Constructing Fully Complete Models of Multiplicative Linear Logic
CONSTRUCTING FULLY COMPLETE MODELS OFMULTIPLICATIVE LINEAR LOGIC ∗ ANDREA SCHALK a AND HUGH P. STEELE ba School of Computer Science, University of Manchester, Oxford Road, Manchester M13 9PL, UK e-mail address : [email protected] b Universit´e Paris 13, Sorbonne Paris Cit´e, LIPN, CNRS, F-93430, Villetaneuse, France e-mail address : [email protected] bstract . The multiplicative fragment of Linear Logic is the formal system in this familywith the best understood proof theory, and the categorical models which best capture thistheory are the fully complete ones. We demonstrate how the Hyland-Tan double glueingconstruction produces such categories, either with or without units, when applied to any ofa large family of degenerate models. This process explains as special cases a number of suchmodels from the literature. In order to achieve this result, we develop a tensor calculus forcompact closed categories with finite biproducts. We show how the combinatorial propertiesrequired for a fully complete model are obtained by this glueing construction adding to thestructure already available from the original category.
1. I ntroduction
Linear Logic [Gir87] is a well-known formal system that has attracted interest from com-puter science as well as logicians. It has a very well behaved proof theory, and categoricalmodels for linear logic also contain a model of the (linear) simply-typed λ -calculus. Fullycomplete [AJ94] models are those that equate precisely those proofs considered equivalentby the proof theory, and which exclusively contain morphisms that are the interpretationof some proof.The best understood fragment of linear logic is that of unit-free Multiplicative LinearLogic , MLL − . To model that one requires a ∗ -autonomous category [Bar79], but not all suchcategories satisfy the desired full completeness property. For example, compact closedcategories [KL80], which are thought of as degenerate models of MLL − , do not satisfy fullcompleteness for the logic. The Chu construction [Chu79] creates ∗ -autonomous categorieswhich generally are not fully complete. Pre-existing studies of ‘good’ models in thisstronger sense are [HO93, Loa94a, DHPP99]. [ Theory of computation ]: Logic—Linear logic / Proof theory; Semantics and reasoning—Program semantics—Categorical semantics.
Key words and phrases:
Linear Logic, ∗ -Autonomous Categories, Compact Closed Categories, Full Complete-ness, Tensor Calculus. ∗ A preliminary version of this article appeared in the proceedings of LICS 2012 [SS12].
LOGICAL METHODS l IN COMPUTER SCIENCE DOI:10.2168/LMCS-11(3:6)2015 c (cid:13)
A. Schalk and H. P. Steele CC (cid:13) Creative Commons
A. SCHALK AND H. P. STEELE
In [Tan97] the double glueing construction is introduced (see also [HS03] for a generalaccount), and it is suggested that fully complete models may be obtained when this isapplied to three particular compact closed categories. The three categories in question arethe category
Rel of sets and relations, the category
FDVec F of finite dimensional vectorspaces over an arbitrary field F of characteristic 0, and the category of Conway games andhistory-free strategies. However, the proof of the second—arguably the most interestingcase—is not completed in the cited work; and the restriction regarding the characteristicof the field turns out to be unnecessary. Furthermore no two of the three proofs lendthemselves to a common unification.In this paper, a greatly expanded version of the extended abstract [SS12], we providean entirely new approach to proving full completeness which can be applied to a largevariety of models provided by tensor-generated compact closed categories with finite biproducts to which the double glueing construction has been applied. In the process we developa ‘tensor calculus’ for such categories, and discuss its combinatorial consequences. Thefull completeness proofs consist of algorithms which calculate the required proof-theoreticstructure for a given natural transformation. As a consequence using the tensor calculushas a very algorithmic flavour, and there is certainly interesting future work to be doneto connect this with other such work, for example in game semantics. Both Rel and
FDVec F belong to this collection of compact closed categories, and so the result is indeeda generalisation of Tan’s work.For the sake of self-containment, we start in Section 2 by o ff ering some backgroundinformation relevant to the theorem being proved. This includes a short introduction tounit-free multiplicative linear logic and its proof theory, as well as its categorical modelsand the double glueing construction which can be placed over them. We then provide aformal description of the ‘tensor calculus’ and its validity for the resulting categories withinSection 3. It is within Section 4 that we provide the proof of our main result, Theorem 4.24,which says that all compact closed categories with finite biproducts satisfying an extremelyweak version of full completeness give rise to fully complete MLL − models via doubleglueing. Using a mixture of results from earlier sections and from [Tan97], we finish bydemonstrating in Section 5 that the same double glueing construction under a slack focusedorthogonality [HS03] can produce fully complete models of MLL − + Mix, the multiplicativefragment of linear logic with the ‘Mix’ rule (Theorem 5.11).The results in this paper are taken from the thesis of the second author [Ste13]. Thethesis, to which we occasionally refer, o ff ers further background and discusses some issuesin more detail than is possible here. 2. B ackground In this section we give a short account of the well-known proof theory of MLL − [DR89]and how this system can be modelled categorically [BFSS90, Blu93]. We repeat the formalnotion of full completeness for such models, and indicate what can be said about themodels provided by compact closed categories. An introduction to double glueing andorthogonalities is also contained within this background survey.2.1. Cut-Free MLL − Proof Nets.
The linear logic fragment MLL − possesses a beautifulproof theory revolving around the notion of proof nets [Mel06, Str06]. Proof nets provide amethod of equating two distinct derivations which di ff er only due to ‘bureaucracy’ [Gir87]. ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 3
Since derivations in MLL − can be normalised confluently, we are interested primarily incut-free MLL − proof nets. It is su ffi cient to consider right-sided sequents of formulae builtfrom literals (which in a derivation are created as pairs, one positive, one negative) usingthe multiplicative conjunction ⊗ and disjunction M . Given a deduction in the system wecan construct a graph by using the parse trees of the final sequent, connecting those literalsthat are created together in said proof via edges known as (axiom) links . This is the proof net that corresponds to the derivation, and we wish to equate those deductions that have thesame proof net.2.1.1. Correctness Criteria.
While every proof net represents a derivation, it is possible tocreate graphs which resemble proof nets but do not correspond to correct derivations.Given a parse tree for a sequent constructed from literals and the MLL − connectives ⊗ and M we use the term proof structure for a graph resulting from connecting matching literals.In this paper, proof structures are written only as the sequent together with its set ofaxiom links connecting appropriate literals above it. An example is given below.(( α ⊗ α ⊥ ) ⊗ α ) M ( α ⊥ ⊗ α ) M (( α ⊥ M α ) ⊗ α ⊥ )It is possible to check whether a given proof structure is, in fact, a proof net [DR89]using certain correctness criteria . A switching of a proof structure is a subgraph created byremoving exactly one of the two argument edges of each M -vertex. A proof structure is aproof net if and only if every one of its switchings is both acyclic and connected .2.1.2. MDNF Proof Structures.
Multiplicative linear logic does not possess all the distribu-tivity laws associated with Boolean logic. However, there are weak distributivity laws which have the e ff ect of “weakening” formulae into a state closer to (and to closure into) adisjunctive form. A ⊗ ( B M C ) ( w LL ) ( A ⊗ B ) M CA ⊗ ( B M C ) ( w LR ) ( A ⊗ C ) M B ( A M B ) ⊗ C ) ( w RL ) B M ( A ⊗ C )( A M B ) ⊗ C ) ( w RR ) A M ( B ⊗ C )From now on, in the appropriate contexts, we use the following notation: [ M , N ] = { n ∈ N : M ≤ n ≤ N } , and [ N ] = [1 , N ]. These are sometimes known as linear distributivity laws.
A. SCHALK AND H. P. STEELE
Definition 2.1.
An MLL − formula A is in multiplicative disjunctive normal form (or MDNF )if A = M Mm = ( N L m l = ( α f ( m , l ) ) φ ( m , l ) ) for literals { α i : i ∈ [ N ] } for some N ∈ N , M , L , . . . , L M ∈ N ,and functions f : P Mm = { ( m , l ) : l ∈ [ L m ] } −→ [ N ] and φ : P Mm = { ( m , l ) : l ∈ [ L m ] } −→ { ǫ, ⊥} indicating the literal and polarity of said literal respectively, where ǫ denotes an emptysuperscript, and therefore positivity. Each subformula N L m l = ( α f ( m , l ) ) φ ( m , l ) of A for a given m is called a block of the formula. An MLL − sequent is considered to be in MDNF if all itsconstituent formulae are in MDNF; and its blocks are the blocks of its formulae.The correctness criteria for proof structures over MLL − sequents which are in MDNFare even further simplified. Fact 2.2. [Ste13] An MDNF proof structure is a proof net for MLL − if and only if its maximal M -free subgraph is a tree.The simplicity of these MDNF proof structures and their correctness criteria is veryuseful to us when proving full completeness in a category. As is seen in Sections 4 and 5,seemingly weaker full completeness theorems concerning only sequents of this form canbe shown to be equivalent to theorems dealing with all sequents.Each block of literals in a MDNF structure written as a sequent with axiom links canbe seen as one large vertex without a ff ecting the acyclicity and connectedness of graph (theswitchings of their parse trees are indeed still trees), and the instances of ‘ M ’ can be ignoredby Fact 2.2. From this perspective, we can check this graph for acyclicity and connectednessvery swiftly. For example, the MDNF proof structure below is clearly incorrect due to thecycle between the two blocks. ( α ⊗ α ⊥ ) M ( α ⊗ α ⊥ )2.1.3. The ‘Mix’ Rule.
Although not a formal part of linear logic, the ‘Mix’ rule is routinelyseen as a part of the multiplicative structure. This is partly due to its insistence to beingrepresented in many standard models, not least the category of coherence spaces [Gir87].The proof theory of MLL − is however not unduly made too di ffi cult by the addition of thisderivation rule. Thanks to [FR94] we know that the only true di ff erence which can occurbetween correct proof structures for MLL − and MLL − + Mix regards the disconnectednessof switchings.
Fact 2.3. [FR94] An MLL − proof structure describes a proof net for MLL − + Mix if and onlyif all its switchings are acyclic.It is also possible to create a ‘Mix’ version of the MDNF criterion of the previous section:
Fact 2.4. [FR94, Ste13] An MDNF proof structure is a proof net for MLL − + Mix if and onlyif its maximal M -free subgraph is a forest. ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 5
Sound Categorical Models.
Sound categorical models of MLL − are found in the formof ∗ -autonomous categories [Bar79, See89]—these are symmetric monoidal categories witha well-behaved self-duality. The underlying idea is quite simple: Each symbol in the logicis interpreted by a functor (of the appropriate arity) on the category; the monoidal structure ⊗ gives conjunction, the duality ( − ) ⊥ allows negation, and to model disjunction these canbe combined to define a De Morgan dual − M − = (( − ) ⊥ ⊗ ( − ) ⊥ ) ⊥ . We use the latter functoralso to interpret the commas separating formulae in a sequent.Hence every sequent in MLL − determines a functor C N × ( C op ) N ✲ C where N ∈ N + . That is to say, each sequent is described by a multivariant endofunctorwith N co- and contravariant arguments respectively. If we look at these functors then wesee that they are built by • applying the duality functor ( − ) ⊥ to each copy of C op , • creating as many copies of the arguments as required, then reordering them appropri-ately, • applying the functors ⊗ and M to get a result in C .We refer to the functors that can be built in this way as MLL − functors . Similarly, functorsthat correspond to MDNF sequents are referred to as MDNF functors .Assume we have a right-sided sequent interpreted by the MLL − functor F . The formularepresenting truth is interpreted by the functor whose value is the constant I , the unit for themonoidal structure. We refer to this functor as K I , and allow ourselves to adjust its sourceas needed. Every proof of the given sequent is interpreted by a dinatural transformation from K I (with the same source as F ) to F , which is a family of arrows ( τ R ∈ C [ I , F ( R , R )]) R ∈ C N forwhich the diagram below commutes for every f = ( f , . . . , f N ) : R ✲ S , where R , S ∈ C N . F ( R , R ) I τ R ✲ F ( S , R ) F ( f , R ) ✲ F ( S , S ) F ( S , f ) ✲ τ S ✲ Definition 2.5. An MLL − (respectively MDNF ) transformation is a dinatural transformationto an MLL − (respectively MDNF) functor from a constant functor K I of appropriate source.It is possible to build MLL − ( + Mix) transformations corresponding to correct one-sidedsequent derivations of the logic in an inductive manner [Ste13] in any ∗ -autonomouscategory. Furthermore, it can be shown that any two derivations which reduce to the samecut-free proof net are represented by the same MLL − transformation in the category [LS06].This suggests that ∗ -autonomous categories are a sensible collection of models throughwhich one can investigate MLL − .Dinatural transformations do not naturally compose with one another. However,they are capable of being composed with transformations natural in all components. Thediagram below demonstrates the dinatural behaviour a composition of natural µ anddinatural τ . A. SCHALK AND H. P. STEELE F ( R , R ) µ ( R , R ) ✲ G ( R , R ) I τ R ✲ F ( S , R ) µ ( S , R ) ✲ F ( f , R ) ✲ G ( S , R ) G ( f , R ) ✲ F ( S , S ) µ ( S , S ) ✲ F ( S , f ) ✲ τ S ✲ G ( S , S ) G ( S , f ) ✲ The equivalent result for precompositions is demonstrated in a dual manner.One can now show that two proofs of the same sequent are interpreted by the samedinatural transformation if and only if they have the same proof net [LS06]. In otherwords, this categorical interpretation of proofs fits very well with the existing proof theoryfor MLL − .With ∗ -autonomous categories being sound categorical models of MLL − , it is expectedthat they should all contain natural transformations which model the weak distributivitylaws discussed in Section 2.1.2. w LL : − ⊗ ( − M − ) −→ ( − ⊗ − ) M − w LR : − ⊗ ( − M − ) −→ ( − ⊗ − ) M − w RL : ( − M − ) ⊗ − −→ − M ( − ⊗ − ) w RR : ( − M − ) ⊗ − −→ − M ( − ⊗ − )It is shown how one can construct each of them in [Ste13].The ‘Mix’ Rule is modelled in a ∗ -autonomous category if and only if there is a naturaltransformation Mix : ( − ) ⊗ ( − ) ✲ ( − ) M ( − ) . This is equivalent to there existing a‘Mix’ morphism mix : ⊥ ✲ I [Tan97].2.3. Full Completeness.
Full Completeness was first defined in [AJ94], and it is meant todescribe the tightest possible connection between the logic and its model. Not only are theinterpretations of two proofs the same if and only if they are equivalent in the proof theory,but the model does not contain any representations of ‘non-proofs’. In the case of the work in this paper, we are considering what this means from adinatural interpretation of proofs. This provides us with the following definition, whichoriginates from the notion as it is set out in [BS96], and is further used in such worksas [Loa94b, Tan97, Hag00]:
Definition 2.6. A ∗ -autonomous category C satisfies MLL − ( + Mix) full completeness if everyMLL − ( + Mix) transformation in the category is the interpretation of a cut-free proof net. The name ‘full completeness’ is derived from its original non-dinatural interpretation sense, in that theproperty establishes the existence of a full functor between the model category and a free ∗ -autonomouscategory. With this analogy in mind, due to every pair of derivations sharing the same cut-free proof netcorresponding to the same MLL − ( + Mix) transformation, the satisfaction of the above definition may be moreaccurately described as full and faithful completeness. However, for the sake of convenience and easy compre-hension, we keep to the originally given name.
ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 7
Compact Closed Categories with Finite Biproducts.
Compact closed categories [KL80]are particular degenerate ∗ -autonomous categories which possess a parameterised adjunc-tion B ⊗ − ⊢ − ⊗ B ∗ (we denote the negation function ( − ) ∗ for historical reasons in thesecategories). The existence of this adjunction induces an invertible ‘Mix’ natural transfor-mation Mix : ( − ) ⊗ ( − ) ✲ ( − ) M ( − ) between the two functors modelling the binaryconnectives. Nonetheless it turns out that they can form the basis for constructing fullycomplete models, as is seen in Sections 4 and 5.The adjunction associated with compact closed categories creates a bijective corre-spondence v : [( − ) ⊗ ( − ) , ( − ) ] ✲ (cid:2) ( − ) , ( − ) ⊗ ( − ) ∗ (cid:3) . This, along with the coherenceproperties of symmetric monoidal categories and the functoriality of tensor, ensure thatthe MLL − transformations can be reproduced in the following manner:(1) Take the right natural isomorphism on each of the basis MLL − functors, ρ i : ( − ) i ⊗ I ✲ ( − ) i , with ( − ) i : C N ✲ C the projection of the i th component of the product category C N foreach i ∈ [ N ].(2) Use the parameterised adjunction associated with these categories with the right iso-morphism to produce dinatural transformations d i = v I , ( − ) i , ( − ) i ( ρ ( − ) i ) : K I −→ ( − ) i ⊗ ( − ) ∗ i for each argument .(3) If the sequent whose derivation is being modelled contains n i positive / negative occur-rences of the literal being modelled with the i th entries of the functors for each i ∈ [ N ],then create the MLL − transformation I λ − ✲ N O i = n i O j = I N Ni = N nij = d i ✲ N O i = n i O j = ( − i ⊗ ( − i ) ∗ )(4) Use the associativity and symmetry isomorphisms α and σ to rearrange brackets andthe ordering of the literals so that those literals are found in the same order and thebrackets in the same place as the functor requires. If the i th and j th literals are supposedto be connected by an axiom link, then it must be ensured that the two literals foundin those positions are two that were created simultaneously by the exact same ‘axiomlink’ dinatural transformation. The resulting dinatural transformation can be thoughtof as a fixed-point-free involution , relating literals to their negations.(5) Use the natural transformation Mix liberally to change all ⊗ functors into M functorswhere necessary to shape the target functor to create the MLL − functor in the category.Unfortunately, this advantage comes with a natural drawback: the unrestricted freedomto generate representations of axiom links within the MLL − transformations means thatunsequentialisable proof structures can be modelled just as easily as genuine proof nets.As such, MLL − full completeness cannot be satisfied by any of these categories.In this paper we are more interested in the case where there is even more structurethan that given above, namely where the compact closed category C has finite biproducts.Recall [Hou06] that C has finite products if and only if it has finite sums if and only if it hasfinite biproducts. In this case the scalars (that is the homset of endomorphisms on the tensorunit C [ I , I ]) form a commutative unital semiring, and due to the biproducts distributing This dinatural transformation is indeed the collection of arrows ( d X : I ✲ X ⊗ X ∗ ) X ∈ C first discussedin [KL80] A. SCHALK AND H. P. STEELE over the tensor product C is enriched over CSMod C [ I , I ] —the category of commutativesemimodules over the semiring of scalars [Heu08].Given a set of dinatural transformations of the same type { τ i : i ∈ I } for some index set I ,we define a linear combination P i s i · τ i of them as the collection of arrows (cid:0)P i s i · ( τ i ) A (cid:1) A ∈ C .It is easy to show that in such a category every linear combination of dinatural transfor-mations is another such, giving a second obstacle to full completeness. Nonetheless, fromthe above limitations, there is a clear concept of a compact closed category with finitebiproducts being ‘as fully complete as it can hope to be’. Definition 2.7.
A compact closed category C with finite biproducts satisfies feeble fullcompleteness if every MLL − transformation for the category is a linear combination ofinterpretations of proof structures over the same sequent.It is known from [CHS01] (proof reproduced in [Ste13]) that finite biproductal compactclosed categories whose tensor unit acts as a separator satisfy feeble full completeness.These models are called tensor-generated , and there are many of them. Examples includeboth Rel and
FDVec F for any field F , and extend beyond these to include categories suchas that of finite-dimensional Hilbert spaces. Every compact closed category with finitebiproducts clearly has a non-trivial full subcategory which is tensor-generated, namely thecategory generated by the object I and the tensor, biproduct and duality functors.2.5. The Double Glueing Construction.
Double Glueing constructions [HS03] operateupon categories. They can be thought of as adding structure to objects in the form of twoarrows, which then has to be preserved by the morphisms of the newly created category.This leaves a trivial forgetful functor U : D ✲ C from the double-glued category to theoriginal. When the added structure arrows are monomorphisms, this has the e ff ect ofgenerating homsets between two objects in the new category which are fundamentallysubsets of the homsets of their underlying objects. [Ste13] That is, (cid:8) U f : f ∈ D [( A , α, ξ ) , ( B , β, ζ )] (cid:9) ⊆ C [ A , B ] . The most commonly seen double glueing construction used on top of models of linearlogic is the so-called Hyland-Tan construction [Tan97], inspired by Loader’s linear logicpredicates [Loa94b]. This double glueing is based upon structure arrows which take theform of injections into homsets in
Set , and so are clear monomorphisms, allowing us touse the above fact. We define this construction formally below.
Definition 2.8.
Given a ∗ -autonomous category C with tensor unit I , and letting ⊥ = I ⊥ ,the category G C is the category described with the following object set and homsets: • Objects —
Ob j ( G C ) = { ( A , U , X ) : A ∈ Ob j ( C ) , U ⊆ C [ I , A ] , X ⊆ C [ A , ⊥ ] }• Arrows — Arrows in G C [( A , U , X ) , ( B , V , Y )] are described by single morphisms f ∈ C [ A , B ] such that: f ◦ U ⊆ V and Y ◦ f ⊆ X In general, when discussing a G C -object ( A , U , X ), we refer to U and X as the object’ssets of values and covalues respectively. We write( A , U , X ) Val = U ( A , U , X ) CoVal = X ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 9
As with any double glueing, the Hyland-Tan construction preserves ∗ -autonomy, and italso removes the degeneracy of compact closed categories. Tensor products and negationsof arrows are immediately inherited from the underlying category — indeed, the functor U : G C ✲ C is ∗ -autonomous. The object assignments are as follows:( A , U , X ) ⊥ = ( A ⊥ , X ⊥ ◦ χ I , U ⊥ )( A , U , X ) ⊗ ( B , V , Y ) = ( A ⊗ B , ( U ⊗ V ) ◦ λ I , Z ) I = ( I , { I } , C [ I , ⊥ ])where Z = ( z ∈ C [ A ⊗ B , ⊥ ] : ∀ v ∈ V , A ρ A ✲ A ⊗ I A ⊗ v ✲ A ⊗ B z ✲ ⊥ ∈ X , ∀ u ∈ U , B λ B ✲ I ⊗ B u ⊗ B ✲ A ⊗ B z ✲ ⊥ ∈ Y ) .2.5.1. Focused Orthogonalities.
Many examples of double-glued structures which have beeninvestigated, particularly ones in which their glued nature is hidden, have restrictionson the objects which they are allowed to contain. For example, the category of ChuSpaces [Chu79] can be thought of as a full subcategory of a generalised version of thecategory
Rel under the influence of a double-glueing [Hug04]. Of course, the constraintsmust be defined sensibly in order to preserve the closure of the operations defined on suchcategories. These constraints are called orthogonalities , and they can come in a variety offorms. In this paper, we wish to look at focused , slack orthogonalities.Given any subset E ⊆ C [ I , ⊥ ], we can construct an orthogonality in which two mor-phisms f ∈ C [ I , A ] and g ∈ C [ A , ⊥ ] are considered orthogonal if and only if they composeto form an arrow in the chosen set. This is called the orthogonality focused on E , and spawnsa focused glueing G E C whose values and covalues must be mutually orthogonal. Definition 2.9.
Given a ∗ -autonomous category C and a set E ⊆ C [ I , ⊥ ], the category G E C is defined as the smallest full subcategory of G C containing every object ( A , U , X ) such that ∀ u ∈ U , x ∈ X , x ◦ u ∈ E . Of course, the tensor unit I of G C is not contained in any of the orthogonally gluedcategories for which the focus is a proper subset of C [ I , ⊥ ] , meaning that the ∗ -autonomousstructure of G C is not preserved per se . However, each one of these subcategories hasanother object within it which acts as legitimate unit for the tensor functor: the unit for G E C is I E = ( I , { I } , E ).With this fact taken into account, and the fact that the categories are closed under thetensor product and negation functors, we know that all of these constructions preserve ∗ -autonomy (with respect to the new unit definitions). Thus, just like the original G -glueing,the glueings generate models of MLL − .2.5.2. Properties.
The Hyland-Tan double glueing construction and its focused orthogonal-ities are particularly well-behaved, and in fact possess some properties which are immedi-ately relevant to the coming results. The categories G E C and G C are clearly the same when E = C [ I , ⊥ ] Given an arbitrary focused glueing E ⊆ C [ I , ⊥ ], the value and covalue sets of an objectare in fact precisely the homsets from I E and to ⊥ = I ⊥ . Fact 2.10. [Ste13] For every object R ∈ Ob j ( G E C ), R Val = G C [ I E , R ] and R CoVal = G C [ R , ⊥ E ],where ⊥ E = ( I E ) ⊥ .We can also claim a large understanding of the MLL − transformations for these double-glued categories. Since homsets in these categories are in some way ‘stripped’ versions ofthose from their underlying categories, it would be reasonable to assume that an analogousstatement could be made about dinatural transformations. This is indeed correct, as canbe deduced from the below proposition. Proposition 2.11. [Tan97, Ste13]
Let τ : F −→ G be a dinatural transformation in G E C . Thenthere is a dinatural transformation ˜ τ : UF −→ UG in C which defines τ ; that is, U τ R = ˜ τ UR forevery R = ( R , . . . , R N ) .Proof. Theorem 1.3.2 of [Tan97] provides the result for E = C [ I , ⊥ ]. As remarked onpage 119 in [Ste13], this proof only requires intermediary objects which are found in allcategories of the given form, thus it extends to all subsets E .The consequences of this proposition are marked. The transformations of feebly fullycomplete compact closed categories with biproducts take the form of linear combinations offixed-point-free involutions [CHS01]. Therefore the same morphisms are used to describedthe transformations in the categories created by applying the glueing construction to them.Finally, given a compact closed category C with finite biproducts, which is assureda morphism ι − = ρ ⊥ ◦ v ( λ I ) ∈ C [ ⊥ , I ] and a separate zero morphism 0 ⊥ , I in the homset C [ ⊥ , I ], we note that every focused glueing G E where { ι } ( E ⊆ C [ I , ⊥ ] stops this morphismbeing found in G E C [ ⊥ E , I E ], but G { ι } C preserves this modelling of the mix rule. Fact 2.12.
For every compact closed category C with finite biproducts and a set E ⊆ C [ I , ⊥ ]containing ι , the category G E C models the mix rule if and only if E = { ι } . (See [Tan97])For shorthand, we say G { ι } C = G C . The category is the subject of Section 5.3. A rrow D ecomposition Every finite-dimensional vector space over a field F can be given a finite basis, meaningthat all arrows in their category FDVec F can take the shape of a matrix, or in fact a tensorif desired, over the underlying field. Although it is not possible to say that all arrows ina given compact closed category with finite biproducts can be reduced to an array-basedform over a single input type, the multilinear representations still appear and can certainlybe of use. We introduce this generalisation of matrix representations of arrows in thecoming section, and show that MLL − transformations and the calculations required indeducing sets of values and covalues in double-glued objects can take a simplified formwhen using this notation. Definition 3.1. An array with index set I over a set X is a function f : I −→ X . An array canbe considered to be N-dimensional if its index set takes the form Q Ni = I i for some I , . . . , I N . A concept found in folklore, and briefly explained in [Hou06].
ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 11
Definition 3.2.
Letting n , . . . , n N ∈ N + , an Q Nk = n k -tensor t over a semiring S is an N -dimensional array, each of whose components contain a value from S . The ( i , . . . , i N ) th entry is written t i , ··· , i N .Indices of tensors are allowed to be separated by commas and semicolons to demarcaterelevant groups of indices. Superscripts may also be used to facilitate writing, though oncea notation style is chosen for a specific tensor it must be adhered to. The number of indicesa tensor requires to be expressed (in this case N ) is called its order . Often tensors are writtenwith general index variables to emphasise that they are indeed multidimensional arrays.Furthermore, long sets of indices can be replaced by bold ‘superindices’. For example, t i , ··· , i M , j , ··· , j N can be rewritten t ij for shorthand, where i = ( i , . . . , i M ) and j = ( j , . . . , j N ).Sometimes we overload the notation of the tensor so as to show the reader informationconcerning relative positions of indices with respect to other tensors or a set position.Calculations involving tensors over semirings are unsurprisingly done in an identicalmanner to those involving tensors over fields in standard multilinear algebra. All thedi ff erences which may occur concern how its entries sum and multiply together. Thestandard algebraic manipulations are found below. • (Addition) — Given two tensors t and u over the same index sets, their sum is clearlyfound in the same semimodule.( t + u ) i , ··· , i M = t i , ··· , i M + u i , ··· , i M . • (Composition) — Let t and u be ( M + N )- and ( N + P )-order tensors respectively with N index positions ranging over the same index sets in both arrays. The composition ofthe two is given by an ( M + P )-order tensor for which each component is found to be asfollows: ( tu ) i , ··· , i M ; k , ··· , k P = X j , ··· , j N (cid:0) t i , ··· , i M ; j , ··· , j N · u j , ··· , j N ; k , ··· , k P (cid:1) . • (Product) — The outer product of two tensors t i , ··· , i M and u j , ··· , j N is given by an ( M + N )-tensor for which each entry is merely a product of entries from its factors.( t ⊗ u ) i , ··· , i M ; j , ··· , j N = t i , ··· , i M · u j , ··· , j N . Examples of the first three manipulation techniques from above are seen ubiquitously inthe forms of addition, multiplication and the trace operation of matrices. It should be notedthat, if desired, one can consider tensor composition as the equivalent of finding the outerproduct of two tensors sharing indices and contracting the result.There are five types of tensor which are seen continuously in various guises within thispaper. Their definitions are given below.
Definition 3.3. (1) A zero tensor , written 0 i for some indices i , is a tensor, all of whose entries contain thescalar 0.(2) Similarly, a one tensor , written 1 i , is a tensor whose entries all contain the scalar 1.(3) A Kronecker delta , written δ , is an ( N × N )-tensor for any N ∈ N such that δ ij = (cid:26) i = j . It is the tensor representation of the identity matrix.We also write δ j ··· j N i ··· i N = N Nk = δ i k j k as shorthand .(4) A (full) M-permutation over [ n ] is an n M -tensor p i ··· i M such that for all k ∈ [ M ], and x l ∈ [ n ]for each l , k , there exists an x k ∈ [ n ] so that p i ··· i M · Q l , k δ i l x l = δ i k x k . We use the notation Perm ( M , n ) to denote the set of all M -permutations over [ n ]. Note that a 1-permutationover [ n ] is simply a tensor of the form δ ix for some x ∈ [ n ].Of particular use are cycle permutation tensors . We define cycle ( L , n , r ), with L , n , r ∈ N + , to be the n L -tensor with entries defined as follows: cycle ( L , n , r ) i ··· i L = (cid:26) P Lj = i j ≡ r mod n . (5) Generalising the above, a partial M-permutation over [ n ] is an n M -tensor p i ··· i M such thatfor all k ∈ [ M ], and x l ∈ [ n ] for each l , k , there exists an x k ∈ [ n ] so that p i ··· i M · Q l , k δ i l x l is equal to δ i k x k or 0 i k . We use the notation PPerm ( M , n ) to denote the set of all M -permutations over [ n ].Because of the number of indices which may be given to each tensor, the calculus of tensorscan become very cluttered. The use of summation symbols adds to the excessive numberof symbols in many expressions, and they can be thought of in many cases as unnecessary.As such, at some points it is useful to use the Einstein summation convention: whenever anexpression containing tensors has two index positions involved within it which are summedtogether over the same index j say, the summation symbol may be omitted without worry.This means that a simple composition rule, for example, can be rewritten ( tu ) ik = t ij u jk .3.1. Tensor Representation.
Decomposition of arrows between objects in the form of ten-sor products of direct sums is possible in a symmetric monoidal category with finitebiproducts, and can be a very useful tool. Suppose that C is such a category, and considerthe arbitrary arrow f : N Nl = L n l j l = A l , j l ✲ N Mk = L m k i k = B k , i k . Due to the preservation of products by the tensor product , we know that the following setof m arrows describe f . ( π i ⊗ M O k = L mki = B k , i ! ◦ f : i ∈ [ m ] ) We can follow the same procedure of decomposing f using the projections1 L m i = B , i ⊗ · · · ⊗ π i k ⊗ · · · ⊗ L mMi = B , i for all k of the object. Furthermore, due to the bi-functoriality of the tensor product, each of the projections is independent of the others.The injections operate similarly; and as such, it is possible to separate f yet further and Note that these tensors are an example of where it is of use to overload the index notation as discussed onpage 11. It facilitates the understanding of which indices are connected by such a relation without having toresort to a yet more cumbersome notation. Since there is a natural isomorphism 1 C (cid:27) ( − ) ∗∗ in any compact closed category, the functor B ⊗ − : C ✲ C in a compact closed category for any object B can be seen as both a left and right adjoint, and as such preservesboth products and coproducts [Mac97]. ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 13 describe it using a set of Q Mk = m k × Q Nl = n l arrows { f i ,..., i M ; j ,..., j N : ∀ k , l i k ∈ [ m k ] , j l ∈ [ n l ] } ,where f i ,..., i M ; j ,..., j N = M O k = π i k ◦ f ◦ N O l = in j l . This decomposition of f is known as a tensor representation . It is generally inconvenientto write down the resultant tensor as a single entity due to the number of dimensionsrequired. This is, however, of little concern to us, because we are able to view the entriesof the tensor on a case-by-case basis.The greatest implication this has is to the freedom of expression one has when decom-posing arrows concerning the tensor product: given two arrows in the prescribed formabove, it is possible to gain a tensor representation of their tensor product. This comesabout almost trivially, with each entry in the new tensor being created by producing thetensor product of one entry from each of the representations of the more primitive mor-phisms. Given arbitrary arrows f ∈ C [ A , B ] and g ∈ C [ C , D ], with A = N N A a = L n A , a k a = A a , j a , B = N N B b = L n B , b j b = B l , j b , C = N N C c = L n C , c k c = C c , i c and D = N N D d = L n D , d j d = D d , i d , we find that( f ⊗ g ) i , ··· , i NC + ND ; j , ··· , j NA + NB = N C + N D O l = π i l ◦ ( f ⊗ g ) ◦ N A + N B O l = in j l = N C O l = π i l ◦ f ◦ N A O l = in j l ! ⊗ N C + N D O l = N C + π i l ◦ g ◦ N A + N B O l = N A + in j l = f i , ··· , i NC ; j , ··· , j NA ⊗ g i NC + , ··· , i NC + ND ; j NA + , ··· , j NA + NB Rather unsurprisingly, addition and composition operate in an almost identical manner asthey do in the matrix algebra. Identity and zero morphisms between objects of the formgiven above are represented by Kronecker deltas and zero tensors respectively. Letting A = N N A l = L n A , l k l = A l , j l , B = N N B l = L n B , l j l = B l , i l , and f , g ∈ C [ A , B ],(0 A , B ) j ,..., j NB ; k ,..., k NA = N B O l = π j l ◦ A , B ◦ N A O l = in k l = N NAl = A l , kl , N NBl = B l , jl ( f + g ) i ,..., i NB ; j ,..., j NA = N B O l = π i l ◦ ( f + g ) ◦ N A O l = in j l = N B O l = π i l ◦ f ◦ N A O l = in j l ! + N B O l = π i l ◦ g ◦ N A O l = in j l ! = f i ,..., i NB ; j ,..., j NA + g i ,..., i NB ; j ,..., j NA (1 A ) i ,..., i NA ; j ,..., j NA = N A O l = π i l ◦ A ◦ N A O l = in j l = N A O l = π i l ◦ N A O l = L nA , lk = A l , k ◦ N A O l = in j l = N A O l = (cid:18) π i l ◦ L nA , lk = A l , k ◦ in j l (cid:19) = N A O l = π i l ◦ n A , l X k = ( in k ◦ π k ) ◦ in j l ! = X k ,..., k NA N A O l = ( π i l ◦ in k l ◦ π k l ◦ in j l ) = δ j ··· j NA i ··· i NA = N NAl A l , il ∀ l ∈ [ N A ] , i l = j l N NAl A l , jl , A l , il otherwise Then taking arbitrary C = N N C l = L n C , l i l = B l , i l and h : B ✲ C ,( h ◦ f ) i ,..., i NC ; k ,..., k NA = ( h ◦ B ◦ f ) i ,..., i NC ; k ,..., k NA = N C O l = π i l ◦ h ◦ X j ,..., j NB N B O l = in j l ◦ N B O l = π j l ! ◦ f ◦ N A O l = in k l = X j ,..., j NB N C O l = π i l ◦ h ◦ N B O l = in j l ! ◦ N B O l = π j l ◦ f ◦ N A O l = in k l ! = X j ,..., j NB h i ,..., i NC ; j ,..., j NB ◦ f j ,..., j NB ; k ,..., k NA There is a specific type of interaction composition and tensor multiplicaton whichought to be noted, namely when a composition passing through an object A ⊗ B contains anarrow of the form f ⊗ B or 1 A ⊗ g for some arrows f , A or g , B . Consider the composi-tion C ⊗ B f ⊗ B ✲ A ⊗ B h ✲ D ; the tensor representations of the composite morphisms arewritten, naturally, as ( f ⊗ B ) j , ··· , j NA , j ′ , ··· , j ′ NB ; k ··· k NC , k ′ , ··· , k ′ NB and h i , ··· , i ND ; j , ··· , j NA , j ′ , ··· , j ′ NB . However,( f ⊗ B ) j , j ′ ; k , k ′ = f j ; k ⊗ δ k ′ j ′ , and the only e ff ect the Kronecker delta on the end representationof the composition is the change of index, which is superficial. In these types of situation,we consider ourselves at liberty to think instead of the composition of the two tensors f j ; k and h i ; j , j ′ , summing over the indices of j only. ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 15
The monoidality of the category C allows for scalar multiplication of arrows to bemodelled by the synonymous operation on tensors.( s · f ) i ,..., i NB ; j ,..., j NA = N B O l = π i l ◦ (( λ B ◦ ( s ⊗ B ) ◦ λ − B ) ◦ f ) ◦ N A O l = in j l = (( λ N NBl = B l , il ◦ ( s ⊗ N NBl = B l , il ) ◦ λ − N NBl = B l , il )) N B O l = π i l ◦ f ◦ N A O l = in j l ! = s · f i ,..., i NB ; j ,..., j NA When the source and target objects of an arrow are tensor products of direct sumsof the tensor unit I , the arrow’s tensor representation acts as a standard tensor over C [ I ⊗ · · · ⊗ I , I ⊗ · · · ⊗ I ]. These homsets are trivially in bijective correspondence withthe set of scalars C [ I , I ], and so we can work purely with tensors over this semiring.Of course, in a compact closed category with finite biproducts, we are not restrictedto building objects over the two functors − ⊗ − and − ⊕ − : the contravariant endofunctor( − ) ∗ is also available. We now show how tensor representations can be given to morphismsbetween objects built with this functor.Consider a morphism f : A ✲ C ⊗ B ∗ , where A = L ak = A k , B = L bj = B j , and C = L ci = C i . The extent to which this arrow can be decomposed using the tensor rep-resentation system given above is less than one would hope. The negation of B means thatit is not possible to break it down without extending the representation. The parameterisedadjunction − ⊗ B ⊣ − ⊗ B ∗ described in Section 2.4 makes this extension possible.The isomorphism v associated with the adjunction relates f to an arrow v − A , B , C ( f ) : B ⊗ A ✲ C . This new arrow is certainly in exactly the shape seen of mor-phisms at the beginning of this section; B is no longer shackled by the negation functor andits projections may be used in a decomposition in the same way as A and C . The set ofarrows { v − A , B , C ( f ) ijk : i ∈ [ c ] , j ∈ [ b ] , k ∈ [ a ] } , where v − A , B , C ( f ) ijk = π k ◦ v − A , B , C ( f ) ◦ ( in i ⊗ in j )determines v − ( f ) uniquely; and since v is an isomorphism the arrows provide a uniquedescription of f as well.Using the isomorphism once more, this time on every arrow in the above set, we areable to understand f by viewing a set of morphisms that have a superficially identical formto it: { v A i , B j , C k ( v − A , B , C ( f ) ijk ) : i ∈ [ c ] , j ∈ [ b ] , k ∈ [ a ] } . The parameterised adjunction o ff ers asimplification of the descriptions of each of these arrows. v A i , B j , C j ( v − A , B , C ( f ) ijk ) = v A i , B j , C j ( π k ◦ v − A , B , C ( f ) ◦ ( in i ⊗ in j )) = ( π i ⊗ ( in j ) ∗ ) ◦ f ◦ in i The principle above, where the duals of injection arrows into B are used in place ofprojections, is entirely generalisable to multiple instances of dual objects in a tensor product.Furthermore, the inherent duality of compact closed categories allows morphisms fromtensor products with dualised elements to be decomposed by substituting injection arrows for duals of projection arrows. Suppose that we have an arrow g : N N A l = ( L n A , l k l = A l , k l ) φ ( l ) ✲ N N B l = ( L n B , l i l = B l , j k ) ψ ( l ) where the functions φ : [ N A ] ✲ { ǫ, ∗} and ψ : [ N B ] ✲ { ǫ, ∗} depict the polarity of a directsum in the tensor product with respect to the negation functor, ǫ denoting an emptysuperscript once more. We define the tensor representation of such an arrow to be the set ofmorphisms (cid:0) g j ; k (cid:1) j ∈ Q NBl = [ n B , l ] , k ∈ Q N A l = [ n A , l ], where g j ,..., j NB ; k ,..., k NA = N B O l = ν l , j l ◦ g ◦ N A O l = µ l , k l ,ν l , j l = (cid:26) π j l ψ ( l ) = ǫ ( in j l ) ∗ ψ ( l ) = ∗ and µ l , k l = (cid:26) in k l φ ( l ) = ǫ ( π k l ) ∗ φ ( l ) = ∗ . We can extend this using a simple induction so that the functions φ and ψ may adopt thenew range of the set of strings using only the letter ∗ , F {∗} as we shall name it at this point ,so that we can apply the duality functor as many times as wished to a direct sum. Thetensor representation of the arrow g remains the same, but the definitions of ν and µ aregeneralised accordingly. ν l , j l = (cid:26) ( π j l ) ψ ( l ) ψ ( l ) ∈ F {∗∗} ( in j l ) ψ ( l ) ψ ( l ) < F {∗∗} and µ l , k l = (cid:26) in k l φ ( l ) ∈ F {∗∗} ( π k l ) ∗ φ ( l ) < F {∗∗} . Of course, the negation of a number of positive and negative direct sums of objectstogether in a tensor product is also possible, and is in fact necessary if the M functor is to bedefined. It would therefore be useful to produce tensor representations for arrows betweenobjects containing such components. The principle being used in the earlier instances ofnegation may be translated directly to produce such entities. Given an arrow f : A α ✲ D δ ⊗ ( B β ⊗ C γ ) ∗ where A = L l A l , B = L j B j , C = L k C k and D = L i D i are direct sums of objects acted onby the negation functor α , β , γ and δ times respectively. There is a bijective correspondence v − A α , B β ⊗ C γ , D δ to C [ A α , D δ ⊗ ( B β ⊗ C γ ) ∗ ] to C [ A α ⊗ ( B β ⊗ C γ ) , D δ ], and every morphism in thistarget set can be decomposed and be represented by a tensor containing arrows in the formshown below. ( v − A α , B β ⊗ C γ , D δ ( f )) i ; l , j , k = ν i ◦ v − A α , B β ⊗ C γ , D δ ( f ) ◦ ( µ l ⊗ ( µ j ⊗ µ k )) . Using the correspondence v once again, we see that each morphism in the aforementionedtensor is related to an arrow which may be placed into a new representation: the relativeof the above arrow is f i , j , k ; l = v A α l , B β j ⊗ C γ k , D δ i (( v − A α , B β ⊗ C γ , D δ ( f )) i ; l , j , k ) = v A α l , B β j ⊗ C γ k , D δ i ( ν i ◦ v − A α , B β ⊗ C γ , D δ ( f ) ◦ ( µ l ⊗ ( µ j ⊗ µ k ))) = ( ν i ⊗ ( µ j ⊗ µ k ) ∗ ) ◦ f ◦ µ l It is more traditionally written with a Kleene star, but the authors believe the resulting expression {∗} ∗ leadsto confusing overloading. ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 17
The arrows of this form bundled into an appropriate set are defined to be the tensorrepresentation of the arrow, and it is once again clear that the concept is generalisable tomorphisms whose sources and targets may contain any number of tensor products a ff ectedby the functor ( − ) ∗ . The self-duality of compact closed categories once more means thata similar line of reasoning can be a ff orded to dealing with negation in the source of anarrow by swapping the roles of injection and projection arrows. A logical extension to theargument can be constructed to deal with tensor products with the duality functor appliednumerous times. Using the symmetry natural isomorphism σ sensibly it is possible toexpress tensor representations for arrows between objects which are tensor products oftensor products of direct sums of primitive objects under any number of instances of theduality functor. Furthermore, due to induction, this style of reasoning is extendable toany arrow between objects built using indecomposable objects and the duality and tensorfunctors. We give new, recursive definitions for the arrow sets µ and ν . These are, however,reliant on the concept of index sets in a category theoretical sense. Definition 3.4. An index set of an instance of an object A in C , written ι ( A ), is definedinductively as follows: • If A = L ni = A i for some instances of objects A , . . . , A n we wish to be considered inde-composable, then ι ( A ) = [ n ]. • If A = B ∗ for some object B , then ι ( A ) = ι ( B ). • If A = N Nl = A l for some objects A , . . . , A N , then ι ( A ) = Q Nl ι ( A l ).Equivalently, the index set of an object A = N Nl = L n l i = A l , i for which we consider itsinstances of A l , i indecomposable for each l and i is the set of N -tuples where for every l ∈ [ N ], the l th component ranges over the number of objects in the l th direct sum.We are now in a position to define µ and ν . Definition 3.5.
The injection and projection arrow functions of an object A built using thetensor product and negation functors over direct sums of indecomposable objects in acompact closed category C with finite biproducts, written µ A and ν A respectively, havedomain ι ( A ) and are defined recursively as follows: • If A = L ni = A i for indecomposable A , . . . , A n , µ A ( i ) = in i and ν A ( i ) = π i . • If A = B ∗ for some B , then µ A ( i ) = ( ν B ( i )) ∗ and ν A ( i ) = ( µ B ( i )) ∗ . • If A = N Nl = A l for some A , . . . , A N , then µ A ( i , . . . , i N ) = N Nl = µ A l ( i l ) and ν A ( i , . . . , i N ) = N Nl = ν A l ( i l ), where each i l ∈ ι ( A l ).The definition above facilitates the description of the tensor representation of every arrowbuilt in the manner that has been discussed earlier. Definition 3.6. A tensor representation of an arrow f : A −→ B , where A and B are objectsbuilt using tensor products and the negation functor over direct sums of instances of objectsconsidered indecomposable, is defined as the set of morphisms { f i , j : i ∈ ι ( B ) , j ∈ ι ( A ) } where f i , j = ν i ◦ f ◦ µ j Note how this definition of tensor representation is consistent with the earlier, morerudimentary forms which do not consider the possibility of negation existing beyond the direct sums. The entries in these tensors are still merely composites containing arrows;and addition of morphisms is a consequence of enrichment over
CMon , so the connectionbetween tensor and arrow addition is maintained, including the zero morphisms. Similarly,scalar multiplication is una ff ected. The extended definition of the identity arrow of anyobject A is created recursively from prior knowledge of the standard diagonal matrixrepresentation of the identity arrow for direct sums of objects and the preservation ofidentity morphisms by functors, and takes the form P I ∈ ι ( A ) ( µ A ( I ) ◦ ν A ( I )). • If A = L ni = A i for indecomposable objects A , . . . , A n , then the representation of theidentity morphism 1 A is(1 A ) i , j = n X k = ( in k ◦ π k ) = X I ∈ ι ( A ) ( µ A ( I ) ◦ ν A ( I )) . • If A = B ∗ for some object B , then1 A = B ∗ = (1 B ) ∗ = X i ∈ ι ( B ) ( µ B ( i ) ◦ ν B ( i )) ∗ = X i ∈ ι ( A ) ( µ B ( i ) ◦ ν B ( i )) ∗ = X i ∈ ι ( A ) (( ν B ( i )) ∗ ◦ ( µ B ( i )) ∗ ) = X i ∈ ι ( A ) ( µ A ( i ) ◦ ν A ( i )) . • If A = N Nl = A l for some objects A , . . . , A N , then1 A = N Nl = A = N O l = A l = N O l = X i l ∈ ι Al ( µ A l ( i l ) ◦ ν A l ( i l )) = X ( i , ··· , i N ) ∈ Q Nl = ι ( A l ) N O l = µ A l ( i l ) ◦ N O l = ν A l ( I l ) ! = X i ∈ ι ( A ) ( µ A ( i ) ◦ ν A ( i )) . Because of the lack of change required in the shape of this arrow from earlier versions, themanner in which composition operates is also preserved.Unlike when we consider only the simpler form of arrow being placed into a multilinearrepresentation, which lacks any use of the duality functor, it cannot be taken for grantedthat the tensor describing an arrow between two objects built over direct sums of I , andtensor and negation functors can be viewed as being a standard tensor over C [ I , I ]. This isbecause the entries take the form of arrows between tensor products over tensor products ofboth I and I ∗ . Fortunately, λ I ∗ ◦ v I , I , I ( ρ I ) : I ✲ I ∗ is an isomorphism; and we can composesuitable tensor products built from it, its inverse, dualities of both these arrows and theidentity morphism 1 I to “remove” the dual instantiations of the unit, and so therefore allowus to view the tensors as being over C [ I ⊗ · · · ⊗ I , I ⊗ · · · ⊗ I ] (cid:27) C [ I , I ]. ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 19
Describing Transformations with Tensors.
The representations described in this sec-tion o ff er a simplification to the form of morphisms between objects built from the threefunctors expected to exist in a compact closed category with finite biproducts. It there-fore follows naturally that certain arrows that are part of an MLL − transformation canbe viewed as tensors. In this section we consider an object F ( A , A ) in a compact closedcategory C satisfying feeble full completeness for an arbitrary MLL − functor F where A = ( n I , . . . , n N I ), where n I = L ni = I , and give the arrows into the object which areconstituents of MLL − transformations.It is known that every MLL − transformation in a compact closed category with finitebiproducts satisfying feeble full completeness is a linear combination of fixed-point-freeinvolutions [CHS01], with an involution being the equivalent of an appropriate numberof instances of the unit dinatural transformation η joined by a tensor product being post-composed with a series of symmetry natural isomorphisms. Each pair of objects createdtogether by an instance of η models a pair of literals joined by an axiom link in a cut-freeproof net. As such, it makes sense to initially provide the tensor representation of η n I forarbitrary n ∈ N . The unit transformation is derived from the bijective correspondenceconnected to C being applied to the identity arrows in C : for every A ∈ C , η A = v I , A , A ( λ A );and because of this the arrow η nI : I ✲ n I ⊗ ( n I ) ∗ is easily shown to have an n × n identitymatrix as its tensor representation.( η n I ) ij = ν ij ◦ η n I ◦ µ ∗ = λ I ◦ (1 I ⊗ χ − ) ◦ ( π i ⊗ ( in j ) ∗ ) ◦ v ( λ n I ) ◦ I = λ I ◦ (1 I ⊗ χ − ) ◦ v ( δ ij ) = (cid:26) χ − ◦ ( λ I ◦ v ( λ I )) = χ − ◦ χ = i = j χ − ◦ ( λ I ◦ v (0 I ⊗ I , I )) = χ − ◦ I , I ∗ = i , j We therefore are at liberty to express an axiom link as a Kronecker delta tensor, with oneindex referring to the object modelling the positive literal and the other the negative literal.This concept may be extended now by tensor multiplication to MLL − functors with morethan one pair of literals to being joined. Letting α be modelled by n I , the arrow modellingthe proof structure(( α ⊗ α ⊥ ) ⊗ α ) M ( α ⊥ ⊗ α ) M (( α ⊥ M α ) ⊗ α ⊥ )is represented by the tensor δ j ··· j M i ··· i M , where the indices i l and j l are connected to the objectsrepresenting the l th positive and negative literals respectively. In a more general form, ifthe literals which are connected via axiom links are less well-ordered so that the l th positiveliteral is connected to the σ ( l ) th negative literal for some permutation σ ∈ S M , then themodelling tensor is δ σ ( j ) ··· σ ( j M ) i ··· i M . Associativity natural isomorphisms are suppressed in this explanation, but this can be perceived as self-evident.
Example 3.7.
The tensor δ j j j j i i i i represents the proof structure below.(( α ⊗ α ⊥ ) ⊗ α ) M ( α ⊥ ⊗ α ) M (( α ⊥ M α ) ⊗ α ⊥ )As discussed in Section 2.4, MLL − functors may model sequents which contain morethan one instance of a single literal; and such entities can have more than one set of axiomlinks attached to them. Furthermore, regardless of the number of repeated literals, scalarmultiplications on arrows, and so natural and dinatural transformations, are always pos-sible. This gives rise to the possibility of transformations describing linear combinationsof sets of axiom links on a single sequent. Such linear combinations are modelled byMLL − transformations which are linear combinations of transformations representing sin-gle sets of axiom links, whose constituent arrows are linear combinations as well. It follows,therefore, that the tensor representation of an arrow τ A for some MLL − transformation τ where A = ( n I , . . . , n N I ) is in the following form: τ j ··· j M i ··· i M = X σ ∈ S M s σ · δ σ ( j ) ··· σ ( j M ) i ··· i M where s σ ∈ C [ I , I ] for every σ ∈ S M . Example 3.8.
The tensor τ j j j j i i i i = δ j j j j i i i i − δ j j j j i i i i represents the linear combination ofaxiom links on the MLL − formula given below.(( α ⊗ α ⊥ ) ⊗ α ) M ( α ⊥ ⊗ α ) M (( α ⊥ M α ) ⊗ α ⊥ ) ××− ff ects of replacing some instances of the functor ⊗ with those of M . Tensorrepresentations of arrows between arbitrary objects are shown to exist in Section 3.1; thisimmediately implies that they exist for ones containing the par functor as well. After all, itis the de Morgan dual of the tensor product using ( − ) ∗ as negation.We can see how involutions are viewed tensorially when certain tensor products arechanged by considering Mix : − ⊗ − ✲ − M − described in Section 2.2 and its represen-tation. In compact closed categories, the ‘Mix’ transformation is not just natural but isomor-phic, and is in fact built from the same correspondence v : C [ − A ⊗− B , − C ] (cid:27) C [ − A , − C ⊗ ( − B ) ∗ ]: Mix A , B = v A ⊗ B , A ∗ ⊗ B ∗ , I ( λ I ◦ ( v − ( λ − B ∗ ) ⊗ v − ( λ − A ∗ )) ◦ ˜ σ ) , where ˜ σ is the composition of σ natural isomorphisms such that˜ σ A , B , C , D : A ⊗ B ⊗ C ⊗ D ✲ D ⊗ B ⊗ C ⊗ A . Via an inductive argument on the size of A and B , we find that the representation of Mix A , B reduces to an identity , as long as both A and B are unit-generated. The natural consequenceof this is that one need not even think of the multiplicative functor being used once the ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 21 tensor calculus is employed: we di ff erentiate between arrows with the same tensors purelyby looking at their signatures.We can take this knowledge and provide an tensorial method to look at the the weak-ening natural transformations. Consider w LL : − ⊗ ( − M − ) −→ ( − ⊗ − ) M − . If thethree inputs to the natural transformation are unit-generated, then we know from abovethat the instantiations of the − M − functor may be viewed in exactly the same manner asthe usages of the − ⊗ − functor. It becomes obvious that, just like the MLL − transformationsabove, w LL is perceived as a tensor in the same way as the associativity isomorphism. Thatis, letting I k and J k be the superindices denoting the k th entries of the source and target func-tors respectively, we find that ( w LL ) I I I J J J = δ J J J I I I . The other weakening transformationsare defined similarly.3.3. Describing Double-Glued Objects with Tensors.
The sets of values and covalues ofan object R in G C contain morphisms from I and to I ∗ respectively. As such, if UR is builtsolely from I , ( − ) ∗ and − ⊗ − in the underlying category, then the elements of R Val and R CoVal can be represented by tensors over the semiring of scalars of C .Given R , S ∈ G C , E ⊆ C [ I , I ∗ ], we provide the shape of the values and covalues createdfrom the ∗ -autonomous structure. One can see particularly how the formulation of thesesets is greatly simplified for the negation and tensor product. • I = (cid:0) I , { I } , λ I ∗ ◦ v I , I , I ( ρ I ) ◦ E (cid:1) • R ⊥ Val = R CoVal ; R ⊥ CoVal = R Val . • ( R ⊗ S ) Val = (cid:8) r i s j : r i ∈ R Val , s j ∈ S Val (cid:9) • ( R ⊗ S ) CoVal = (cid:8) z ij : ∀ r i ∈ R Val , z ij r i ∈ S CoVal , ∀ s j ∈ S Val , z ij s j ∈ R CoVal (cid:9)
4. MLL − F ull C ompleteness for G C For the purposes of this section, we let C be an arbitrary compact closed category withbiproducts satisfying feeble full completeness (Definition 2.7). Feeble full completeness al-lows us to assume that every MLL − transformation (Section 2.2) in this category is describedby a linear combination of fixed-point-free involutions, each involution representing an ax-iom link connecting the two literals being modelled. It is shown in [Tan97, HS03] that G C must be ∗ -autonomous, and so it is already known that all dinatural transformationsmodelling correct MLL − proof structures are found in the glued category. Fact 2.11 tellsus that the dinatural transformations in the glued category are also described by linearcombinations of fixed-point-free involutions. What still remains to be shown, however, isthat the double glueing construction not only preserves the feeble full completeness of C ,but removes enough arrows to ensure that the only linear combinations of proof structureswhich are still modelled are not linear combinations at all, but denotations of single proofnets.With such strong constraints on the arrows within transformations already placed, thestrategy to do this becomes in essence remarkably simple. For an MLL − transformation τ : K I −→ UF in C to be found in G C , it must be the case that τ U R is found as an arrowfrom I to F ( R , R ) for every R ∈ ( G C ) N , i.e. the set of values for the G C -object F ( R , R ).Alternatively, it may be said that τ does not translate into G C if there is a tuple of G C -objects R where τ U R does not belong to F ( R , R ) Val . We provide tuples which expose howsome arrows describing incorrect proof structures and impure linear combinations do not find themselves in all the sets of values needed to ensure they remain transformations in G C .Every compact closed category with finite biproducts has a full subcategory closedunder all three of the characteristic functors which is generated solely by its tensor unit I .By choosing ‘test objects’ for the tuples from this subcategory, we ensure the proof is asgeneral as possible. Furthermore, if the tuple R consists of test objects whose underlying C -objects are of the form n I for some n ∈ N + , the shape of the object F ( R , R ) must be akinto those discussed in Section 3.1, and tensor representations of arrows may be consideredinstead, simplifying the process noticeably.The structure of the coming proof can be viewed as follows:( § G C model correct proofnets, a property which we call MDNF Full Completeness .( § { A n : n ∈ N } and { C n : n ∈ N } ,and calculate the sets of values for the objects given by MDNF functors wheninstantiated using a chosen object from either one of these categories. Thesevalue sets are dependent on the families of full and partial permutations givenin Definition 3.3.( § G C only contains MDNF transformations modelling linear com-binations of proof structures whose scalar multiples sum to 1 (Proposition 4.2).In particular, MDNF transformations containing only zero morphisms ( zero trans-formations ) are not found in the glued category.( § G C models linear combinations ofacyclic proof structures is given. • Consider a transformation τ : K I ✲ UF in C modelling a linear combinationcontaining a cyclic proof structure. • Use Algorithm 4.4 to produce partial permutations for all the blocks of a givencyclic structure. • Proposition 4.9, together with technical lemmas 4.6 and 4.8, are used to showthat the generated partial permutations can be used to prove that he tensorrepresentation (Section 3.2) of the transformation instantiated with a singleobject n I for large enough n cannot be found in the instantiation of F whereall arguments are A n . This disproves the possible existence of the equivalenttransformation in G C .( § G C models linear combinations of correct proof structures (i.e.connectedness is proved). • We consider a transformation τ : K I ✲ UF in C modelling a linear combi-nation of acyclic, disconnected proof structures, noting all the structures mustbe disconnected (Lemma 4.10). • We create appropriate full permutations for all blocks except one using Algo-rithm 4.11 and Lemma 4.13. • Proposition 4.14 explains how these permutations when composed with atensor representation of appropriate dimensions of the linear combination ofdisconnected proof structures produces a zero tensor. This proves that therepresentation could not exist in the instantiation of F where all arguments are C n , meaning that the transformation cannot exist in the double-glued category.( § G C . ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 23 • The method is nearly identical to those seen above, considering a transforma-tion in C to F modelling a true linear combination of proof nets (that is, onecontaining at least two di ff erent proof nets whose scalars are non-zero). • Algorithm 4.15 generates partial permutations for all except one block contain-ing exactly one literal (which is a leaf in each MDNF proof net). • Proposition 4.18, which lightly makes use of the full form of Proposition 4.2,shows that the tensor representation of this linear combination cannot be foundin the set of values of the functor when instantiated using an appropriate A n once again, and so MDNF full completeness holds. ( § Finally, we show that the full completeness of MDNF transformations extends tothat for all MLL − transformations. • Algorithms 4.21 and 4.22 take advantage of the natural isomorphisms and weakdistributivity natural transformations in ∗ -autonomous categories to create naturaltransformations which compose with general MLL − transformations to give ones forMDNF functors. These transformations are used in Theorem 4.24 that MDNF fullcompleteness implies MLL − full completeness, thus giving the desired result.It should be noted that the lemmata used in this full completeness proof find themselvesin an unusual order. In previous MLL − full completeness results, particularly those re-quiring glueing constructions [Loa94b, Tan97, Hag00], it is generally proved first that onlyMLL − proof structures are modelled in the category, and their correctness is shown af-terwards. In the coming proof we do the exact reverse. The existence of cyclic proofstructures in modelled linear combinations is disproved, and disconnectedness of mod-elled proof structures in the glued category is shown to be unallowed. Only then do weprove that ‘impure’ linear combinations and scalar multiples of proof nets are not repre-sented by transformations in G C . Without assuming acyclicity the most natural approachto show the ‘purity’ (or ‘uniqueness’ as we refer to it from now on) of the allowed linearcombinations required; the results of following this train of thought is seen in Section 5.Most parts of the coming proof method require the use of an algorithm to produce anumber of permutation tensors, and these are given in each of the corresponding subsec-tions. The intuition behind how and why the permutations are of use are given alongsidethese algorithms before providing each formal, generalised proof.4.1. Test Objects and Permutations in MDNF Functors.
It turns out that only two funda-mental types of test object are required for full completeness to be proved. We define themboth for each n ∈ N + as follows: • A n : = ( n I , { in x ∈ C [ I , n I ] : x ∈ [ n ] } ∪ { I , n I } , { π x ∈ C [ n I , I ] : x ∈ [ n ] } ∪ { n I , I } ) • C n : = ( n I , { in x ∈ C [ I , n I ] : x ∈ [ n ] } , { π x ∈ C [ n I , I ] : x ∈ [ n ] } )As discussed in Section 3.3, we can represent these objects using sets of tensors insteadof collections of arrows. Their new form is given below. • A n : = ( n I , { δ ix : x ∈ [ n ] } ∪ { i } , { δ ix : x ∈ [ n ] } ∪ { i } ) • C n : = ( n I , { δ ix : x ∈ [ n ] } , { δ ix : x ∈ [ n ] } )It is clear, particularly from the tensor representations, that A n and C n are self-dual —that is, A n (cid:27) A ⊥ n and C n (cid:27) C ⊥ n — for all positive natural numbers n . As a consequencethe sets of tensor representations of values and covalues of the objects F ( R , R ) and F ′ ( R , R )are the same if F ′ is the MLL − functor representing the same sequent as F , only with all instances of negative literals converted to positive instances, and R is filled only with objectssuch as A n and C n . Because of this, in these situations we are at liberty to pretend that allinstances of literals are positive in the formulae when viewing arrows solely through theirmultilinear arrays.Tensor powers of these objects (and therefore tensor products only containing positiveand negative instances of them) reproduce the sets of higher-order permutations, bothfull and partial as defined in Definition 3.3. This can be demonstrated using a standardinductive argument. We use the notation X ⊗ N for the N th tensor power of an object X . Lemma 4.1.
For each N ∈ N + , C ⊗ Nn = (cid:16) ( n I ) ⊗ N , { δ x ··· x N i ··· i N : ( x , . . . , x N ) ∈ [ n ] N } , Perm ( N , n ) (cid:17) .Similarly, A ⊗ Nn = (cid:16) ( n I ) ⊗ N , { δ x ··· x N i ··· i N : ( x , . . . , x N ) ∈ [ n ] N } ∪ { i ··· i N } , PPerm ( N , n ) (cid:17) .Proof. The claim is trivially true for N = (cid:12)(cid:12) A ⊗ Nn (cid:12)(cid:12) = | A n | ⊗ N = ( n I ) ⊗ N = | C n | ⊗ N = (cid:12)(cid:12) C ⊗ Nn (cid:12)(cid:12) . To prove that the values and covalues are as desired for any positive N , we assume it istrue for a natural number M and show that it remains true for M +
1. Section 3.3 containsthe standard calculation rules that are used in this proof. Starting with the values of thetensor power, we find( A ⊗ M + n ) Val = (cid:0) A ⊗ Mn ⊗ A n (cid:1) Val = (cid:8) u i ··· i M · v i M + : u i ··· i M ∈ ( A ⊗ Mn ) Val , v i M + ∈ ( A n ) Val (cid:9) = { δ x ··· x M i ··· i M · δ i M + x M + : ( x , . . . , x M ) ∈ [ n ] M , x M + ∈ [ n ] }∪{ δ x ··· x M i ··· i M · i M + : ( x , . . . , x M ) ∈ [ n ] M }∪{ i ··· i M · δ i M + x M + : x M + ∈ [ n ] } ∪ { i ··· i M · i M + } = { δ x ··· x M + i ··· i M + : ( x , . . . , x M + ) ∈ [ n ] M + } ∪ { i ··· i M + } . Finally, the covalues are evaluated to be as required.( A ⊗ M + n ) CoVal = (cid:0) A ⊗ Mn ⊗ A n (cid:1) CoVal = (cid:8) z i ··· i M + : ∀ u i ··· i M ∈ ( A ⊗ Mn ) Val , z i ··· i M i M + · u i ··· i M ∈ ( A n ) CoVal , ∀ v i M + ∈ ( A n ) Val , z i ··· i M i M + · v i M + ∈ ( A ⊗ Mn ) CoVal (cid:9) = (cid:8) z i ··· i M + : ∃ y ∈ [ n ] , z i ··· i M + · i ··· i M ∈ { δ i M + y , i M + } , ∀ x ∈ [ n ] M , ∃ y ∈ [ n ] , z i ··· i M + · δ x ··· x M i ··· i M ∈ { δ i M + y , i M + } , ∃ y ∈ [ n ] M , z i ··· i M + · i M + ∈ { δ y ··· y M i ··· i M , i ··· i M } , ∀ x M + ∈ [ n ] , ∃ y ∈ [ n ] M , z i ··· i M + · δ i M + x M + ∈ PPerm ( M , n ) (cid:9) = z i ··· i M + : ∀ M + j = ∃ y ∈ [ n ] , z i ··· i M + · Y k , j δ i k x k = { δ i j y , i j } = PPerm ( M + , n )The inductive evaluations of the values and covalues for C ⊗ M + n are simpler but essen-tially identical. ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 25
If a functor F is in multiplicative disjunctive normal form, then the object F ( R , R ) takesthe form M Mm = ( N L m l = R φ ( m , l ) )for some M and L , . . . , L M , where R = ( R , . . . , R ) for some object R = ( n I , U , X ), and φ : S Mm = { ( m , l ) : l ∈ [ L m ] } −→ { , } indicates the number of times the negation functoris applied to one of the instances of the object R . If R is self-dual like each all objects ofthe form A n and C n , and we continue to use tensor representations, then the function φ becomes irrelevant and may be ignored to all intents and purposes. The sets of valuesand covalues in tensor representation form of the object F ( R , R ) are then the same as thoseof the object M Mm = R ⊗ L m . Knowing the values and covalues of the tensor powers of theobjects A n and C n for each n allows us to calculate F ( R , R ) Val and F ( R , R ) CoVal concretelyfor R = A n = ( A n , . . . , A n ) or C n = ( C n , . . . , C n ). As in the previous chapter, to aid clarity ofarguments, the names of the indices used in these sets are based upon the polarity of theliteral to which they are tied: ‘ i ’-indices represent positive literals, and ‘ j ’-indices negativeones. In a block m , we say that there are P m positive and N m negative literals; and in totalthere are L m = P m + N m literals. F ( A n , A n ) Val = n z j (1 , ··· j ( M , NM ) i (1 , ··· i ( M , PM ) : ∀ k ∈ [ M ] , ∀ m , k a m ∈ PPerm ( L m , n ) , ∃ ( x , . . . , x P k , y , . . . , y N k ) ∈ [ n ] L k z ji · Q m , k ( a m ) j ( m , ··· j ( m , Nm ) i ( m , ··· i ( m , Pm ) = δ x ··· y Nk i ( k , ··· j ( k , Nk ) or 0 i ( k , ··· j ( k , Nk ) o F ( A n , A n ) CoVal = ( M Y m = ( a m ) j ( m , ··· j ( m , Nm ) i ( m , ··· i ( m , Pm ) : a m ∈ PPerm ( L m , n ) ) F ( C n , C n ) Val = n z j (1 , ··· j ( M , NM ) i (1 , ··· i ( M , PM ) : ∀ k ∈ [ M ] , ∀ m , k c m ∈ Perm ( L m , n ) , ∃ ( x , . . . , x P k , y , . . . , y N k ) ∈ [ n ] L k z ji · Q m , k ( c m ) j ( m , ··· j ( m , Nm ) i ( m , ··· i ( m , Pm ) = δ x ··· y Nk i ( k , ··· j ( k , Nk ) o F ( C n , C n ) CoVal = ( M Y m = ( c m ) j ( m , ··· j ( m , Nm ) i ( m , ··· i ( m , Pm ) : c m ∈ Perm ( L m , n ) ) The derivation of these sets follows from a simple induction on the number of blocks oftensor powers which exist in an MDNF formula, using the same principles as demonstratedin Lemma 4.1 and Section 3.3.If an MDNF transformation τ in C representing a linear combination of sets of axiomlinks over an MDNF sequent modelled by a functor F is to exist in the glued category, then τ R must belong to both F ( A n , A n ) and F ( C n , C n ) when R = ( n I , . . . , n I ) ∈ Ob j ( C ) N . The restof this section is dedicated to showing how we can always find permutations (either fullor partial) that expose the inability of the aforementioned arrow to belong to at least oneof these sets if the transformation models an unwanted axiom link combination. Zero Transformations.
A zero transformation is an MLL − transformation found inevery compact closed category with finite biproducts whose constituent arrows are all zeromorphisms. They could be viewed from a certain perspective as the representation ofthe statement of a sequent being provable with no evidence, even if the sequent is in factunprovable by standard linear logic reasoning. There is no place for such transformationsin fully complete models of MLL − , and so it is fortunate that their absence from the category G C can be deduced with minimal e ff ort.It is actually worthwhile proving a stronger result than merely the zero transformationsbeing excluded from the glued category. We show that a linear combination of proofstructures cannot possibly be modelled by an MLL − transformation in G C unless the sumof the scalars given to each of the proof structures totals exactly 1. Although this is not aparticularly strong result—in particular in categories such as GRel , where the result is infact equivalent to stating that zero transformations are forbidden—it is necessary for thefinal full completeness proof to be finished. The test object which provides the clearestproof of the desired result is C = ( I , { } , { } ). The object is in fact the tensor unit I of thesub- ∗ -autonomous category G C defined in Section 2.5.1, and the properties its position asa unit of a model of MLL − + Mix bestows upon it are pivotal in the lemma below.
Proposition 4.2.
Every MLL − transformation in G C models a linear combination of proof struc-tures where the scalars applied to the constituent linkings sum to .Proof. Using induction, we observe that, for every MLL − functor F , the arrows in the valuesand covalues of F ( C , C ) have the tensor representation 1. • The base case is trivially true: C = C , and ( C ) Val = { } = ( C ) CoVal . • R ⊥ = (( UR ) ∗ , R CoVal , R Val ) = ( R ∗ , { } , { } ). • R ⊗ S = ( U ( R ⊗ S ) , R Val ⊗ S Val , { z : 1 · z ∈ { } , z · ∈ { }} ) = ( U ( R ⊗ S ) , { } , { } ). • R M S = ( R ⊥ ⊗ S ⊥ ) ⊥ , and so this step may be deduced from the previous two.A linear combination of proof structures is modelled by a tensor P β s β · δ j β (1 , ··· j β ( M , PM ) i (1 , ··· i ( M , PM ) ,summing over bijections β from the set of indices for positive indices to the set of indicesfor negative ones, with s β ∈ C [ I , I ] for all β , when the underlying C -objects of the inputsto F are of the form nI for some n ∈ N + . When n =
1, as is the case when C is the only G C -object being used in F , the deltas become trivial (all the indices must be given the value1), meaning that the tensor becomes P β s β ; that is, the proof structures are modelled by thesum of the scalars associated with the proof structures.The only scalar found in F ( C , C ) is 1 regardless of the form of F , meaning morphismsfor any sum of scalars not equalling 1 do not lift to the homset G C [ I , F ( C , C )]. Assuch, because not all the arrows required to form them in the glued category can be seen,transformations describing linear combinations of proof structures whose scalars do notadd up to 1 in their semiring cannot translate into G C either. Corollary 4.3.
No zero transformation K I , F : K I −→ F in C exists in the category G C . Acyclicity.
It is first shown that each proof structure that is part of a linear combinationbeing modelled in G C satisfies the acyclicity criterion of Danos and Regnier [DR89]. Theconcept behind how this is done can be understood by considering a few simple examples.The most basic example of a cyclic proof structure involves a single axiom link: ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 27 L ⊗ L ⊥ Modelling L with A n for some n >
1, the values of the resultant object are square ( n × n )-matrices with at most one entry being non-zero, and if such an entry exists it must be thenumber 1. However, the tensor representation of the proof structure desired is δ ij , whichfor this representation of L has n non-zero entries. There are too many entries containingnon-zero positions. This fault still remains if we consider scalar multiples of the proofstructure. The same problem occurs with larger cycles containing more than one axiom link andmore than one block. ( L ⊗ L ⊥ ) M ( L ⊗ L ⊥ ) M ( L ⊗ L ⊥ )The proof structure above is described by δ j j j i i i ( i k and j k relating to the positive andnegative literals in the k th block as is the norm). One of the criteria a tensor z j j j i i i must satisfyif [[ L ]] = A n for large enough n in order for it to belong to the values of the correspondingobject is for every pair of partial (2-)permutations over [ n ], a i j and b i j say, to create an( n × n )-matrix with at most one non-zero entry when composed with it. If we say that a i j = δ i j + δ i j and b i j = δ i j + δ i j — which are indeed partial permutations — we find a i j · b i j · δ j j j i i i = ( δ i j + δ i j ) · ( δ i j + δ i j ) · δ j j j i i i = ( δ i j · δ i j · δ j j j i i i ) + ( δ i j · δ i j · δ j j j i i i ) + ( δ i j · δ i j · δ j j j i i i ) + ( δ i j · δ i j · δ j j j i i i ) = δ j
251 2 i + δ j
261 4 i + δ j
453 2 i + δ j
463 4 i = δ i j + δ i j The resulting matrix clearly has two non-zero entries: when i = j =
1; and when i = j =
3. As such, this particular cyclic structure is shown to be represented neitherin the set of values of ( A n ⊗ A ⊥ n ) M ( A n ⊗ A ⊥ n ) M ( A n ⊗ A ⊥ n ), nor the MLL − transformations of G C .The reason why this argument is possible is based on the fact that every block involvedin a cycle has two literals incident to axiom links in the cycle. The two literals in the firstblock may be given tensors containing two non-zero positions which compose with theproof structure tensor because they are in the same block, and ( A n ⊗ A ⊥ n ) CoVal contains allpartial permutations; the same is true of those in the second block. If they were not, aswould be the situation in the proof structure below, only one non-zero entry would be ableto be found: tensors in ( A n M A ⊥ n ) CoVal only have at most one value of 1, with the rest beingzeroes. Multiplying the structure by zero naturally removes the problem. However, this produces the zeromorphism; the existence of a zero transformation has already been contradicted in Proposition 4.2. L M L ⊥ M ( L ⊗ L ⊥ ) M ( L ⊗ L ⊥ )The objective of the coming proof is to generate partial permutations with two non-zeropositions as above for all bar one of the blocks connected to the axiom links in a chosencycle; and these tensors should compose with the Kronecker delta tensor representing anincorrect proof structure in a linear combination being considered to create a tensor withmore than one entry not equalling zero. Of course, the scenarios o ff ered so far have beencurtailed in two ways: all of their axiom links are involved in the cycle (and there is onlyone cycle), and linear combinations of two or more distinct proof structures are absent. Thealgorithm below deals with both of these problems, and this is discussed in more detailafter its description. Algorithm 4.4.
Input: A cyclic MDNF proof structure with linking λ containing M blocks,the m th of which containing L m literals, and one of its minimal cycles ˆ λ .Output: A number n ∈ N + ; tensors a , . . . , a M such that ( a m ) i ··· i Lm ∈ PPerm ( L m , n ) for each m .(1) Let i =
0, and note that, at this point, none of the links in λ has been dealt with.(2) Find a link l ∈ λ \ ˆ λ which has not been dealt with yet.(a) If one should exist, then assign the number i + l .Increment i , and go to Step 2.(b) If one does not exist, then move to Step 3.(3) Find a link l ∈ ˆ λ which has not been dealt with yet.(a) If one should exist, then assign both the numbers i + i + l . Increase the value of i by 2, and restart Step 3.(b) If one does not exist, then move to Step 4.(4) For each tensor product of literals which does not contain a literal incident to a linkwithin ˆ λ , place the values assigned to each literal into a tuple in the same order as theirliterals appear in the subformula. This tuple ‘belongs’ to that subformula.(5) For each tensor product of literals which does contain a literal incident to at least onelink within ˆ λ , create two tuples as follows:(a) Place the lowest values assigned to each literal into a tuple in the same order astheir literals appear in the subformula.(b) Place the highest values assigned to each literal into a tuple in the same order astheir literals appear in the subformula. (6) Set n = i ; and for each m ∈ [ M ], define an element ( a m ) i ··· i Lm of PPerm ( L m , n ) as follows:( a m ) i ··· i Lm = (cid:26) i , . . . , i L m ) is a tuple for block m λ only ever contain one non-zero entry, and that entry is 1. In this type of situation it is abundantly clear that there areonly 1s and 0s as entries, and there cannot be two 1s in the same column. For those blocks If a literal has been assigned only one number, then this number is indeed considered both the highestand lowest value.
ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 29 connected to the cycle, there are two 1s in their tensors, but since the two tuples whichdefine their positions in the tensor di ff er in more than one component, they cannot exist inthe same column, and so they can exist together in a partial permutation. An example ofhow the algorithm functions with a typical input is given below. Example 4.5.
Consider the sum of the two linkings λ and λ seen in the following diagram: L M L ⊥ M ( L ⊗ L ⊥ ) M ( L ⊗ L ⊥ ⊗ L ) M L ⊥ λ λ The tensor representation of this linear combination of axiom links is δ j (2 , j (4 , j (3 , j (5 , i (1 , i (3 , i (4 , i (4 , + δ j (4 , j (2 , j (3 , j (5 , i (1 , i (3 , i (4 , i (4 , .We apply Algorithm 4.4, choosing λ to be λ , and the subset of axiom links drawn below to be ˆ λ (with representation δ j (4 , j (3 , i (3 , i (4 , )L M L ⊥ M ( L ⊗ L ⊥ ) M ( L ⊗ L ⊥ ⊗ L ) M L ⊥
2. Step 2 from the algorithm is iterated twice. Starting with the left most axiom linkfrom λ \ ˆ λ , the literals, along with the numbers to them are as follows: L M L ⊥ M ( L ⊗ L ⊥ ) M ( L ⊗ L ⊥ ⊗ L ) M L ⊥ L M L ⊥ M ( L ⊗ L ⊥ ) M ( L ⊗ L ⊥ ⊗ L ) M L ⊥ L M L ⊥ M ( L ⊗ L ⊥ ) M ( L ⊗ L ⊥ ⊗ L ) M L ⊥ λ λ [1] [1] [3 , 5] [5 , 3 , 2] [2][4 , 6] [6 , 4 , 2]
6. The tensors a , . . . , a are then created from the tuples above.( a ) i (1 , = δ i (1 , ( a ) j (2 , = δ j (2 , ( a ) i (3 , j (3 , = δ i (3 , j (3 , + δ i (3 , j (3 , ( a ) i (4 , j (4 , i (4 , = δ i (4 , j (4 , i (4 , + δ i (4 , j (4 , i (4 , ( a ) j (5 , = δ j (5 , In the tuples underneath each of the blocks in the MDNF formula given in Part 4 / λ . This can be realised by noting that Steps 2 and 3 areactually, in principle, numbering the axiom links of λ , and the literals, and only the literals,incident to an axiom link take the numbers associated with that link. This principle allowsus to di ff erentiate the chosen λ from all other linkings, as can be seen in the coming claims. Lemma 4.6.
Suppose that we have a linear combination of proof structures including a non-zeroinstance of a linking λ which contains a minimal cycle ˆ λ , and choose a block k through which ˆ λ passes. Then a tensor modelling a linking λ ′ when composed with all the tensors except a k createdby Algorithm 4.4 (when using λ and ˆ λ as inputs) will result in a zero tensor if λ and λ ′ do not havean identical set of axiom links not connected to block k.Proof. Suppose there is a link l ∈ λ ′ \ λ not connected to block k . Then the tensor modelling λ ′ must take the form ¯ λ ′ ij ij = δ ij ω ij for some tensor ω , with i and j being indices associatedwith the literals connected via l . For the blocks m i and m j containing the literals allocatedthe index i and j , the algorithm creates partial permutations built from summations ofone or two tensors of the form δ ix γ i ′ and δ jy χ j ′ , with x , y ∈ N + and γ and χ products ofKronecker deltas whose details are of little relevance.Since the two literals connected to l are not linked in λ , the numbers given to thoseliterals in the tuples created by the algorithm are di ff erent, and therefore we know x , y for each of the parts of the partial permutations a m i and a m j of the form δ ix γ i ′ and δ jy χ j ′ . Wetherefore find that δ ij δ ix δ jy = δ xy =
0, meaning ( δ ij ω ij )( δ ix γ i ′ )( δ jy χ j ′ ) = ( δ ij δ ix δ jy ) ω ij γ i ′ χ j ′ = a m i a m j ¯ λ ′ = λ ′ · Q m , k a m = k =
4, which is certainly in the cycle ˆ λ in λ = λ , then we see that the link in λ fromthe left-most negative literal to the positive literal directly to its right, represented by thetensor δ i (3 , j (2 , , is neither in λ nor incident to block 4. The algorithm provides two partialpermutations ( a ) j (2 , = δ j (2 , and ( a ) i (3 , , j (3 , = δ i (3 , j (3 , + δ i (3 , j (3 , which are intended to becomposed directly with δ i (3 , j (2 , , and they produce a zero tensor as desired. δ i (3 , j (2 , · δ j (2 , · ( δ i (3 , j (3 , + δ i (3 , j (3 , ) = δ j (3 , + δ j (3 , = j (3 , ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 31
The e ff ects of the tensor relating to λ are therefore eradicated when composed with all of a , a , a and a .There are examples of linear combinations of proof structures and choice of block k ,unlike the one given above, where there is a linking λ ′ which shares all of its axiom linksnot incident with block k with λ . Lemma 4.6 is rendered useless in these situations whenattempting to di ff erentiate all other sets of axiom links from λ . The example below typifiessuch a dilemma. Example 4.7.
Consider the sum of the three linkings λ , λ and λ seen in the diagrambelow. L ⊥ M ( L ⊗ L ) M ( L ⊥ ⊗ L ⊥ ⊗ L ) λ λ λ Their corresponding tensor representation is δ j (1 , j (3 , j (3 , i (2 , i (2 , i (3 , + δ j (1 , j (3 , j (3 , i (3 , i (2 , i (2 , + δ j (1 , j (3 , j (3 , i (3 , i (2 , i (2 , . Choosing λ to be λ , and taking the subset of links given in the figure below to be ˆ λ (with representation δ j (3 , j (3 , i (2 , i (2 , )L ⊥ M ( L ⊗ L ) M ( L ⊥ ⊗ L ⊥ ⊗ L )ˆ λ
2. Step 2 from the algorithm is only used once. We provide the single pair of literalsconnected to the sole link in λ \ ˆ λ with the number 1. L ⊥ M ( L ⊗ L ) M ( L ⊥ ⊗ L ⊥ ⊗ L )1 13. Step 3 is repeated twice. Starting from the left, the assign numbers as follows: L ⊥ M ( L ⊗ L ) M ( L ⊥ ⊗ L ⊥ ⊗ L )2 4 4 23 5 5 3 L ⊥ M ( L ⊗ L ) M ( L ⊥ ⊗ L ⊥ ⊗ L ) λ λ λ [1] [2 , 4] [4 , 2 1][1] [3 , 5] [5 , 3 , 1]6. The tensors a , . . . , a are then created from the tuples above.( a m ) j (1 , = δ j (1 , ( a m ) i (2 , i (2 , = δ i (2 , i (2 , + δ i (2 , i (2 , ( a m ) j (3 , j (3 , i (3 , = δ j (3 , j (3 , i (3 , + δ j (3 , j (3 , i (3 , Then ¯ λ a a , , ¯ λ a a .Fortunately, we are able to find another claim which provides enough information todi ff erentiate any such rogue sets of axiom links from the chosen linking λ su ffi ciently. Lemma 4.8.
Suppose that we have a linear combination of proof structures including a non-zeroinstance of a linking λ which contains a minimal cycle ˆ λ , and choose a block k through which ˆ λ passes. Let ( x , y ) and ( x , y ) be the two tuples created for block k by Algorithm 4.4 with λ and ˆ λ over their associated MLL − formula as its inputs, reordered so the l th positions of x and x relate tothe l th positive literal (given index i ( k , l ) ), and y and y to the l th negative literal with j ( k , l ) . If a set ofaxiom links λ ′ in the linear combination is similar enough to λ that their axiom links not connectedto block k are the same, then the tensor ¯ λ ′ when composed with all the tensors created by Algorithm4.4 except a k , and either one of the tensors δ x y i ( k , − ) j ( k , − ) and δ x y i ( k , − ) j ( k , − ) , the result is the scalar if λ ′ = λ ,and otherwise.Proof. We start by showing that zero is created in the case that λ , λ ′ . In this situation, thereis an axiom link in λ ′ incident to block k which is not found in λ . Without loss of generality,we assume that this link is connected to block k by a positive literal given index i ( k , l i ) in thetensor representation; it is adjacent to a negative literal with index j ( m , l j ) for some m and l j .The tensor representation of the axiom link is therefore δ j ( m , lj ) i ( k , li ) , and the representation of λ ′ has this as a factor.The value (or values) found in the ( l j ) th position of the tuple (or tuples) associated withblock m in this case are not the same as x pl i for p = λ :the tensor ( a m ) i ( m , − ) j ( m , − ) has form δ y lj j ( m , lj ) · Q l , l j δ y l j ( m , l ) or δ y lj j ( m , lj ) · Q l , l j δ y l j ( m , l ) + δ y lj j ( m , lj ) · Q l , l j δ y l j ( m , l ) , andwe find that ¯ λ ′ ji · δ x y i ( k , − ) j ( k , − ) · ( a m ) i ( m , − ) j ( m , − ) = j ′ i ′ ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 33 for appropriate superindices i ′ and j ′ , from which this part of the claim is a trivial conse-quence. This is because, for both r = δ j ( m , lj ) i ( k , li ) · δ y lj j ( m , lj ) · δ x ri ( k , li ) i ( k , li ) = δ y lj x ri ( k , li ) = . Now we consider the case when λ ′ = λ . Suppose that two literals, one of each polarity,given indices i ( m i , l i ) and j ( m j , l j ) are connected by an axiom link in λ . Then ¯ λ ji has δ j ( mj , lj ) i ( mi , li ) as afactor.If blocks m i and m j are not connected to ˆ λ , then it must be the case that δ x ( mi , li ) i ( mi , li ) and δ x ( mi , li ) i ( mi , li ) are factors of ( a m i ) i ( mi , − ) and ( a m j ) j ( mj , − ) respectively for the x ( m i , l i ) and x ( m j , l j ) defined in thealgorithm for these index position. We know that x ( m i , l i ) = x = x ( m j , l j ) for some x for bothconstant tensors due to the axiom link being in λ . δ j ( mj , lj ) i ( mi , li ) · δ xi ( mi , li ) · δ xj ( mj , lj ) = δ = λ ji · δ x ( mi , li ) i ( mi , li ) · δ x ( mj , lj ) j ( mj , lj ) produces Kronecker deltas representing an MDNF proof structureidentical to that of λ , but with the literals incident to the chosen axiom link removed(together with said axiom link). Following this line of argument to its logical conclusion,we find, when Γ ( ˆ λ ) means all blocks not adjacent to ˆ λ , that¯ λ ji · Y m < Γ ( ˆ λ ) ( a m ) i ( m , − ) j ( m , − ) · Y q δ z ( q ) q reduces to the tensor representation of the axiom links solely in the neighbouring blocks ofˆ λ , with the q in the product being indices for each literal found in the blocks connected tothe links in ˆ λ which are adjacent via axiom links to blocks not connected to the cycle.We now consider tuples { b m : m ∈ Γ ( ˆ λ ) } for the remaining indices in the blocks con-nected to ˆ λ . That is, for each of these m ,( a m ) i ( m , − ) j ( m , − ) = ( b m ) i ′ ( m , − ) j ′ ( m , − ) · Y q l δ z l q l for appropriate superindices.Suppose that there is an axiom link connected to blocks connected to the cycle ˆ λ butnot connected to block k . Then its representation takes the form δ j ( mj , lj ) i ( mi , li ) for some m i , m j , l i , l j .It cannot be the case that m i = m j , since we have chosen for ˆ lambda to be a minimal cycle in λ (as prescribed by Algorithm 4.4). If it were, it would actually be the entire cycle. We aretherefore left with the only possibility that m i , m j .If m i , m j , then there will be a factor of the original composition which is equivalentto the following: δ j ( m , lj ) i ( m , li ) · ( b m i ) i ′ ( mi , − ) j ′ ( mi , − ) · ( b m j ) i ′′ ( mj , − ) j ′′ ( mj , − ) . The tensor δ j ( m , lj ) i ( m , li ) is acting as the representation of an axiom link between the two literalsrepresented by the indices i ( m , l i ) and j ( m , l j ) . The two tuples b m i and b m j are partial permutations containing exactly two non-zeropositions. This is seen by how they are found by the blocks a m i and a m j : the ‘ a ’ blocks arepartial permutations with exactly two non-zero positions, and since they can be formedfrom the two ‘ b ’ tuples by use of a single Kronecker delta the same must be true of theirfactors. By assumption, and without loss of generality regarding which tuple contains thepositive literal of the axiom link and which contains the negative, we know that the tuples b m i and b m j are tuples which take the forms below.( b m i ) i ′ ( mi , − ) j ′ ( mj , − ) = δ p q i ′′ ( mi , − ) j ′ ( mi , − ) δ r i ( mi , li ) + δ p q i ′′ ( mi , − ) j ′ ( mi , − ) δ r i ( mi , li ) ( b m j ) i ′ ( mi , − ) j ′ ( mj , − ) = δ s t i ′ ( mi , − ) j ′′ ( mi , − ) δ r j ( mj , lj ) + δ s t i ′ ( mi , − ) j ′′ ( mi , − ) δ r j ( mj , lj ) for some appropriately sized constant superindices p , p , q , q , s , s , t , t and constants r , r , whilst letting i ′′ ( m i , − ) and j ′ ( m j , − ) be the free superindices which are i ′ ( m i , − ) and j ′ ( m j , − ) with i ( m i , l i ) and j ( m j , l j ) removed respectively. We know this because Algorithm 4.4 ensuresthat the tuples created for two blocks of tensors will have entries which match each otherif and only if there is an axiom link in λ connecting the literals whose positions they arerepresenting. Since the axiom links of λ ′ and λ which are not connected to block k are the,this crosses over to this linking.Once these representations have been created, it becomes clear that the tensor com-position from before reduces to nothing more than another partial permutation with twonon-zero entries. δ j ( m , lj ) i ( m , li ) · ( b m i ) i ′ ( mi , − ) j ′ ( mi , − ) · ( b m j ) i ′′ ( mj , − ) j ′′ ( mj , − ) = δ j ( m , lj ) i ( m , li ) · (cid:18) δ p q i ′′ ( mi , − ) j ′ ( mi , − ) δ r i ( mi , li ) + δ p q i ′′ ( mi , − ) j ′ ( mi , − ) δ r i ( mi , li ) (cid:19) · (cid:18) δ s t i ′ ( mi , − ) j ′′ ( mi , − ) δ r j ( mj , lj ) + δ s t i ′ ( mi , − ) j ′′ ( mi , − ) δ r j ( mj , lj ) (cid:19) = δ p q i ′′ ( mi , − ) j ′ ( mi , − ) δ r i ( mi , li ) δ s t i ′ ( mi , − ) j ′′ ( mi , − ) δ r j ( mj , lj ) δ j ( m , lj ) i ( m , li ) + δ p q i ′′ ( mi , − ) j ′ ( mi , − ) δ r i ( mi , li ) δ s t i ′ ( mi , − ) j ′′ ( mi , − ) δ r j ( mj , lj ) δ j ( m , lj ) i ( m , li ) + δ p q i ′′ ( mi , − ) j ′ ( mi , − ) δ r i ( mi , li ) δ s t i ′ ( mi , − ) j ′′ ( mi , − ) δ r j ( mj , lj ) δ j ( m , lj ) i ( m , li ) + δ p q i ′′ ( mi , − ) j ′ ( mi , − ) δ r i ( mi , li ) δ s t i ′ ( mi , − ) j ′′ ( mi , − ) δ r j ( mj , lj ) δ j ( m , lj ) i ( m , li ) = δ p q i ′′ ( mi , − ) j ′ ( mi , − ) δ s t i ′ ( mi , − ) j ′′ ( mi , − ) δ r r + δ p q i ′′ ( mi , − ) j ′ ( mi , − ) δ s t i ′ ( mi , − ) j ′′ ( mi , − ) δ r r + δ p q i ′′ ( mi , − ) j ′ ( mi , − ) δ s t i ′ ( mi , − ) j ′′ ( mi , − ) δ r r + δ p q i ′′ ( mi , − ) j ′ ( mi , − ) δ s t i ′ ( mi , − ) j ′′ ( mi , − ) δ r r = δ p q i ′′ ( mi , − ) j ′ ( mi , − ) δ s t i ′ ( mi , − ) j ′′ ( mi , − ) + δ p q i ′′ ( mi , − ) j ′ ( mi , − ) δ s t i ′ ( mi , − ) j ′′ ( mi , − ) ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 35
The representation could be said to be equivalent to the block of literals which would becreated by merging to two blocks to which b and b are connected and removing the linkedliterals. The same idea applies to the two tuples that are created.This is certainly a repeatable process for all axiom links not connected to block k . Fol-lowing this procedure as far as possible, we find that the answer to the original compositionbecomes the same as c i ( k , − ) j ( k , − ) · δ x r y r i ( k , − ) j ( k , − ) · Y q δ z ( q ) q . The tensor c is the partial permutation described by two tuples providing the locations ofthe non-zero entries of block k : the concatenation of all the first tuples of blocks which arenot k , with tuple positions relating to indices describing literals not connected to block k byan axiom link deleted; and the same with the second tuples. In Example 4.7, taking that k =
3, it happens that c i (3 , − ) j (3 , − ) = ( a ) j (3 , j (3 , j (3 , .Due to the algorithm ensuring that certain tuple positions are kept equal to one another, c i ′′ ( k , − ) j ′′ (3 , − ) · Y q δ z ( q ) q = δ x y i ( k , − ) j ( k , − ) + δ x y i ( k , − ) j ( k , − ) . It then follows trivially that( δ x y i ( k , − ) j ( k , − ) + δ x y i ( k , − ) j ( k , − ) ) · δ x r y r i ( k , − ) j ( k , − ) = δ x y x r y r + δ x y x r y r = . Example 4.7 fits the scenario where Lemma 4.8 is used neatly. Assuming block 3 is thechosen block k , it is clear that every axiom link in λ = λ not incident to that block is foundin λ and vice versa. However, multiplying ¯ λ with a and a gives the following tensor:¯ λ j (1 , i (2 , i (2 , j (3 , j (3 , i (3 , ( a ) j (1 , ( a ) i (2 , i (2 , = δ j (1 , j (3 , j (3 , i (3 , i (2 , i (2 , δ j (1 , ( δ i (2 , i (2 , + δ i (2 , i (2 , ) = δ j (3 , j (3 , i (3 , + δ j (3 , j (3 , i (3 , Multiplying the resulting tensor with either δ j (3 , j (3 , i (3 , or δ j (3 , j (3 , i (3 , will immediatelyresult in producing a 0 as an output.( δ j (3 , j (3 , i (3 , + δ j (3 , j (3 , i (3 , ) · δ j (3 , j (3 , i (3 , = δ + δ = δ j (3 , j (3 , i (3 , + δ j (3 , j (3 , i (3 , ) · δ j (3 , j (3 , i (3 , = δ + δ = δ j (3 , j (3 , i (3 , produces two tensors being summedtogether which either have the right numbers to use in the entry positions but in the wrongorder as in the first (due to the axiom links connecting with the block in a di ff erent manner),or the wrong numbers altogether like in the second (due to using the tensor created fromthe information from the upper tuple rather than the lower). The dual of this remark canbe found by considering composition with δ j (3 , j (3 , i (3 , .Armed with the two claims created from the earlier simple principle, we are in aposition to express why MDNF transformations from C only translate to G C if the linearcombination of proof structures they model do not contain a cyclic proof structure. Proposition 4.9.
Every MDNF transformation in G C models a linear combination of acyclic proofstructures. Proof.
Suppose that the functor [[ F ]] is in MDNF, and therefore for any n ∈ N + [[ F ]]( A n , A n ) = M Mm = ( N L m l = A φ ( m , l ) n ) . The tensor representations of the values of the above object are given in Section 4.1Let τ = ( τ R ∈ C [ I , | [[ F ]] | ( R , R )]) R ∈ C N be an MDNF transformation in C , modelling a linearcombination of proof structures, one of which is cyclic. For the object R = ( n I , . . . , n I ), thecomponent τ R is represented by the tensor τ j (1 , ··· j ( M , NM ) i (1 , ··· i ( M , PM ) = P β s β · δ j β (1 , ··· j β ( M , PM ) i (1 , ··· i ( M , PM ) . One of the bijections, ζ say, is such that s ζ ,
0, and there is a cycle between the blocks.We show that τ ji < [[ F ]]( A n , A n ) Val = G C [ I , [[ F ]]( A n , A n )] for some choice of n ∈ N + . Wechoose n and define a set of partial permutations { a m : m ∈ [ M ] } , one for each block oftensor products of literals, using Algorithm 4.4; and we nominate any one of the blocksthat is part of the chosen cycle, calling it k . We know from Lemmas 4.6 and 4.8 thatthere are two distinct entry tuples ( x , y ) = ( x k , , . . . , x k , P k ) , y k , , . . . , y k , N k ) ) and ( x , y ) = ( x k , , . . . , x k , P k ) , y k , , . . . , y k , N k ) ) for a k where ( a k ) x , y = = ( a k ) x , y that, for l ∈ { , } , δ x l y l i ( k , − ) j ( k , − ) · δ x l ( k , ··· x l ( k , Pk ) y l ( k , ··· y l ( k , Nk ) i ( k , ··· i ( k , Pk ) j ( k , ··· j ( k , Nk ) · τ ji · Q m , k ( a m ) i ( m , ··· i m , Pm j ( m , ··· j ( m , Nm ) = s ζ , . The tensor τ ji , when multiplied with these partial permutations, produces a tensor withat least two non-zero entries, meaning it fails the criterion desired of it to belong to[[ F ]]( A n , A n ) Val . As such τ R is not an arrow in G C [ I , [[ F ]]( A n , A n )], and consequently τ cannot be seen as an MLL − transformation in G C .4.4. Connectedness.
The previous subsection holds the proof that MDNF transformationsin G C only describe linear combinations of acyclic proof structures. This section is devotedto proving the proof structures are connected as well. Once again we can see the intuitionbehind the coming proof using a couple of small examples.Consider the simplest disconnected proof structure. L M L ⊥ M L M L ⊥ If we say [[ L ]] = C n for some n >
1, it is quickly observed that the values of[[ L M L ⊥ M L M L ⊥ ]] are described by the 4-permutations over [ n ]. These are all tensorsof the form z j j i i ( i l and j l being linked to the l th positive and negative literals respectively)which, when composed with any three constant tensors, produce another constant tensor.The proof structure is described by δ j j i i . It is already known that this is not a full 4-permutation, and therefore does not belong to the set of values. However, we require a moregeneralisable perspective in order to the learn from the example. If we compose δ j i with δ i and δ j , which are both 1-permutations over [ n ], we see that δ j i · δ i · δ j = δ =
0, meaning δ j j i i · δ i · δ j = · δ j i = i j . Remembering the description of values of MDNF objects builtsolely from C n in Section 4.1, it becomes clear that δ j j i i cannot belong in ( C n M C ⊥ n M C n M C ⊥ n ) Val : ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 37 the zero tensor does not belong to ( C ⊥ n ) Val , and composing any tensor from the non-empty( C n ) CoVal with 0 i j produces 0 j .In the above example it was possible to find (1-)permutations for both blocks in the left-hand component which, when composed with the Kronecker delta modelling the axiomlink incident to their associated literals, produce the scalar 0. The generalisability of thisidea becomes further evident when a more complicated proof structure is considered. L ⊥ M ( L ⊗ L ) M ( L ⊥ ⊗ L ⊥ ⊗ L ) M L M L M L M L ⊥ The above structure contains far more axiom links, and some blocks have more than oneliteral within them. Its links are described by ¯ λ = δ j (1 , j (3 , j (3 , j (3 , j (7 , i (2 , i (2 , i (5 , i (4 , i (6 , , with the left-mostcomponent’s links given by ¯ λ l = δ j (1 , j (3 , j (3 , j (3 , i (2 , i (2 , i (5 , i (4 , .The first five blocks constitute the left-most component, and five permutations whichcan annihilate the above tensors are found below.( c ) j (1 , = δ j (1 , ( c ) i (2 , i (2 , = cycle (2 , n , i (2 , i (2 , ( c ) j (3 , j (3 , j (3 , = cycle (3 , n , j (3 , j (3 , j (3 , ( c ) i (4 , = δ i (4 , ( c ) i (5 , = δ i (5 , ( c ) j (1 , ( c ) i (2 , i (2 , ( c ) j (3 , j (3 , j (3 , ( c ) i (4 , ( c ) i (5 , ( ¯ λ l ) j (1 , j (3 , j (3 , j (3 , i (2 , i (2 , i (4 , i (5 , = δ j (1 , ( c ) i (2 , i (2 , ( c ) j (3 , j (3 , j (3 , δ i (4 , δ i (5 , δ j (1 , j (3 , j (3 , j (3 , i (2 , i (2 , i (5 , i (4 , = cycle (2 , n , i (2 , i (2 , cycle (3 , n , j (3 , j (3 , j (3 , δ j (3 , j (3 , j (3 , i (2 , i (2 , = ( cycle (2 , n , i (2 , i (2 , δ i (2 , ) · ( cycle (3 , n , j (3 , j (3 , j (3 , δ j (3 , j (3 , i (2 , ) · δ j (3 , = ( δ i (2 , · cycle (2 , n , i (2 , j (3 , ) · δ j (3 , = δ j (3 , j (3 , = δ = c ) j (1 , ( c ) i (2 , i (2 , ( c ) j (3 , j (3 , j (3 , ( c ) i (4 , ( c ) i (5 , ( c ) i (6 , ( ¯ λ ) ji = j (7 , for all c ∈ Perm (1 , n ), and so ¯ λ is not a value, meaning that λ is not modelled in G C by anMLL − transformation.The lesson to be learned from the arithmetic above is that although cycle permutationsare being used for the permutations c , . . . , c above , this is only due to their ease incomprehension. What is most important is that certain positions in the permutations havevalue 1. The proof structure figure below shows a choice of positions in the permutations It should be remembered that cycle (1 , n , x ) i = δ xi . corresponding to each block in the proof structure from above which ensure that theircomposition with the tensor ¯ λ yields a zero tensor. L ⊥ M ( L ⊗ L ) M ( L ⊥ ⊗ L ⊥ ⊗ L ) M L M L M L M L ⊥ [1] [1 , 4] [4 , 5 , 2] [2] [3]Notice that each pair of literals in the proof structure are only given the same number fortheir entry positions if they are connected by an axiom link, and that all bar one of thelinked pairs share a number, one of the rogue pair being the leaf which is the 5 th block.Because of this, it is possible to follow a chain of compositions of permutations with ¯ λ in anorder such that the permutation of a block m is not considered until those of all the blockswhose unique path to the 5 th block in the block graph passes through m have been (leaving c to last). This ensures that the product of all the permutations corresponding to blocksgreater than or equal to m in the block graph of the structure with respect to the partialtree ordering induced by the vertex representing the 5 th block with the primitive Kroneckerdeltas of ¯ λ having at least one index in common with one of the permutations reduces toa Kronecker delta δ xi , where i is the sole remaining index relating to the literal connectedvia an axiom link to a literal not in the set of blocks greater than or equal to m , and x is thenumber in the tuple placed underneath the same literal. The inevitable consequence is thatthe final Kronecker delta associated with the axiom link connected to the right-most leaf ofthe component having its indices substituted for two distinct numbers, which is equal tozero.Of course, we are not in a position to assume we are only dealing with singular proofstructures and their scalar multiples — linear combinations of proof structures have notyet been discounted. Lemma 4.9 allows us to assume that all proof structures in a linearcombination being considered are acyclic. This has a rather useful consequence, namelythat if any one proof structure in a linear combination of them is disconnected, then all theothers are as well. Lemma 4.10.
If a sequent S can be bestowed with a valid set of axiom links which induce an acyclicyet disconnected proof structure, then all possible valid linkings are also disconnected.Proof.
Given a set of proof structures over S , we know that all the structures have the samenumber of edges. The number of axiom links in a single structure is equal to the numberof pairs of literals which exist in S , and so independent of the position of the links; andsince the proof structures are built over the same parse forest (namely the one describedby S ), the number of edges which are not axiom links are equal as well. The number of M connectives is trivially only dependent on S , and so the number of M -vertices in eachstructure does not vary. The number of edges in a switching of a proof structure is equalto the number in the entire proof structure minus the number of M -vertices in it, meaningthat every switching of every proof structure over S has the same number of edges.The number of vertices in a proof structure over S (and therefore in each of its switch-ings), n say, is equal to the total number of literals and connectives in the sequent; for agraph over this number of vertices to be connected the number of edges must be greaterthan n −
1. If there is an acyclic and disconnected proof structure, its switchings musthave strictly fewer than n − n −
1, and the
ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 39 disconnectedness discounts the possibility of the number being exactly n −
1. Therefore all switchings of all other proof structures have strictly fewer than n − − proof structure. Di ff erent proofstructures clearly lead to di ff erent components, and this could be viewed as creating amoving target. However, the claim above ensures that no problems are caused. We chooseone leaf in the example to be the block not to be given a permutation to compose with theMDNF tensor, and let ourselves be prepared to add restrictions to permutations for any ofthe other blocks if they are in a di ff erent component from that leaf for any one of the proofstructures being modelled in the linear combination.The first task is to create a set of permutations over some [ n ] for an unacceptablelinear combination of proof structures over a sequent F which makes it possible to derivea disproof of its associated tensor’s existence in the set of values [[ F ]]( C n , C n ). Algorithm 4.11.
Input: A linear combination of acyclic, disconnected MDNF proof struc-tures of the same sequent containing M blocks.Output: A number n ∈ N + ; tensors b , . . . , b M such that ( b m ) i ··· i Lm ∈ PPerm ( L m , n ) for each m .(1) Let { λ , . . . , λ K } be the set of axiom links in a linear combination of acyclic yet discon-nected MDNF proof structures which are multiplied by a non-zero scalar. For eachof the blocks containing exactly one literal ( leaves ) assign distinct values 1 , . . . , B . Weassign variables v , . . . , v M to each of the M blocks, with v m = L m as the initial setting.We call these variables valencies . Let i = B + k = λ k and the M -free subgraph of the parse tree, and consider the component of thegraph containing the lowest numbered component which does not contain block 1.Choose the leaf with the highest number in that component and change its valency to0.(3) Choose the first block m in the component such that v m = v m =
0. If it is not adjacent to another leaf inthe graph, then assign the number given to that leaf to the literal to which it isconnected and decrement the valency of the block of the connected literal. RestartStep 3.(b) If one does exist but it is not a leaf, then mark the sole literal not yet allocated anumber as an ‘exit’ literal and go to Step 4.(c) If there are no more blocks of valency 1 and k < K , create a new L m -tuple x for each block m in the component, and for every i ∈ [ L m ] let x i be thelabel given to the i th literal of the block. If the i th literal is the exit of the block, thenmark x i as an exit entry. Delete duplicate tuples associated with each block, andremove all the labels and marks from all the literals except the leaves. Increment k and return to Step 2.(d) Otherwise, terminate the algorithm after declaring that n = i − m ∈ [2 , M ],( b m ) i ··· i Lm = (cid:26) i , . . . , i L m ) is a tuple for m (4) Check to see whether the numbers given to each of the non-exit literals correspondexactly to those of a tuple x = ( x , . . . , x L m ) already associated with the block (meaningthat if u i is the number given to the i th literal in the block, that u i = x i for every i forwhich u i is defined).(a) If so, then assign the final unused number in x to the exit literal.(b) If not, assign i to the exit literal and increment i .Assign the number to the literal with which the exit literal shares an axiom link, unlessthat literal is a leaf and already has been assigned a number. Decrement the valenciesof both block m and the block with which the exit literal of block m is linked by anaxiom link in λ k . Go to Step 3. Example 4.12.
Consider the sum of the three linkings λ , λ and λ provided below in red,blue and green respectively. L M L ⊥ ⊗ L ⊥ M ( L ⊗ L ) ⊗ ( L ⊥ ⊗ L ⊥ ⊗ L ⊥ ) M L M L The linear combination’s tensor representation is¯ λ j (2 , j (3 , j (5 , j (5 , j (5 , i (1 , i (4 , i (4 , i (6 , i (7 , = δ j (2 , j (3 , j (5 , j (5 , j (5 , i (1 , i (4 , i (4 , i (6 , i (7 , + δ j (3 , j (2 , j (5 , j (5 , j (5 , i (1 , i (4 , i (4 , i (6 , i (7 , + δ j (2 , j (3 , j (5 , j (5 , j (5 , i (1 , i (4 , i (4 , i (6 , i (7 , . We apply Algorithm 4.11.1. We first attach values to the leaves in each of the proof structures described. Thevalue i reaches 6. L M L ⊥ ⊗ L ⊥ M ( L ⊗ L ) ⊗ ( L ⊥ ⊗ L ⊥ ⊗ L ⊥ ) M L M L λ λ λ [1] [2] [3] [4] [5]We make sure all the other components have valency the same as the number ofliterals within them: v = v = v = v = v = λ . With this set of axiom links the first block not in the samecomponent as the first is the third. The component containing block 3 contains the4 th , 5 th , 6 th and 7 th blocks as well. We change the valency of block 7 to 0.3 & 4. We must repeat the processes contained in Steps 3 and 4 four times (once for eachaxiom link in λ connected to the component containing block 3). • Block 3 is a leaf and connects to the first literal of block 4. That literal is thereforegiven the same number allocated to block 3 (3), and the second literal is markedas an exit. Block 4 now has valency 1, and block 3 has valency 0. • Block 4 is now the first block to have valency 1. Its second literal, is given i = i is incremented to 7. ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 41 • Block 6, a leaf, is now the first block with valency 1. It has already been givenvalue 4, so we give this to its adjacent literal — the third literal in the fifth block.We decrement the block valencies. • Block 5 is now the only one with valency 1 in the component. Its only emptyposition — the second — is allocated the value i =
7. However, it is connected tothe last leaf, and so we do not give its adjacent literal the same number (it alreadyhas been given the entry 5).We are left with the following tuples, with the numbers in squares emphasising exitliterals for that particular linking: L M L ⊥ ⊗ L ⊥ M ( L ⊗ L ) ⊗ ( L ⊥ ⊗ L ⊥ ⊗ L ⊥ ) M L M L λ λ λ [1] [2] [3] [3 , 6 ] [6 , 7 , 4] [4] [5]2. The process is then repeated with λ and λ . Particular points to note are thefollowing: • The component containing block 1 can change (note that blocks 1 and 2 are notrespected with respect to λ , but the two are adjacent with when the linkings λ and λ are used). • In the third iteration for λ , block 4 is reached at the same time in the same wayas in λ . As such, the number 6 is reused for the tuple entry for its second literal.We now have three sets of tuples for each block (possibly repeated): L M L ⊥ ⊗ L ⊥ M ( L ⊗ L ) ⊗ ( L ⊥ ⊗ L ⊥ ⊗ L ⊥ ) M L M L λ λ λ [1] [2] [3] [3 , 6 ] [6 , 7 , 4] [4] [5][1] [2] [3] [2 , 8 ] [8 , 9 , 4] [4] [5][1] [2] [3] [3 , 6 ] [ 10 , 6 , 4] [4] [5]3. The algorithm finished, leaving the following set of tensors:( b ) j (2 , = δ j (2 , ( b ) j (3 , = δ j (3 , ( b ) i (4 , i (4 , = δ i (4 , i (4 , + δ i (4 , i (4 , ( b ) j (5 , j (5 , j (5 , = δ j (5 , j (5 , j (5 , + δ j (5 , j (5 , j (5 , + δ
10 6 4 j (5 , j (5 , j (5 , ( b ) i (6 , = δ i (6 , ( b ) i (7 , = δ i (7 , The above algorithm provides a set of partial permutations with entries containing thevalue 1 at positions which are of use in the final proof of the necessity of connectednesshaving to be satisfied. However, they are still partial permutations, and in order to discussobjects of the form F ( C n , C n ) full permutations must be used. It is therefore necessaryto complete the permutations. Completing permutations is non-trivial, but by virtue of anumber of properties bestowed on each of the partial permutations described in Algorithm4.11, we are indeed capable of performing such a task. Lemma 4.13.
Every partial higher-order permutation ( b m ) i where m ∈ [2 , M ] which is an outputtensor of Algorithm 4.11 can be completed to form a full higher-order permutation ( c m ) i .Proof. It is known from Definition 3.3 that for any L and n it is always possible to find thetensor cycle ( L , n , i ··· i L in Perm ( L , n ). It is also well established that using a permutationover [ n ] on the numbering used on a single index of a permutation in Perm ( L , n ) for any L ∈ N + always produces another equally valid permutation in the same set.Suppose that b i ··· i L ∈ PPerm ( L , n ) is one of the partial permutations generated byAlgorithm 4.11 after being given appropriate inputs. Then we can refer back to the tuples x , . . . , x K ′ created in the middle of the algorithm which describe the positions where itsentries are equal to 1, and also the positions in each of the tuples with an ‘exit marker’.From the definition of the procedure, we can be assured that if the j th position of the k th tuple is marked, then l kj , l k ′ j for all k , k ′ . In other words, the number found in the positionof a tuple’s exit marker is not used again in the same position of another tuple. With thisinformation, we define a set of partial functions in N + { α l : l ∈ [ L ] } in the following manner:(1) Let i = k = l = α m ( x kl ) has not yet been assigned a value and the tuple position in question was notmarked an exit entry in Algorithm 4.11, then we say that α m ( x kl ) = i and then increment i . Go to Step 3.(3) If l < T , increment l and go to Step 2; otherwise increment k , set l back to 1 and go toStep 4.(4) If k ≤ K , then go to Step 2; otherwise, go to Step 5.(5) Reset k and l .(6) If α l ( x kl ) has not yet been assigned a value, then set α l ( x kl ) = n − P j , l α j ( x kj ) and go toStep 7. Otherwise increment l and repeat Step 6.(7) If k < K , increment k and let l =
1, and return to Step 6. Otherwise, terminate thealgorithm.This algorithm is well defined, and each partial function α l is injective with all definedsource and target values in [ n ]. As such, it is possible to restrict each of the partialfunctions to act upon [ n ], and then to extend them to full 2-permutations over (i.e. bijectiveendomorphisms on) [ n ]. One way in which such an extension of a partial function α m couldbe formed is inductively: by assigning the lowest value in [ n ] not already in the image of α m to be the image of the lowest number in [ n ] not in domain of definition, and continuingsimilarly with the new definition of the partial function until it becomes total.We now define the tensor c i · i L for b i ··· i L in the following manner: c i ··· i L = (cid:26) P Lj = α j ( i j ) ≡ n ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 43
The tensor is created by using 2-permutations on each of the entry positions of cycle ( L , n , Perm ( L , n ). Furthermore, since c x ··· x T = x , . . . , x T )where b x ··· x T = b i ··· i L , and as such we have satisfied theoriginal statement as desired.Much in the same manner as with Lemma 4.9, we are finally in a position to prove thatthe proof structures being modelled in linear combinations in the double-glued categoryare always proof nets. Proposition 4.14.
Every MDNF transformation in G C modelling a linear combination of acyclicproof structures is modelling a linear combination of cut-free proof nets.Proof. Let F be an MDNF formula as in Lemma 4.9, and let τ = ( τ R ∈ C [ I , | [[ F ]] | ( R , R )]) R ∈ C N be an MDNF transformation in C modelling a linear combi-nation of acyclic yet disconnected proof structures. We consider [[ F ]] ( C n , C n ) Val where n isthe output integer given by Algorithm 4.11 if the linear combination described by τ is usedas an input. From Section 4.1 we know the criteria the tensor representations of the valuesof this object must satisfy.As seen in earlier deductions, for the integer n acquired from the algorithm, we havethe tensor representation τ j (1 , ··· j ( M , NM ) i (1 , ··· i ( M , PM ) = P β s β · δ j β (1 , ··· j β ( M , PM ) i (1 , ··· i ( M , PM ) . Algorithm 4.11 and Lemma 4.13 makes it possible to obtain a set of (full) permutations { c m : m ∈ [2 , M ] } for the coming argument.Take any set of axiom links described by a bijection, ζ say, such that s ζ ,
0, and considerthe tensor δ j ζ (1 , ··· j ζ ( M , PM ) i (1 , ··· i ( M , PM ) . It is possible to factorise this tensor into a product of Kroneckerdeltas, one for each component in the switching of ζ . Multiplying the first of these smallercomponent tensors not containing the indices concerning the first block, δ j ζ ( θ (1) , ··· j ζ ( θ ( M ′ ) , P θ ( M ′ )) i ( θ (1) , ··· i ( θ ( M ′ ) , P θ ( M ′ )) say, with each of the permutations b θ (1) , . . . , b θ ( M ′ ) corresponding to the blocks within thecomponent in question, the zero scalar is produced. Claim.
Using the definitions of ζ and θ given above, δ j ζ ( θ (1) , ··· j ζ ( θ ( M ′ ) , P θ ( M ′ )) i ( θ (1) , ··· i ( θ ( M ′ ) , P θ ( M ′ )) · Q M ′ m = (cid:0) c θ ( m ) (cid:1) j ( θ ( m ) , ··· j ( θ ( m ) , N θ ( m )) i ( θ ( m ) , ··· i ( θ ( m ) , P θ ( m )) = Proof.
It is easy to show that there is a partial ordering induced by every tree andevery choice of node within it. Given a tree G and vertex w , we can create an ordering ≤ ( G , w ) defined as follows: if the unique path starting from a vertex u and finishing at w passes through the vertex v , then we say that u ≤ ( G , w ) v . The point w is thereforemaximal in this ordering. This is a partial order if we close this under reflexivity.We consider the blocks with numbers in the image of θ (that is, the blocks of thesecond component) observing the tree partial ordering induced by the leaf with thehighest block number θ ( q ). With block θ ( q ) being a leaf, the tensor c θ ( q ) = δ Lk ( θ ( q ) , for some constant L , where the k -index replaces either an i - or j -index depending onthe polarity of the literal in the block. It should also be noted that, by design of thealgorithm, L is a smaller number than any of the numbers given exit markers within the component which are not in a leaf block, and greater than all other numbers givento leaves in the component. We use an inductive argument on the blocks in this orderto demonstrate that, for any block θ ( m ) except the reference leaf, a composition of thetensors in { c θ ( m ′ ) : θ ( m ′ ) ≤ G θ ( m ) } with all the Kronecker deltas pairs whose indicesare shared with two permutations in that set reduces to a constant tensor on the indexwith the exit marker created at this point in Algorithm 4.11 and the number with saidmarker. That is, Q m ′ < ( G ,θ ( q )) θ ( m ) δ j ζ ( m ′ , ··· j ζ ( m ′ , Pm ′ ) i ( m ′ , ··· i ( m ′ , Pm ′ ) · Q m ′ ≤ ( G ,θ ( q )) θ ( m ) (cid:0) c m ′ (cid:1) j ( m ′ , ··· j ( m ′ , Nm ′ ) i ( m ′ , ··· i ( m ′ , P θ ( m ′ )) = δ xk where k is the sole index not to be composed, and x is the number connected to theexit marker in the tuple generated by the algorithm the appropriate linking. • The base case occurs when block θ ( m ) is a leaf. The first product of Kroneckerdeltas is then empty, and the second only contains the permutation c θ ( m ) . Sinceblock θ ( m ) is a leaf, it must be the case that there is only one literal in the block, andtherefore the permutation is a 1-permutation, i.e. a constant tensor of dimension1. The algorithm defines the non-zero entry to occur at the x th position, where x isthe single value given to the block in Algorithm 4.11, as desired. • If not, then the composition can be split further. Let H ( m ) denote the neighbour-hood of block m in the graph and E = (cid:8) m ′ ∈ H ( θ ( m )) : m ′ < ( G ,θ ( q )) (cid:9) , ∅ . We canrewrite the composition above as follows: Y m ′ ∈ E δ l ξ m ′ k m ′ Y m ′′ < ( G ,θ ( q )) m ′ δ j ζ ( m ′′ , ··· j ζ ( m ′′ , Pm ′′ ) i ( m ′′ , ··· i ( m ′′ , Pm ′′ ) · Q m ′′ ≤ ( G ,θ ( q )) m ′ (cid:0) c m ′′ (cid:1) j ( m ′′ , ··· j ( m ′′ , Nm ′′ ) i ( m ′′ , ··· i ( m ′′ , Pm ′′ ) (cid:19) · (cid:16) c θ ( m ) (cid:17) j ( θ ( m ) , ··· j ( θ ( m ) , N θ ( m )) i ( θ ( m ) , ··· i ( θ ( m ) , P θ ( m )) By the induction hypothesis, the inside of the bracket can be simplified, leaving Y m ′ ∈ E δ x m ′ k m · δ l ξ ( m ′ ) k m ′ ! · (cid:16) c θ ( m ) (cid:17) j ( θ ( m ) , ··· j ( θ ( m ) , N θ ( m )) i ( θ ( m ) , ··· i ( θ ( m ) , P θ ( m )) = Y m ′ ∈ E δ x m ′ l ξ ( m ′ ) · (cid:16) c θ ( m ) (cid:17) j ( θ ( m ) , ··· j ( θ ( m ) , N θ ( m )) i ( θ ( m ) , ··· i ( θ ( m ) , P θ ( m )) = (cid:16) c θ ( m ) (cid:17) x π ζ − θ ( m ) , ··· x π ζ − θ ( m ) , y )) ··· x π ζ − θ ( m ) , Nm )) x π ζ ( θ ( m ) , ··· i ( θ ( m ) , y ) ··· x π ζ ( θ ( m ) , Pm )) or (cid:16) c θ ( m ) (cid:17) x π ζ − θ ( m ) , ··· j ( θ ( m ) , y ) ··· x π ζ − θ ( m ) , Nm )) x π ζ ( θ ( m ) , ··· x π ζ ( θ ( m ) , y )) ··· x π ζ ( θ ( m ) , Pm )) Since c θ ( m ) is a full ( P m + N m )-permutation, it must be the case that our finaltensor, with constants taking the places of all bar one of the entry positions, isa 1-permutation as desired. Furthermore, the Kronecker deltas which have beenfed into the equation are in agreement with exactly one set of tuples that is providedby the algorithm for c θ ( m ) . We therefore know that the number that the remainingfree variable for the tensor must equal the final unused number from the tuple,which is the value at the exit marker. Our equation reduces to the form δ xk withfree index k and constant x as desired. ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 45
Now we refer back to the original claim. The composition δ j ζ ( θ (1) , ··· j ζ ( θ ( M ′ ) , P θ ( M ′ )) i ( θ (1) , ··· i ( θ ( M ′ ) , P θ ( M ′ )) · Q M ′ m = (cid:0) c θ ( m ) (cid:1) j ( θ ( m ) , ··· j ( θ ( m ) , N θ ( m )) i ( θ ( m ) , ··· i ( θ ( m ) , P θ ( m )) = δ j ζ ( θ (1) , ··· j ζ ( θ ( M ′ ) , P θ ( M ′ )) i ( θ (1) , ··· i ( θ ( M ′ ) , P θ ( M ′ )) · Q m , q ( c θ ( m ) ) j ( θ ( m ) , ··· j ( θ ( m ) , N θ ( m )) i ( θ ( m ) , ··· i ( θ ( m ) , P θ ( m )) · δ Lk reduces to the very simple δ xk ( θ ( q ) , δ L ′ k ( θ ( q ) , = δ Lx , where the k -index is a substitute foran i - or j -index, depending on whether the literal for the component θ ( q ) is positiveor negative as before. By the design of the algorithm, this final value x is assured ofbeing not equal to L ; and therefore it must be the case that δ Lx = δ j ζ (1 , ··· j ζ ( M , PM ) i (1 , ··· i ( M , PM ) · Q Mm = ( c m ) j ( m , ··· j ( m , Nm ) i ( m , ··· i ( m , Pm ) = j (1 , ··· j (1 , N i (1 , ··· i (1 , P . Since τ j (1 , ··· j ( M , NM ) i (1 , ··· i ( M , PM ) is merely a linear combination of tensors like δ j ζ (1 , ··· j ζ ( M , PM ) i (1 , ··· i ( M , PM ) , and ζ wasarbitrary, we know that τ j (1 , ··· j ( M , NM ) i (1 , ··· i ( M , PM ) · Q Mm = ( c m ) j ( m , ··· j ( m , Nm ) i ( m , ··· i ( m , Pm ) = j (1 , ··· j (1 , N i (1 , ··· i (1 , P . This means that τ j (1 , ··· j ( M , NM ) i (1 , ··· i ( M , PM ) < [[ F ]] ( B n , B n ) Val , and so τ R does not meet the criteria to befound in G C [ I , [[ F ]] ( B n , B n )]. The MLL − transformation τ in C is therefore unable to betranslated into G C either.4.5. Uniqueness.
Non-simple linear combinations of two or more proof structures cannever be proof nets themselves. Because of this, their representations must be demonstratedto have been eradicated from the categorical model by the double glueing constructionbefore we can declare MDNF full completeness proved.The sections above allow us to take as fact that every MLL − transformation in the gluedcategory G C is not only the representation of a linear combination of proof structures, buta linear combination of proof nets . Furthermore, it is also known that the scalars that eachof the proof nets being modelled is multiplied by must sum to the semiring multiplicativeunit 1. Interestingly, the information here is su ffi cient to provide MDNF full completenessresults for a number of categories of the form G C for some category C : namely those whosesemiring of scalars have the property that if a sum P i s i happens to equal 1, then there isexactly one x such that s x =
1, and every other s i =
0. An example of this type of categoryis the category of semimodules over N .However, such grandiose claims cannot be written about the majority of compactclosed categories with finite biproducts. In particular, the categories Rel and
FDVec F ,which are the examples investigated in [Tan97] in detail, do not benefit from having thisproperty. A more comprehensive proof for uniqueness is required, and this unsurprisinglytakes the same combinatorial form as seen throughout this chapter. We show the intuitionto the proof below.For the sake of the coming argument, we assume that we do not have the result fromLemma 4.2 (we use it in a more restricted form later). Suppose we have an MLL − transfor-mation C which can be thought of as representing the sum of the two sets of axiom linksover the formula below, itself being described by a functor F . L M ( L ⊥ ⊗ L ⊥ ) M L ⊥ λ λ Both linkings successfully describe proof nets for the formula, and so we are lookingat a scenario concerning a linear combination of proof nets. The tensor representationsof the two linkings λ and λ , given by the red and blue linkings respectively, usingthe standard indices for each literal, are ¯( λ ) j (2 , j (2 , i (1 , i (3 , = δ j (2 , j (2 , i (1 , i (3 , and ¯( λ ) j (2 , j (2 , i (1 , i (3 , = δ j (2 , j (2 , i (1 , i (3 , respectively. The description of the linear combination of proof nets being discussed is¯ Λ j (2 , j (2 , i (1 , i (3 , = δ j (2 , j (2 , i (1 , i (3 , + δ j (2 , j (2 , i (1 , i (3 , .It is simple enough to find partial permutations for both of the first two blocks in theformula of some order which produces a constant tensor when composed with ¯ λ j (2 , j (2 , i (1 , i (3 , yet creates a zero tensor when composed with ¯ λ j (2 , j (2 , i (1 , i (3 , . If we let ( u ) i (1 , = δ i (1 , and( u ) j (2 , j (2 , = δ j (2 , j (2 , , then¯( λ ) j (2 , j (2 , i (1 , i (3 , ( u ) i (1 , ( u ) j (2 , j (2 , = δ j (2 , j (2 , i (1 , i (3 , δ i (1 , δ j (2 , j (2 , = δ i (3 , = δ i (3 , ¯( λ ) j (2 , j (2 , i (1 , i (3 , ( u ) i (1 , ( u ) j (2 , j (2 , = δ j (2 , j (2 , i (1 , i (3 , δ i (1 , δ j (2 , j (2 , = δ i (3 , = i (3 , By symmetry, it must also be possible to reverse this e ff ect by letting ( v ) i (1 , = δ i (1 , and( v ) j (2 , j (2 , = δ j (2 , j (2 , . ¯( λ ) j (2 , j (2 , i (1 , i (3 , ( v ) i (1 , ( v ) j (2 , j (2 , = δ j (2 , j (2 , i (1 , i (3 , δ i (1 , δ j (2 , j (2 , = δ i (3 , = i (3 , ¯( λ ) j (2 , j (2 , i (1 , i (3 , ( v ) i (1 , ( v ) j (2 , j (2 , = δ j (2 , j (2 , i (1 , i (3 , δ i (1 , δ j (2 , j (2 , = δ i (3 , = δ i (3 , There are two significant points in the development of these partial permutations.Firstly, letting ( u ∪ v ) i = u i · v i (not using Einstein notation), the unions w = u ∪ v and w = u ∪ v are both partial permutations, where u I ∪ v I = u I = = v I and 1 otherwise.Secondly, the constant tensors created by the compositions ¯( λ ) j (2 , j (2 , i (1 , i (3 , ( u ) i (1 , ( u ) j (2 , j (2 , and¯( λ ) j (2 , j (2 , i (1 , i (3 , ( v ) i (1 , ( v ) j (2 , j (2 , are not identical. The e ff ect that these properties togetherhave is that we can find a pair of partial permutations which act on the tensor sum¯( λ ) j (2 , j (2 , i (1 , i (3 , ( v ) i (1 , ( v ) j (2 , j (2 , + ¯( λ ) j (2 , j (2 , i (1 , i (3 , ( u ) i (1 , ( u ) j (2 , j (2 , , and the resulting tensor is thesum of two constant tensors where the constants are di ff erent (meaning they cannot inter-fere with one another and cancel each other out).¯ Λ j (2 , j (2 , i (1 , i (3 , ( w ) i (1 , ( w ) j (2 , j (2 , = δ j (2 , j (2 , i (1 , i (3 , ( u ) i (1 , ( u ) j (2 , j (2 , + δ j (2 , j (2 , i (1 , i (3 , ( v ) i (1 , ( v ) j (2 , j (2 , = δ i (3 , + δ i (3 , ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 47
There are clearly two non-zero entries in the resultant tensor, which means we can inferthat the tensor ˆ Λ cannot belong to the set of values F ( A n , A n ) for any number n ≥ − transformation τ in C modelling a non-simple linearcombination of proof nets, we can show it does not manifest itself in any form in G C . Analgorithm is given which takes linkings being described by τ and creates partial permu-tations for all bar one of the blocks in the formula such that composing them with thetensor representation of two axiom links leaves a constant tensor, minimising interferencebetween tensor representations of each linking involved. Composing these permutationswith the a tensor representation of the linear combination of all the axiom links gives atensor containing more than one non-zero entry, and therefore the tensor τ n I is not in F ( A n , A n ) Val for some n , thus τ cannot be found in G C . We start by giving the algorithmwhich begins this process. Algorithm 4.15.
Input: A non-trivial linear combination of MDNF proof nets for a sequentcontaining M blocks, the m th of which having size T m for each m .Output: A number n ∈ N + ; tensors c , . . . , c M such that c mi ··· i Tm ∈ PPerm ( T m , n ) for each m except for one leaf.(1) Let λ and λ be two distinct sets of axiom links in the given linear combination ofMDNF proof structures. We set the valencies v , . . . , v M such that v m = T m for all m ∈ [ M ], except for the last leaf, block l say, for which v l =
0. We let i = k = m with v m = T m tuple x for each block m in the component, and let x i = u i for every i ∈ [ T m ], where u i is the label given tothe i th literal of the block. Remove all the labels. If k =
1, increment k , reset thevalencies and restart Step 2; otherwise go to Step 4.(3) Check to see whether the numbers given to each of the non-exit literals in the blockcorrespond exactly to those of a tuple t = ( t , . . . , t T m ) already associated with the block.(a) If so, then assign the final unused number in t to the exit literal.(b) If not, increment i and then assign the new value of i to the exit literal.Assign the number to the literal with which the exit literal shares an axiom link in λ k .Decrement the valencies of both block m and the the block with which the exit literalof block m is linked by an axiom link in λ k . Restart Step 2.(4) Declare that n = i , and we state that for each m ∈ [ M ],( c m ) i ··· i Tm = (cid:26) i , . . . , i T m ) is a tuple for m λ and λ being acyclic, there is always a blockwith valency 1 at every stage of the algorithm for a single value of k when one is neededuntil all the blocks bar leaf l have been visited. This assures that each block is considered inStep 2 exactly twice, and therefore at most two entries are found in each tensor of a block m , l . On top of that, at the point where all bar one of the positions in a new tuple havebeen decided when k = k =
1. If they do then the second tuple is completed to be a clone of the first; if not then a completely new number is used in the unfilled position. This ensures thatthere is never a situation where the two tuples di ff er in exactly one coordinate, and as suchthe tuples c , . . . , c M (excluding c l ) have to be partial permutations.The partial permutations created need to have the e ff ect of annihilating tensor represen-tations of all possible axiom links on a sequent other than the two that have been selectedfor use in Algorithm 4.15. Fortunately, the algorithm does do this naturally, and in factmakes sure that the representations of λ and λ only make use of one non-zero entry ineach of the permutations. Example 4.16.
Consider the sum of the three linkings λ , λ and λ (given in red, blue andgreen respectively) in the diagram below. L M ( L ⊥ ⊗ L ⊥ ) M ( L ⊗ L ⊗ L ⊥ ) M L M ( L ⊥ ⊗ L ⊥ ) M L λ λ λ The tensor representation of this linear combination is δ j (2 , j (2 , j (5 , j (3 , j (5 , i (1 , i (3 , i (3 , i (4 , i (6 , + δ j (2 , j (2 , j (5 , j (3 , j (5 , i (1 , i (3 , i (3 , i (4 , i (6 , + δ j (2 , j (2 , j (5 , j (3 , j (5 , i (1 , i (3 , i (3 , i (4 , i (6 , . We apply Algorithm 4.15, choosing λ and λ as the two (synonymous) input linkings.
1. Block 6 is the right-most block containing only one literal, so the valencies of eachblock are given as follows: v = , v = , v = , v = , v = , v =
02 & 3. We deal first with λ ( k = k = L M ( L ⊥ ⊗ L ⊥ ) M ( L ⊗ L ⊗ L ⊥ ) M L M ( L ⊥ ⊗ L ⊥ ) M L λ [1] [1 , 2] [2 , 4 , 3] [3] [4 , 5] [5]2 & 3. The same is done for k =
2, i.e. for the linking λ , only making sure tuples di ff eringin exactly one position from the ones created when considering λ do not occur(which in this case only occurs at leaf blocks 1 and 3). We obtain the following: L M ( L ⊥ ⊗ L ⊥ ) M ( L ⊗ L ⊗ L ⊥ ) M L M ( L ⊥ ⊗ L ⊥ ) M L λ [1] [6 , 1] [6 , 7 , 3] [3] [7 , 8] [8]4. Joining the two sets of tuples together, we get ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 49 L M ( L ⊥ ⊗ L ⊥ ) M ( L ⊗ L ⊗ L ⊥ ) M L M ( L ⊥ ⊗ L ⊥ ) M L λ λ λ [1] [1 , 2] [2 , 4 , 3] [3] [4 , 5] [5][1] [6 , 1] [6 , 7 , 3] [3] [7 , 8] [8]Therefore the tensors u , . . . , u can now be formed( u ) i (1 , = δ i (1 , ( u ) j (2 , j (2 , = δ j (2 , j (2 , + δ j (2 , j (2 , ( u ) i (3 , i (3 , j (3 , = δ i (3 , i (3 , j (3 , + δ i (3 , i (3 , j (3 , ( u ) i (4 , = δ i (4 , ( u ) j (5 , j (5 , = δ j (5 , j (5 , + δ j (5 , j (5 , ( u ) i (6 , = δ i (6 , + δ i (6 , Composing the five partial permutations with the representation of λ ends with a constanttensor.¯( λ ) j (2 , j (2 , j (3 , j (5 , j (5 , i (1 , i (3 , i (3 , i (4 , i (6 , · Y m = u m i m j m = δ j (2 , j (2 , j (5 , j (3 , j (5 , i (1 , i (3 , i (3 , i (4 , i (6 , · δ i (1 , ( δ j (2 , j (2 , + δ j (2 , j (2 , )( δ i (3 , i (3 , j (3 , + δ i (3 , i (3 , j (3 , ) δ i (4 , ( δ j (5 , j (5 , + δ j (5 , j (5 , ) = δ i (6 , + δ i (6 , + δ i (6 , + δ i (6 , + δ i (6 , + δ i (6 , + δ i (6 , + δ i (6 , = δ i (6 , With similar equations it is easily shown that ¯( λ ) j (2 , j (2 , j (3 , j (5 , j (5 , i (1 , i (3 , i (3 , i (4 , i (6 , · Q m = ( u m ) i m j m = δ i (6 , , andthat ¯( λ ) j (2 , j (2 , j (3 , j (5 , j (5 , i (1 , i (3 , i (3 , i (4 , i (6 , · Q m = ( u m ) i m j m = i (6 , . The components a ff ected by the tensor rep-resentations of λ and λ are di ff erent from one another, meaning they do not interfere withone another. Similarly, the tensor representation for λ and partial permutations reduceto the zero morphism, and therefore has no e ff ect on any component. The permutationscomposed with the tensor representation for the sum of all the linkings is therefore foundto be δ i (6 , + δ i (6 , , which has two non-zero entries. This proves that the linking combinationcannot be modelled in G C .It can also be seen that this lack of interference between di ff erent proof nets in the finalresult means that scalar multiples of the three linkings in question in the example can besummed together in a linear combination and the same concept of proof still holds. Ifwe consider an MLL − transformation τ ′ in the underlying category of such a form, thencomposing τ ′ n I with the same five partial permutations for suitable n results in a tensor withtwo non-zero entries, with the fifth and eighth positions filled by the scalars multipled to λ and λ repectively. The exact values within the entries are irrelevant—it only mattersthat two are non-zero. Lemma 4.17.
Let τ : K I −→ F be an MDNF transformation in a compact closed category C withfinite biproducts modelling a linear combination of at least two distinct proof nets, { λ a : a ∈ [ K ] } say,over the MDNF formula modelled by F containing M blocks, the l th of which is the last leaf. Letting λ and λ be the two primary inputs to Algorithm 4.15, producing M − partial permutations { ( u m ) i ( m , − ) j ( m , − ) : m ∈ [ M ] \{ l }} , we find that ( ¯ λ ) ji · Y m , l ( u m ) i ( m , − ) j ( m , − ) = δ xk , and ( ¯ λ ) ji · Y m , l ( u m ) i ( m , − ) j ( m , − ) = δ yk where x and y are the first and second values o ff ered to leaf l in the algorithm, and k is the index i ( l , or j ( l , , depending on the polarity of the literal denoted by leaf l. Furthermore, for every a > , ( ¯ λ a ) ji · Y m , l ( u m ) i ( m , − ) j ( m , − ) = k Proof.
The tensor representation of a linking of a proof net is represented by a product ofKronecker deltas, where two indices are in the same delta if the literals to which they areassociated are linked by an axiom link. As such, if two indices i ( m i , l i ) and j ( m j , l j ) associatedwith literals connected by an axiom link in λ are forced to be given di ff erent values p and q say, then we know that ( ¯ λ ) ji · δ pi ( mi , li ) · δ qj ( mj , lj ) = j ′ i ′ for appropriate index sets i ′ and j ′ .The partial permutations, containing either one or two non-zero entries, can be mul-tiplied together and the distribution law can be used to give a sum of 2 M ′ products ofconstant tensors ( M ′ being the number of blocks given partial permutations with two non-zero entries), with each index being given a number except k . These 2 M ′ assignments ofnumbers are determined by the combinations of first and second tuples associated withthe sets of indices for each block—Example 4.16 provides a good example of this.Of these combinations, only two are capable of being composed with a tensor repre-sentation of a set of axiom links to create a non-zero tensor: the one where all the constantsare chosen to be from the top tuples of each block in the algorithm; and the one from allthe bottom tuples. This is because the algorithm continuously uses new numbers when anarbitrary number must be selected (Step 3(b)), and so mixing up the top and bottom tuplesleaves more than one number given to only one index, thus ensuring it cannot be part of anaxiom link pairing without creating a zero tensor. An example of this argument in actioncan be seen in Example 4.16 at the start of Step 4: we see that (1 ,
2) of block 2 and (6 , ,
3) ofblock 3 only have one value 2 within the two of them together.For the tensor representation of λ , only the combination developed from the top tuplesgives a non-zero tensor. The second tuples are defined so that two literals are only giventhe same number in their tuples if they share an axiom link in λ ; and since λ and λ are distinct, there must be at least one pair of literals joined by an axiom link in λ whichdoes not exist in λ , thus leading to a zero tensor. The one index from ¯ λ which doesnot have a number assigned to it by a partial permutation is clearly k , which is connectedto the sole literal in a block other than l assigned the number x in the top tuple. Thecomposition of ( ¯ λ ) ji with the contsant tensors for relating to the top tuples from the M − l therefore reduces to a product of Kronecker deltas of the form ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 51 δ zz for various z and δ xk . Since δ zz = z , this becomes simply δ xk ; and so it follows that( ¯ λ ) ji · Q m , l ( u m ) i ( m , − ) j ( m , − ) = δ xk as desired.The argument for the representation of λ is fundamentally identical, replacing thebottom tuples for the top tuples throughout the above paragraph. For every other λ a in thelinear combination when composed any one of the 2 M ′ − λ and λ before. Each of them isalso distinct from both λ and λ by definition, meaning that neither the combination of allthe top tuples nor that of the bottom tuples induce constant tensors which compose with¯ λ a to give a non-zero tensor due to the same principle as why the bottom tuples do not for¯ λ . We are now able to give the proof for a lemma proving the simplicity of the linearcombinations of proof nets capable of being modelled in G C . Proposition 4.18.
Every MDNF transformation in G C modelling a linear combination of proofnets is modelling a unique proof net with scalar .Proof. Let F be an MDNF formula, and consider [[ F ]]( A n , A n ) for arbitrary n , where A n isdefined as in Section 4.1. Again, we use the criteria also given in Section 4.1 that the tensorrepresentation of an arrow f ∈ C [ I , | [[ F ]]( A n , A n ) | ] must satisfy in order to be found in G C .Suppose that τ is an MDNF transformation in C which models a linear combination of twoor more proof nets. Then, for R = ( n I , . . . , n I ), τ R once again takes the tensorial form τ j (1 , ··· j ( M , NM ) i (1 , ··· i ( M , PM ) = P β s β · δ j β (1 , ··· j β ( M , PM ) i (1 , ··· i ( M , PM ) and we know that there are two bijections, ζ and ζ say, corresponding to axiom linksets λ and λ respectively, where s ζ , , s ζ . We use these two sets of axiom links inAlgorithm 4.15 to produce partial permutations { c m : m ∈ [ M ] } for all of the blocks, and tofix the required size of the number n ∈ N + to prove the lemma.Let x and x be the first and second tuple values associated with the last leaf (block l )from the algorithm. Then we find that for k ∈ { , } δ x k r ( l , · τ ji · Q m , l ( c m ) j ( m , ··· j ( m , Nm ) i ( m , ··· i ( m , Pm ) = s ζ l , r ( l , equals i ( l , or j ( l , depending on whether the literal of leaf l is positiveor negative.We have shown that there are at least two non-zero entries in τ ji · Q m , l ( c m ) j ( m , ··· j ( m , Nm ) i ( m , ··· i ( m , Pm ) , and so τ ji < [[ F ]]( A n , A n ) Val . The arrow τ R ∈ C [ I , | [[ F ]]( A n , A n ) | ] doesnot therefore exist in G C [ I , [[ F ]]( A n , A n )], and so τ is not an MLL − transformation.From the above, we know that every MLL − transformation in G C models a scalarmultiple of a cut-free proof net. Lemma 4.2 ensures that those scalar multiples with scalarnot equal to 1 are not possible, and so we have proved the result. Theorem 4.19. If C is a compact closed category with biproducts satisfying feeble full completeness,then G C satisfies MDNF full completeness. Extending to MLL − Full Completeness.
The previous subsections contain resultswhich combine to produce a full completeness result for MLL − transformations to targetfunctors describing sequents in the multiplicative disjunctive normal form. Although astrong connection between any family of MLL − transformations and proof nets in MLL − is advantageous, it is certainly no substitute for a ‘complete’ MLL − full completenesstheorem. One of the remarkable properties of the categories in which we are interested isthat the earlier MDNF full completeness proof can be extended using one lemma and acouple of supplementary algorithms so that it works for all MLL − functors, giving the fullcompleteness theorem originally proposed.As stated in Chapter 2.2, in every ∗ -autonomous category D there are natural transfor-mations w LL = ( w LLA , B , C : A ⊗ ( B M C ) → ( A ⊗ B ) M C ) A , B , C ∈ D w LR = ( w LRA , B , C : A ⊗ ( B M C ) → ( A ⊗ C ) M B ) A , B , C ∈ D w RL = ( w RLA , B , C : ( A M B ) ⊗ C → B M ( A ⊗ C )) A , B , C ∈ D w RR = ( w RRA , B , C : ( A M B ) ⊗ C → A M ( B ⊗ C )) A , B , C ∈ D , and these are canonically isomorphic to appropriate compositions of the associativity andsymmetry isomorphisms α and σ if D happens to be compact closed, making them bijec-tions (no two distinct MLL − transformations compose with the same weak distributivitytransformation to give the same result). The tensor representations of these natural isomor-phisms are both described solely by Kronecker deltas as demonstrated in Section 3.2. Dueto the small diagrammatic argument on page 5, an MLL − transformation τ in C may becomposed with a sequence of natural transformations built from the weak distributivitytransformations and the associativity and symmetry isomorphisms of C to produce anMDNF transformation τ ′ . Furthermore, τ describes an MLL − transformation in G C only if τ ′ does. Proposition 4.20.
Let τ : K I ✲ F be an MLL − transformation in C , and let ¯ w : F ✲ G bea natural transformation built from the weak distributivity, associativity and symmetry naturaltransformations of C . Then τ does not exist in G C if ¯ w ◦ τ does not.Proof. The transformation ¯ w is in G C , and so we know that if τ is dinatural in the gluedcategory then so does ¯ w ◦ τ , since both are well-defined in G C . The statement of theproposition is the contrapositive of this fact.It is always possible to find a natural transformations which preserves the cyclicity ofat least one modelled cyclic proof structure between MLL − transformations. As such it isalso possible to find a composition of natural transformations which not only preserves theexistence of a cyclic proof structure modelled but ensures at least one of these cycles neverpasses through a M -vertex in its switching . Algorithm 4.21.
Input: An MLL − transformation τ : K I −→ F describing a linear combi-nation of proof structures over a common sequent with at least one structure breaking theacyclicity criterion.Output: An MLL − transformation τ ′ describing a linear combination of proof structuresover a common sequent with at least one structure containing a switching cycle passingthrough no M -vertices. In fact, if the cycle does not pass through a M -vertex the cycle exists in all switchings of the proof structure ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 53 (1) Select one of the proof structures associated with τ which fails the acyclicity criterion.Choose a cycle of minimum length from one of its switchings. We let τ = τ and F = F ,and we name the cycle C . Let n =
0, and whenever we refer to subformulae called X , Y and Z , we name their lowest vertices as in a parse tree x , y and z respectively.(2) Search for a position in F n of the form X ⊗ ( Y M Z ) for some subformulae X , Y and Z ,the corresponding edge ⊗ − M of which is in the cycle C n .(a) If one should exist, and C n passes through y , then let τ n + = (cid:0) · · · w LL · · · (cid:1) ◦ τ n . The functor F n + is defined to be the target functor of τ n + , which is the same as F n except that the subformula X ⊗ ( Y M Z ) is replaced by ( X ⊗ Y ) M Z . The cycle C n + passes through the same vertices as those in C n in the identical sections of F n + and F n . If C n contains the path x − ⊗ − M − y , then the cycle C n + is completed by adding x − ⊗ − y ; otherwise, M − ⊗ − y is used to connect the ends together. Increment n and restart Step 2.(b) If one should exist, but C n passes through z instead, let τ n + = (cid:0) · · · w LR · · · (cid:1) ◦ τ n . The functor F n + is defined similarly to before: X ⊗ ( Y M Z ) is replaced by ( X ⊗ Z ) M Y .The cycle C n + passes through the same vertices as those in C n in the identicalsections of F n + and F n . If C n contains x − ⊗ − M − z , then the cycle C n + is completedby adding the path x − ⊗ − z ; otherwise, M − ⊗ − z . Increment n and restart Step 2.(c) If one does not exist, then move to Step 3.(3) Search for a position in F n of the form ( X M Y ) ⊗ Z for some subformulae X , Y and Z ,the corresponding edge M − ⊗ of which is in the cycle C n .(a) If one should exist, and C n passes through x , then let τ n + = (cid:0) · · · w RL · · · (cid:1) ◦ τ n . The functor F n + is defined to be the target functor of τ n + , which is the same as F n except that the subformula ( X M Y ) ⊗ Z is replaced by Y M ( X ⊗ Z ). The cycle C n + passes through the same vertices as those in C n in the identical sections of F n + and F n . If C n contains the path x − M − ⊗ − z , then the cycle C n + is completed by adding x − ⊗ − z ; otherwise, x − ⊗ − M is used to connect the ends together. Increment n and restart Step 2.(b) If one should exist, but C n passes through y instead, let τ n + = (cid:0) · · · w RR · · · (cid:1) ◦ τ n . The functor F n + is defined similarly to before: ( X M Y ) ⊗ Z is replaced by X M ( Y ⊗ Z ).The cycle C n + passes through the same vertices as those in C n in the identicalsections of F n + and F n . If C n contains y − ⊗ − M − z , then the cycle C n + is completedby adding the path y − ⊗ − z ; otherwise, M − ⊗ − z . Increment n and go back to Step2.(c) If one does not exist, then define τ ′ = τ n , F ′ = F n and C ′ = C n Intuitively it is simple to see why this algorithm works. Suppose, without loss of generality,that one of the scenarios expected for Step 2(a) unfolds (dual arguments can be given foreach of the other four situations). We can think of the proof structure with the cycle C n forsome n taking the form of one of the left-hand diagrams in Figures 7.1 and 7.2 below. Figure 1: The e ff ect of Algorithm 4.21 X Y Zx y z M ⊗ X Y Zx y z ⊗ M Figure 2: The e ff ect of Algorithm 4.21 X Y Zx y z M ⊗ X Y Zx y z ⊗ M The diagrams highlight the two possible key paths from C n in F n (which also happento the the edges which still exist in one of the switchings causing the cycle). • In the first situation the path is composed of the following: the switching path linkingthe subformulae X and Y , which may pass through any number of other subgraphs of aswitching; the edge from x to the ⊗ -vertex; the switch edge from the M -vertex to y ; andthe edge between the ⊗ - and M -vertices denoting the synonymous connectives writtenexplicitly in X ⊗ ( Y M Z ). It is trivially true that there must be a path from the literalsconnected to the highlighted axiom link through their respective subformulae X and Y to x and y .The natural transformation w LL converts that proof structure from Figure 7.1 and thecycle described within it to the one to its right. The weak distributivity transformationdoes not a ff ect which literals are connected to one another via an axiom link, and alsodoes not a ff ect any subformulae of F n except X ⊗ ( Y M Z ). The only change is a ‘shu ffl ing’of the ⊗ - and M -vertices in the primary subformula being considered. The cycle itselfcan be thought of as having been reduced by the algorithm so it contains all the samevertices except possibly for the M -vertex, which may have been removed. • In the second scenario we have a slightly longer path: the switching path linking thelowest ⊗ -vertex in the subgraph back to Y at the top, which may pass through anynumber of axiom links; the switch edge from the M -vertex to y ; and the edge between the ⊗ - and M -vertices denoting the synonymous connectives written explicitly in X ⊗ ( Y M Z ).The natural transformation w LL converts the left proof structure of Figure 7.2 to theone on its right, with the cycle highlighted in the second diagram still existing. In thissituation the length of the cycle has not been reduced, but a ⊗ -vertex has e ff ectively been ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 55 moved further up the proof structure, which creates new possibilities for scenarios suchas that seen in the previous point to be found.The lowest vertices in a cycle in the switching of a proof structure must be ⊗ -vertices, withthe cycle flowing through both of its argument edges. Whenever we meet such a situationas seen in the first case, the length of the cycle is reduced by one: a M -vertex is removed.In the second case the length of the cycle is not changed, but the alteration in the form ofthe cycle in e ff ect pushes the M -vertex further down the structure until it inevitably meetsa ⊗ -vertex of maximal depth, at which point the vertex is removed.Since the other four steps 2(b), 3(a) and 3(b) of Algorithm 4.21 are identical in conceptto Step 2(a) they also steadily remove these vertices. Since the algorithm stops once thereare no M -vertices in the cycle, and eventually in the worst case all M -vertices will findthemselves under the ⊗ -vertices, this principle assures termination.It is also always possible to produce an MDNF transformation from an MLL − transfor-mation using an algorithm such as the one below. Algorithm 4.22.
Input: An MLL − transformation τ : K I −→ F describing a linear combina-tion of proof structures over a common sequent.Output: An MDNF transformation τ ′ describing a linear combination of proof structuresover a common sequent.An algorithm which takes an MLL transformation describing a linear combinationof proof structures over a common sequent, and produces another MLL transformationrepresenting a linear combination of proof structures, only over an MDNF sequent. Thenew sequent will preserve the number of tensor and par operators within the formulae.(1) Let τ = τ : K I −→ F and F = F . Let n =
0, and whenever we refer to subformulaecalled X , Y and Z , we name their lowest vertices as in a parse tree x , y and z respectively.(2) Search for any position in F n of the form X ⊗ ( Y M Z ) for some subformulae X , Y and Z .(a) If one should exist, then let τ n + = (cid:0) · · · w LL · · · (cid:1) ◦ τ n . The functor F n + is definedto be the target functor of τ n + . Increment n and go back to Step 2.(b) If one does not exist, then move to Step 3.(3) Search for any position in F n of the form ( X M Y ) ⊗ Z for some subformulae X , Y and Z .(a) If one should exist, then let τ n + = (cid:0) · · · w RR · · · (cid:1) ◦ τ n . The functor F n + is definedto be the target functor of τ n + . Increment n and go back to Step 2.(b) If one does not exist, then let ˆ τ = τ n , ˆ F = F n and ˆ C = C n .Algorithm 4.22 is far simpler than Algorithm 4.21, because there is no need to considerthe preservation of anything: we are merely composing an MLL − transformation with anumber of natural transformations which lead to the creation of an MDNF transformation.However, it is useful to make the following observation. Observation 4.23.
Let τ : K I −→ F be an MLL − transformation modelling a linear combi-nation of proof structures over a sequent, at least one of which fails the acyclicity criterion.Then applying Algorithms 4.21 and 4.22 in that order produces an MDNF transformation τ ′′ modelling a linear combination of proof structures, with at least one being cyclic.It is now possible to prove MLL − full completeness for G C . Theorem 4.24. If C is a compact closed category with finite biproducts satisfying feeble fullcompleteness then G C satisfies MLL − full completeness. Proof.
Let τ be an MLL − transformation in C modelling a linear combination of proofstructures, at least one of which is cyclic. Using Algorithms 4.21 and 4.22, we obtain a familyof arrows τ ′′ modelling a linear combination of MDNF proof structures, with at least one ofthese containing a switching cycle. By Proposition 4.19, τ ′′ is not an MDNF transformationin G C ; and therefore it immediately follows from Proposition 4.20 that τ is not either.If we instead assume that τ models a linear combination of acyclic yet disconnectedMLL − proof structures, Algorithm 4.22 produces an MDNF transformation τ ′ when intro-duced to τ , and this new transformation is also a linear combination of acyclic, disconnectedproof structures. Proposition 4.19 once again allows us to say that τ ′ —and consequently τ —does not exist in G C . The same principle can be used for the case where τ is a non-simple linear combination of proof nets, and so no MLL − transformations not representinga unique proof net is found in G C .We can in fact improve on the theorem above greatly with near enough no e ff ortby making a simple observation. For any choice of category C , the only objects usedin the lemmata in this section are in the sets { A n : n ∈ N + } and { B n : n ∈ N + } ; and allof these are found in the collection of objects for any orthagonality category G E C withfocus S ⊇ = { I , ⊥ , ι } . The values and covalues of a G E C -object Z also describe the set ofmorphisms between the category’s tensor unit I S = ( I , { I } , S ) and Z and between Z and I ⊥ S respectively (Fact 2.10). Because of this, the arguments from previous chapters and earlierin this chapter using the properties of the values and covalues may be replicated verbatimto give an MLL − full completeness theorem for all categories of this more restricted form. Theorem 4.25. If C is a compact closed category with finite biproducts satisfying feeble fullcompleteness then G E C satisfies MLL − full completeness for any S ⊇ { I , ⊥ , ι } . Corollary 4.26.
For a tensor-generated compact closed category with biproducts C the category G C satisfies MLL − full completeness. Note that biproducts are necessary for this result despite the fact that we are notaddressing the additive connectives here. Indeed, it is very easy to produce a (tensor-generated) compact closed category without biproducts which does not create a fullycomplete model under the double glueing construction (the full subcategory of
FDVec F containing only the tensor unit R for example).5. MLL − + M ix F ull C ompleteness for G C The theorem given in the previous section has the e ff ect of generalising the primary resultsfound in first four chapters of [Tan97], which concern the categories GRel and
GFDVec F forarbitrary field F of characteristic 0. However, the manner in which Tan proves her theoremfor GRel actually has a positive side e ff ect that the proof above is unable to replicate. Thelemmata proving acyclicity and uniqueness are not stated with an assumption of an alreadyderived connectedness lemma; and the objects used in the proofs of the acyclicity anduniqueness are found in the collection of objects in the orthogonality subcategory G Rel with focus the singleton { ι } for ι = λ I ∗ ◦ v I , I , I ( ρ I ), which is defined earlier in Section 3.2.The category G Rel is known to be a categorical model of MLL − + Mix, and every dinaturaltransformation in G Rel is inherited from
Rel . All these facts together allow us to concludethe following:
ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 57
Fact 5.1. [Tan97] The category G Rel satisfies MLL − + Mix full completeness .This result can in fact also be generalised, though not the same extent as that seen inSection 4. In this section we demonstrate that the category G C satisfies MLL − + Mix fullcompleteness if C is, like Rel , a zero-sum-free compact closed category with biproducts.
Definition 5.2.
A semiring S is zero-sum-free if its additive unit 0 is its sole element with anadditive inverse. A compact closed category with finite biproducts is zero-sum-free if it isenriched over a category SMod S for which the S is zero-sum-free.Though the use of counterexample test objects to show certain MLL − transformationsfound in C cannot be seen in the glued category is still abundantly present, the flavour ofthe coming proof resembles the deduction of Tan more closely than the earlier proofs:( § − transformations which are found in both C and G C must model simple linear combinations of proof structures. This is done byusing reducing the problem to one for MLL − functors built using no tensor functors(Proposition 5.3). By necessity the proof of uniqueness (Proposition 5.6) takes aslightly less combinatorial shape than before and has a more algebraic feel, with itsbasis being simultaneous equations.( § − + Mix proof netcriterion. Algorithm 5.8 reduces the target to the most simple standard form, leavinga comparartively simple argument for Proposition 5.10. The theorem immediatelyfollows.In spite of the small change in proof style, the index notation for values and covalues ofthe objects of the form ( n I , U , X ) that has been employed greatly in this chapter continuesto be utilised ubiquitously.5.1. Uniqueness.
In this full subcategory of G C in which we now reside, we have restric-tions on the sets of values and covalues which we may choose for our test objects to provecertain transformations from the underlying category do not become transformations inour new location. Every value must compose with every covalue to produce the morphism ι : I ✲ ⊥ and vice versa. This is remarkably restrictive, primarily because it becomesharder to find valid test objects which are self-dual and still useful, such as the objects A n and B n described in Section 4.1 in G C . However, this is counterbalanced by the factthat categorical models of MLL − + Mix are equipped with the ‘Mix’ natural transformation
Mix : − ⊗ − ✲ − M − , which permits great simplifications to the set of MLL − formu-lae which must be considered to produce the desired results. The proof of the simplicityof all linear combinations of proof structures modelled in G C demonstrates both thesechanges to the playing field in the clearest manner.Proposition 4.20 in the previous section allowed us to simplify the entire of theMLL − full completeness proof of G C to one for MDNF full completeness. Now usingthe mix natural transformation, we can produce a more extreme version of this propositionwhose proof follows from an identical concept, which is possible due to Mix being anisomorphism in compact closed categories. The category in question was actually named S in [Tan97]. Proposition 5.3.
Let τ : K I ✲ F be an MLL − transformation in C , and let ¯ w : F ✲ G bea natural transformation built from the ‘Mix’, weak distributivity, associativity and symmetrynatural transformations of C . Then τ does not translate into G C if ¯ w ◦ τ does not. For the uniqueness proof the actual connectives that are in each formula being modelledbecome irrelevant—only the number of linkings is important. Because of this, a naturaltransformation eradicating all uses of the tensor product ‘ ⊗ ’, leaving a formula of the form M Mm = ( L φ ( m ) M L ⊥ φ ( m ) ), is a reasonable suggestion for ¯ w in the above proposition. Corollary 5.4.
Every MLL − + Mix transformation in G C models a single proof structure if andonly if every MLL − + Mix transformation to a functor leading to objects of the form
Par M ( L , L ) = M Mm = ( L φ ( m ) M L ⊥ φ ( m ) ) for some φ : [ M ] −→ [ N ] does. We now follow the same procedure of choosing a single family of objects in G C , { S n : n ∈ N + } , where US n = n I for each n , so that the set of values of the object Par M ( S n , S n )does not contain the tensor representation of an MLL − transformation modelling a non-simple linear combination of axiom links. We define each of the S n as follows: S n : = ( n I , { δ ix : x ∈ [ n ] } , { i } ) , where 1 i is the tensor with entries equalling 1 for all i . The values and covalues of Par M can be found from this definition for any M easily. Lemma 5.5.
For each M ∈ N + , • Par M ( S n , S n ) Val = n z j ··· j M i ··· i M : ∀ ( x , . . . , x M ) ∈ [ n ] M ∀ k ∈ [ M ] z ij · Q m , k ( δ x m i m j m ) ∈ Perm (2 , n ) o • Par M ( S n , S n ) CoVal = nQ Mm = δ x m i m j m : ( x , . . . , x M ) ∈ [ n ] M o Proof.
We follow a similar argument to that seen in Lemma 4.1, which is based aroundinduction. We start by considering the covalues. The proof that the covalues of
Par ( S n , S n )is near enough trivial. Par ( S n , S n ) CoVal = ( S n M S ⊥ n ) CoVal = { y i · v j : y i ∈ ( S n ) CoVal , ( S ⊥ n ) CoVal } = { δ xi j : x ∈ [ n ] } It now takes minimal e ff ort to show that the result desired is true for Par M + ( S n , S n ) = Par M ( S n , S n ) M Par ( S n , S n )whilst assuming the claim is true for Par K ( S n , S n ) for all K ≤ M . Par M + ( S n , S n ) CoVal = ( Par M ( S n , S n ) M Par ( S n , S n )) CoVal = { t ji · u j M + i M + : t ji ∈ Par M ( S n , S n ) CoVal , u j M + i M + ∈ Par ( S n , S n ) CoVal } = n(cid:16)Q Mm = δ x m i m j m (cid:17) · δ x m i M + j m : ( x , . . . , x M ) ∈ [ n ] M , x M + ∈ [ n ] o = nQ M + m = δ x m i m j m : ( x , . . . , x M + ) ∈ [ n ] M + o ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 59
Now we deal with the values of these object amalgamations. The base case shows usthat the values of
Par ( S n , S n ) are the permutations over [ n ]. Par ( S n , S n ) Val = ( S n M S ⊥ n ) Val = { z ji : ∀ y i ∈ ( S n ) CoVal z ji y i ∈ ( S ⊥ n ) Val , ∀ v j ∈ ( S ⊥ n ) CoVal z ji v j ∈ ( S ⊥ n ) Val } = { z ji : ∃ y ∈ [ n ] z ji i = δ jy , ∀ y ∈ [ n ] z ji δ jy = z yi = i } = Perm (2 , n )The inductive case follows. Par M + ( S n , S n ) Val = ( Par M ( S n , S n ) M Par ( S n , S n )) Val = { z j j M + i i M + : ∀ w ji ∈ Par M ( S n , S n ) CoVal z j j M + i i M + w ji ∈ Par ( S n , S n ) Val , ∀ u j M + i M + ∈ Par ( S n , S n ) Val z j j M + i i M + u j M + i M + ∈ Par M ( S n , S n ) Val } = { z j j M + i i M + : ∀ ( x , . . . , x M ) ∈ [ n ] M z j j M + i i M + · Q Mm = δ x m i m ∈ Perm (2 , n ) , ∀ x M + ∈ [ n ] z j j M + i i M + δ m M + i M + j M + ∈ Par M ( S n , S n ) Val } = { z j j M + i i M + : ∀ x ∈ [ n ] M z j j M + i i M + δ xi ∈ Perm (2 , n ) , ∀ x M + ∈ [ n ] z j j M + i i M + δ m M + i M + j M + ∈ { q ji : ∀ x ∈ [ n ] M ∀ k ∈ [ M ] q ij · Q m , k ( δ x m i m j m ) ∈ Perm (2 , n ) }} = { z j ··· j M + i ··· i M + : ∀ ( x , . . . , x M + ) ∈ [ n ] M + ∀ k ∈ [ M + z ij · Q m , k ( δ x m i m j m ) ∈ Perm (2 , n ) } Having access to the form of the values of
Par M ( S n , S n ) for each value of M and n makes itsimple enough to prove uniqueness for G C as wished. Proposition 5.6.
Every MLL − transformation in G C for zero-sum-free C models a unique proofstructure.Proof. We show that transformations are as desired for the MLL − functor Par M ; Corollary5.4 forces the result to work for all MLL − functors. At this point we notice that Z = ( I , { } , { } )utilised primarily in Section 4.2 is actually S , and so is an object in G C . We can thereforeuse Lemma 4.2 to say that the sum of every scalar involved in a linear combination of proofstructures modelled in the category totals 1.The formula Par ( L ) is equal to L M L ⊥ , meaning only one valid set of axiom links canbe placed on its formula, namely the linking containing only one axiom link connecting theonly two literals. Since every MLL − transformation in the category must model a linearcombination of proof structures, this means that a transformation to Par has to model ascalar multiple of this one axiom link, and therefore have a tensor representation s · δ ij forsome s ∈ C [ I , I ]. Lemma 5.5 tells us that Par ( S n , S n ) Val = Perm (2 , n ) for all n ∈ N + , whichonly contains s · δ ij if s = I . The conclusion of this is that non-identity scalar multiples ofthe axiom link are not modelled by a transformation in the glued category. Now we consider the more general case of when M ≥
2. It is possible to follow theargument through for any n ≥ M , so for the sake of simplicity we take Par M ( S M ). Lemma 5.5states that the tensors in Par M ( S M ) Val are those which create full permutations when any M − M i - j -index pairs are composed with matrices of the form δ xi j for each x . Thisleads to many tensor equations, but due to the symmetry and addition properties notedin Section 3.2 we may choose to restrict the equations somewhat without loss of generality.We only consider equations for which each δ xi k composing with the value tensors are suchthat x = k . This leaves M simultaneous equations to be solved over the semiring C [ I , I ]: foreach k ∈ [ M ] there must be a vector of values y ∈ [ n ] M such that ω j ··· j k ··· j M ··· k ··· M · Y m , k j m ∈ δ j k y k . Again simplifying using the ideas from Section 3.2, we can look solely at the entries wherethe one remaining i -indexed position, the k th position, also considered at k . This means thatthe only entries of ω which are now of interest are of the form P p ∈ S M c p · δ p (1) ··· p ( M ) j ··· j M , whereeach p : [ M ] −→ [ M ] is a permutation and c p a corresponding scalar. Entries where the j -indices do not form a permutation of [ M ] are therefore always zero.For some set of y ∈ [ n ] M , X p c p · δ p (1) ··· p ( k ) ··· p ( M ) j ··· j k ··· j M · Y m , k j m = δ j k y k This leads to M linear equations of consequence, all bar M of which being sums whichhave to equal 0. In the context of a zero-sum-free semiring, as C [ I , I ] is from our originalassumption, each of the values in the sums of these ( M − M ) equations must equal 0.Let f : [ M ] −→ [ M ] be the function which describes the vector y , i.e. for every k , f ( k ) = y k . Focusing on an arbitrary k , we find that X p ∈ S M c p · δ p (1) ··· p ( M ) j ··· j M · Y m , j m = j k , y k . Therefore, by zero-sum-freeness, for every permutation p whenever j k , y k , c p · δ p (1) ··· p ( M ) j ··· j M = . By symmetry, we find that for all p ∈ S M , c p · δ p (1) ··· p ( M ) j ··· j M = j k = y k for all k ∈ [1 , M ], which forces us to conclude that ω j ··· j k ··· j M ··· k ··· M = c · δ y ··· y M j ··· j M for some c ∈ C [ I , I ].By the form of ω , we already know then that f ( x ) = y k must form a permutation, andthat ω j j M i i M = c p · δ i f (1) ··· i f ( M ) j ··· j M . The sum of the scalars must be unity, so c =
1; and so the arrow which ω describes is aninstantiation of an MLL − transformation which represents a unique proof structure. ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 61
The proof above can be elucidated somewhat through the use of an example.
Example 5.7.
We view the proof of Lemma 5.6 for the MLL − formula Par ( S , S ) in moredetail.We assume that ω j j j is the tensor representation of a linear combination of linkingson the formula modelled by Par . Lemma 5.5 is provides us with the three tensor equationswhich must be satisfied for some numbers y , y and y found in { , , } . ω j j j · j j = δ j y ω j j j · j j = δ j y ω j j j · j j = δ j y Suppose that y k = k for each k , making each constant distinct from one another and thefunction f proposed in the proof of the earlier lemma a permutation. This creates 3 = ω + ω = ω + ω = ω + ω = ω + ω = ω + ω = ω + ω = ω + ω = ω + ω = ω + ω = ω = ω = ω = ω = ω = . Substituting these assignments into the 9 equations leaves the six sums totalling zero astrivial, and equations (1), (5) and (9) simplify to become ω = y , y and y coincide with the order of the three superscriptindex numbers of ω . This is to be expected, since the y k = k makes it necessary for the k th superscript index position to be k in order to for it to be in the equation for which the other( M − j -indices are changed in are added together and equate to 1. As such, the givenposition is the only one which fits this criteria for all k .Due to ω being a linear combination of Kronecker deltas representing axiom linksbetween positive and negative literals, the only MLL − transformation which can be said tosatisfy the criteria given here is the one modelling the unique proof structure given below. L M L ⊥ M L M L ⊥ M L M L ⊥ Symmetry in permutations allows us to ascertain that all the possible linkings are possibleby changing the permutation that f describes to each of the other 3! − =
5, but nonon-simple linear combinations have been shown to be possible.Now, we choose values of y , y , y so that the induced function f is not a permutation.Let us first consider y k = k ; these induce the following linear equations: ω + ω = ω + ω = ω + ω = ω + ω = ω + ω = ω + ω = ω + ω = ω + ω = ω + ω = ω j j j that can be non-zero is the one where j = y . Bearing this in mind, we can expect no valuations of the entries of interest satisfy theabove equations. Hypothetically, only ω would be allowed to be non-zero; but by theconstraints set upon it it is 0 by definition. By similar arguments, no tensor representationof an MLL − transformation ω satisfies the equations when two or more of y , y , y areequal.For a tensor representation of an MLL − transformation to be in the values, it mustsatisfy the 9 equations for at least one of the combinations of values y , y and y can take.From the evidence above we know that such an ω j j j i i i can only do that if only one tuple( j , j , j ) provides a non-zero value for ω j j j , and that position must equal zero. The onlyconclusion is, therefore, that ω j j j = ω j p (1) j p (2) j p (3) i i i for some permutation p over 3. That is,no non-simple linear combinations of proof structures are modelled by a tensor in Par ( S ).5.2. Acyclicity.
Now that is has been established that G C only contains MLL − transfor-mations modelling unique proof structures, we can now attempt to show that those proofstructures modelled are also acyclic. Once this has been shown, we will have provedMLL − + Mix full completeness: the connectedness condition of the Danos-Regnier criteriais rendered redundant once the ‘Mix’ rule has been installed in the logic.Simplifying the problem down to dealing solely with the MLL − transformations Par M for di ff erent values of M is not an option here. If we wish to prove that a cyclic structurecannot be modelled in the glued category, we need to make sure that any natural transfor-mations do not create a transformation modelling an acyclic structure. As such, at leastone of the ⊗ -vertices must be preserved. However, ignoring the existence of Mix would bea waste of the available attributes of the category.
ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 63
The methods utilised in Section 4 to allow us to only consider the easier-to-handleMDNF transformations are once again of use here. There is an algorithm given belowwhich takes MDNF transformations modelling unique cyclic proof structures and producesother MDNF transformations modelling a formula containing the minimum number of ⊗ -vertices possible whilst preserving exactly one cycle. Algorithm 5.8.
Input: An MDNF transformation τ : K I −→ F describing a unique proofstructure, with M blocks say, breaking the acyclicity criterion.Output: An MDNF transformation τ ′ : K I −→ F ′ built by composing τ with natural trans-formations which describes a unique proof structure still breaking the acyclicity criterionwith the minimum number of ⊗ -vertices within it.(1) Choose a cycle from the proof structure described by τ passing through the minimumnumber of axiom links. We let n = l = m = τ = τ , and we name the cycle C (referring only to the literals). For every n , F n is the target functor of τ n (2) Search for the block furthest to the left in the formula modelled with literals within C , calling it block m . Let ˜ σ be a composition of the associativity and symmetryisomorphisms for the monoidal bifunctor − ⊗ − which modifies the block so it takes theform (( X ⊗ Y ) ⊗ Z ), where X and Y are the positions signifying the literals in C and Z is a tensor product containing the rest of the literals. Define τ n + = M m − m = B m M ˜ σ MM Mm = m + B m , where B m is the m th block. If X and Y are connected via an axiom link goto Step 4; otherwise go to Step 3.(3) Increment l . Search for the block (which we name block m l ) which contains the literal, X l , connected to literal Y l − via an axiom link. Let ˜ σ be a composition of associativityand symmetry isomorphisms which modifies the block so it takes the form (( X l ⊗ Y l ) ⊗ Z ),where Y l is the other literal position in the block which is in C . Define τ n + = M m l − m = B m M ˜ σ M M Mm = m l + B m . If Y l is connected via an axiom link to X then go to Step 4; otherwiserepeat Step 3.(4) Consider block m , B m = L , m ⊗ L , m ⊗ L m where L , m and L , m are the functors representingthe first and second literal positions. We set the natural transformation MIX F themaximal composition of ‘mix’ transformations on an MLL − functor F which eradicatesall instances of ⊗ and replaces them with M whilst maintaining the order and bracketingof the inputs of the functor.(a) If it does not contain a literal in C , then let µ m = MIX B m .(b) If it does, then let µ m = Mix ◦ ((1 G C ⊗ G C ) ⊗ MIX L m )If m < M , then increment the number, and repeat Step 4. Otherwise let τ n + = M Mm = µ m ◦ τ n , increment n and go to Step 5.(5) Let ˜ β be a natural isomorphism built using the associativity and symmetry isomor-phisms corresponding to the bifunctor − M − so that all the blocks in the proof structuredescribed by τ n containing only one literal are to the left of all those containing twoliterals, and the two-literal blocks maintain the same order with respect to each otheras found in τ n . We let τ ′ = ˜ β ◦ τ n , and terminate the algorithm.The algorithm above certainly terminates—after all, each step changing an MLL − trans-formation can only deal with a finite set of finite blocks. Equally, the output dinaturaltransformation does indeed describe a cyclic proof structure: in every switching there isa cycle X − ⊗ − Y − · · · − X l − ⊗ − Y l , and the ⊗ -vertices in the cycle are the only oneswhich appear in the structure. Together with Algorithms 4.21 and 4.22, we can find natural transformations from C which continue to exist in G C which take any MLL − transforma-tion describing a unique cyclic proof structure in C and create another one which modelsa proof structure with shape much like the proof structure given below. L M L ⊥ M · · · M L M L ⊥ M ( L ⊗ L ⊥ ) M · · · M ( L ⊗ L ⊥ )This is a very regular form of MLL − formula being modelled. The formulae can be separatedinto two distinct parts. If we instantiate the functor with tuples containing the same self-dual object built over a C -object n I for some n ∈ N + in every entry, D say, we can saythe tensor representations of the values and covalues in the resulting object take the form F ( D , D ) = Γ g ( D ) M ∆ d ( D ) for the MDNF functors Γ g and ∆ d for some g ∈ N , d ∈ N + , where Γ g ( D , D ) = D M g ∆ d ( D , D ) = ( D ⊗ ) M d As has been the standard throughout this chapter, a single test object is required forall inputs of the total resulting MLL − transformation to show that acyclic proof structuresare not modelled in G C . An object which is su ffi cient for the task whilst remaining in theconfines of G C is D : = (2 I , { δ ix : x ∈ { }} , { δ ix : x ∈ { }} ) . The values and covalues of objects F ( D , D ) = Γ g ( D , D ) M ∆ d ( D , D ), where the N -dimensional tuple is D = ( D , . . . , D ) ∈ ( G C ) N and F is the target functor of an MDNF trans-formation which is an output of Algorithm 5.8 can be found easily by using inductivemethods, as seen in the claim below. Lemma 5.9.
For every n ≥ , Γ n ( D , D ) = ((2 I ) M n , Ξ n , { δ ··· ··· i ··· i n j ··· j n } ) , where Ξ n = { w i ··· i n : w ··· = ∀ x containing ( n −
1) 1 -entries , w x = } . Similarly, ∆ n ( D , D ) = (((2 I ) ⊗ ) M n , { z ji : ∀ k ∀ m , k z ji w i m j m ∈ Ξ , z ji · Y m , k w i m j m = δ i j } , n Y m = Ξ ) .Proof. The object Γ n ( D , D ) = D M n , and so we show that D M n has the form of Γ n ( D ) is as seenabove, only with 2 n replaced by n , starting with n = D M D = (2 I M I , { w ij : w j = δ j , w i = δ i } , { u i v j : u i , v j ∈ D CoVal } ) = ((2 I ) M , Ξ , { δ i j } )The inductive case is proved as follows: D M ( n + = D M n M D = ((2 I ) M n M (2 I ) , { w i ··· i n j : w i ∈ Ξ n , w ··· j = δ j } , { δ ··· i ··· i n · δ j } ) = ((2 I ) M n + , Ξ n + , { δ ··· i ··· i n j } )The result is exactly as desired, and so the claim concerning Γ n ( D , D ) is solved by corollary.We now move to demonstrate ∆ n ( D , D ) takes the form wanted. We can conclude thefollowing due to the functor − ⊗ − being the de Morgan dual of the functor − M − : ∆ ( D , D ) = (2 I ⊗ I , { δ i j } , Ξ )The final result immediately follows. ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 65
Proposition 5.3, in collaboration with Algorithms 4.21 and 5.8, means the followingstatement is true: if we can show that MLL − transformations in the form seen in the abovefigure are not capable of existing in G C , then no cyclic proof structures may be modelledin the category C under the orthogonalised glueing. Proposition 5.10.
Every MLL − transformation in G C modelling a unique proof structure modelsan acyclic proof structure.Proof. Due to the earlier discussion it is only necessary to consider MLL − transformationsof the shape seen in Figure 5.2 due to Algorithms 4.21 and 5.8 and Proposition 5.3. Supposethat F = Γ M ∆ be such a functor, with Γ containing all the one-literal blocks and ∆ all thetwo-literal blocks. We use the test object D = (2 I , { δ ix } , { δ ix } ), meaning that we are able touse the calculations given in Lemma 5.9 to find the values of F ( D , D ) = Γ g ( D , D ) M ∆ d ( D , D )for some g ∈ N , d ∈ N + .Suppose that Γ is a functor describing a subformula containing no literals, making F ( D , D ) = ∆ d ( D , D ) for a positive integer d . The values of this object are therefore exactlyas seen in Lemma 5.9. To simplify matters, instead of making i - and j -indices correspondto positive and negative literal positions respectively as earlier, the indices i m and j m nowrelate to the first and second literals in the m th block in ∆ d ( D ) for all m ∈ [ d ]. We cantherefore assume that the proof structure being discussed, which we say has linking λ , hastensor representation ¯ λ j ··· j d i ··· i d = δ j d j ··· j d − i i ··· i d .For d =
1, the situation is clear: ∆ ( D , D ) Val = { δ i j } , which clearly does not contain δ i j as required for the cyclic proof structure to be modelled. For larger d , we need to usecovalues from D ⊗ D .( D ⊗ D ⊥ ) CoVal = Ξ = { w ij : w = , w = = w } For every block m ∈ [2 , d ], we use δ j m i m , which indeed does belong to the set of covalues. Wefind that ¯ λ j ··· j d i ··· i d · d Y m = δ j m i m = δ j d j ··· j d − i i ··· i d · δ j ··· j d i ··· i d = δ j i The tensor δ j i does not equal δ j i , and so it does not belong to the set of values for D ⊗ D .As such, ¯ λ does not satisfy the criteria expected of all elements of the set of values for theobject ∆ d ( D , D ). Because of this, the MDNF transformation to F does not lift to G C .We extend this result so that it works for an non-empty Γ . In this scenario we expect F ( D , D ) = Γ g ( D , D ) M ∆ d ( D , D ) for positive g and d , and by standard tensor calculations wenote that, when k and l act as superindices for the positive and negative literals respectively, F ( D , D ) Val = { z jki : ∀ γ k ∈ Γ g ( D , D ) Coval z ljki γ k ∈ ∆ d ( D , D ) , ∀ w ji ∈ ∆ d ( D , D ) CoVal z ljki w ji ∈ Γ g ( D , D ) Val } Lemma 5.9 tells us that there is only covalue for Γ g ( D , D ), namely δ ··· k ··· l g . The tensorrepresentation of the transformation is¯ λ l jki = δ l ··· l g j d j ··· j d − k ··· k g i i ··· i d Composing the two tensors presented above together, we find¯ λ l jki · δ ··· k ··· l g = δ l ··· l g j d j ··· j d − k ··· k g i i ··· i d · g Y m = δ k m l m = δ ··· j d j ··· j d − ··· i i ··· i d = δ j d j ··· j d − i i ··· i d We know from the base case when g = ∆ d ( D , D ), irrespective of the size of d . Wetherefore conclude that ¯ λ is not found in the values of F ( D , D ); and so its correspondingMDNF transformation from C does not lift into G C . Theorem 5.11.
Let C be a zero-sum-free compact closed category with finite biproducts satisfyingfeeble full completeness. Then the category G C satisfies MLL − + Mix full completeness.
The Necessity of Zero-Sum-Freeness.
The satisfaction of full completeness for thelogic MLL − + Mix by degenerate categorical models under the orthogonalised glueing con-struction with focus { ι } is quite a strong result, but the form of the scalars required for theproof to operate is an unfortunate stumbling block. The proof given earlier in this sectionis designed to show that a compact closed category with finite biproducts satisfying feeblefull completeness creates a full complete model of MLL − + Mix when the ‘ G ’-glueing isused if the homset C [ I , I ] acts as a zero-sum-free semiring.We now make clear the insurmountable hurdle which stops Tan’s method from beingfurther generalised. Lemma 5.6 is insu ffi cient for compact closed categories enriched oversemimodules over semirings containing even a single additive inverse. This is confirmedby the algorithm below. Example 5.12.
We view the limitations of the proof of Lemma 5.6 by viewing the MLL − for-mula Par ( S , S ) in more detail.Suppose that C = FDVec R , and let λ be the linear combination of the following sets ofaxiom links. L M L ⊥ M L M L ⊥ M L M L ⊥ − − − ×××××× Letting S n and A n denoting the symmetric and alternating groups of n elementsrespectively, the tensor representation of the MLL − transformation is written¯ λ j j j i i i = X p ∈ S s p δ j p (1) p (2) p (3) i i i , where the scalars in the set { s p : p ∈ S } are as follows: ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 67 s p : = p = (1 2 3)1 if p ∈ A \ (1 2 3) − . Note that this means the following essential entries have the following values within them: • ¯ λ = • ¯ λ = ¯ λ = • ¯ λ = ¯ λ = ¯ λ = y , y and y found in { , , } : ¯ λ j j j · j j = δ j y ¯ λ j j j · j j = δ j y ¯ λ j j j · j j = δ j y Taking the non-trivial situation from Example 5.7, where y k = k for each k , the significantequations may be found. ¯ λ + ¯ λ = λ + ¯ λ = λ + ¯ λ = λ + ¯ λ = λ + ¯ λ = λ + ¯ λ = λ + ¯ λ = λ + ¯ λ = λ + ¯ λ = λ from earlier, andso it is deduced that ¯ λ j j j i i i ∈ Par ( S , S ) Val . As stated in Lemma 5.6, other simultaneousequations using other entry positions in the tensor ¯ λ reduce to linear combinations of thosefound above using the rules from Section 3.2. Furthermore, other choices of n for the testobject S n do not provide any more information when it comes to producing equations whichcontradict the existence of a tensor representation of ¯ λ in the set of values of Par ( S n , S n ).It follows that the proof, or any obvious minor alterations to it, is unable to be used todisprove that ¯ λ describes a transformation in the double-glued category.It comes as no surprise that there is more than one solution once negatives are added,especially when the values in the entries are in the genuine ring, once it is realised thatthere are 6 variables and the maximal linearly independent set of equations has 5 elements.Having fewer linear independent equations than variables within them means that the kernel of the equations has positive dimension, and so a myriad of solutions may be found.( ¯ λ + ¯ λ ) − ( ¯ λ + ¯ λ ) + ( ¯ λ + ¯ λ ) − ( ¯ λ + ¯ λ ) =
0( ¯ λ + ¯ λ ) − ( ¯ λ + ¯ λ ) + ( ¯ λ + ¯ λ ) − ( ¯ λ + ¯ λ ) =
0( ¯ λ + ¯ λ ) − ( ¯ λ + ¯ λ ) + ( ¯ λ + ¯ λ ) − ( ¯ λ + ¯ λ ) =
0( ¯ λ + ¯ λ ) + ( ¯ λ + ¯ λ ) + ( ¯ λ + ¯ λ ) − ( ¯ λ + ¯ λ ) − ( ¯ λ + ¯ λ ) − ( ¯ λ + ¯ λ ) (cid:27) = Par n ( S n , S n ) the larger n becomes. Asstated in the proof of Lemma 5.6, the number of variable positions uniquely describing anMLL − transformation’s tensor representation is n !, whilst the number of equations whichmay be considered linearly independent from one another even in the absence of negativeelements is n ; and clearly n ! > n for every n ≥ − + Mix models. After all, only one testobject has been considered; all objects in the focused glued category must be shown to stillcontain the morphisms associated to a rogue MLL − transformation from C to show that itremains a one in the glued category. Fortunately, it is possible to extend the principle fromthe above example to a result on the level of MLL − transformations for all the non-zero-sum-free categories. Proposition 5.13.
Let C be a compact closed category with finite biproducts satisfying feeble fullcompleteness. Then G C does not satisfy MLL − + Mix full completeness if C is not zero-sum-free.Proof. Suppose C is such a compact closed category, meaning that there exists at least onearrow s ∈ C [ I , I ] with an additive inverse ( − s ). We assume without loss of generalitythat we are in a strict compact closed category, suppressing the usage of the isomorphism ι : I −→ I ∗ . However, at times when associativity and unit isomorphisms are silently beingused are noted for the sake of clarity, either in the equation or as a side comment. Considerthe MLL − transformation ¯ λ : K I −→ − M − M − M ( − ) ⊥ M ( − ) ⊥ M ( − ) ⊥ found in C modellingthe following linear combination of proof structures: L M L ⊥ M L M L ⊥ M L M L ⊥ − s − s (1 + s ) − sss ×××××× Up to isomorphism the transformation can be written(1 + s ) · ( ˜ σ ◦ ( η ⊗ η ⊗ η )) + ( − s ) · ( ˜ σ ◦ ( η ⊗ η ⊗ η )) + ( − s ) · ( ˜ σ ◦ ( η ⊗ η ⊗ η )) + ( − s ) · ( ˜ σ ◦ ( η ⊗ η ⊗ η )) + s · ( ˜ σ ◦ ( η ⊗ η ⊗ η )) + s · ( ˜ σ ◦ ( η ⊗ η ⊗ η ))where the natural transformation ˜ σ p (1) p (2) p (3)1 2 3 for given permutation p ∈ Perm (2 ,
3) is thecomposition of associativity and symmetry natural transformations expected of symmetric
ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 69 monoidal categories containing arrows with source and target in the form below.( ˜ σ p (1) p (2) p (3)1 2 3 ) A , A , A , B , B , B : A ⊗ B ⊗ A ⊗ B ⊗ A ⊗ B −→ A ⊗ A ⊗ A ⊗ B p (1) ⊗ B p (2) ⊗ B p (3) For short, we can write ¯ λ = P p ∈ S s p · ˜ σ p (1) p (2) p (3)1 2 3 , where s p : = s + p = (1 2 3) s if p ∈ A \ (1 2 3) − s otherwise . Let A = ( R , U , X ) be an arbitrary object in G C , meaning that x ◦ u = ι = I for every u ∈ U and x ∈ X . We define the following notation: • z ( l ) is the tuple z missing the l th entry; • (cid:10) g ( j ) , f (cid:11) j = f ◦ (cid:16)N i < j g i ⊗ ⊗ N j < i ≤ n g i (cid:17) ◦ Λ n , where Λ : I −→ I ⊗ n is the natural compo-sition of the unit isomorphism.We see that, Par ( A , A ) Val = { f ∈ C [ I , | Par ( A , A ) | ] : ∀ k = u ∗ k ∈ U ∀ x k ∈ X , ∀ l D (( x ( l ) , u ∗ )) , f E l ∈ U , D ( x , ( u ∗ ) ( l ) ) , f E l + ∈ X ∗ } . If either U or X = ∅ then it is trivially true that ¯ λ A is in the set of values. If both sets are non-empty, let u ∗ , u ∗ , u ∗ ∈ U ∗ and x , x ∈ X be arbitrary, not necessarily distinct morphisms.Noting (and suppressing usage of the isomorphism ι ) that ( x ⊗ R ∗ ) ◦ η R = λ − R ∗ ◦ x ∗ and(1 R ⊗ u ∗ ) ◦ η R = ρ − R ◦ u for all suitable arrows u and x in C due to dinaturality, and that x i ◦ u j = ι = I for any choice of i , j , the composition D ( x , x , u ∗ , u ∗ , u ∗ ) , ˜ σ p (1) p (2) p (3)1 2 3 ◦ ( η ⊗ η ⊗ η ) E canbe found for each permutation p .(1 ⊗ x ⊗ x ⊗ u ∗ ⊗ u ∗ ⊗ u ∗ ) ◦ ˜ σ p (1) p (2) p (3)1 2 3 ◦ ( η ⊗ η ⊗ η ) = ˜ σ p (1) p (2) p (3)1 2 3 ◦ (1 ⊗ u ∗ p − (1) ⊗ x ⊗ u ∗ p − (2) ⊗ x ⊗ u ∗ p − (3) ) ◦ ( η ⊗ η ⊗ η ) = ˜ σ p (1) p (2) p (3)1 2 3 ◦ (((1 ⊗ u ∗ p − (1) ) ◦ η ) ⊗ ((1 ⊗ u ∗ p − (1) ) ◦ η ) ⊗ ((1 ⊗ u ∗ p − (1) ) ◦ η )) = ˜ σ p (1) p (2) p (3)1 2 3 ◦ (( λ − I ∗ ◦ u p − (1) ) ⊗ λ − I ∗ ⊗ λ − I ∗ ) = u p − (1) ∈ U By symmetry and a similar argument we also find D (( x ( l ) , u ∗ )) , ˜ σ p (1) p (2) p (3)1 2 3 ◦ ( η ⊗ η ⊗ η ) E l = u p − ( l ) ∈ U D ( x , ( u ∗ ) ( l ) ) , ˜ σ p (1) p (2) p (3)1 2 3 ◦ ( η ⊗ η ⊗ η ) E l + = x ∗ p ( l ) ∈ X ∗ Finding the compositions D (( x ( l ) , u ∗ )) , ˜ σ p (1) p (2) p (3)1 2 3 ◦ ( η ⊗ η ⊗ η ) E l and D ( x , ( u ∗ ) ( l ) ) , ˜ σ p (1) p (2) p (3)1 2 3 ◦ ( η ⊗ η ⊗ η ) E l + now becomes a simple case of taking the correct linear combination of answers from the above calculations. D ( x ( l ) , u ∗ ) , ¯ λ R E l = * (( x ( l ) , u ∗ )) , X p ∈ S s p · ( ˜ σ p (1) p (2) p (3)1 2 3 ◦ ( η ⊗ η ⊗ η )) + l = X p ∈ S s p · D (( x ( l ) , u ∗ )) , ¯ λ R E l = D (( x ( l ) , u ∗ )) , ˜ σ p (1) p (2) p (3)1 2 3 ◦ ( η ⊗ η ⊗ η ) E l = X p ∈ S s p · u p − ( l ) This final sum is a linear combination of the arrows u , u , u . One finds by inspectionthat the scalars added together in this sum for u k for some k are s ( i i i ) and s ( j j j ) , thetwo scalars for which i k = l = j k . Since ( i i i ) and ( j j j ) are distinct but they sharea value in exactly one position in this situation, it must be the case that one is an evenpermutation (that is, it belongs to A ), and one is odd (and therefore is not a member of A ). For the sake of convenience, let the former be the even permutation. If i k , k , j k ,then we know that neither of the two permutations are (1 2 3), and so we deduce that s ( i i i ) + s ( j j j ) = s + ( − s ) =
0. If however, i k = k = j k , then one of them does equal (1 2 3),meaning s (1 2 3) + s ( j j j ) = (1 + s ) + ( − s ) = + ( s + ( − s )) =
1. We therefore conclude that D ( x ( l ) , u ∗ ) , ¯ λ R E l = u l ∈ U By a dual argument, we can find that D ( x , ( u ∗ ) ( l ) ) , ¯ λ R E l + = x ∗ l ∈ X ∗ The arrows u , u , u ∈ U and x , x , x ∈ X are arbitrary throughout the above argument,meaning that any triple of values and covalues of A = ( R , U , X ) could be chosen. Thus¯ λ R ∈ ( A M A M A M A ⊥ M A ⊥ M A ⊥ ) Val . Furthermore, we may say that this for any choiceof A , since A is considered arbitrary in this proof as well. Naturally this means that the C -arrows in the collection ¯ λ = { ¯ λ R : R ∈ C } are found in all the required homsets in G C ,and therefore the MLL − transformation ¯ λ is also in G C . The linear combination of proofstructures modelled by this transformation is non-simple, and so the category does notsatisfy MLL − + Mix full completeness.
Corollary 5.14.
Let C be a compact closed category with finite biproducts satisfying feeble fullcompleteness. The category G C satisfies MLL − + Mix full completeness if and only if C is zero-sum-free.
6. C onclusions
In this paper we show that there are two simple, elegant methods of producing categoricalmodels for both MLL − or MLL − + Mix. The Hyland-Tan double glueing construction isseemingly the perfect semantic description of the deductive system when applied to tensor-generated compact closed categories with finite biproducts. Certainly the Danos-Regnierdescription of proof nets has a strongly combinatorial flavour, and this paper shows that
ONSTRUCTING FULLY COMPLETE MODELS OF MULTIPLICATIVE LINEAR LOGIC 71 the same combinatorial restrictions are imposed on the categorical model by the doubleglueing construction.It is notable that the tensor representations of MLL − transformations are precisely the isotropic tensors of even power over the semiring of scalars. Viewed as vectors in Euclideanspace, this means that such representations are exactly those which are invariant underbasis change. It is therefore reasonable to suggest that the more combinatorial proofs givenin this paper may be replaceable by geometric arguments. Such a methodology couldunveil a di ff erent perspective on the categorical models and MLL − itself.An obvious possibility for future work is applying these techniques to the (unitless)multiplicative additive fragment of linear logic, MALL − . Since we start from compactclosed categories with finite biproducts we know that our models, after double glueing,have both finite sums and products [HS03] and so are models of MALL − .However, it is known that the Hyland-Tan construction alone cannot produce fullycomplete models of MALL − [Ste13]: none of the resulting categories satisfy Joyal’s soft-ness property on the dinatural level as is required. Categories which accurately modelMALL − proof nets are in short supply: the only example whose full proof has been pub-lished is GHypCoh — the category of hypercoherence spaces under the ‘ G ’-construction— found in [BHS05].We note that HypCoh is another example of a double-glued category, but with a tightorthogonality upon it. We are writing an article giving the appropriate generalisationof the definition of orthogonality found in [HS03] in order to allow the essence of thisconstruction to be extracted. It is hoped that this will provide a starting point for finding afull completeness result for MALL − which is as general as ours for MLL − . One might notein passing that the result of Blute, Hamano and Scott is based upon the MALL − proof netcriteria of Girard [Gir96] based around monomial weights, whereas there is currently nocomparable result for the Hughes-van Glabbeek proof net criteria [HvG05]. We hope thatour methods are suitable to address that situation.A cknowledgements The authors would like to thank the anonymous reviewers of this paper for their con-structive criticism of our originally submitted draft. Thanks also to Luke Ong and HaroldSimmons, examiners of the thesis from which this article was originally drawn, for theirsuggestions. The commutative diagrams were drawn using the diagrams package of PaulTaylor.The second author was partially supported by the ANR projects LOGOI (10-BLAN-0213-02)and COQUAS (12-JS02-006-01), and an EPSRC studentship.R eferences [AJ94] S. Abramsky and R. Jagadeesan. Games and full completeness for multiplicative linear logic.
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