Noncommutative coherence spaces for full linear logic

by   Sergey Slavnov, et al.

We propose a model of full propositional linear logic based on topological vector spaces. The essential feature is that we consider partially ordered vector spaces, or, rather, positive cones in such spaces. Thus we introduce a category whose objects are dual pairs of normed cones satisfying certain specific completeness properties, such as existence of norm-bounded monotone suprema, and whose morphisms are bounded (adjointable) positive maps. Norms allow us distinct interpretation of dual additive connectives as product and coproduct; in this sense the model is nondegenerate. Also, unlike the familiar case of probabilistic coherence spaces, there is no reference or need for preferred basis; in this sense the model is invariant. Probabilistic coherence spaces form a full subcategory. whose objects, seen as posets, are lattices. Thus we get a model fitting in the tradition of interpreting linear logic in linear algebraic setting, which arguably is free from the drawbacks of its predecessors. We choose the somewhat show-offy title "noncommutative coherence spaces", hinting of noncommutative geometry, because the passage from probabilistic coherence spaces with their preferred bases to general partially ordered cones seems analogous to the passage from commutative algebras of functions on topological spaces to general, not necessarily commutative algebras. Also, natural non-lattice examples of our spaces come from self-adjoint parts of noncommutative operator algebras. Probabilistic coherence spaces then appear as a "commutative" subcategory.



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1 Introduction

From the very beginning of linear logic it has been implicit in notation and terminology that it should be considered as similar to (multi-)linear algebra.

Indeed, the multiplicative operations, i.e. tensor, linear implication and linear negation are intuitively interpreted as, respectively, the tensor product of vector spaces, vector space of linear maps and vector space duality; and the additive ones are often understood as corresponding to the product and coproduct of normed spaces. As for the exponentials, they resemble a “Fock space” of power series, i.e. some completion of the free symmetric algebra (as discussed in

[2], see also [11]).

However, putting this intuition into solid mathematical form turned out to be rather challenging. Different fragments of the system require different structures on vector spaces, and it seems not so easy to put them all together.

In the seminal work [11] Girard proposed to interpret dual formulas as dual pairs of Banach spaces, calling them coherent Banach spaces. This gave a perfectly natural interpretation of the multiplicative-additive fragment, however failed for the exponentials.

For exponentials it was necessary to consider some space of power series, i.e. analytic functions, which should be a Banach space. A natural candidate was the space of functions analytic in the unit ball, but there were issues with the behaviour at the boundary. Unfortunately, functions arising from linear logic proofs tend to map the interior of the unit ball to its closure, and there is no reasonable way to extend a general analytic function from the open ball to its boundary. In particular, composing these functions was problematic.

Another model was developed in [5] in the context of Koethe sequence spaces. This gives a successful interpretation of full linear logic. However, since Koethe sequence spaces do not have any norms, there is no distinction between products and coproducts, and dual additive connectives are identified. Thus, the model of additives is degenerate. A similar degeneracy occurs in the category of convenient vector spaces [1], which models differential linear logic (see [7], [8]), where exponential connectives play a crucial role.

Finally it turned out that the category of probabilistic coherence spaces (PCS), initiated in [12] and developed in [4], provides a nondegenerate model of the full system, including the additives. Omitting some technicalities, a PCS can be described as a dual pair of (real) Banach spaces, which are spaces of sequences on the same index set. In a more algebraic language, these are Banach spaces with a fixed basis. Having a basis allows us speaking about positive (or, rather, non-negative) vectors, represented as non-negative sequences, and positive maps, represented as non-negative matrices. PCS morphisms can be described as bounded positive maps.

In such a setting, formulas are interpreted as sequence spaces, and proofs, as positive elements (sequences and matrices). Exponential connectives correspond to spaces of power series with positive coefficients converging in the unit ball (which still have an interpretation as real analytic functions). Somewhat remarkably, in this case we do not encounter problems with exponentials typical for coherent Banach spaces. A conceptual reason for that is that analytic functions defined in the open unit ball by positive power series have canonical extensions to the boundary (although not necessarily continuous).

A major drawback of the PCS model is that it is very non-invariant. All spaces come with fixed bases. On the other hand, it has eventually become apparent that what makes the model work is not the choice of bases, but the structure of positivity, i.e. of a partial order. In fact, the preferred bases of PCS themselves are determined by the partial order. Sequence spaces, seen as posets (with elementwise ordering) are vector lattices, and preferred bases consist of “minimal” positive elements (elements such that implies for some ).

This suggests that the whole construction can be reformulated in an invariant fashion using only the language of partially ordered vector spaces. Or, since eventually we are interested only in positive elements and maps, in the language of abstract positive cones (in the sense of [16]). However it soon becomes evident that the lattice structure, characteristic for PCS, is, in fact, not necessary.

In particular, in [6] a coordinate-free generalization of PCS was proposed for modeling probabilistic functional programming, where more general positive cones are considered. These cones are equipped with norms, similar to vector space norms, and satisfy a variant of completeness property: every norm-bounded monotone nondecreasing sequence must have a least upper bound. It turns out that such cones can be organized into a cartesian closed category with specific stable cone maps as morphisms. (This gives already a model of intuitionistic logic, but in order to interpret probabilistic computation further refinements were proposed, leading to the category of measurable cones and measurable stable maps). The question of modeling classical linear logic in such a setting remained open.

In this work we propose to consider normed positive cones in dual pairs of partially ordered vector spaces, or, simply, dual pairs of normed cones. Duality alone would allow us to interpret the multiplicative-additive fragment of linear logic along the lines of [11]. But in order to model the exponential fragment we, similarly to [6], introduce certain completeness property. However our version of completeness is different: any norm-bounded monotone sequence has a weak limit. (It turns out, though, that completeness in our sense implies completeness in the sense of [6].) Here weak limit is understood in the following sense. If is a dual pair of cones, and is a sequence in , then is the weak limit of iff for any it holds that

Completeness then gives us control over convergence of power series needed for exponentials.

A certain exercise is to define a tensor product of such cones, which must be complete itself. This leads us to studying possibilities of cone completion. A crucial observation is that, if is a dual pair of cones with complete, then there is a minimal cone containing such that the pairing with extends from to and both members of the dual pair are complete. For constructing we note that the cone embeds into the complete cone of norm-bounded linear functionals on . The completion is then the minimal complete subcone of containing .

In this way we obtain a well-defined sound model of the exponential fragment in terms of real analytic functions and distributions, similar to the case of PCS. The category of PCS itself is identified as a proper full subcategory. Relations with stable and measurable maps of [6] remain to be clarified.

To conclude the introduction, let us comment on some amusing analogies with noncommutative geometry (probably not too strong).

A standard example of a non-lattice partially ordered vector space is the space of self-adjoint operators on a Hilbert space. In fact, a classical result [18] states that the self-adjoint part of a -algebra is a lattice iff the algebra is commutative. Our passage from vector lattices, which have preferred bases, to general partially ordered vector spaces resembles passage from “commutative” spaces to general noncommutative “pointless” spaces, i.e. general algebras. In particular, probabilistic (“commutative”) coherence spaces have natural representations as subspaces of sequence spaces, and these are commutative algebras. There are however genuinely non-lattice (“noncommutative”) objects, especially coming from spaces of self-adjoint operators on a Hilbert space. Let us mention that for finite dimensions such a noncommutative construction was hinted by Girard a while ago under the name of quantum coherence spaces [12].

1.1 Notation and background

We assume that the reader is familiar with linear logic (see [9], [10] for an introduction), as well as with its categorical interpretation. The standard reference on linear logic and -autonomous categories is [15]. For a modern treatment of categorical semantics of linear logic, especially of the exponential fragment, we refer to [13].

We denote the monoidal product in a -autonomous category as and call it tensor product, and we denote duality as a star . The dual of tensor product is called cotensor and denoted as , i.e. . Monoidal unit is denoted , with its dual denoted . Product and coproduct are denoted as, respectively, and . The internal homs functor is denoted , as usual.

We also assume that the reader is familiar with basic notions of locally convex vector spaces. We use [14] for reference. All vector spaces in this paper are real. “Topological vector space” in this paper always means locally convex vector space.

2 Dual pairs and reflexivity

The first thing for interpreting linear logic in the context of vector spaces is to get involutive duality. In the finite-dimensional case we have the usual duality of vector spaces. However, when we come to infinite dimensions, there are different ways to generalize this duality, which, in general, produce non-involutive operations. One of the most direct ways to deal with this problem is to use dual pairs.

Recall that a dual pair is a pair of vector spaces equipped with a bilinear pairing

such that for any with there exists with , and for any with there exists with (i.e., the pairing is nondegenerate).

For finite-dimensional there is only one (up to a natural isomorphism) potential partner to form a dual pair, namely the dual space . In the infinite-dimensional case there are many different choices for the dual space, and a dual pair is a way to fix one such choice.

A map of dual pairs , is a pair of linear maps

satisfying for all , the adjointness condition


Obviously, such maps compose, and dual pairs can be organized in a category.

The dual of a dual pair is the dual pair equipped with the same pairing, written with the opposite order of arguments. Obviously, this duality is involutive . It also extends to maps in the obvious way: , which gives us a contravariant functor from dual pairs to dual pairs.

2.1 Weak topologies and reflexive spaces

Of course, a dual pair is just a vector space with explicitly specified dual. Always keeping this dual explicit may lead to somewhat cumbersome formulations and expressions (at least, from notational point of view). We can hide duals from notation by using topology.

Indeed, any dual pair gives rise to particular locally convex topologies on its members and .

The -weak (or simply weak) topology on the space determined by the dual pair is the topology of pointwise convergence on elements of (where is identified with a subspace of functions on ). The -weak topology on the second element is defined similarly, with the roles of and interchanged.

In the sequel, unless this leads to confusion, we will speak loosely about weak topology without mentioning explicitly the dual pair that determines it.

A crucial property of weak topology determined by a dual pair is the following (see, for example, [14], Theorem 3.10).

Lemma 1

For a dual pair the space of weakly continuous functionals on is .

Now let be an arbitrary topological vector space (TVS).

We define the dual TVS as the space of continuous functionals on , with the topology of pointwise convergence. We will understand this topology on as -weak and, accordingly, we will call it weak. (It is often called *-weak in literature though.)

We say that a TVS is reflexive if the natural inclusion

is a topological isomorphism.

Reflexivity of means that topology on is that of pointwise convergence on . In particular, if is a dual pair, then the weak topology makes a reflexive TVS.

The natural notion of TVS map is of course that of a continuous linear map.

Now let , be TVS.

A linear map is adjointable iff there exists an adjoint map defined by the equation


Equation (2) extends duality of TVS to morphisms thus making it a contravariant functor. In fact, Lemma 1 implies the following.

Note 1

A linear map of reflexive TVS is continuous iff it is adjointable.

The above discussion, especially Lemma 1 and the preceding Note, readily give the following.

Note 2

There is a natural correspondence between reflexive TVS and dual pairs given by

The correspondence preserves duality.

3 Adding positivity

Our ultimate goal is to interpret the exponential fragment of linear logic, which eventually involves analytic maps given by power series, and we need to control their convergence, which may be problematic at certain points. A possible solution is to use positive power series, whose behavior is much simpler. This, in turn, leads us to considering positivity and partial order.

An extensive introduction to theory of partially ordered topological vector spaces can be found in [17], Chapter V. We will present necessary bits in a mostly self-contained form.

3.1 Positive cones and positive dual pairs

3.1.1 Cones and partial order

Let be a vector space.

Recall that a cone in the vector space is any subset satisfying

A more fancy way to define a cone in a vector space is to note that any vector space is a module over the semi-ring and to say that a cone in is an -submodule.

A proper cone in is a cone such that

A proper cone gives rise to a partial order on .

Recall that a partially ordered vector space (POVS) is a vector space equipped with a partial order , such that if then

  • for all ;

  • for all .

Note 3

For any POVS the set


is a proper cone. Conversely, any proper cone in determines a POVS structure on by:

The set is called the positive cone of the POVS . Elements of are called positive, and elements that are differences of positive elements are called regular.

A POVS is regular if it is spanned by its positive elements, i.e., if all its elements are regular.

A linear map between POVS is positive, if implies .

A natural notion of a POVS morphism is that of a positive map.

3.1.2 Cones abstractly

Eventually we will be interested only in positive elements and positive maps. Therefore it may be more convenient sometimes to speak directly about cones without reference to ambient vector spaces (which may be non-unique). We will consider abstract cones in the sense of [16].

We say that an (abstract) cone is a module over the semi-ring , which embeds, as an -submodule into some vector space becoming a proper cone in .

A cone map is a map of -modules.

Any abstract cone has an intrinsically defined partial order. Namely if there exists such that . Of course, if we see as a positive cone in a vector space , then the induced partial order on extends the intrinsic partial order on .

It is rather obvious that for a cone there is a “minimal” embedding in a vector space. Basically, we pick some embedding in a vector space and restrict to the subspace of spanned by .

More intrinsically, the enveloping vector space of is the -module quotiented by the equivalence


where multiplication by is defined by , making it an -module, i.e. a vector space. The original cone embeds in by , which makes a regular POVS.

Note 4

There is a natural correspondence between cones and regular POVS given by

3.1.3 Positive dual pairs

As in the case of vector spaces, we need also an involutive duality for cones or for POVS, and the most direct way is to mimic the construction of dual pairs.

A reasonable adaptation of the definition of a dual pair to the partially ordered case is the following.

A POVS dual pair is a dual pair, whose members are regular POVS, such that

  • for any it holds that iff ;

  • for any it holds that iff .

A map of POVS dual pairs , is a map where the maps , are positive. (In fact it is sufficient to require positivity of any one of the two maps.)

Since regular POVS are essentially just abstract cones it may be reasonable to consider directly dual pairs of cones.

A cone dual pair is a pair of cones together with an -bilinear pairing

such that

  • for any with there exists with ;

  • for any with there exists with ;

  • for any it holds that iff ;

  • for any it holds that iff .

The first two conditions in the above definition just mean that the pairing is nondegenerate.

A map of cone dual pairs , is a pair of maps , satisfying adjointness condition (1).

In view of Note 4, the two above versions of dual pair are representations of the same structure, and the relation between them is very transparent. Therefore we often will be rather loose and mix the two notions, using the term positive dual pair.

Now, as in the case of ordinary dual pairs, we will try to hide explicit duals from notations and define reflexive POVS and reflexive cones. Again, we will use topology.

In particular, as for ordinary dual pairs, we define for a cone dual pair the -weak, or simply, weak topology on as the topology of pointwise convergence on .

3.2 In the topological language

3.2.1 Topology and partial order

Let be a TVS, and be a cone in .

The dual cone of is the subset of the topological dual space defined by

The bidual cone is the subset of defined by

Remark There is an ambiguity in notation for , because we could interpret it also as a subset of the double dual space . In our case this is harmless, because we always consider reflexive .

Hahn-Banach separation theorem (see [14], Theorem 3.10 for reference) readily yields the following result, well known in convex geometry (see , for example, [19], Theorem 1.1.9).

Lemma 2

The bidual of a cone in the topological vector space coincides with the closure of in .

The dual cone is closed in .

Now let be a cone in . The cone is spanning or generating if the span of is dense in .

It is easy to see that if is spanning in , then is proper in . Furthermore, using Hahn-Banach theorem, it can be shown that if is proper and closed in , then is spanning in . This suggests the following definition.

A partially ordered TVS (POTVS) is a TVS together with a closed spanning proper cone .

Then Lemma 2 implies the following.

Note 5

If is a POTVS which is reflexive as a TVS then is isomorphic to both as a POVS and as a TVS.

We will be interested in the more restricted case of regular POTVS and corresponding topological cones.

3.2.2 Topological cones

We say that a topological cone is a cone equipped with a topology such that there exists a continuous embedding of into a TVS, making a closed proper cone.

A morphism of topological cones is a continuous morphism of cones.

The dual of an abstract topological cone is the space of continuous -linear functionals . It is topologized by the weak topology, which is the topology of pointwise convergence on .

The correspondence between cones and regular spaces extends to the topological case.

Given a topological cone we topologize the enveloping space as the quotient space , where the equivalence is given by (4).

Lemma 3

With notation as above, is a (locally convex) POTVS, in particular, is closed in . Furthermore, any continuous cone map extends uniquely to a positive continuous map .

Proof We need to check that the above defined topology on is locally convex.

We know that embeds continuously into some TVS . Then identifies with the span of in . We have the following commutative diagram. V×V&& V
& &
P×P&^π_P&EP Here the upper horizontal arrow is the subtraction map , the lower horizontal arrow is the quotient map and vertical arrows are the obvious embeddings. We already know about all arrows except the right vertical one that they are continuous. Now, open sets in are precisely the quotients of open sets in . But a set is open in iff it is a preimage of an open set in , and a set is open in iff its image under the subtraction map is open in . It follows from commutativity of the above diagram that a set is open in iff it is a preimage of an open set in . Hence the topology of is that of a subspace of . In particular it is locally convex. Also, is closed in , since it is closed in .

Now let be a continuous cone map. Obviously extends (uniquely) to a positive linear map of enveloping spaces. We need to check that this extension is continuous. We have the following commutative diagram. P×P&^L×L& Q×Q
^π_P& &_π_Q
EP&^EL&EQ Here vertical arrows are the quotient maps. We know about all arrows except the lower horizontal one that they are continuous. Let be open in . Since the left vertical arrow is surjective, it follows that

and the set on the righthand side is open since the maps involved are continuous, and takes opens to opens.

Thus is continuous.

3.2.3 Reflexive cones

A reflexive cone is a topological cone such that the natural map

is a continuous isomorphism.

Let us see what does it mean in the language of regular POVS.

For any topological cone , the topology on is that of pointwise convergence on , which is the same as pointwise convergence on , since spans . If, furthermore, is reflexive, then we get that topology on is that of pointwise convergence on , hence on . It follows that and form a dual pair and are equipped with the corresponding weak topologies. In particular, .

Let us say that a reflexive POTVS is a POTVS which is reflexive as a TVS and whose dual is regular.

Then the above discussion together with Note 4, Note 5 and Lemma 3 amount to the following.

Note 6

There is a natural correspondence between reflexive cones and reflexive POTVS. The correspondence preserves duality.

In particular, we get the following analogue of Note 1.

Note 7

A map of reflexive cones is continuous iff there exists an adjoint map defined by (2).

3.2.4 Positive dual pairs again

Now let be a positive dual pair, represented as a pair of POVS equipped with the corresponding weak topology.

The positive cones in and are by definition duals of each other. It follows that they are closed in the ambient spaces, hence and are POTVS. Obviously they are also reflexive as TVS and regular by definition. This leads to the following continuation of Note 6.

Note 8

There is a natural correspondence between reflexive cones, reflexive POTVS and positive dual pairs.

4 Adding norms

It is well agreed that in the setting of vector spaces the additive connectives of linear logic correspond to norms.

We are now going to consider normed cones.

Remark Unfortunately we are unable to work with the more familiar normed vector spaces. In fact there are different ways to extend norms from cones to ambient POTVS, and there seems no preferred choice in general. We do not know if the constructions of this paper can be adapted to the setting of normed POTVS rather than just normed cones. It might be interesting to study this question.

A cone is a normed cone if it is equipped with a norm

satisfying the usual properties

  • ;

  • ;

  • iff .

A norm on is monotone if implies .

If is a normed cone, we say that the positive functional on is norm-bounded, or simply bounded, iff there exists


The norm dual cone is the cone of norm-bounded functionals on (we use in the superscript to distinguish it from the topological dual) equipped with the dual norm defined by (5). Note that the dual norm is necessary monotone.

Similarly, a cone map is bounded iff


Morphisms of normed cones are bounded cone maps.

Bounded map of norm less or equal to is a contraction.

4.1 Dual pairs and reflexive norms

A normed positive dual pair is a cone dual pair , where , are normed cones, and the pairing is such that for all , it holds that


Again, note that norms on members of a normed positive dual pair are necessary monotone.

If is a positive dual pair map, then it is immediate that is norm-bounded in the sense of (6) iff is, moreover, in this case .

We say that is map or a morphism of normed positive dual pairs if (hence ) is bounded. We say that it is a contraction if is a contraction.

In the language of reflexive cones an equivalent structure is described readily.

A norm on the topological cone is reflexive if any functional in is norm-bounded on , and the natural map is norm-preserving.

A reflexive normed cone is a reflexive cone equipped with a reflexive norm.

Again there is a natural correspondence between normed positive dual pairs and reflexive normed cones.

Remark The same structure can be described in the language of regular POVS, but definitions become rather awkward and not very useful.

4.2 Completeness

Unlike the case of vector spaces, a normed cone does not have any intrinsic metric; a metric would require extending the norm to the whole enveloping space. Thus there is no analogue of Banach (i.e. metric complete) spaces in the setting of normed cones. Yet there is a specific notion of completeness, which will be crucial for our purposes.

We say that a normed topological cone is complete, if any norm-bounded monotone non-decreasing sequence in has a limit. (It would be more accurate, but also more cumbersome, to use some term like sequential order-completeness.)

Remark Our definition of completeness differs from the one proposed in [6], where it is required only that norm-bounded non-decreasing sequences have least upper bounds. However, it is not hard to see that completeness in our sense implies completeness in the sense of [6].

Note 9

If is a complete cone, then any norm-bounded monotone non-decreasing sequence has a least upper bound in .

Proof By definition there exists .

Then for any we have . It follows from Lemmas 3 and 2 that the difference belongs to the closure of in , which is . Hence for all .

Now if is some other upper bound of , then for any it holds that . Hence . Thus .

Our main use of positivity is in dealing with power series when modeling the exponential fragment. Completeness then gives control on their convergence.

However, prior to interpreting exponentials, we need a multiplicative structure; i.e. we will need tensor products. Since algebraic tensor products, in general, will not be complete, the following construction will be crucial.

4.2.1 Completing reflexive cones

It turns out that reflexive normed cones have canonical completions.

Let be a reflexive normed cone.

We denote as the space of norm-bounded positive functionals on , where is equipped with dual norm (5).

We equip with topology of pointwise convergence on . Note that is reflexive.

It is easy to see also that is complete.

Now a subcone of is a topologically closed -submodule of . Equipped with the subspace topology and the restriction of -norm, a subcone is itself a topological normed cone.

We note that the set of complete subcones of is closed under intersection, and that , being reflexive, itself identifies with a subcone of . Therefore there exists the smallest complete subcone containing . We call it the completion of .

Note that the dual of coincides (as a cone) with . Indeed, , and coincides with (again, as a cone). Since topology of is that of pointwise convergence on , it follows that is reflexive.

Theorem 1

With notation as above, the completion satisfies the following universal property.

For any continuous cone map

where is a complete reflexive normed cone, there is a unique continuous

with , making the following diagram commute. ¯P&^¯L&X
P&& Here is the inclusion map.

Proof Let be as in the formulation.

There is a double adjoint map defined by

where , . The map is itself adjointable (with the adjoint ), hence continuous by Note 7. It is also bounded with .

Since is reflexive, it is identified as a subcone of . Let .

If is a norm-bounded non-decreasing sequence in , then is a norm-bounded non-decreasing sequence in . Since is complete, we have that . Since is continuous, it takes to . Hence . It follows that the cone is complete, hence . Then the restriction of to is a well defined map to . Its restriction to coincides with .

Theorem 2

With notation as above, if is complete in the topology of pointwise convergence on , then it remains complete in the topology of pointwise convergence on .

Proof It is enough to show that -weak convergence and -weak convergence, when restricted to monotone sequences, coincide on .

Let be the subcone of elements satisfying the following property:

  • for any monotone non-decreasing sequence in with -weak limit it holds that

We are going to show that is a complete subcone of .

So let be a norm-bounded monotone non-decreasing sequence in . Then there exists .

Let be a monotone non-decreasing sequence in with -weak limit . Note that all and are continuous on , hence on .

We have

Thus , hence is complete.

But , hence as well. And this is precisely what we needed to show.

The moral of the above is that if is a normed positive dual pair where is complete in the corresponding weak topology, then it has a completion to a normed positive dual pair, whose both members are complete. The completion is minimal and canonical as is seen from the universal property of Theorem 1.

5 Coherent cones.

We are ready to introduce our category for modeling linear logic.

Definition 1

Coherent cone is defined by one of the following equivalent structures:

  • A complete reflexive normed cone whose dual is also complete;

  • a normed positive dual pair such that , are complete for respective weak topologies.

Unless otherwise stated, we will use the first representation and inderstand coherent cones as complete reflexive cones.

Morphisms of coherent cones are continuous contractions.

Since contractions compose, and identities are contractions, it follows that coherent cones can be organized into a category, which we denote as .

It follows from the very definition that the duality equips with a contravariant functorial involution.

Now let us make sure that coherent cones indeed exist and show some examples.

5.1 Examples

Finite dimensions Let be a regular finite-dimensional POTVS equipped with a vector space norm .

Then is also regular (the subspace of regular elements is dense in , but in finite dimensions this means ). We equip the cone with a cone norm


This induces the dual norm on . (Note that the new norm is not a vector space norm, and, in general, .) We claim that the two cone norms make a normed positive dual pair.

We need to check that for it holds that


For we have . Hence for we have

On the other hand for , we have .

This proves (9).

That , are complete is immediate from finite-dimensionality. The topology is unique, and any closed norm-bounded set is compact, hence, any norm-bounded monotone sequence converges.

Sequence spaces Recall that the (real) space is the space of real sequences with the norm

The space is the space of real sequences with the norm

For , , the pairing is given by

These spaces are partially ordered in the obvious way, with the spanning positive cones , consisting of nonnegative sequences. The normed positive dual pair is immediately seen to define a coherent cone.

Note that the cones , , seen as posets are rather specific, because they are lattices. In particular they have preferred bases (consisting of “minimal vectors” , for which implies for some ). In this sense these spaces can be called “commutative”.

A generalization of this example is given by probabilistic coherence spaces (PCS). This will be discussed in a greater detail a little bit later.

Bounded operators A genuinely noncommutative, non-lattice example comes from the space of bounded operators on a Hilbert space . This requires some background, see for example [3].

The space is equipped with the norm


which makes it a (complex) Banach space.

The subspace of trace-class operators consists of operators with the norm


The above norm also makes a complex Banach space, and becomes the Banach dual of under the Hermitian pairing


The space is also a dual; it is the Banach dual of the subspace of compact operators.

Now, there are real subspaces

of self-adjoint operators, which are real Banach spaces. In fact, the ambient spaces , are complexifications of those real ones:

and is the real Banach dual of . The self-adjoint spaces are partially ordered, with the positive cones consisting of positive operators , i.e. those for which for all .

Then it is easy to check that the cones , with norms (10), (11) form a normed positive dual pair. They are also complete in the corresponding weak topologies.

Indeed, is complete simply because is the norm-dual of . Furthermore , and the cone gets identified as the completion of for -weak topology (see Section 4.2). Since is obviously complete for -weak topology, by Theorem 2 it remains complete in -weak topology

This example can be seen as a “noncommutative” version of the -example.

It might be interesting to try generalizing this example by considering other normed spaces of self-adjoint (or essentially self-adjoint) operators and pairing (12). A version of such a construction for finite-dimensional Hilbert spaces was hinted by Girard a while ago under the name of quantum coherent spaces [12].

It is worth noting that the preceding examples of , and, in general, probabilistic coherence spaces, also have ready interpretation as coming from spaces of self-adjoint operators, when the ambient operator algebras are commutative.

5.1.1 Probabilistic coherence spaces

We briefly recall the notion of probabilistic coherence space (PCS). For more details see [4].

Let be an at most countable index set.

For any denote as the sequence indexed by , all whose elements with index other than are zero, and the element with the index is .

Let be a set of real nonnegative sequences indexed by .

The polar of consists of all real nonnegative sequences on the index set such that for any , where the pairing of sequences is defined by


A probabilistic coherence space (PCS) is a pair, where is an at most countable index set, and is a set of real nonnegative sequences on , satisfying the following properties.

  • ;

  • For any there exist such that and .

The dual PCS of is then the pair . (It can be shown that this is indeed a PCS.)

If is a PCS then the set

is a cone, and we have the norm on defined as the Minkowski functional

making the “positive unit ball”, i.e. the set of elements with norm less or equal to .

It is easy to see that is then a normed positive dual pair, and the cones are complete.

Thus a PCS gives rise to the dual pair of coherent cones , which we will loosely denote with the same letter .

Remark Just as in the previously discussed example, the coherent cones coming from PCS have a specific property that their constituent cones, seen as posets, are lattices. It can be shown that in the finite-dimensional case this property characterizes PCS in the class of coherent cones completely. In infinite dimensions, however, this property alone apparently is not sufficient. It might be interesting to work out the missing conditions.

Also, the lattice structure determines for PCS preferred bases, and in this sense they are “commutative” spaces. In particular, they can be seen as subspaces of commutative algebras, namely, algebras of sequences with pointwise multiplication.

We now discuss morphisms of PCS.

Let , be PCS.

A PCS morphism is a double sequence (a matrix) indexed by , such that for any the sequence , where is defined by


(It is implicit in the definition that the series in (14) converges for all .)

Now, since and are just positive unit balls in the corresponding cones, it is immediate from definition that, for a PCS map , formula (14) defines, in fact, a contraction of cones . Furthermore, this contraction has the adjoint given by

(note that the series in the above formula is convergent for all ), hence it is continuous. Thus, a PCS map induces also the map of the corresponding coherent cones.

We are going to show that the converse is true as well: any coherent cone map induces a map of corresponding PCS.

So let be a normed positive dual pair map of norm less or equal to .

Define the matrix by


Let and . Then

Thus is defined by matrix in the sense of formula (14). Also , since , and is a contraction.

It follows that the coherent cone map induces the map of corresponding PCS by means of matrix (15), and we have the theorem.

Theorem 3

The category of PCS and PCS maps is a full subcategory of .

It can be shown that the model of linear logic in described in this paper induces the model of [4] when restricted to PCS. We will not go into these routine details here.

5.2 Tensor product

We now show that is a -autonomous category, that is, we define tensor and cotensor products and internal homs.

Let , be coherent cones.

The algebraic internal homs cone is the set of bounded continuous maps from to equipped with standard norm (6).

This cone can also be described as a cone of bilinear functionals.

Let us say that an -bilinear functional is bounded, if


Consider the cone of bilinear separately continuous functionals on bounded in the above norm (the algebraic cotensor product).

Note 10

The normed cones and are isomorphic.

Proof The isomorphism sends a map to the functional

In the opposite direction, if , then for any the functional is continuous on , hence it is represented as an element of . This gives us a linear map from to , . This map is adjointable, with the adjoint given by the similar map , hence it is continuous by Note 7.

Let us topologize our cones.

We equip the cone with the topology of pointwise convergence on .

Accordingly, the cone gets the topology of pointwise convergence on under the identification .

Note 11

The cones , are complete.

Proof Indeed, if is a norm-bounded non-decreasing sequence in , then for any the sequence is non-decreasing and bounded in norm by

(since norms on reflexive cones are monotone). Hence it has a limit in . Thus we define

Obviously, this is adjointable with the adjoint given by the similar formula

hence it is continuous. It is also bounded with norm less or equal to .

So . But , seen as a functional on , is precisely the pointwise limit of .

The cone will be eventually the cotensor product of , . But we need the second element of the dual pair.

Consider the algebraic tensor product of -modules over .

We have the pairing between and , given for any , where all , , and any functional by the formula

We equip the cone with the norm

and the topology of pointwise convergence on .

Now, the topology on (of pointwise convergence on ) is just the topology of pointwise convergence on . So , and is reflexive.

We define the tensor product of coherent cones , as the completion of ,

The cotensor product of , is the cone equipped with the topology of pointwise convergence on . The cone is complete, so by Theorem 2 the cone is complete as well. So is a coherent cone.

Finally, the internal hom-space is defined as the cone equipped with topology of .

Theorem 4

The category is -autonomous, i.e. there is a natural bijection .

Proof By pure algebra there is a natural bijection

in the category of reflexive normed cones. The theorem follows then from Theorem 1.

5.3 Additive structure

The main use of norms is that they allow defining (non-degenerate) additive structure.

The product and, respectively, the coproduct of normed cones and are defined as the set-theoretic cartesian product , equipped respectively with norms




The trivial coherent cone comes from the trivial cone (or vector space) .

It is immediate that the above operations define indeed categorical product and coproduct in . Furthermore we have

and is not isomorphic to unless or .

6 Exponentials

Recall that linear logic exponential connectives allow one to recover the expressive power of intuitionistic logic.

In particular the -modality allows multiple use or waste of a formula in a proof, thus making a bridge between “linear” multiplicative conjunction and “intuitionistic” additive or context-sharing , which is summarized in the exponential isomorphism

Also, the intuitionistic implication inside linear logic defined as

provides an embedding of intuitionistic logic.

This is reflected on the semantic side. A corresponding functor on a -autonomous category produces a model of intuitionistic logic, whose maps from to come from -maps from to .

An accurate formulation is in terms of a linear-nonlinear adjunction. We briefly describe this structure; however, see [13] for full details.

6.1 Abstract model theory

Recall that a linear-nonlinear adjunction consists of the following data:

  • a -autonomous category (i.e. a model of multiplicative linear logic);

  • a category with a cartesian product and a terminal object ;

  • two functors and , which are monoidal, i.e. there are natural transformations


    and moreover are lax symmetric monoidal, which means that the above natural transformations commute in a reasonable way with various combinations of associativity and symmetry morphisms for - and - monoidal structures;

  • an adjunction between and , i.e. a natural bijection


    which is a symmetric monoidal adjunction.

The last condition, i.e. that adjunction (21) is symmetric monoidal is equivalent (see [13], Section 5.17) to saying that the monoidal functor is strong, i.e. that natural transformations (20) are isomorphisms.

Theorem 5

[13] In the above setting the endofunctor equips the category with a model of the linear logic -connective.

We now go to the concrete case of coherent cone. As expected, the exponential connectives will be interpreted as spaces of power series, which can be understood as analytic functions and analytic distributions. Using these ingredients, we will construct a category of analytic maps between coherent cone. This will be the second member of a linear-nonlinear adjunction providing us with a model of the exponential fragment.

6.2 Symmetric tensor and cotensor

At first we discuss symmetric (co)tensor algebra in . Let be a coherent cone.

For any