRel(A), and σ R the associated closure operator, then σ R satisfies the following:• σ R is positive.• σ R is total: ∀x ∈ σ •, either x has polarity P , or N .

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In particular, being positive entails that σ is civil, meaning that for every counter-strategy τ , ⟨σ ;τ ⟩ , ⊤. This was establishedin lemma B.1. We call the second condition totality as it will , as proven below, entails that the element reached through ainteraction with a counter-opponent is maximal, and hence implies totality as defined in the original paper on concurrentgames [2]. Assuming the lemma, we prove the proposition.

Proof. Let us note that if we reach a N -position, then this position is maximal and the proposition is proven. As σ and τ are Oand P-closure operators respectively, ⟨σ ,τ ⟩ , ⊤. Furthermore, as σ is total, ⟨σ ,τ ⟩ is either a P or N position. Similarly, as τ istotal (but negative), ⟨σ ,τ ⟩ is either a O or N -position. Hence ⟨σ ,τ ⟩ is a N -position, and hence it is maximal.

We now endeavour to prove the lemma.

Proof. We start by proving positivity. Given a partial additive resolution x seen as an element of the domain, and consideringits sub-partial &-resolution neg(x), as R is (P1), there are some elements of R above every &-resolution and in particularabove neg(x). Hence σ R (neg(x)) , ⊤. Given any element x ∈ ⟦A⟧, either x is incompatible with R , that is, σ R (x) = ⊤, orthere are some positions of R above it. We write A x for A Ψ where Ψ is the partial additive resolution corresponding to x .Let us consider the set of &-occurrences of A x . As R is (P1), for each partial &-resolution of it, there is an element on it. Inparticular, each O-prime p starting from x correspond to selecting the direct left or right sub-formula of a direct sub-formulaC & D of A x . Consequently, there is a position y1 ∈ R such that y1 C , ∅ and a position y2 ∈ R such that y2 D , ∅.Hence p @ y1 ⊓ y2 and σ R will not bring it. So, as a result, x

P↠ σ (x), establishing the second point of positivity. The proof

that σ R (neg(x)) ⊔ pos(x) = σ R (x) follows the same lines, establishing that the P-primes from x brought by σ R depend solelyon neg(x).

Now, let us focus on totality. Totality is immediate once we notice the following property:O corresponds to strict incoherence,P to strict coherence, and N to a position that is both coherent and incoherent. More precisely, given a set of maximal positionsxi such that

.xi : O , (respectively P , N ), then xi is strictly hyper-incoherent (respectively strictly hypercoherent,

respectively hyper-coherent and hyper-incoherent, that is, a singleton). That is, there is an equivalence between the polarityof the intersection and the hyper-coherence.

This is proven by induction on the formula A. If A = X , then there is only a N -position, that is both coherent and incoherent.If A = B &C , then if

.xi is in B, or C , the property follows by induction. Otherwise it is ⊥A, and hence it implies that there

are some xi in B and some in C . So by definition of hypercoherence xi ∈ Γ∗(A), and ⊥A : P as expected. A similar reasoningworks for the ⊕-case. For the multiplicative cases, we first deal with ⊗. Let us consider a set pi such that the intersectionin A ⊗ B is P . It means that

⋂xi A =

⋂xi A is P or N , and similarly for B, with at least one of the both being P . By

induction hypothesis, pi A ∈ Γ(A), and pi B ∈ Γ(B), with at least one of them in Γ∗. Henceforth, xi ∈ Γ∗(A ⊗ B). Theother cases are proven similarly.Therefore, given a partial additive resolution x , either σ (x) is incompatible with x , in which case σ (x) = ⊤. Or σ (x)

corresponds to the intersection of the set of elements of R that are above x . And these ones are, by definition, hypercoherent.Hence σ (x) : P if there is more than one element above it, and σ (x) : N is there is exactly one.

Finally, we conclude the section with the desired property to prove that HypPermRel forms a category.

Proposition E.4. Let R ∈ HypPermRel(A,B), Q ∈ HypPermRel(B,C), then given ΨA a ⊕-resolution of A, ΨC a &-resolutionof C , there exists a unique additive resolution ΦB such that R ΨAΦB , ∅, and Q ΨCΦB , ∅.

That is, R ΨAΦB is a partial relation, and so is Q ΦBΨC . The proof of this property is a simple combination of all theresults of this section. The unicity simply comes from noticing that if there were two, then there would be two maximalelements in the set σ •

R∩ τ •

Q, which would contradict ⟨σ

R,τ

Q⟩ being minimal.

E.2 Proof that HypPermRel is a star-autonomous categoryLemma E.5. HypPermRel forms a category.

Proof. First, we shall explicit what is the identity. A &-resolution A ⊸ A = A⊥ `A, is the data of a &-resolution Φ on A, and a&-resolution Ψ on A⊥, that is a ⊕-resolution on A. Let us note that a &-resolution together with a ⊕-resolution give rise to aunique additive-resolution, and this additive resolution is on these both partial resolutions. The identity morphism on this&-resolution is the identity partial relation on the unique additive resolution coming from Φ,Ψ on A.We prove that this morphism indeed acts as the identity. We do the proof only for the right-action, the left-one being

dealt with symmetrically. We consider a morphism R : A → B1, and the identity relation idB : B1 → B2. We shall provethat R ; idB = R . Let us pick a &-resolution Ψ on B2, and a ⊕ resolution ϒ of A. The set of ⊕-resolutions Φi on B⊥

1 suchthat idB : B1 Φi → B2 Ψ , ∅ is exactly Ψ by definition. Therefore, the composition ϒ R ; idB Ψ is equal to

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ϒ R Ψ;Ψ idB Ψ. There is a unique ⊕-resolution Λ ofA ϒ ⊸ B1 Ψ such that ϒ R Ψ is onA ϒΛA ⊸ B ΨΛB .Completing arbitrarily ΛB as a ⊕-resolution of B, then ΛB idB Ψ : B1 ΛB → B2 Ψ is the identity partial relationΛBΨ id ΛBΨ : B1 ΛB1Ψ → B2 ΛBΨ. Therefore, we have the following sequence of equations:

ϒ R ; idB Ψ = ϒ R Ψ;Ψ R Ψ

= ϒΛA R ΨΛB ;ΛBΨ id ΛBΨ

= ϒΛA R ΨΛB

= ϒ R Ψ

Therefore, on each &-resolution R and R ; idB agree. As R is perfectly determined by its partial relation on each &-resolution, following the (P1) condition, we conclude that R = R ; id.

We now deal with composition more generally. We need to prove that given two partial relations R : A → B and Q : B → Csatisfying (P1), namely, that on each &-resolution they give rise to a non-empty partial relation, then so does their composite.This was proven in the previous section. Furthermore, we need to prove that R ;Q is a clique (that is, each subset of it ishypercoherent). This follows from the definition of Γ(A ⊸ B).

Lemma E.6. HypPermRel is star-autonomous with products.

The star-autonomy follows from the one of hypercoherences and the one of PermNomRel. The product follows from A & Bbeing a product of A,B in the category of hypercoherences.

Therefore, HypPermRel forms a sound model ofMALL. We will establish some properties of the morphisms of HypPermRelin the next sections. First, we prove strong softness. This allows us to prove that HypPermRel is fully compete for a fragmentofMALL, and allows us to map a proof-structure to each morphism of HypPermRel.

F Proofs of section VIIProposition F.1. Every morphism R : 1 → l1 ` ... ` lm ` (A1,1 ⊕ A1,2)` ... ` (An,1 ⊕ An,2) in HypPermRel is strongly soft.

Proof. We assume that R is not strongly soft and prove a contradiction. This means that for every Ai,1, and Ai,2, thereare two elements xi,1,xi,2 ∈ R such that xi,1 Ai,1 , ∅, and xi,2 Ai,2 , ∅. Furthermore, we pick xi,1,xi,2 such that∀j, if 1 ≤ j ≤ m then we have xi,1 lj = xi,2 lj = ⊥. Let us consider the set R ⊇ s = ∪i xi,1,xi,2, then for every` block ∆i = Ai,1 ⊕ Ai+2, s ∆i ⊇ xi,1,xi,2 ∆i and therefore is strictly incoherent. Now among the literals, wehave s l1, ..., lm ∈ Γ(l1, ..., lm) ∩ Γ⊥(l1, ..., lm), that is, they project onto a singleton. Therefore, ∀li .s li ∈ Γ⊥(li ), and∀∆i .s ∆i ∈ Γ⊥∗(∆i ). Hence s ∈ Γ⊥,∗(l1 ` ... ` lm ` (A1,1 ⊕ A1,2)` ... ` (An,1 ⊕ An,2)). Hence ∃s ⊇ R such that s is strictlyincoherent. This is a contradiction, and R factors through one of the ⊕.

Theorem F.2. HypPermRel is fully complete for PALL +MIX.

The key of the proof is the softness of hypercoherences. With this in mind, we prove the theorem.

Proof. We do the proof for HypPermRel(I , ⟦Γ⟧), where Γ is a sequent, by induction on the number of additives. If there is none,then as it is a partial relation, it is of the form:

⊢ X1,X⊥1 ,X2,X

⊥2 , ...,Xn ,X

⊥n

And this can be obtained as the denotation of a sequence of axioms and MIX-rules.If there is a conclusion C that is of the form C = C1 &C2. That is, the sequent can be written:

⊢ Γ,C1 &C2

Then the let us write R 1 for the elements x ∈ R ,x C1 , ∅ and R 2 for the elements x ∈ R ,x C2 , ∅. Then R 1 ∩ R 2 = ∅

and R = R 1 ∪ R 2. Furthermore, R 1, R 2 are morphisms made out of partial relations, are cliques (any finite subset of themis hypercoherent), and satisfy the (P1) property on &-resolutions. Therefore, R 1 ∈ HypPermRel(Γ,C1), and similarly for R 2.Therefore, R is the denotation of the following proof, where ⟦π1⟧ = R 1, ⟦π2⟧ = R 2. :

π1⊢ Γ,C1

π2⊢, Γ,C2 &

⊢ Γ,C1 &C2

Similarly, if R is a morphisms of HypPermRel(1, Γ), and there is a ` at the bottom-level in the sequent, we just can forgetabout it, since it will not change the denotation of Γ.

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⊢ Γ,A,B `⊢, Γ,A` B

By applying the two above rules, we can progressively decompose the relation R by disposing of the &-connectives andthe `-connectives at the bottom level of the sequent. We then obtain a relation R that is in the denotation of a sequent of theform:

⊢ l1, ..., lm ,A1,1 ⊕ A1,2, ...,An,1 ⊗ An,2

As HypPermRel is strongly soft, there is a i such that R factors through one of the injection, that is, R is derived from thefollowing ⊕-rule, where the ⊕ is either a ⊕1 or a ⊕2. We present the ⊕1 case above:

⊢ l1, ..., lm ,A1,1 ⊕ A1,2, ...,Ai,1, ..,An,1 ⊕ An,2⊕1

⊢ l1, ..., lm ,A1,1 ⊕ A1,2, ...,Ai,1 ⊕ Ai,2, ..,An,1 ⊕ An,2

We now have dealt with all the cases, and are able to interpret R as the denotation of a proof of PALL. This concludes thefull completeness proof.

Assuming that there is a proof structure coming from a morphism of HypCDGPermRel with a cycle, we get, after manysteps, to the final argument (presented in G.4) where we have obtained a morphism of HypPermRel encoding a proof structureof the type :

⊢ F1, ..., Fnand Fi = αi,m ⊗ (Bi1 & Bi2),Ni [α

⊥i+1,1,α

⊥i+1,1],αi+1,1 ⊗ α⊥

i+1,2, ....,αi+1,m−1 ⊗ α⊥i+1,m ,Ξi

Where Ξi = Ei11 ⊕ Ei12, ...,Eim−1 ⊕ Eim , li1, ..., lir and Ni is a context made out exclusively of literals, ⊕ and `, and such thatthe two α⊥

i do not appear simultaneously in an ⊕- resolution. That is, Ni = Mi [Ci,1[α⊥i ] ⊕Ci,2[α

⊥i ]]: the first α

⊥i appears inC1,

and the second in C2. Furthermore, the linking is such that the choice of B1 by the opponent entails the choice for proponentof an additive resolution that selects the left α⊥

i , and respectively for B2 and the right one, while otherwise letting the theother links that appear in the cycle (defined later) intact. Locally, the linking is as follows:

αi,m−1 ⊗ α⊥i,m αi,m ⊗ (Bi1 &i Bi2) Mi [Ci,1[α⊥i,1] ⊕Ci,2[α⊥i,1]] αi,2 ⊗ α⊥i,2 ... αi,m−1 ⊗ α⊥i,m Ξ Fi+1

axi,1

axi,2

and therefore leads to a cycle as displayed in Figure 5, where the cycle is displayed in red, and the axiom-links that are not partof the cycle in black. So one can see that there are 2n additive resolutions on which the cycle belongs (by switching the n &i s).We define Gi to be the part of the sequent:

Gi = (Bi1 &i Bi2),αi+1,1 ⊗ α⊥

i+1,2,αi+1,2 ⊗ α⊥i+1,3,αi+1,n−1 ⊗ α⊥

i+1,n ,αi1,n ,Ξi

We assume for contradiction that there is a partial hypercoherent relation R that encodes this proof-structure. We consider aset S of n3 elements S = (x)l of R , indexed by lists l = (l1, ..., ln) such that li ∈ 1, 2, 3. Furthermore, these elements aresuch that:

• x(....,li , ...) is on the &-additive resolution that picks Bi,1 (and hence picks the axi,1) if li ∈ 1, 3• x(....,li , ...) is on the &-additive resolution that picks Bi,2 (and hence picks the axi,2) if li = 2

All these x ′s will be ⊥ on all literals, except precisely those that appear in the cycle, that are those written αi, j . We refer toFigure 5 for a better understanding. We write x ...,i :k, ... to indicate that we speak about an element whose index-list is suchthat its ith position is k . The elements x ’s are chosen such that for every two lists l , l ′ that agree on the ith element, li = l ′ithen xl αi, j = xl ′ αi, j . Therefore x ....,i :k, .... αi, j is a singleton and we write xi :k αi, j .In this paragraph, we will be working within the context Gi , and will simply write xi :k , to speak broadly about all those

elements xl Gi such that li = k . Formally, xi :k = xl | li = k Gi , and we can write xi,k A for any sub-formula A of G todesign the projection of this family to A. The family of sets xi :1,xi :2, ,xi :3 will be designed such that xi :1 ∪ xi :2 ∪ xi :3 αi,n ∈

Γ⊥,∗(αi,n). Furthermore, we recall that xi :1,xi :3 choose the left &-resolution on &i , (and hence, the left ⊕ one as well onMi )and xi :2 the right one. Furthermore, the only literals l of Gi such that xi :1,xi :2,xi :3 l , ⊥ are related by axiom-links to otherliterals appearing in Gi . So the elements of xi :1 will be of the following form:

⊥Bi,1 ,⊥Mi [⊥Ci,1 [ai,1]],ai,1 ⊗ ai,2, ...,ai,m−2 ⊗ ai,m−1,ai,m−1 ⊗ ai,m ,ai,m ,⊥Ξi .

where we write ⊥B11 for a minimal ⊥ element of ⟦B11⟧, and ⊥M [⊥C1 [a11]] for an element of ⟦M⟧ that is ⊥ everywhere exceptin αi,m where it is a11, and picks the ⊕-resolution that chooses C1. We recall that xi :2 is picking the right resolution, and

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N1[α⊥1,1, α

⊥1,1] α1,1 ⊗ α1,2 α⊥

2 ⊗ α1,3 ... α⊥1,m−1⊗ αi,mΞ1 α⊥

1,m ⊗ (B1,1&1B1,2)

N2[α⊥2,1, α

⊥2,1] α2,1 ⊗ α2,2 α⊥

2,2 ⊗ α2,3 ... α⊥2,m−1⊗ α2,mΞ1 α⊥

2,m ⊗ (B2,1&2B2,2)

Ξ3 N3[α⊥3,1, α

⊥3,1] α3,1 ⊗ α3,1

...

α⊥i−1,m ⊗ (Bi−1,1&i−1Bi−1,2)

Ni [α⊥i,1, α

⊥i,1] αi,1 ⊗ αi,2 α⊥

2 ⊗ αi,3 ... α⊥i,m−1⊗ αi,mΞi α⊥

i,m ⊗ (Bi,1&iBi,2)

Ξi+1 Ni+1[α⊥i+1,1, α

⊥i+2,1] αi+1,1⊗αi+1,1

...

α⊥n,m ⊗(Bn,1&nBn,2)

Figure 5. Global cycles in the final proof-structure

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(Bi1 & Bi2) Mi [C1[α⊥i,1] ⊕ C2[α

⊥i,1]] αi,2 ⊗ α⊥

i,2 ... αi,m−1 ⊗ α⊥i,m αi,m Ξi

P O O P O P O N/O

O O

Figure 6. Local coherence for Si on Gi

αi−1,m ⊗ (Bi−1,1 &Bi−1,2) M[C1[α⊥i,1] ⊕ C2[α⊥i,1]] αi,2 ⊗ α⊥i,2 ... αi,m−1 ⊗ α⊥i,m Ξi αi,m ⊗ (Bi,1 &Bi,2)

O P O O P O P N /O O P

O O O O

Figure 7. Local coherences of S in Fi

furthermore we settle to make them select different names than xi :1. That is, the elements of xi :2 are of the following form,where ∀j .1 ≤ j ≤ m, bi, j , ai, j .

⊥Bi2 ,⊥Mi [⊥Ci,2 [bi,1]],bi,1 ⊗ bi,2, ...,bi,m−2 ⊗ bi,m−1,bi,m−1 ⊗ bi,m ,bi,m ,⊥Ξi

Finally, the third elements xi :3 are designed to make the whole set xi :1∪xi :2∪xi :3 strictly incoherent on all direct sub-formulasof Gi , except in Bi,1 & Bi,2 where we are unable to prevent it from being coherent. Formally, if αi,k is positive, then we takeci,k ∈ ⟦αi,k⟧ to be a third name. That is, writing X for the atomic formula such that αi,k = X , then ci,k ∈ AX \ ai,k ,bi,k .Otherwise, we take ci,k = ⊥. That way, ai,k ,bi,k , ci,k ∈ Γ⊥,∗(αi,k ). Then the elements of xi :3 are as follows:

⊥Bi :1 ,⊥Mi [⊥Ci,1 [ci,1]], ci,1 ⊗ ci,2, ..., ci,m−2 ⊗ ci,m−1, ci,m−1 ⊗ ci,m , ci,m ,⊥Ξi

In this paragraph, we study the local coherences of the set Si = xi :1 ∪ xi :2 ∪ xi :3 on the sub-formulas of Gi . As theelements x are designed so that xi :k αi, j is a unique element, we then get that xi :1 αi, j = ai, j , xi :2 αi, j = bi, j andxi :3 αi, j = ci, j . Therefore Si αi,k ∈ Γ⊥∗(αi, j ) by design, and consequently Si α

⊥i, j ∈ Γ∗(α⊥

i, j ). On the first formulaof Gi we have S (Bi,1 & Bi,2) ∈ Γ∗(Bi,1 & Bi,2) since the S Bi,1 , ∅ and S Bi,2 , ∅. At last we deal with the Mipart. Here, the reader should remember that xi :k Mi is, in the general case, not a singleton. Indeed, nothing preventstaking two elements x ...,i :k, .. and x ′

...,i :k, ... from choosing different additive resolutions in Mi , except for the distinguished⊕, in which they have to agree by choosing either Ci,1 or Ci,2 by definition. However, the key point is that M is a contextmade exclusively of ` and ⊕. Therefore, two different ⊥ elements in it are automatically hypercoherent. FurthermoreSi Ci,1[α

⊥i,1] ⊕ Ci,2[α

⊥i,1]] ∈ Γ⊥,∗(Ci,1[α

⊥i,1] ⊕ Ci,2[α

⊥i,1]) since Si Ci,1[α

⊥i,1] , ∅ and S2 Ci,2[α

⊥i,1] , ∅. Finally, we get to

Si Mi [Si Ci,1[α⊥i,1] ⊕ Ci,2[α

⊥i,1]] ∈ Γ⊥,∗(M[[Ci,1[α

⊥i,1] ⊕ Ci,2[α

⊥i,1]]). A similar reasoning leads us to conclude that Si Ξi ∈

Γ⊥(Ξi ), although it might be that Si is not strictly incoherent on Ξi .So, we get the following figure 6, displaying local coherence (or incoherence) of Si , where we write P for Γ∗, O for Γ⊥∗, and

N for Γ ∩ Γ⊥,So now let us look at the global picture by studying the coherence of the family xl on each sub-formula of Fi . Let us recall

that (xl ) Ai = Si Ai for every sub-formula Ai of Gi , by definition of Si . This leads us to the following figure 7.That is, for all formulas A of the sequent Γ, we got S A ∈ Γ⊥(A), and, for some formulas A, S A ∈ Γ⊥,∗(F ). And therefore,

as each xl ∈ R , there is a set S , such that S ⊆ R and S ∈ Γ⊥,∗(Γ). Consequently, the relation R is not a clique, and such aproof structure is rejected. It entails that to all our morphisms can be associated a valid proof net of MALL−, as summed up inthe following proposition.

Proposition F.3. HypCDGPermRel is a fully complete model ofMALL−.

Let us note that for the proof we make full use of the structure of ⟦X⟧, that is, our proof would not work if we limitedourselves to the Sierpinski domain. The simplest domain we could have work with would be a three-elements domain (withone ⊥ and two non-compatible primes, as the one presented in [4]), however, we find it more natural to work with names, as itgives some intentional meaning to the elements of the domain.

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We are furthermore in position to make a stronger claim.

Theorem F.1. There is a one-to-one correspondence between the morphisms of HypCDGPermRel onMALL− and the equivalenceclasses of MALL− proofs of linear logic.

As we do not deal with the units, it is a bit dubious to speak about a free-category at this stage, and that is the reason whywe refrain from doing so. To do it we would need to define what is a star-autonomous category without units and with binaryproducts but no final elements. The theorem is based on the following propositions, all proven by Dominic Hughes and RobVan Glabbeek in their paper on proof-nets for MALL− [7, 9], and transposed verbatim here.

Proposition F.4. Each linking is a proof-net if it is the denotation of a proof.

Proposition F.5. Two MALL− proofs translate to the same proof net if and only if they can be converted into each other by aseries of rule commutations.

Proof of theorem F.1. Let us consider a morphism ofHypCDGPermRel(A,B). It denotes a unique linking onA⊥`B. This linkingis the denotation a proof, by full-completeness. Hence this linking leads to a proof net by F.4. Furthermore, the morphisms ofHypCDGPermRel are precisely defined by their linkings. That is, two morphisms are equal if and only if they have same linking,and consequently, same proof net. Therefore, there is a bijection between the set of morphisms of HypCDGPermRel(A,B),where A,B ∈ MALL− and the proofs nets of ⊢ A⊥,B and, by the proposition F.5 the equivalence classes of proofs of MALL− of⊢ A⊥,B.

G Translation of the proofG.1 Hypercoherent partial relations and proof structuresThe whole point of this section is to repeat the arguments of [3] and check that they adapt smoothly within the context ofhypercoherent partial relations (that is, the category HypPermRel), instead of dinatural transformations. We start by recallingthe definitions of Girard proof structures. We then proceed to show that to each hypercoherent partial relation a set of proofstructures can be associated, and that this set can be narrowed down to a subset of canonical proof structures that satisfyadditional desirable properties. We use these to reduce the cases that are relevant to prove a full completeness property.As noted above, this section is a clear plagiarism from [3], simply replacing dinatural transformations with morphisms of

HypPermRel when needed. We present it for sake of completeness. Some proofs will be omitted.

G.1.1 Girard proof structuresThe original definition of Girard proof structure can be found in [5]. The goal of the original paper was to find a suitablecharacterisation of proof structures that arise as denotations of MALL− proofs.

Definition G.1. A proof structure Θ consists of the following :• Occurrences of formulas and links. Each occurrence of link takes its premise(s) and conclusion(s) among the set ofoccurrences of formulas.

• A set of eigenweights p1, ...., pn associated to L1, ...,Ln where L1, ...,Ln is the list of occurrences of &-links appearingin Θ. Each pi is a boolean variable, that is, it can take value in the set 0, 1.

• For each occurrence of formulaA or occurrence of link L a weightw(A) (orw(L)), where the weight is a non-zero elementof the boolean algebra generated by the pi .

Furthermore, the proof structure has to satisfy the following conditions.

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• Conditions on Links:Link L premise(s)

Lconclusion(s)weights of L and its premises

Axiom Link(.)

axA A⊥

`-link A B `A` B

w(L) = w(A) = w(B)

⊗-link A B⊗

A ⊗ Bw(L) = w(A) = w(B).

&-link A B &iA & Bw(A) = pi .w(L) w(B) = (¬pi ).w(L)

⊕1-linkA ⊕1

A ⊕ Bw(A) = w(L)

⊕2-linkB ⊕2

A ⊕ Bw(B) = w(L)

• We require that for each occurrence of a formula A,w(A) =∑

k Lk where (Lk ) is the family of links with conclusion A.• Moreover, if L1,L2 are two distinct links sharing the same conclusion A, thenw(L1).w(L2) = 0.• Given an occurrence A of a formula that is not the premise of any link (that is, a conclusion of the proof structure), thenw(A) = 1.

Furthermore, a proof structure satisfies the two following conditions :• dependency: Every weight appearing in the proof structure is monomial, that is, is a product of eigenweights andnegations of eigenweights.

• technical : For every weight v appearing in Θ, if v depends on pi then v ⊆ w(Li ) (and the weight of Li does not dependon pi ).

A formula occurrence is a conclusion of Θ if it is not a premise of any link. A link L is terminal if it has one conclusion,this conclusion is a conclusion of Θ, and if furthermorew(L) = 1. If a link is terminal, it might be removed, leading to one ortwo new proof structures. The removal of terminal links for proof structures is defined below. Let Θ be a proof structure. Wedenote CL(Θ) the set of conclusions of Θ.

• If L is a terminal ⊗-link with conclusion A ⊗ B, and premises A, B. Then the removal of ⊗ (when possible) consists inpartitioning the formula occurrences of Θ \ A ⊗ B into two sets X ,Y , such that A ∈ X and B ∈ Y . The partitioningmust be done in such a way that whenever a link L′ has a conclusion in X (respectively Y ), then all the premises andconclusions of L′ belong to X (respectively Y ). It therefore leads to two new proof structures X and Y , that have A for X ,B for Y among their conclusion(s).

• If L is a `-link with premises A,B and conclusion A` B, then the removal of L consists in removing A` B and the linkL from θ . It leads to a new proof structure with conclusions CL(θ ) \ A` B ⊎ A,B.

• If L is a ⊕1-link with premise A and conclusion A ⊕ B, then the removal of L consists in removing L together with itsconclusion A ⊕ B, leading to a proof structure with conclusion(s) CL(Θ) \ A ⊕ B ⊎ A.

• If L is a ⊕2-link with premise B and conclusion A ⊕ B, then the removal of L consists in removing L together with itsconclusion A ⊕ B, leading to a proof structure with conclusion(s) CL(Θ) \ A ⊕ B ⊎ B.

• If L is a &-link (call it Li ) with premisesA,B and conclusionA&B, then the removal of Li consists in removing the link Litogether with its conclusion A& B, and then forming two proof structures ΘA and ΘB . The proof structure ΘA is formedby making the replacement pi = 0 and keeping only those formulas and links whose weights are non 0. Respectively, ΘBis formed by taking pi = 1 and keeping only those formulas and weights that are non 0.

• In the case where we consider theMIX rule, then there is a 0-removal, that removes no link, but splits the proof structureinto two new ones. That is, let us suppose that Θ can be partitioned into two proof structures Θ1,Θ2 such that whenevera link L has a conclusion in Θ1 (resp Θ2), then all its premises and conclusions belong in Θ1 (resp Θ2), then the MIXsplitting consists in splitting the proof structure Θ into Θ1,Θ2.

Definition G.2. A proof structure is MALL− sequentialisable if it can be reduced to a set of axiom links by an iteration oflink-removals. Furthermore, it is MALL− +MIX sequentialisable if it can be reduced to a set of axiom links by an iteration oflink removals and MIX-splittings.

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By definition, if a proof structure is sequentialisable, then there is a proof π that can be canonically associated to it, as eachlink-removal basically corresponds to a rule of MALL−, and the MIX-splitting corresponds to the use of the MIX rule. On theother hand, to each proof π of a MALL− sequent, one can associate a proof structure. However, this proof structure is notunique.

G.1.2 The criterionJust as when dealing with MLL− proof structures, we look at switchings in order to establish if a MALL− proof structureis sequentialisable, though the notions of switchings differ between the two cases. We define switchings for MALL− proofstructures.

Definition G.3. • A switching for a MALL− proof structure consists in:1. A choice of a valuation ϕS which is a function p1, ..., pn → 0, 1. Therefore ϕS defines a function from weights tothe boolean, that we also note ϕS by abuse of notation. The slice of the proof structure sl(ϕS (Θ)) corresponds to therestriction of Θ to those formulas and links such that ϕS (w) = 1.

2. For each `-link L of sl(ϕS (Θ)), the choice S(L) ∈ L,R.3. For each &-link Li of sl(ϕS (Θ)), the choice of a formula S(Li ) in sl(ϕS (Θ)), called jump of L. This jump must dependon pi in the sense that it must be the conclusion of a link L such that pi ⊆ w(L). A jump is normal if S(L) is a premiseof L. Otherwise, it is proper.

• A normal switching is a switching without proper jumps.• For a proof structure Θ together with a switching S , we define the undirected graph ΘS as follows:1. The vertices of ΘS are the formulas of sl(ϕS (Θ)).2. For each axiom link in sl(ϕS (Θ)), one draws an edge between the corresponding literals conclusions of the link.3. For each ⊗-link, one draws two edges from the conclusion A ⊗ B of the link onto its premises.4. For a`-link, one draws an edge from its conclusion to its occurrence chosen by the switching. That is, if the conclusionof the `-link L is A` B, and S(L) = l (respectively S(L) = r ), then one draws an edge from A` B to A (respectively B).

5. For a ⊕1-link, one draws an edge between its premise occurrence A and its conclusion A ⊕ B.6. For a ⊕2-link, one draws an edge between its premise occurrence B and its conclusion A ⊕ B.7. For a &-link Li , one draws an edge between its conclusion A & B and its jump S(Li ).

With this definition, we are in position of establishing the correctness criterion. We define a proof net as a proof structurethat is sequentialisable.

Theorem G.4. [5]• A proof structure Θ isMALL− sequentialisable if and only if for all switchings S the graph ΘS is connected and acyclic.• A proof structure Θ isMALL− +MIX sequentialisable if and only if for all switchings S the graph ΘS is acyclic.

In our case, we would like to use a slightly different characterisation, that will turn out to be more adapted for us.

Proposition G.5. [3] A proof structure Θ isMALL− sequentialisable if for all switchings S the graph ΘS is acyclic, and for allnormal switchings S the graph ΘS is connected.

Indeed, the graph of a normal switching is approximatively the same graph as the one of a multiplicative switching (that is,the graph coming from a switching of a MLL− proof structure). Hence, while working with a relation R ∈ HypCDGPermRel,the fact that for all &-resolutions Ψ we get that R Ψ ∈ CDGPermNomRel, and the Chu partial relations are MLL complete,will entail that for each normal switching S the graph ΘS (where Θ is a proof structure associated with R ) is acyclic andconnected. Therefore, what will remain to be proven is the acyclicity of the proof-structure for all proper switchings.

G.1.3 HypPermRel and proof structuresThe gal of this section is to attribute to each morphism of HypPermRel a non-empty set of proof structures.

Lemma G.6. In a Girard proof structure:• if we replace an occurrence of a ⊗-link with a`-link, and we replace every occurrence of ⊗ in vertices (formulas) hereditarilybelow it in the proof structure with the corresponding `, then it remains a valid proof structure.

• if we replace an occurrence of a `-link with a ⊗-link, and we replace every occurrence of ⊗ in vertices hereditarily below itin the proof structure with the corresponding ⊗ , then it remains a valid proof structure.

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We writeMIX the operation on proof structure that replaces an occurrence of ⊗ with a ` as defined in the above lemma.We recall that PALL is the fragment ofMALL− without ⊗. Each proof structure of PALL, that is, without ⊗-link, is PALL+MIX

sequentialisable. That is, to each proof structure of PALL can be associated a proof of PALL +MIX. This follows from thefollowing lemma, that was first proven in [6].

Lemma G.7. [6] Softness of MALL-proof structure: Let Θ be a MALL proof structure, such that Θ has at least, one occurrenceof a &-link. There there is, at least, one &-link that has weight 1.

Corollary G.8. Each proof structure of PALL is sequentialisable.

The proof of the corollary follows approximatively the same lines as the proof of PALL full completeness for morphisms ofHypPermRel ??. The full proof can be found in [6].Therefore, one can devise a way to associate some Girard proof structures to each morphism of HypPermRel. First, we

compose it with all the necessarymix rules to remove all the ⊗, and obtain a morphism of type PALL. This is the denotation ofa morphism π of PALL +MIX (since HypPermRel is fully complete) and we associate to it a set of proof structures. In theseproof structures, we replace all the necessary ` with ⊗ in order to fit the type of the original relation. This remains a proofstructure, this time ofMALL−. We define this procedure properly in the next paragraphs.To do that formally, we first have to define the other direction, namely that to eachMALL− +MIX sequentialisable proof

structure Θ, then Θ gives rise to a unique morphism of HypPermRel. That follows also from the fact that HypPermRel is astar-autonomous category with products (that accepts the mix-rule), and hence soundly model MALL +MIX. For recreationalpurposes, we re-prove it here in the context of proof-structure.

Proposition G.9. Let π be a MALL− + MIX sequentialisable proof structure Θ. Then Θ determines a unique morphism ofHypPermRel, [Θ], such that [Θ] is the denotation of aMALL +MIX proof. Therefore we have a mapping:

[ . ] : MALL− +MIX sequentialisable proof Structures of type Γ ⇒ HypPermRel(I , Γ),where the type of a proof structure is the

˙-tensored type of its conclusion(s), seen as a list of formulas. That is, the type of Γ is˙

CL(Γ), where we assume there is a canonical ordering on the formulas of CL(Γ).

Proof. The proof is done by induction on the number of &-links. If there is none, then Θ can be identified with a proof ofM ⊕ LL +MIX, and then therefore it is simply a set of axiom links. These can be faithfully interpreted within the category ofhypercoherent partial relations. In case there is a &-link, then in particular, by softness of proof structures, there is one withweight 1. Name it Li . Therefore, it corresponds to its namesake in the sequent. Consider the two proof structures obtainedby takingwi = 0, 1 respectively. Then to each of them can be assigned a unique hypercoherent partial relation by inductionhypothesis. Then, by combining them following the sequentialisation of the proof structure, one obtains a hypercoherentpartial relation associated with π .

We extend this mapping to a wider class of proof structures. We call extension of a partial function, another function thathas greater or equal domain of definition, and such that the two functions agree on their common domain.

Given [ . ]k a function from proof structures to partial relations, we define [ . ]k+1 as follows:• If Θ < Dom([ . ]k ) but there is a ⊗-link in Θ such that MIX Θ ∈ Dom([ . ]k ), where MIX is applied to that occurrenceof an ⊗-link .

• If, furthermore, given a relation R such that R is of type Θ, and such thatmix R = [MIX Θ]k , (where, again,mix isapplied to the relation R in accordance with the MIX from the first point), then Θ ∈ Dom([ . ]k+1).

• Since mix is monic (in our case, it is actually the identity), this R is uniquely defined and we set [ Θ ]k+1 = R .• Since the applications of mix are commutative, the mapping [ . ]k is well-defined.

Note that in the case of hypercoherent partial relations, mix is the identity. In the following, we take [ . ]1 = [ . ], thefunction from proof structures to relations previously defined. Note that [ . ]k+1 is indeed an extension of [ . ]k . We denote[ . ]∗ = ∪k [ . ]

k .

Lemma G.10. Lifting property. Let R , R ′ two relations such that R ′ = mix R . Let Θ,Θ′ be proof structures such thatΘ′ = MIX Θ. Then if [Θ′]∗ = R ′ and Θ, R are of the same type (that is, the mix, MIX are applied to the same part of theformula), then [Θ]∗ = R .

Given a morphism R of HypPermRel, we can speak of its weakly assigned proof structures to be :WPS( R ) = Θ | [Θ]∗ = R .

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Our goal is to show that this function [ . ]∗ is surjective, that is, that to each morphism R of HypPermRel its setWPS( R ) isnon-empty. Consider a hypercoherent relation R , of type ∆. Then replace all the ⊗ in ∆ by `, leading to a new type ∆`. ThenR can also be seen as a morphism of type ∆`. As the hypercoherent relations are fully complete for PALL +MIX, there existsa proof π such that ⟦π⟧ = R , and therefore a proof structure Θ` such that [Θ`] = R . Now, we apply back theMIX-rules andwe obtain a proof structure Θ of type ∆ such that [Θ]∗ = R , by the lifting property.

Corollary G.11. Given R a morphism of HypPermRel,WPS( R ) , ∅.

Lemma G.12. A relation R denotes a MALL−-proof net, and hence, a proof, if and only if the set of weakly associated proofstructuresWPS( R ) contains a proof net Θ.

Proof. The only if part is direct, as if R is a denotation of a proof, then there exists a proof net Θ such that [Θ] = R . So let ustackle the other direction, and assume there is a Θ ∈WPS(R) such that Θ is a proof net. Then R = [Θ]∗ by definition. In thatcase, Θ is in the domain of [ . ] since Θ is a proof net. Therefore R is the denotation of a MALL− proof by soundness.

G.1.4 Strongly canonical proof structuresWe add certain requirements to the domain of [.]∗, in order to obtain a subset PS( R ) ⊆WPS( R ) that enjoys some desirableproperties.

Definition G.13. Semantical splitting of a hypercoherent partial relation. Given a hypercoherent partial relation R

ofMALL−-type we define a ⊗,`,&, ⊕,mix-splitting as follows:• A ⊗ (respectively &,mix) splitting of R is written R 1 ⊗ R 2 (respectively R 1 & R 2, R 1mixR 2), when R of type∆1,∆2,A1 ⊗ A2 (respectively ∆,A & B, and ∆1,∆2) is split into R 1 ⊗ R 2 (respectively R 1 & R 2, and R 1mixR 2) withR i of type ∆i ,AI (respectively ∆,Ai and ∆i ).

• A `-splitting (respectively ⊕1, ⊕2) of R of type Γ,A` B (respectively Γ,A ⊕ B, and Γ,A ⊕ B) is a relation R 1 of typeΓ,A,B (respectively Γ,A and Γ,B) such that R = R 1 (respectively R = inl( R 1) and R = inr( R 1)).

Each splitting corresponds to a MALL +MIX rule.A total splitting is a sequence of splittings such that no further splitting can be done. A total splitting terminates if the final

relations obtained after the splittings are identities on literals. A total splitting is represented as a tree, where the root is theoriginal relation, the binary nodes are the binary splittings and the unary nodes the unary ones.

Definition G.14. A total splitting α is legal if for every &-splitting appearing in α , the &-splitting happens provisio that itwas impossible to execute any ⊗,mix,`, ⊕1, ⊕2-splitting to the relation at this stage.

That is, a legal splitting happens to split the transformation with a &-splitting only when no other splitting is possible.Then of course, if a hypercoherent partial relation has a total splitting that terminates, then it has a total legal splitting thatterminates. Notably, each denotation of a proof has a total legal splitting.

Definition G.15. A proof structure is said to satisfy the :• Unique Link Property (UL): If L in θ is either a ⊗-link, `-link or a &-link with conclusion an occurrence D, then it isthe only link whose conclusion is that occurrence of D.

• No duplicate axiom-link property (NDAL): There occurs in Θ no distinct axiom links ax1, ..., axn whose two conclu-sions coincide and whose sum of weights is 1.

We try to rely on legal splittings to relate morphisms of HypPermRel and proof-structures that satisfy UL and NDAL. Forthat, we rely on the following lemmas, that have been first proven in [6].

Lemma G.16. Suppose that a MALL− +MIX proof π is obtained from proof(s) πi by means of a MALL− +MIX-rule α . Thenfor any UL proof structure Θi whose sequentialisations are πi , one can construct a canonical proof structure Θ such that itssequentialisation is π , its splitting corresponding to α (that is, removing of the terminal link corresponding to α ), yields the proofstructure(s) Θi , and furthermore Θ satisfies the UL.

Lemma G.17. Given a proof structure, one can replace sets of axioms links sharing the same conclusions and such that the sumsof their weights is 1 with unique axiom links. This process is confluent, terminates, and gives rise to a new proof satisfying theNDAL. Furthermore, calling Θ the original proof, and Θ the newly obtained one, if [Θ] = R then [Θ] = R .

Proposition G.18. Let R be a morphism of HypPermRel. Then every terminating total legal splitting α of R can be canonicallyinterpreted by a unique MALL +MIX proof net Θα that satisfies UL and NDAL and such that [Θα ] = R .

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The proof relies heavily on propositions G.16 and G.17.

Proof. The proof is done by induction on the size of α . We simply do it when the last rule of α is a &. Then there is two totalsplitings αi for two relations R i such that R = R 1 & R 2. By induction hypothesis, this leads to two proof structures Θithat are proof nets satisfying the UL and NDAL. So we can get a proof structure Θ1 & Θ2, such that it satisfies UL followingG.16. Furthermore, following the lemma G.17, as Θi is sequentialisable, so is Θ = Θi & Θ2. Finally, we replace possible sets ofaxiom links with same conclusions as in lemma G.17 until we get a proof structure Θα that satisfies NDAL. Following lemma,[Θα ] = R .

Given a total legal splitting α of a morphism R of HypPermRel we write Θα for its associated proof structure.Let denote the function [ . ]− defined by:

[ . ]− : Proof structures→ relationssuch that [ . ] is an extension of [ . ]−.

Θ ∈ Dom([ . ]−) if Θ = Θα for some legal splittings of α of [Θ].Let R be in the image of [ . ]. Then R is a denotation of a proof. Hence there exists a terminating legal splitting α . Therefore,

one can associate to R a proof structure Θα as in the proposition above. Finally, R = [Θα ]. What it tells us is that the imageof [.] and [.]− is the same. We define [ . ]∗− from [ . ]− the exact same way we defined [ . ]∗ from [ . ]. Then once again [ . ]∗−has the same image as [ . ]∗, but every proof structure in its domain satisfies UL and NDAL.

As a result, we define:PS( R ) = Θ | [Θ]∗− = R

Then PS( R ) ⊆ WPS( R ) andWPS( R ) , ∅ ⇒ PS( R ) , ∅. In particular, to each hypercoherent partial relation, one canassign to it a non-empty set PS(Θ) of proof structures satisfying UL and NDAL.

These proof structures can be built almost the exact same way as described in the paragraph above G.1.3. We start with anhypercoherent relation R . We replace all the ⊗ by ` in the type of R . By full completeness, it is the denotation of a proof πof PALL− +MIX. This proof π leads to a sequentialisation α . This sequentialisation can be chosen to be legal. Hence we canassign a proof structure Θα as in proposition G.18. This proof structure satisfies the UL and the NDAL. Now, we replace theappropriate `-links with ⊗-links, in order to get back a proof structure of the appropriate type.

Proposition G.19. A relation R denotes aMALL− proof if and only if there is a Θ ∈ PS( R ) such that Θ is a proof net.

The proof is the exact same as the one of proposition G.12.

G.1.5 Canonical splittingThe goal of this section is to strengthen the above proposition. We would like to prove that a relation R denotes a proof if andonly if all the proof structures in PS( R ) are proof nets. That way, one can consider any of them.

Proposition G.20. Suppose that a hypercoherent partial relation R is such that PS( R ) is not empty, and R can be split via a@-splitting, @∈ ⊗,mix,`, ⊕1, ⊕2,&. Then every Θ ∈ PS( R ) has the corresponding @-splitting.

The proof is exactly the same as in [3], simply replacing the word dinatural transformation by partial relation.

Definition G.21. We say that a proof structure Θ has a cycle C if there is a switching S such that C appears in ΘS . We saythat a partial relation R yields a cycle if PS( R ) , ∅ and ∃Θ ∈ PS( R ) such that Θ has a cycle.

The next corollary is fundamental.

Corollary G.22. Suppose that a HypPermRel morphism R can be split into hypercoherent partial relation(s) R i . Then if Ryields a cycle, then so does R i , for at least one of the i .

Proof. Suppose that a relation R can be split by @ into Ri . Suppose moreover that R yields a cycle, that is, there is aΘ ∈ PS( R ), such that Θ has a cycle. Then Θ can also be split via @ into Θi . Hence if Θ has a cycle, then one of Θi must haveit as well. Since Θi ∈ PS( R i ), we conclude.

Theorem G.23. Let R be a Chu partial hypercoherent relation, that is R ∈ HypCDGPermRel. Then R denotes a proof if andonly if ∀Θ ∈ PS( R ), Θ is a proof net.

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Proof. We only need to prove the only if part. Let R be a relation that is the denotation of a proof, and suppose that thereexists Θ ∈ PS( R ) such that Θ is not a proof net. As R is the denotation of proof, every additive resolution of Θ (that is, everynormal switching of Θ), is connected. Hence, Θ is not a proof net only if Θ yields a cycle. Let @ be a splitting of R , thenit is also a splitting of Θ and this splitting yields proof structure(s) with cycle. So let α be a total splitting of R such that αterminates (it exists since R is the denotation of aMALL proof). We can apply α to Θ, and we obtain a set of proof structure(s),and at least one of them has a cycle. But each of this proof structure is an axiom on literals, and hence is acyclic. This is acontradiction.

Corollary G.24. Let R be a morphism of HypCDGPermRel. Then R is the denotation of a proof if and only if ∀Θ ∈ PS( R ),Θ yields no cycle.

The proof follows from the theorem above.

G.2 Narrowing down the cyclesThis section aims at summing up the results obtained in [3] on cycles in proof structures. Unless deemed relevant for theunderstanding of the chapter, no proofs will be given. We start by laying some definitions. The canonical orientation of jumpedges is from their conclusion A &i B to their jump S(Li ). A cycle in a graph ΘS is oriented if there is an orientation of thewhole cycle that respects the orientation of the jumps.

Lemma G.25. Let Θ be a proof structure that is connected for every normal switching S0. Then given a switching S , every cycle ofΘS can be transformed into an oriented cycle of ΘS , where S is another switching that has same valuation as S .

A cycle C is furthermore canonical if:• Every proper jump on the cycle is the conclusion of an axiom-link• If A,B formulas on the cycle are nested in the syntactic formula tree, then the orientation of C induces a directed pathfrom A to B or vice-versa. Then the unique path from A to B (or from B to A) in the cycle is the one corresponding totheir nesting in the syntactic formula tree.

Lemma G.26. For an arbitrary proof structure Θ and a switching S , every cycle of ΘS can be transformed into a canonical circleof ΘS ′ , for some switching S ′. If the cycle of ΘS was oriented, then so is the new canonical cycle.

Therefore, it is enough to consider canonical oriented cycles. Given a canonical cycle, we denote &i , 1 ≤ i ≤ n the occurrencesof &-links on which the cycle properly jump, and axi+1 the jump of &i .

We go on narrowing down the class of cycles one should consider. A cycleC in a graph ΘS is called simple if, for every linkK whose conclusion is a proper jump S(Li ) lying on C (so K is an axiom in the case of canonical cycles), thenw(K) = ϵ .pi .v ,where ϵ ∈ 1,¬, pi is the weight associated with Li , and v does not depends on any eigenweight associated with a &-linkwhose conclusion lies onC . In particular, a canonical cycleC is simple if for all i such that axi+1 is the axiom whose conclusionis a jump from Li lying in C , thenw(axi+1) = ϵ .pi .v where v does not depend on pi (1 ≤ i ≤ n).

Lemma G.27. • For a simple cycle C ∈ ΘS , let &k (1 ≤ k ≤ m) denotes the list of all &-links whose conclusions lie on C .Thenw(&k ) does not depend on pi , 1 ≤ i ≤ n.

• Every oriented cycleC of ΘS can be transformed into a simple oriented cycleC ′ of ΘS . Furthermore, ifC is canonical, then sois C ′.

G.2.1 Global cyclesWe add a final characterisation to the cycles. We want them to be “global”. Unfortunately, a proof-structure might have a cyclewithout having a global one. However we will prove that if the category has relations that yield cyclic oriented proof-structures,then it has some that yield global oriented cycles. First, we fix some terminology. A cycleC passes trough L, (where L is a link),means that the conclusion(s) of L lie(s) in C . A cycle is global if it passes through all the &-links whose weights are 1.

Lemma G.28. Given a simple canonical oriented cycle C , C global entails:• For all &i occurrences of &-links that cause proper jumps on C ,w(&i ) = 1.• For the weightsw(axi+1) = ϵ pi .vi , if the vi depends on a eigenweight pr , thenw(&r ) depends on pi .• Given 1 ≤ i , j ≤ n, ifw(axi+1) = ϵ pi .vi andw(axj+1) = ϵ pj .vj then the sets of eigenweights on which vi and vj dependsare disjoint.

We are interested in proving some properties about the way axiom-links behave in global cycles.28

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Definition G.29. Let Θ be a NDAL proof structure and α a literal in Θ. We say that a valuation ϕ yields two distinctaxiom-links with relation to an eigenweight p and a literal α if the following property holds. Let ϕ ′ be the same as ϕ forall eigenweights except for p where ϕ ′(p) = ¬ϕ(p). Then α ∈ sl(ϕ(Θ)), α ∈ sl(ϕ ′(Θ)), and the axiom links L ∈ sl(ϕ(Θ)) andL′ ∈ sl(ϕ ′(Θ)) that have conclusion α are different (that is, they have different conclusions).

That entails that the two axiom links L,L′ depends on p.One can prove this fundamental property for global cycle.

Proposition G.30. Suppose a proof structure Θ has a global simple oriented cycle C living in a valuationw such that ∀i .1 ≤ i ≤n.w(αi+1) = 1, where αi+1 is the conclusion of axi+1 that lies on the cycle. Then there exists a switching S such thatC ∈ ΘS and ϕSyields two distinct axiom-links with respect to pi and αi+1 for all i ∈ 1, ...,n.

G.3 Reduction to &-semi-simple sequentsDefinition G.31. A context is a sequent generated from distinguished holes, noted ⋆i , literals and MALL connectives, suchthat each hole appears at most once within the sequent. It is denoted Γ[⋆1, ...,⋆n]. A hole has a multiplicative occurence in thecontext if all the connectives in the parse tree from a root to this hole are multiplicative.

We may substitute any hole with a formula in the context. This substitution is written Γ[A1, ..,An]. AM ⊕ LL context is acontext where all the connectives are restricted to the M ⊕ LL fragment ofMALL, that is, without &.

Definition G.32. AM ⊕ LL sequent ∆ is semi-simple if it can be written ∆ = Γ[l11 ⊗ ... ⊗ l1k , ..., ln1 ⊗ ... ⊗ lnm] and Γ is aM ⊕ LL context without ⊗ (that is, a ` ⊕ LL-context).A MALL sequent ∆ is &− semi-simple if it can be written ∆ = Γ[A11 & .... & A1k , ....,An1 & ... & Anm] and Γ is a M ⊕ LLsemi-simple context (that is, if we replace holes by literals, it is a semi-simple sequent).

Proposition G.33. Suppose Γ is a M ⊕ LL sequent. Then there exists a list of M ⊕ LL semi-simple sequents ⊢ Γ1, ..., ⊢ Γn suchthat ⊢ Γ is provable (within M ⊕ LL +MIX) if and only if for all i then Γi is provable (within M ⊕ LL +MIX).

This follows from two observations .

LemmaG.34. Let Γ := Γ[A⊗(B`C)] aMALL sequent. Let Γ1 = Γ[(A⊗B)`C] and Γ2 = Γ[(A⊗C)`B]. Then Γ is provable if andonly if Γ1 and Γ2 are provable. Furthermore, Γ[(A⊕B) ⊗ (C ⊕D)] is provable if and only if Γ[(A⊗C) ⊕ (A⊗D) ⊕ (B ⊗C) ⊕ (B ⊗D)]are provable.

Proposition G.35. Suppose Γ is aMALL sequent. Then there exists a list ofMALL semi-simple sequents ⊢ Γ1, ..., ⊢ Γn such that⊢ Γ is provable within MALL if and only if for all i Γi is provable.

The proof is the exact same as the one of the proposition before, but by replacing literals with & formulas.

G.3.1 &-semi-simple sequents and global cyclesThe goal is to prove that one can consider only oriented canonical global cycles.

Proposition G.36. Consider the set of hypercoherent partial relations R of &-semi-simple types such that there is a Θ ∈ PS( R )

such that Θ has a oriented cycle. If this set is non-empty, then there is one hypercoherent partial relation R such that there existsΘ ∈ PS( R ) such that every oriented cycle in Θ is global.

Proof. Let take R of T type minimal such that R has a oriented cycle, where the order on types is defined as follows:

(number of ⊗ , number of `,&, ⊕)Then R cannot be split using any rule of MALL (otherwise, its proof structure would split as well, and one of them would

have a oriented cycle). The resulting R ′ would have a lesser type in the above hierarchy, contradicting the minimality of R .Therefore, ∀Θ ∈ PS( R ), Θ has no terminal ⊗-splitting link, no terminal &,`, ⊕1, ⊕2-link and is not the union of two proofstructures.Let us consider a Θ ∈ PS( R ), and a &-link L within it such thatw(L) = 1. Then there is no &-link below it (otherwise its

weight would not be 1). Now, if all links below it would be ⊕,`, then the proof structure would split. Hence there exists a⊗-link below it. Now suppose that ⊗ is not directly below the &-link. If a ⊕ is below the &-link, then it has weight 1, andtherefore the R comes from an injection, which is excluded. On the other hand, if a ` is just below the &, it means that thereis a ` above the ⊗ in the parse tree, which contradicts the &-semi-simplicity of R (since the &-link L is minimal).We know that Θ has a oriented cycle. Let us assume for contradiction that it has a non-global cycle. So there is a oriented

cycle C ∈ Θ, and a &-link L of weight 1 such that C does not pass through L. By the discussion above, this link has a ⊗ right29

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below it. Since the cycle does not pass through L, this ⊗ can be replaced by a ` in Θ (and all occurrences below it), and theproof structure Θ′ hence obtained still have the oriented cycle. Now note that the corresponding R ′ has a type lesser thanR . Finally, we note that pushing the newly created connective ` above using lemma G.34 in order to obtain a semi-simplesequent does not change its number of connectives, hence we obtain a semi-simple sequent of lesser type, contradicting theminimality of the type of R .

G.4 Final form for the decisive argumentWe start by assuming that the category of Chu hypercoherent partial relations is not fully complete. Therefore, this entails theexistence of a morphism R 1 of HypCDGPermRel that has a canonical proof structure with a cycle. As explained above, wecan reduce it to another morphism R 2 of HypCDGPermRel of &-semi-simple type. Now, as every normal switching of Θ R 2

is connected, it can be transformed into an oriented cycle. Therefore, seeing it as a proof-structure coming from a morphismof HypPermRel, proposition G.36 entails that there is a morphism R 3 of HypPermRel, whose canonical associated proofstructure Θ R 3 has a oriented cycle, and such that every oriented cycle of it is global. In particular, it has a simple, orientedcanonical global cycle.

So, as explained in [3], the shape of the cycle around the &i is as follows:

αi,mα⊥i,m

Wi

⊗i

&i

α⊥i+1,1 αi+1,1

Wi+1

We eachWi is shaped as below, without any &-jump.

α⊥i,1 αi,1

⊗i,1

α⊥i,2 αi,2

⊗i,2

α⊥i,3 αi,m−1

⊗i,m−1

α⊥i,m αi,m

Furthermore, theWi path does not contain any proper jump, and bounces on ⊗-links occurrences. Furthermore, by semi-simplicity, all the links between α i,k and ⊗i,k are ⊗-links, and therefore w(αi,k ) = w(⊗i,k ). Finally, note that there are alsoonly ⊗-links between α i1 and ⊗i+1. Also there are only ⊗-links between ⊗i+1 and &i+1. As the cycle is global,w(&i+1) = 1. Thisentails thatw(⊗i+1) = 1, and finallyw(α i1) = 1. Finally, we obtain thatw(α ik ) = 1 for all k and i .

However, if we change the valuation of ϕ we can make the axiom axi , between (α im)⊥ and α im disappears. On the other hand,

asw(α im) = 1, it stays within the proof structure. So in the proof structure there must exists two axioms whose conclusionsare α im . One is axi , of weight pi−1 (or ¬pi−1 but we pick pi−1 without loss of generality), and the other one ax′i has weight¬pi−1.vi . Let us show that ax′i has weight exactly ¬pi−1. Let us note that we can draw a jump from &i−1 to ax′i . Now, with anew valuation only changing pi−1, all the &j−1 and axj remain present (for j , i), hence the simple oriented cycle remainsalmost unchanged. Furthermore, this cycle is global andw(ax′i ) = ¬pi . Therefore, it looks like as displayed in the followingfigure.

αi,mα⊥i,m

Wi

⊗i

&i

α⊥′

i+1,1 α⊥i+1,1 αi+1,1

Wi+1

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Furthermore, picking a different &j , the cycle remains unchanged between the two valuations. That is, for the two differentvaluations switching pi , the cycle around &j remains as follows:

α j,mα⊥j,m

Wj

⊗j

&j

α⊥j+1,1 α j+1,1

Wj+1

Now, since the hypercoherent partial relations satisfy the mix rule, we apply mix to all the ⊗ occurrences in the type of Rthat do not appear on the cycle. Furthermore, by associativity and commutativity, we may assume that ⊗ appears just belowthe two literals. Hence the cycle remains. We then obtain a sequent of the following form :

⊢ F1, ..., Fnwhere Fi = αi,m ⊗ (Bi1 & Bi2),Ni [α

⊥i+1,1,α

⊥i+1,1],αi+1,1 ⊗ α⊥

i+1,2, ....,αi+1,m−1 ⊗ α⊥i+1,m ,Ξi

where m depends on i , where Ni [⋆1,⋆2] is either Ni,1[⋆1] ⊕ Ni,2[⋆2] or (Ni,1[⋆1] ⊕ N′

i,1) ` (Ni,2[⋆2] ⊕ N′

i,2), with all theconnectives in Ni being `. Ξi represents the remaining formulas after having apply the mix transformations to every ⊗ thatwere not part of the cycle. That is Ξi = E11 ⊕ E12, ...,Em1 ⊕ Em2, l1, .., lk where li are literals, and all connectives in Ei j being ⊕

and `.So we have 2n global circles whose structures are displayed in figure 5 of page 20, where the cycles are displayed in red, and

the axiom links not part of the cycle in black.So to conclude about full-completeness, one only needs to prove that there is no morphism of HypPermRel of this form.

References[1] S. Abramsky and R. Jagadeesan. New foundations for the geometry of interaction. In LICS, 1992.[2] S. Abramsky and P-A. Melliès. Concurrent games and full completeness. LICS’99.[3] R. Blute, M. Hamano, and P. J. Scott. Softness of hypercoherences and MALL full completeness. Ann. Pure Appl. Logic, 131, 2005.[4] P.Di Giamberardino. Proof nets and semantics: coherence and acyclicity. PhD thesis, 2004.[5] J-Y. Girard. Proof-nets: The parallel syntax for proof-theory. In Logic and Algebra, pages 97–124. Marcel Dekker, 1996.[6] M. Hamano. Softness of MALL proof-structures and a correctness criterion with Mix. Archive for Mathematical Logic, 2004.[7] D. J. D. Hughes and R. J. Van Glabbeek. Proof Nets for Unit-free Multiplicative-Additive Linear Logic. In LICS 2003.[8] G. Kahn and D. Macqueen. Coroutines and Networks of Parallel Processes. Research report, 1976.[9] Rob J. van Glabbeek and Dominic J. D. Hughes. MALL proof nets identify proofs modulo rule commutation. 2016.

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