A MALL Geometry of Interaction Based on Indexed Linear Logic

We construct a geometry of interaction (GoI: dynamic modeling of Gentzen-style cut elimination) for multiplicative-additive linear logic (MALL) by employing Bucciarelli-Ehrhard indexed linear logic MALL(I) to handle the additives. Our construction is an extension to the additives of the Haghverdi-Scott categorical formulation (a multiplicative GoI situation in a traced monoidal category) for Girard's original GoI 1. The indices are shown to serve not only in their original denotational level, but also at a finer grained dynamic level so that the peculiarities of additive cut elimination such as superposition, erasure of subproofs, and additive (co-) contraction can be handled with the explicit use of indices. Proofs are interpreted as indexed subsets in the category Rel, but without the explicit relational composition; instead, execution formulas are run pointwise on the interpretation at each index, w.r.t symmetries of cuts, in a traced monoidal category with a reflexive object and a zero morphism. The sets of indices diminish overall when an execution formula is run, corresponding to the additive cut-elimination procedure (erasure), and allowing recovery of the relational composition. The main theorem is the invariance of the execution formulas along cut elimination so that the formulas converge to the denotations of (cut-free) proofs.


Introduction
The indexed multiplicative additive linear logic MALL(I), introduced by Bucciarelli-Ehrhard [BucEhr00], is a conservative extension of Girard's MALL in which all formulas and proofs come equipped with sets of indices. The usual MALL is stipulated to be the restriction to the empty set. The status of the indexed syntactical system is noteworthy as it stems from the denotational semantics of Rel, a simple, yet pivotal categorical model of MALL. With the enabling of an explicit notion of location in linear proof theory, the indices can enumerate the locations of formulas and proofs, corresponding to denotational interpretations of MALL. The notion of location becomes a requirement for the additives, although it is redundant for the multipicatives, for which the singleton { * } suffices. To work with parallelism, which the additives bring intrinsically, different locations need to be handled rather than the sole location * . In the context of parallelism, superpositions are known to typically arise under the syntactic additive &-rule. Indices allow one to deal with superpositions by identifying multiple occurrences of formulas in the different indices and by enlarging (or restricting) the indices.
The original motivation for indexed logic was to provide a bridge between a truth-valued semantics (for provability) for MALL(I) and the denotational semantics of (nonindexed) MALL. By means of this bridge, Bucciarelli-Ehrhard obtained a new kind of denotational completeness theorem in [BucEhr00] for MALL and later extended it to the exponential in [BucEhr01].
This paper investigates indexed MALL from the perspective of a dynamic semantics for cut elimination, a topic that-to the best of our knowledge-has remained untouched aside from the precursory work of Duchesne [Duc09] since the original work of Bucciarelli-Ehrhard [BucEhr00].
The dynamic semantics is the Girard project of Geometry of Interaction (GoI), whereby cutelimination is modelled, using operator algebras [Gir89] and more generally traced monoidal categories [JSV96]. The GoI project was successful [Gir89,HS06] for MLL with the exponential, and inspired a new model of computation for β reduction of λ-calculus [DR95]. This paper aims to initiate an exploration of how to combine the two notions of location, which the indexed logic brings, and of dynamics, which GoI brings to cut-elimination. The combination is important in understanding additive cut-elimination. For this goal, we employ the indices to construct a GoI model for (non-indexed) MALL. We combine the Haghverdi-Scott categorical GoI situation [HS06] with the indices in such a way that that the original MLL GoI situation represents a collapse to the singleton index { * }. The dynamics of cut-elimination is captured by a feedback mechanism determined by traces of morphisms interpreting proofs. We further augment the situation with two kinds of actions, identical and zero, over the symmetries interpreting the cut-rule. These two actions provide representations of matches and of mismatches among locations. These come into play during a Gentzen style cut-elimination procedure, in which one encounters noncommunication of individual proofs, due to the additive parallelism. Crucial instances of GoI situation such as Rel + and Hilb 2 [HS06] are directly lifted to our framework, the latter of which is the operator algebraic origin of the Girard project.
We study Girard's execution formula [Gir89] in the general categorical setting of a traced symmetric monoidal category. The execution formula accommodates indices, and faithfully simulates MALL cut-elimination by a hybrid method relating the indexed syntax to the relational semantics. Each location in the relation interpreting a proof is first assigned an endomorphism on a reflexive object U . The cut-rule before execution is interpreted as a tensor product of two premise morphisms, more loosely than their composition. This interpretation allows extraction of the dynamical meaning of the cut, which the usual categorical composition hides by virtue of its static approach. In the loose interpretation, there remain redundant indices when interpreting rules: however, they are shown to disappear, while running the Execution formula in terms of the categorical trace structure. The disappearance of indices is modelled by zero morphisms, which exist in the traced monoidal categories for GoI. Proof-theoretically, the zero morphisms allow us to interpret discarding subproofs specific to additive cut-elimination, and in the way of theory of indices, they provide a way to to interpret mismatches among locations. In traced monoidal categories, the zero morphisms are supposed to act partially on symmetries for cut-formulas, and also to act partially on retractions and co-retractions of the reflexive objects. The latter action arises via tracing along the zero morphisms which takes feedback into account along with the zero. We prove zero-convergence which means that execution formulas converge to zero when two proofs interact with mismatched locations. Thus the execution formulas terminate to the denotational interpretations of proofs, while diminishing sets of indices in order to recover the relational composition. This is realized by properly coupling indices to trace axioms, especially for "generalized yanking" and "dinaturality". The former axiom directly designates that traces are primitive enough to retrieve the categorical composition in a monoidal category, and the latter axiom concerns the interaction of bidirectional flow of morphisms.
In contrast to the precursory work of Duchesne [Duc09] concerning both indices and GoI, the present paper accommodates the indices directly in GoI semantics in order to simulate (nonindexed) MALL cut elimination. The diminution of sets of indices is a typical dynamic aspect our GoI captures using the zero morphisms of our traced monoidal category. The precursor in [Duc09], on the contrary, first accommodates the index into the static category of Rel, using the semantic method of localisation, of which the indexed syntax provides a precise description. Then (nonindexed) dynamic GoI action over the indexed denotational semantics is investigated to characterise the static fix points as the denotational interpretation.
We prove two main result: (i) (Invariance of the execution formula during MALL normalization): The execution formula in our dynamic categorical modelling is shown to converge to the denotational interpretation of proofs in the static categorical model. This characterises the normalization of proofs by categorical invariants. (ii) (Diminution of indices while running the execution formula): The execution formula may converge to 0, making the redundant indices disappear. Part (i) is seen as a pointwise collection of invariants, as previously established for the multiplicatives [Gir89,HS06]. Part (ii) is specific to the additives: Proof-theoretically. it reflects erasure of subproofs as well as additive (co) contraction and superposition, in cut-elimination. Category-theoretically, it ensures that our categorical ingredient (the execution formula) is fine grained enough to retrieve a static monoidal category as well as a relational category handling indices.
The rest of this paper is organized as follows: Section 2 introduces a syntax MALL [c] (I) for indexed MALL with a cut list as well as its relational counterpart Rel [c] . A fundamental lemma is proved, which connects a provable MALL [c] (I) sequent to an indexed subset of the interpretation of a MALL proof with cuts. In Section 3, MALL proof reduction for cut elimination is lifted to MALL(I) proof transformation with diminishing sets of indices. Section 4 concerns our MALL GoI interpretation by means of the indexed system in a traced symmetric monoidal category with zero morphism. Execution formulas are run indexwise, and the main theorem is proved.
2 MALL(I) with cut list and relational semantics 2.1 MALL(I) with cut list (Inference rules of MALL [c] with cut formulas) We accommodate a stack to record cut formulas into the syntax of the multiplicative-additive linear logic MALL. To stress this, the system is written as MALL [c] . To accommodate the stack into the additive fragment, one has to work with superpositions that arise inside the stack as well as in the conclusion (outside the stack).
A MALL [c] sequent ⊢ [∆], Γ with a cut list is a set Γ of formula occurrences together with pairwise-dual formulas occurrences ∆ inside the brackets. Each dual pair in ∆ is written A, A ⊥ .
Sequents are proved using the following rules: Note: In the &-rule, not only is Γ superposed in the conclusion, but so is Σ in the stack. The superposition among cut formulas inside the stack causes the well-known additive (co-) contraction that arises in MALL cut elimination. The formulas occurrences Σ is not deterministically chosen in the premises, so that Σ in general is neither empty (i.e., never superimpose cuts) nor maximal (i.e., superimpose as many cuts as possible). Thus, the &-rule has several possible instances depending on the choice of Σ. The exchange rule is eliminated under the assumption that formula occurrences are implicitly tracked by the premises and the conclusion of a rule. We extend the above accommodation of cut lists to Bucciarelli-Ehrhard indexed system MALL(I) [BucEhr00]. To stress this, the system is written as MALL [c] (I). The extension stipulates that a set of indices is consistently associated with each formula (including cut formulas) and sequent (including cut lists).
We fix an index set I, once and for all. Each formula A of MALL(I) is associated with a set d(A) ⊆ I, called the domain of A.
(MALL(I) formulas and domains) Formulas in the domain J are defined by the following grammar: 0 ∅ and T ∅ are formulas of the domain ∅. For any J, K, L ⊆ I such that K ∩ L = ∅ and K ∪ L = J, For any MALL(I) formula A with d(A) = J, its negation A ⊥ with d(A ⊥ ) = J is defined using the De Morgan duality for the MALL formula. (Restriction) For a MALL(I) formula A with d(A) = J and K ⊆ J, the restriction A ↾ K of A by K is defined to be a MALL(I) formula in the domain J ∩ K as follows: We augment MALL [c] with indices. This makes it possible to accommodate a stack recording cut formulas to the original MALL(I) [BucEhr00]. Although this is straightforward for the multiplicative fragment, careful treatment is required for the listing of cut formulas in the additive fragment. Two kinds of sequences of formulas are considered in our MALL [c] (I)-syntax: One is a sequence Ξ whose all formulas occurrences have a same domain J uniformly, which is denoted by d(Ξ) = J. The other is a sequence Ξ whose any formula occurrence has a domain contained in J, which is denoted by d

MALL [c] (I) and Relational Semantics Rel [c]
It is well known that the category Rel of sets and relations constitutes a denotational semantics of MALL, that is, the interpretation is invariant, (π [∆],Γ ) * = (π ′ [∆ ′ ],Γ ) * , for any reduction π [∆],Γ ✄ π ′ [∆ ′ ],Γ of MALL cut elimination. In particular, the denotation of π is equal to that of a cutfree π ′ when ∆ ′ is empty. The cut rule is interpreted by relational composition in Rel, and this interpretation makes the semantics denotational.
Γ is interpreted as a subset of an associated set of the conclusion (without the cut list), Note that in the interpretation, the cut formulas ∆ become invisible by virtue of the relational composition: Two relations The interpretation π * of (1) is known in [BucEhr00] as the relational interpretation of MALL proofs in a compact closed category Rel × with biproducts +. The interpretation is specified as follows accordingly to the MALL rules, for which we refer to the above MALL [c] -rules ridden of the cut-lists. First, every formula A is interpreted as a set |A|, and every sequence Γ = A 1 , . . . , A n as |Γ| = |A| × · · · × |A n |: a 2 )) | (γ 1 , a 1 ) ∈ π * 1 and (γ 2 , a 2 ) ∈ π * 2 } ⊆ |Γ, A 1 ⊗ A 2 | (`-rule) π * = {(γ, (a 1 , a 2 )) | (γ, a 1 , a 2 ) ∈ π ′ * } ⊆ |Γ, A 1`A2 | Our aim in this paper is to investigate a dynamics of cuts hidden in such a static categorical composition. We begin by interpreting proofs in Rel but without performing cuts by means of relational composition. To stress this interpretation with the unexecuted cuts, the categorical framework is denoted by Rel [c] , in which the cut list [∆] is interpreted explicitly.
To deal with the additives in Rel Consequently, we interpret (2) as an object in Rel in terms of the disjoint union of each interpretations of the sublists: [ sl(∆)] = S⊆{1,...,m} in which |∆ S |, for a nonempty sequence, is the usual interpretation of the sequence in Rel and |∆ ∅ | := { * } = |⊥|. The disjoint sum Σ is taken over different S's.
In what follows in this paper, when S is clear from context, a sublist ∆ S is often abbreviated by ∆, whose hat indicates a pairwise deletion of some cut formulas.
Lemma 2.5 If ∆ = ∆ 1 , . . . , ∆ k such that ∆ i are lists of pairwise-dual formulas, then In what follows in Sections 2 and 3, ∼ = denotes an iso modulo the symmetry of the set-theoretical cartesian product. In the sequel, the symmetry corresponds to the exchange of formula occurrences. As the exchange is always clear from the context and fixed, we use the terminologies ⊆∼ = and ∈∼ = consistently as follows: A ⊆∼ = B (resp. a ∈∼ = B) means that A is a subset (resp. a member) of σ(B) where σ is the exchange for ∼ =.
which is defined inductively and in the same manner as in Definition 2.4, except for the cut rule to make the interpretation differ from the standard (1) in that ∆ is visible without performing the relational composition. (cut rule) The symmetry used for the definition is the exchange between the conclusion of the cut and merging those of π i 's. In obtaining the last inclusion, Lemma 2.5 is used because the two lists ∆ 1 and ∆ 2 are disjoint.
We extend Bucciarelli-Ehrhard translation of Definition 2.7 to Definition 2.8 to accommodate cut formulas inside the stack.
To any MALL formula A and a family a ∈ |A| J , a formula A a of MALL(I) is associated, with domain J so that A a ↾ ∅ is A. For a sequence Γ = A 1 , . . . , A n of MALL formulas, every γ ∈ |Γ| J is written uniquely as γ = γ 1 × · · · × γ n with γ m ∈ |A m | J , and we set Γ γ = A 1 γ 1 , . . . , A n γ n .
-For A = 0 or A = T , if J = ∅, we have |A| J = ∅, and A a is undefined. If J = ∅, |A| J has exactly one element, namely, the empty family ∅, and we set 0 ∅ = 0 and T ∅ = T .
Here b × c denotes the mediating morphism of the set-theoretical cartesian product.
(Notation) Let X be a set and J = J 1 + J 2 . Every x ∈ X J yields the restrictions x i = x ↾ Ji ∈ X Ji with i = 1, 2. Conversely, the two restrictions allow us to recover x. We write this as x = x 1 ⌢ x 2 .
Then, by employing Definition 2.7, for a given MALL sequent ⊢ Note that all the formulas in Γ γ have domain J, while each formula in ∆ δ has a domain contained in J. The ∆ δ 's inside the stack become a list of pairwise-dual MALL(I) formulas in which each pair has the same domain.
The translations commute with restriction of indices, and Lemma 2.2 can be restated:

Fundamental lemma
Indexed linear logic arises essentially due to its tight connection to the relational semantics. The connection is realized by a fundamental lemma due to Bucciarelli & Ehrhard (proposition 20 of [BucEhr00]) establishing a correspondence between indexed sets in Rel and indexed sequents in MALL(I). The former is semantic in MALL, while the latter is syntactic in MALL(I). The fundamental lemma is shown to be preserved under our extended syntax and semantics, designed to accommodate cut formulas in MALL [c] (I) and in Rel [c] , respectively.
Proposition 2.10 (Fundamental lemmaà la Bucciarelli-Ehrhard) For ν ∈ (|sl(∆)| × |Γ|) J with J ⊆ I, the following two statements are equivalent and induce a relationship ρ ↾ ∅ = π between π of (i) and ρ of (ii):   Restricting the two by cut-elimination. Schematically, this can be written: The transformation ρ ◮ I ρ ′ is called a lifting of π ✄ π ′ . The lifting simulates a MALL proof reduction for cut elimination in terms of MALL(I) proof transformation.
The lifting in Definition 3.1 is not unique for a given MALL [c] reduction, as any subset J ′′ of J ′ , ρ ◮ I ρ ′ ↾ J ′′ obviously becomes a lifting for ρ and ρ ′ under this definition.
Example 3.2 The following is a MALL [c] (I) reduction with diminishing sets of indices whose restriction to ∅ is a Gentzen reduction eliminating the pairwise-dual additive connectives & and ⊕ in the cut formulas: The sets of the indices are diminished from J 1 + J 2 to J 1 as a result of erasing the subproof π 2 within the proof transformation.
ρ and ρ ′ are MALL proofs ensured by the fundamental lemma (Proposition 2.10) for the sequents respectively. Hence, we can also denote the lifting by Note: There is no straight connection between ν and ν ′ such as the former is the restriction to the latter.
Proof. For every kind of reduction ✄, we can directly construct ν ′ together with J ′ . There are three crucial cases: (with A and B being different occurrences of the same formula) reduces to , a j ∈ |A| and λ j ∈ |Γ|. Note that ν j = (δ j , b j , a j , b j , λ j ). We define (Crucial case 2) This case is the MALL reduction arisen by Example 3.2 above, when restricting to the empty domain ∅ and identifying MALL(∅) with MALL.
Note that in the last &-rule of π ′ , Ξ and Ω inside the stack are chosen to be superposed. Our categorical framework is a minimal part of the Haghverdi-Scott GoI situation [HS06] with a reflexive object U in a traced symmetric monoidal category C with tensor unit I. Ours in addition requires that C has zero morphisms, in particular, a zero endomorphism 0 U on U : k denotes a pair of morphisms j and k respectively from U 2 to U and the other way around. j and k are called respectively co-retraction and retraction for the reflexive U when k is denoted by ⋆ m both for object ⋆ or morphism ⋆. The trace structure will be introduced later in (14). We require the commutativity of the pair (j, k) and the zero 0 U ; Note: The zero morphism 0 U is absorbing with respect to composition, but not with respect to tensor. That is f ⊗ 0 U and 0 U ⊗ f are not in general 0 U 2 for any endomorphism f on U .
Proof. The first assertion is derived by the condition (6). The general assertion is by iterating The zero endomorphism 0 U acts on the symmetry s as follows.
Definition 4.2 (zero-action (s U,U ) 0 ) 0 action on the symmetry s U,U on U 2 is defined to annihilate the symmetry s U,U to the zero endomorphism on U 2 .
(s U,U ) 0 := 0 U⊗U Alternatively, the action is defined to be the following decomposition in terms of conjugation (both precomposing and composing): We abbreviate s U,U and (s U,U ) 0 as s and s 0 , respectively. To avoid collapsing the categorical framework whose GoI interpretation becomes the degenerate zero, we assume s is nonzero; that is, s and s 0 are distinct endomorphisms on U 2 in C.
This is a technical assumption for the main theorem (Theorem 4.16) of this paper to characterise the diminution of the index set in terms of the convergence to the zero, distinguishable from the other morphisms.
The zero morphism, which is required in our framework, exists in crucial examples of GoI situations: (i) Rel + is Rel with the disjoint union + of sets as ⊗ and a reflexive object N. The empty relation on N is the zero morphism. Furthermore 0 N + 0 N = 0 N+N sufficient to the condition (6). (ii) The monoidal subcategories Pfn and PInj of Rel + , both in which resides the zero morphism.
PInj is known to be equivalent to the original category Hilb 2 of Hilbert spaces and partial isometries for Girard's GoI 1 [Gir89]. Note: The above examples of GoI situations happen to be sum-style monoidal structures [HS11], whose ⊗ is given by the disjoint union. The style is known to capture the notion of feedback as data flow in terms of streams of tokens around graphical networks. However our categorical framework (5) in the present paper is the general one, hence does not assume that the monoidal product is sum-style.
The i-th constituent x i of x = (x 1 , . . . , x ℓ ) ∈ |π [∆],Γ | of Definition 2.6 corresponds, in terms of the membership relation, to the i-th occurrence of formulas ∆, Γ with a unique sublist ∆.
,Γ | interpreting π in Definition 2.6 belongs to | ∆| × |Γ| with a unique sublist ∆ of ∆. That is, for x = (x 1 , . . . , x ℓ ), there exists a unique sublist ∆ such that the i-th constituent Note the sublist ∆ is determined not only by π but also by x, as shown in the construction (8) in the following proof.
While all the formulas in the sequence Γ x ′′ of MALL(I) formulas has the domain { * }, each formula in the sequence ∆ x ′ has the domain either { * } or ∅. Thus the unique sublist ∆ is determined by the following two steps: (i) Ridding ∆ x ′ of all the formulas D such that d(D) = ∅, which yields the subsequence of ∆ x ′ . (ii) ∆ is defined to be the subsequence of (i) restricted to ∅ (i.e., forgetting the domain) in order to obtain non-indexed formulas.
To be short for (i) and (ii), the sublist ∆ is the unique one making ∆ x ′ agree with ∆ x ′ ridden of all the formulas whose domain is ∅.
Note whenever a cut formula occurs in ∆, so does its dual formula, hence x ′ = (x 1 , . . . , x 2m ) for a natural number m, then x ′′ = (x 2m+1 , . . . , x ℓ ). By Definitions 2.7 and 2.8, the membership relation for the assertion of x i and the i-th formula in ∆, Γ follows. ✷ In what follows, we make permutations (x τ (1) , . . . , x τ (ℓ) ) among the constituents implicit so that x is up to the permutation. This is because the permutation corresponds to the exchange rule eliminated from our syntax. The permutations will be reflected by the symmetry of monoidal product of C, in the following interpretation x , which we denote by ∼ =.
Definition 4.4 Endomorphism x on tensor folding U 's and tensor folding σ x of symmetry s and of zero s 0 ,Γ | is interpreted as an endomorphism x π on the tensor product U ℓ together with an endomorphism σ π x on a subfactor U 2m of U ℓ . The endomorphism σ π x interprets cut rules in π [∆],Γ and is an m-ary tensor folding of morphisms which are either s 1 = s or s 0 = 0 (cf. Definition 4.2) both on U 2 : To distinguish each i-th component U of U ℓ , we label each component U xi with x i by abuse of notation, since the label x i is always clearly specified to designate the i-th constituent of x. Then x i ∈ |A i | so that A i is the tag of x i . Under this labelling, η(i) of s η(i) on U a ⊗ U a ′ is defined to be the Kronecker delta δ a,a ′ , where the tags of a and a ′ are pairwise dual formulas in ∆. That is, where a and a ′ range so that their tags are pairwise (10) dual cut formulas in ∆.
We define ( x π , σ π x ) by induction on the construction of the proof π. • We simultaneously define that a component U xi such that the tag of x i is a formula in Γ is contracted by the induction on π 1 . Since U xi appears both in the co-domain and in the domain of x , every contracted component in the domain (resp. co-domain) of x is a domain (resp. co-domain) of a unique retraction (resp. co-retraction), called associated retraction (resp. associated co-retraction) 2 . We simply say i is contracted when so is U xi .
In the definition, π 1 and π 2 denote the two premise proofs of the binary rules, and π ′ of the unary rules. (Axiom) x = ( * , * ) ∈ |A ⊥ , A| with A = 1 and A ⊥ =⊥. We define x π to be a symmetry s U * ,U * on U * ⊗ U * of C. Because π is cut-free, σ π x is empty by definition. x has no contracted component so that neither U * nor U * are contracted. (Cut rule) w 1 , a) and x 2 = (v 2 , a ′ , w 2 ) belong respectively to |π 1 | and to |π 2 |. We define x π ∼ = x 1 π1 ⊗ x 2 π2 and σ π x is σ π1 x1 ⊗ σ π2 x2 ⊗ (s Ua,U a ′ ) δ a,a ′ That is, if a = a ′ (resp. a = a ′ ), then σ π x on U a ⊗ U a ′ is s (resp. s 0 ), and σ π x on the remaining components is σ x1 ⊗ σ x2 . Note the definition makes sense because σ π1 x1 ⊗ σ π2 x2 acts on the domain distinct both from U a and U a ′ .
We say the cut (of the last rule of π) matches (resp. mismatches) in x if a = a ′ (resp. otherwise). (`-rule) w 1 , a) and x 2 = (v 2 , w 2 , b) are respectively from |π 1 | and |π 2 |. Note A ⊗ B is the tag of (a, b) in π, while A (resp. B) is the tag of a (resp. b) in π 1 (resp. in π 2 ). The endomorphism x is obtained directly from x 1 π1 ⊗ x 2 π2 on U ℓ+1 = U ℓ × U (a,b) by the retraction U a ⊗ U b ✁ U (a,b) . That is, We also define σ π x by σ π1 x1 ⊗ σ π2 x2 .
In the above both multiplicatives rules (⊗ and`), the introduced U (a,b) in the domain (resp. co-domain) is a contracted component, and the assoc-ret (resp. assoc-coret) is k : U (a,b) ). Other contracted components are those of (v, a, b) for`and (v 1 , w 1 , a) and (v 2 , w 2 , b) for ⊗ distinct from the components U a and U b . Note that (v, a, b) ∈ |premise of`| and (v 1 , w 1 , a) ∈ |left premise of ⊗| and (v 2 , w 2 , b) ∈ |right premise of ⊗|. (&-rule) x is either (v, (1, a)) or (v, (2, a)), so that (v, a) are either from |π 1 | or |π 2 |, respectively. Note A 1 &A 2 is the tag of (i, a) in π, while A i is the tag of a in π 1 or in π 2 when i = 1 or i = 2, respectively. We define x π = (v, a) πi by relabelling the component U a either by U (1,a) or U (2,a) for the domain (equally for the codomain) of x π . We also define σ π x by σ πi (v,a) . (⊕ i -rule) Same as &-rule but using the unique premise π ′ deterministically.
In the above both additives rules (& and ⊕ i ), contracted components are those of (v, a) under the relabelling U a by U (i,a) for the component of the domain (equally of the codomain). Note that (v, a) belongs to one of |left premise| and |right premise| in &-rule depending on i = 1 or i = 2, and obviously to |the unique premise| in ⊕ i -rule.
In the sequel, the pair of Definition 4.4 is simply written ( x , σ x ) by omitting π, since the proof π will be always specified clearly from the context.

(Remark on Def 4.4 ) [The endomorphism x as an I/O box]
The endomorphism x is seen as an input/output (I/O) box on the (n + 2m)-ary tensor folding of U , whose inputs/outputs are the formulas occurring in Γ, ∆, in which Γ contains n occurrences of formulas, and a sublist ∆ contains 2m occurrences of (pairwise dual) formulas. The formulas are the tags of x i s where x = (x 1 , . . . , x ℓ ) ∈ |π [∆],Γ |. The endomorphism σ x is seen as a more special box consisting of m-ary tensor folding of {s, s 0 } for the I/O formulas in the sublist ∆. See Figure 1 below for ( x , σ x ).
A characterisation of contracted component is derived: Lemma 4.5 U xi is a contracted component of x for x ∈ π if and only if x i 's tag contains a multiplicative connective (i.e., ⊗ or`).
Proof. Straightforward accordingly to the inductive step of Definition 4.4, in which the retraction and the co-retraction are used only for multiplicatives-rules (`or ⊗) so that x is constructed x ′ (j,k) or ( x 1 ⊗ x 2 ) (j,k) , respectively. ✷ (labelling associated retractions ✄ i and co-retractions ✁ i ) Every associated retraction ✄ (resp. associated co-retraction ✁) is by Definition 4.4 uniquely labelled with a contracted i such that the tag of x i is a formula in Γ. The labelling is written ✄ i (resp. ✁ i ). In what follows, the labelling is made implicit except when an explicit labelling makes an explanation easier to understand. (assoc-rets and assoc-corets in I/O box x ) When the endomorphism x is seen as the I/O box, the assoc-rets and the assoc-corets are those ✄'s and ✁'s whose domains and co-domains lie respectively among the inputs and the outputs of x . By the construction of x , they lie pairwise in the inputs and the outputs. See Figure 1 for x depicting the occurrence of the assoc-rets ✄'s and the assoc-corets ✁'s.
(Convention omitting Id U s) When an indicated occurrence of a contracted component U is clear from the context, ✄ • x (resp. x • ✁) is an abbreviation for the composition (Id U ⊗ · · · ⊗ Id U ⊗ ✄ ⊗ Id U ⊗ · · · ⊗ Id U ) • x (resp. x • (Id U ⊗ · · · ⊗ Id U ⊗ ✁ ⊗ Id U ⊗ · · · ⊗ Id U ), where the domain of ✄ (resp. the codomain of ✁) is the contracted U . This abbreviation is generalised for plural indicated occurrences of contracted components U 1 , . . . U r in U ℓ as follows: (⊗ r ✄) • x (resp. x • (⊗ r ✁)) stands for the composition (resp. precomposition) to x by the morphism tensoring ✄ (resp. ✁) on the contracted components indicated and Id U on the remaining components. Note because r ≤ ℓ, the abbreviation is for omitting identities on U s.
x with assoc-rets ✄ and assoc-corets ✁ σ x whose each cross is either s or s 0 Figure 1: ( x , σ x ) of Definition 4.4 and assoc-rets and assoc-corets of Definition 4.4 In the sequel, the two abbreviations are pictured as in the left hand respectively. Using a notation of the r-ary tensor folding of ✄ (resp. ✁), it is also written by ✄ r • x (resp. x • ✁ r ). This convention is equally employed when indicated occurrences of assoc-(co)rets are clear from the context.
Under this convention, for any contracted component in the co-domain (resp. domain) of x , it holds; That is, ✁ • ✄ composes (resp. precomposes) with any retraction (resp. co-retraction) as the identity. In other word, ✁ • ✄ is a projector on a contracted component in the codomain (resp. domain) by composition (resp. precomposition). Pictorially, In Equation (11), the assoc-coret (resp. assoc-ret) occurs explicitly as the last composed ✁ (resp. the first precomposed ✄). Thus, the left most ✁ (resp. right most ✄) in (11) is seen labelled ✁ i (resp. ✄ i ) such that the tag of x i is a formula occurrence in Γ, where x = (x 1 , . . . , x ℓ ) ∈ |π [∆],Γ |. Since every contracted component occurs pairwise in the co-domain and the domain of x , the two equations can be written successively all at once; The co-domain and the domain U ℓ of x have in general several contracted components U s labelled with x i s, where i ranges in the set r of the contraced is. Thus, for all the several pairs of contracted components in the domain and the co-domain of x , the parallel compositions and precompositions with ✁ • ✄s to each contracted components act as the identity on x : where r is the cardinality of r, hence is a number of the contracted components, and the third equality is by (✁ • ✄) r = ✁ r • ✄ r , as (−) r is the r-ary tensor folding. Since the last composed ✁ r (resp. the first precomposed ✄ r ) in the rightmost expression of (12) are the explicit occurrences of the assoc-rets (resp. assoc-corets), the endomorphism x is written so that all the assoc-rets ✄ r and the assoc-corets ✁ r can be made explicit as follows: Roughly speaking, x o is x stripped of all the assoc-rets and assoc-corets. which is depicted in the following Figure 3: This subsection is concerned with defining the action ǫ x (Definition 4.8) over the associated retractions (resp. co-retractions) in Definition 4.4 above. The action ǫ x arises from σ x of (9) when the feedback on the trace of C is taken into account, and annihilates, using the zero morphism 0 U , a certain class of retractions and co-retractions. This class is defined in Definition 4.8 in terms of zero input and output.
In what follows, we shall see how feedback stemming from Gentzen cut-elimination for a MALL proof π acts on the assoc-rets and the assoc-corets of x for x ∈ |π [∆],Γ |. The action is stipulated in terms of the zero morphism added in our framework. First, in a categorical framework of Girard's GoI project, the feedback is modelled by the trace structure (cf. [HS06]) defined by the seven axioms below: There are three kinds of naturality axioms: naturality in X and naturality in Y , and dinaturality in Z. The other axioms are vanishing I,II, superposing and yanking. See Appendix A.1 for the seven axioms: the three naturalities and the four axioms. In our setting of Definition 4.4, the endomorphism x is on U n+2m so that n and 2m are the numbers of formulas respectively in Γ and in a sublist ∆, and σ x is on the subfactor U 2m . Then the feedback is calculated by; See Figure 4. Note that when x ∈ |π [∆],Γ | comes from a proof π of the multiplicative fragment, the equation is exactly the GoI interpretation of the proof π [∆],Γ (cf. [HS06]). This is because in the multiplicative fragment, the index set I becomes redundantly the singleton { * }, thus |π [∆],Γ | = {x}, whereby σ x is a simple tensor folding of the symmetry s (free of 0 morphism).
By the naturalities of traces, the assoc-corets (resp. the assoc-rets) of x commute with Tr U 2m U n ,U n , hence taking a trace of (11) composed with Id ⊗σ x yields for any i such that i-th com-ponent of x = (x 1 , . . . , x ℓ ) is contracted.
Thus all the assoc-rets and assoc-corets of x are written explicitly; where r is the set of all contracted is and r is the cardinality of r.
Equation (16) is depicted in Figure 5, in which the dotted squares are the scopes of the traces and the shifting of the scopes are naturalities of the ✄s and the ✁s; While inside the sole x , the assoc-rets and the assoc-corets (written explicitly in (13)) do not interact with zero morphisms because the construction of x of Definition 4.4 is free from the zero morphisms. Remember that the zero morphisms reside only in σ x as subfactors (cf.(9)). However when they are put inside the context Tr U 2m U n ,U n ((Id ⊗σ x ) • −) (written explicitly in (16)), they may interact with zero morphisms arising from σ x via the feedback of the trace. That is, the trace in a monoidal category takes feedback into account, hence makes the zeros stemming from σ x interact with the assoc-rets and the assoc-corets of x . This yields a certain action ǫ x on the assoc-(co)rets of x , as defined in Definition 4.8 below.
Definition 4.6 zero input (resp. output) of assoc-coret (resp. assoc-ret) w.r.t the interpretation σ x of cuts (zero input of assoc-coret ✁) An assoc-coret ✁ of x is said to have zero input w.
Why do we use the terminology zero input (resp. output) ? The object U ⊗ U of ✁'s domain (resp. ✄'s co-domain) can be regarded as having two inputs (resp. outputs), one left component U and the other right one. Then the decomposition in each case says that one of two inputs (resp. outputs) is zero.
for the zero output.
Then x has the unique pair of assoc-ret and assoccoret both interpreting the ⊗-rule. See the left-hand dotted rectangle representing x with the assoc-ret and the assoc-coret. The pair of assoc-ret and the assoc-coret appears explicitly in the second and the third of the following equations (in which σ x = s 0 U,U ): x without the assoc-ret and the assoc-coret. See the above figure whose LHS and RHS are the first and the last equations, respectively, The assoc-coret (resp. assoc-ret) has zero input (resp. output) because the right picture depicts ✁ (resp. ✄) having a zero input (resp. output) from the northwest (resp. to the northeast). Hence (0 U ⊗ U ) composes (resp. precomposes) to s 0 U,U ⊗ s U,U trivially. Definition 4.8 (action ǫ x on assoc-rets and assoc-corets of x ) The endomorphism σ x of Definition 4.4 for x ∈ |π [∆],Γ | yields the following action ǫ x on the assoc-rets and the assoc-corets of x . The action ǫ x acts on each assoc-ret and assoc-coret as either zero or the identity by (pre)composition on them, as follows: where zero actions ✄ 0 and ✁ 0 are defined respectively as follows: That is, the zero annihilates the pair of assoc-ret and assoc-coret j : U ⊗ U ✁ U : k to the pair of the zero morphisms j 0 : U ⊗ U ✁ 0 U : k 0 , where j 0 = ✁ 0 and k 0 = ✄ 0 .
The action ǫ x of Definition 4.8 is by definition conjugate on the pairwise tensor foldings (⊗ i∈r ✄ i , ⊗ i∈r ✁ i ) = (✄ r , ✁ r ) of the assoc-rets and the assoc-corets represented in (13), where r = {i | i is contracted with x = (x 1 , . . . , x ℓ )}. Hence we define to formulate the action on x by conjugation: Pictorially, This action of ǫ x , by naturalities, extends to the action ǫ x on the corresponding retractions and co-retractions in (16): by (17) Note by (15) that the LHS of the first equation is ex (σ, x) ǫx . Recall that r is the number of the assoc-rets ✄ i (equally the assoc-corets ✁ i ) of x such that i is contracted.

The Execution formula
Definition 4.9 (Execution formula Ex (σ, x) for x ∈ |π [∆],Γ |) For every x ∈ |π [∆],Γ |, the endomorphism Ex (σ, x) is defined by where ( x , σ x ) is the pair of the endomorphism on U n+2m and on the subfactor U 2m in Definition 4.4 and ǫ x is the action in Definition 4.8 on the assoc-rets and the assoc-corets of x . The domains (resp. the co-domains) of the assoc-rets (resp. the assoc-corets) lie among the subfactor U n in the domain (resp. the co-domain) of x . See Figure 6.

Zero Convergence of Execution Formula
This subsection concerns the main proposition (Proposition 4.15), which says that communicating two proofs via mismatched pair yields zero convergence of Ex. We start with the tracing zero lemma derivable from some trace axioms.
Proof. First by Lemma 4.1 0 U m = (0 U ) m for any natural number m. Then by superposing Tr U U n+1 ,U n+1 (0 U n+2 ) = 0 U n ⊗ Tr U U,U (0 U 2 ), it suffices to prove the assertion for n = 1. Second, observe the equation 3 Thus Lemma 4.14 : ex denotes ex(σ, x). We prepare the following Lemma 4.14, which will directly entail the main Proposition 4.15.
, allowing interaction of the assoc-rets and the assoc-corets of x with the two zeros 0 U in the context. Zero input (resp. zero output) of assoc-ret (resp. assoc-coret) in this context is defined in the same manner, yielding the action, say δ x , on the assoc-rets and the assoc-corets of x same as in Definition 4.8 (but simpler without the feed back): That is, the morphism ✄ δx is defined to be Lemma 4.14 (lemma for Prop 4.15) where ex (σ, x) δx is ex (σ, x) of (15) whose x is replaced by x δx using the action δ x of Definition 4.13. See Figure 7 (upper-right) depicting the equation. The lemma holds up to the the permutations τ on U n so that the left ex (σ, x) is read by τ −1 • ex (σ, x) • τ . Hence the assertion is independent of the choice of U for the 0 U . The choice is of one formula occurrence from Γ, as each occurrence is interpreted by the distinct U .
Proof. Induction on the construction of π for x in Definition 4.4. In the proof, Equation (19) in the proof of Lemma 4.12 is used. In the following, for i = 1, 2, x i are the premises of x (i.e., x 1 = y and x 2 = z in Definition 4.4), and ex i denotes ex (σ, The last equation is by I.H.'s on x 1 and x 2 . (⊗-rule) (case 2) other than case 1: In this case, the U for the 0 U of Id ⊗0 U is a factor from the (co)domain of x i . We assume without loss of generality that i = 1. Then, (Id ⊗ 0 U ) • ex 1 • (Id ⊗ 0 U ) = 0 U n 1 by I.H on x 1 . This directly implies that the co-retraction and the retraction (j, k) interpreting the ⊗-rule are acted by δ as zero, denoted by (j 0 , k 0 ), since j's output and k's input both on x 1 are zeros by the I.H. Hence, when (j, k) is written by (✁, ✄), (cut-rule) By the rule, ex(σ, x) = Tr U 2 U n ,U n ((Id 1 ⊗s U,U ⊗ Id 2 ) • (ex 1 ⊗ ex 2 )) We assume without loss of generality that the U for the 0 U of the Id ⊗ 0 U is a factor from the (co)domain of x 1 . Then, LHS of the assertion is equal to ,Γi,Ai | so that a i ∈ |A i | with a 1 = a 2 and A 1 and A 2 are pairwise dual formulas. Then Ex (σ a1,a2 , Ex (σ, x 1 ) ⊗ Ex (σ, x 2 )) = 0 U n 1 +n 2 where σ a1,a2 = s δa 1 ,a 2 Note that the LHS of the assertion is, by Definition 4.9, the following, in which ǫ is the action arising from σ a1,a2 = s 0 Ua 1 ,Ua 2 of Definition 4.8: Tr U 2(1+m 1 +m 2 ) U n 1 +n 2 ,U n 1 +n 2 (Id ⊗σ a1,a2 ⊗ σ x1 ⊗ σ x2 ) = Tr U 2(1+m 1 +m 2 ) U n 1 +n 2 ,U n 1 +n 2 ((Id ⊗σ a1,a2 ) • ((Ex (σ, x 1 ) ⊗ Ex (σ, x 2 )) ǫ ) by nats and vanish IIs In the above x j is an endomorphism on U 2(mj +1)+nj with the subfactor U 2mj for σ xj . Before the proof of Proposition 4.15, let us observe a general equation derivable from certain trace axioms (dinaturality and yanking), where f : X ⊗ U −→ Y ⊗ U and 0 U,I (resp. 0 I,U ) is the zero morphism from U to the tensor unit I (resp. the other way around).
Note first that the zero morphisms 0 U,I and 0 I,U above are derivable from 0 U using the trace: Figure 7 (lower-right) depicting Equation (21). (proof of (21)) By the decomposition 0 U = 0 I,U • 0 U,I , the LHS is Tr I X⊗I,Y ⊗I ((Id ⊗ 0 U,I ) • f • (Id ⊗ 0 I,U )), by dinaturality, which is equal to the RHS by vanishing.
(end of proof of (21)) Finally we go to: For this, it suffices to show the following stronger equation, as ex i is Ex i ridden of the zero action on the assoc-rets and the assoc-corets (cf. Definition 4.8): By superposing (after the distribution of ǫ over ⊗), the LHS is equal to On the other hand, Lemma 4.14 and (21) say for all i = 1, 2 Since the two actions ǫ and δ coincide again by (21), the formula (22) becomes equal to 0 U n 1 ⊗ 0 U n 2 = 0 U n . ✷

Main Theorem
This section concerns the main theorem of this paper.

Theorem 4.16 (Ex is invariant and diminishes sets of indices)
(ii) In particular, when π ′ is cut-free so that ∆ ′ is empty, then We prove (i) according to the cases of Proposition 3.3, since (ii) follows directly from (i) as follows: For a cut-free π ′ , σ ν ′ is empty, hence Ex (σ, ν j ) = ν ′ j , where ν ′ j = 0 holds directly both from the construction of Definition 4.4 and from the non-collapsing assumption (7).

Conclusion and Future Work
This paper offers two main contributions: (i) Presenting an indexed MALL system for stacking cut formulas and its relational counterpart to simulate MALL proof reduction of cut elimination.
(ii) Constructing an execution formula for the interpretation of MALL proofs equipped with indices. The MALL proof reduction is characterised by the convergence of the execution formula to the denotational interpretation. Furthermore, the zero convergence of the execution formula characterises the diminution of indices, which is specific to additive cut elimination.
Our explicit use of indexed-syntactical manipulations directly overcomes known difficulties in additive GoI. We hope that this paper, from the perspective of indexed linear logic, will shed light on an approachable understanding of the preceding literature on additive GoI, from precursory ones [Gir95,Duc09] to more recent developments [Gir11,Sei16].
We discuss some future directions. For a genuine MALL GoI without bypassing via indexed logic, a syntax-free counterpart is required to replace the indices. We construct such a genuine GoI [Ham17] using an algebraic ingredient: a scalar extension of Girard's * -algebra of partial isometries over a boolean polynomial semi-ring. The genuine GoI may help us connect our syntactic manipulation of indices to Girard's semantic use of clauses for predicates in the precursory MALL GoI [Gir95].
In a syntactic direction, the status of Gentzen cut elimination for MALL(I) remains open since the present paper only concerns lifting the image to the indices of MALL cut-reduction. The status will complement the reduction-free cut elimination, known to be derivable from the Fundamental lemma 2.10 (cf. [BucEhr00,BucEhr01,HamTak08]).
Extending the present paper to the exponentials is challenging to use Bucciarelli-Ehrhard LL(I) [BucEhr01] for modelling GoI. This will involve extending our methodology of a traced monoidal category with a zero morphism to the whole GoI situation [HS06,HS11], compatibly with the multisets interpretation of the exponential connective in Rel. The explicit accommodation of the indices to GoI will give a novel approach to the (non indexed) GoI modelling for the exponentials.

C Indices and Additive Cut Elimination
This appendix elucidates the fundamental idea of the paper. The appendix may read as a prologue of the paper by readers yet familiar with MLL GoI interpretation on a reflexive object in a traced monoidal category. Consider a sequence π 1 ✄ π 2 ✄ π 3 of cut eliminations for proofs in the additive fragment of MALL. In our sequent notation, pairwise cut formulas, if present, are stored inside a stack [·] in a sequent. The first reduction, intrinsic to the additives, eliminates a & in a cut, whereby the subproof ax 2 is pruned. The second reduction eliminates a redundant cut against an axiom: Step 1 (Interpretation |π| in Rel with unperformed cuts and indices for additives) We begin by interpreting proofs in Rel but without relational composition. For this, the cut rule is interpreted same as the tensor rule. This interpretation is consistent with the syntactic convention, starting from Girard's GoI 1, which puts the cut formulas into a stack. For simplicity and in accordance with the fact that the multiplicative dual elements 1 and ⊥ are interpreted in Rel as the singleton set, we take A = 1 and, dually, A ⊥ = ⊥, so that |1| = | ⊥ | is the singleton, whose unique element is denoted * or * (obviously, * = * ), depending on whether it comes from |A| or |A ⊥ |, respectively.