May 31, 1991 - ... Jean-Yves Girard, Carolyn Brown, Doug Gurr, Harold Schllinx, Andy .... It is clear, for those who know Chu's construction, Barr'79], that the ...
Categorical Multirelations, Linear Logic and Petri Nets (DRAFT) Valeria de Paiva May 31, 1991
Abstract This note presents a category of multirelations, which is, in a loose sense a generalisation of both our previous work (the categories GC, [dP'89]) and of Chu's construction AN C [Barr'79]. The main motivation for writing this note was the utilisation of the category GC by Brown and Gurr [BG90] to model Petri Nets. We wanted to extend their work to deal with multirelations, as Petri Nets are usually modelled using multirelations pre and post. That proved easy enough and people interested mainly in concurrency theory should refer to our joint work [BGdP'91]; this note deals with the mathematics underlying [BGdP'91]. The upshot of this work is that we build a model of Intuitionistic Linear Logic (without modalities) over any symmetric monoidal closed category C with a distinguished object (N; ; ; e ?) { a closed poset. Moreover, if the category C is cartesian closed with free commutative monoids, we build a model of Intuitionistic Linear Logic with a non-trivial modality `!' over it.
Introduction
This note extends the treatment of relations in the category GC { which is an interesting model of (Full Intuitionistic) Linear Logic [dP89], see sequent presentation in the appendix { to multirelations over a category C where C is a category with a distinguished (ordered) object (N; ). In particular we discuss the case of multirelations [Wins88] in the category Sets, where (N; ) is the set of natural numbers N with the opposite of its usual ordering. The main motivation for writing this note was the utilisation of the category GC by Brown and Gurr [BG90] to model Petri Nets. The idea of using category theory to model Petri Nets originates with Glynn Winskel, who attributes some of the insights to Mike Fourman. Brown and Gurr following Winskel's lead and also Girard's dictum that Linear Logic ought to relate nicely to Concurrency, used the category GC to model Petri Nets, but as GC dealt with relations, they could only account for particular Petri Nets, called in some of the literature elementary Petri Nets. Elementary Petri nets are nets where the relations pre and post have multiplicities restricted to 0 ? 1. It seemed to us that it should be easy to extend the treatment in [BG90] to modelling Petri nets with multiplicities; that turned Small corrections were made in August 1991. I would like to thank Pino Rosolini and Thomas Streicher for their comments.
1
to be the case and people interested mainly in concurrency theory should consult [BGdP'91]. This note deals with the mathematics underlying [BGdP'91], which could not be all explained in that paper. The construction described here can also be seen as a common generalisation of the constructions of the category GC cf.[dP89] and of GAME K of Yves Lafont [YL'88], [LS'91]. Note that, as the category GAME K can be seen as a special case of Chu's construction AN C - cf. the appendix of -Autonomous Categories [Bar79] - this note compares GC and Chu's category. In the rst section we present our variation of Chu's construction, which we call the category MN C and compare the two constructions. We also state a proposition interesting from the abstract viewpoint, but not explicitly used anywhere in the paper and whose proof, by Dominic Verity, would make this note even longer. This proposition shows that one of Chu's main results, that his category was enriched over the base category C, is also true of our construction MN C. In the second section, to motivate the richer structure on the category MN C, we restrict ourselves to the case where C is the category Sets and N is the set of the natural numbers with the opposite of its usual order. (Lawvere's seminal work on metric spaces [Law] makes this order the sensible one to consider.) That is the interesting case for Petri Nets applications. In the third section we describe the multiplicative and additive structures of MN C in the general case. In the fourth and longest section we discuss Linear Logic modality `!' for MN C under the strong assumption that C is cartesian closed with free commutative monoids. This section is a straightforward generalisation of our previous results for GC, but the calculations are slightly more complicated and quite lengthy. Finally, in the last section we describe some of the possible generalisations of MN C under investigation and their possible applications. This work was rst presented at the Edinburgh Workshop in Concurrency, Petri Nets and Linear Logic in April 1990 and subsequently at the CLICS review meeting in Paris, September 90. Many thanks to these audiences, in particular to Martin Hyland, Jean-Yves Girard, Carolyn Brown, Doug Gurr, Harold Schllinx, Andy Pitts and Dominic Verity. Thanks also to Peter Dybjer, who made me write about the relationship between MN C and GAMEK .
1 Chu's Construction Revisited
In this section we de ne a category MN C and compare it to Chu's original construction in [Barr'79]. To construct MN C we assume that C is a symmetric monoidal closed category and N is a distinguished object of C equipped with a partial order. The objects of the category MN C are triples (U; X; ) where U and X are N is a morphism in C. We write this triple as (U ( 7 X ) objects of C and U X ! ) and call it the object A. To de ne the morphisms in MN C, rst note that the order in the object (N; ) of C induces an order on the homset C(U; N ) given by for f; g: U ! N f g
i
8u 2 U f (u) g(u)
7 X ) and B = (V ( Then for objects A = (U ( ) ) 7 Y ) in MN C, say that a morphism from A to B corresponds to a pair of morphisms in C, (f; F ), f : U ! V and F : Y ! X such that, in the following diagram,
2
U F U Y ???????????! U X
j
j jj #
f Y jj
#
V Y ??????????! N
we have (U F ) (f Y ) as morphisms in C(U Y; N ). Diagramatically we have:
U ???7??? X
j
f jj
#
+
" jj F j
8u y 2 U Y (u Fy) (fu y)
V ???7??? Y
The data above can be collected in the following de nition. De nition 1 Given a symmetric monoidal closed category C with a distinguished object (N; ) the category MN C consists of:
7 X ), where U X ! N is a objects are triples (U; X; ) written as (U ( ) morphism in C; morphisms are pairs of maps (f; F ) in C, f : U ! V and F : Y ! X , such
that in the following diagram
U F U Y ???????????! U X
j
j jj #
f Y jj
# V Y ??????????! N
we have (U F ) (f Y ) as morphisms in C(U Y; N ). Identities in MN C are identities of C in each coordinate, composition is given by composition in each coordinate and associativity comes from the associativity in C. Thus we have the following proposition.
Proposition 1 The description above de nes a category MN C. The only thing to check is composition of morphisms (f; F ): A ! B and (g; G): B !
C , which is easily done using the diagram:
3
U ???7??? X
#
" jj F j
8u y 2 U Y (u Fy) (fu y)
j g jj #
" jj G j
8v z 2 V Z (v Gz ) (gv z )
j
f jj
V ???7??? Y
W ???7??? Z
We clearly have composite morphisms gf : U ! W and FG: Z ! X in C. Now for u 2 U and z 2 Z , we have (u FGz ) (fu Gz ) (gfu z ), which shows that (gf; FG) is a morphism in MN C. 2 Note that composition really corresponds to the following diagram:
U G U F U Z ???????????! U Y ???????????! U X
j j j jj f Y f Z j = # # V Z ??????????! V Y V G j @@ g Z jj @& # W Z ?????????????? ????????????!
jjj j j j jj #
N
1.1 First Comparison
It is clear, for those who know Chu's construction,[Barr'79], that the category MN C is { in one sense { an easy generalisation of it. To keep this note self-contained we recall the de nition of Chu's category, which he calls AN { A for autonomous { . There is a small clash of notation, as Chu uses X for the distinguished object, which we call N and he does not mention the base category C, whereas we want to have it explicit. Thus we write Chu's category as AN C. The category AN C can be constructed over any symmetric monoidal closed category C with pullbacks, see de nition below.
De nition 2 Given a symmetric monoidal closed category C with pulbacks and a chosen object N , the category AN C consists of Objects are triples (U; X; ), where U X ! N is a morphism in C; Morphisms in AN from an object U X ! N to an object V Y ! N are pairs of morphisms in C, (f; F ) f : U ! V and F : Y ! X such that the following diagram commutes.
4
U F U Y ???????????! U X
j
j jj #
f Y jj
#
V Y ??????????! N
The commutativity of the diagram above means that for all u y in U Y one has (u Fy) = (fu y) in AN C, when we only ask in MN C that (u Fy) (fu y), so MN C is a generalisation of AN C. On the other hand, MN C `looks like' a special case of Chu's construction, as we need a partial order on the object N , whereas for Chu's construction any object in C will do. But one could say that Chu is using the trivial partial order in N , the one which says n m i n = m. To compare Chu's construction with GAMEK note that the category Sets is trivially symmetric monoidal closed - as it is cartesian closed. Also instead of writing N for the distinguished object write K to make explicit the analogy with vector spaces. Thus GAMEK is AK Sets in our notation, but note that GAMEK arises from a more restricted case in Linear Algebra, where U is a vector space over a eld K and X is the collection of linear functionals on U , U . To compare MN C with GC, rst recall that we can write usual relations on Sets either as a subset of a product A ,! U X or as a function into 2, U X ! 2. Thus to talk about relations in a general categorical set-up one can use either subobjects of a (possibly tensor) product A ,! U X - call it the `subobject' 2 if your category has an object 2 - call it approach - or maps of the form U X ! the `span' approach. In our previous work with GC [dP89] we used the subobject approach. Thus objects in GC are (classes of equivalences of) monics A ,! U X and morphisms are pairs of maps in C satisfying the same conditions as the ones for MN C. Each approach to `categorical relations' has advantages and disadvantages. For instance, using the `subobject' approach one can talk about `decidable' and `undecidable' relations, if your ambiance category C has enough structure. This notion of `decidability' comes from topos theory, [PTJ page 162]. There is an interesting problem to look at if the constructions of this paper are carried inside a realisability universe, for instance the e ective topos, where other (recursion-theoretic) notions of decidability could be used. On the other hand, using the `span' approach and looking for instance at M C - where C is a ccc with coproducts - all your objects are decidable. Given a relation : U X ! 2, you can always produce : U X ! 2, saying that u x i it is not the case that u x. 2
Thus we can draw the following diagram
5
MN C
j jj #
ANj C
@
@@ &
jj #
GC
GAMEK and summarize as follows: The category GAMEK is AN C, where AN (?) denotes Chu's construction; N is the set K and C is the category of Sets. The categories AN C and MN C have the same objects, but there is an inclusion of the set of morphisms. The morphisms of AN C are contained in the morphims of MN C, thus for any pair of objects A and B AN C(A; B ) MN C(A; B )
The morphisms of MN C and GC are the same for any two objects A and B , MN C(A; B ) = GC(A; B ) but the objects of GC can only be compared with the objects of M C. For objects in M C we have jM Cj jGCj, as all objects in M C are decidable. Both categories AN C and MN C are symmetric monoidal closed categories 2
2
2
2
with nite products and coproducts. The symmetric monoidal closed structures are very similar, but di erent - more about that in section 3.
We now state a proposition, not necessary for the rest of this note, whose proof can be found in [Ver91]. The reason for mentioning the proposition is that its analogue for AN C was one of Chu's main results in [Barr'79]. Note, however, that to prove this proposition, we assume that C has pullbacks as well as being a symmetric monoidal closed category with nite products. Proposition 2 The category MN C is enriched over the category C. The next step is to de ne more structure in MN C. Given objects A and B 7 X ) and (V ( respectively, (U ( ) ) 7 Y ) in MN C, we want to de ne an internal hom ?7 U Y ), but the problem [A; B ]. The object [A; B ] should look like (V U X Y ( ) is to de ne a map ? : V U X Y U Y ! N with good properties. By good properties it is meant mean that we should be able to de ne an adjoint tensor product to the internal hom. That can be dicult in the general case, but if C is Sets and N really is N the set of natural numbers it is easy. We start with the easy case in the next section and then do the more general one in section 3. 6
2 The Category MSets
If we consider the construction of MN C where C is the category of sets and usual maps Sets and N is the set of natural numbers N , then our objects (U ( )7 X ) N cf. [Wins]. Recall that Sets is not only correspond to multirelations U X ! cartesian closed with coproducts, but a topos. A morphism (f; F ): A ! B in MN Sets corresponds to a condition on multirelations and saying that 8u 2 U; 8y 2 Y (u; Fy) (fu; y) as natural numbers, or equivalently that 8u 2 U; 8y 2 Y ? (u; Fy) + (fu; y) 0. Moreover the set of natural numbers N has some extra structure, apart from its usual order, that allows us to de ne a symmetric monoidal closed structure in MN Sets. For a start we can add natural numbers, so we can de ne a tensor product in MN Sets, which we call from now on MSets. Before de ning the tensor product, we summarize the discussion above in a de nition. De nition 3 The category MSets consists of: 7 X ), where U X ! N is a Objects are triples (U; X; ) written as (U ( ) function in Sets, that is a multirelation. 7 X ) to an object Morphisms in MSets from an object U X ! N or (U ( ) V Y ! N or (V ( ) 7 Y ) are pairs of morphisms in Sets, (f; F ) where f : U ! V and F : Y ! X are such that
U ???7??? X
j
f jj
#
" jj F j
+
V ???7??? Y
That means that 8u 2 U; 8y 2 Y ? (u; Fy) + (fu; y) 0.
This de nition is just an instance of the de nition of MN C in the rst section, so we have a category. Also recall that both (u; Fy) and (fu; y) are natural numbers, hence the condition above makes sense. We use the usual addition of natural numbers to de ne a tensor product in MSets: 7 X ) and (V ) De nition 4 Given two objects (U ( ) ( 7 Y ) in MSets we de ne A B
their tensor product as the following object:
A B = (U V ????7?????? X V Y U )
where the multirelation \ " is given by (u; v; f; g) = (u; fv) + (v; gu). To give a formal de nition of , consider the composition:
\hev; evi" + U V X V Y U ?????????????! U V X Y ???????????! N N ??????! N 7
Note that we are using the cartesian closed structure of Sets - to write X V and Y U - and the monoidal structure `+' of N to de ne . This operation clearly de nes a bifunctor, which is a tensor product. Associativity and commutativity are straightforward and the object I = (1 ( )7 1) - where the multirelation 1 1 ! N `picks' the 0 of N - is the identity for this tensor product. Actually this odd-looking tensor product, analogous to the one in GC, is the right one to prove monoidal-closedness of MSets with respect to a (reasonably) intuitive internal-hom, which we proceed to de ne. 0
0
7 X ) and (V ) De nition 5 Given two objects (U ( ) ( 7 Y ) in MSets we de ne [A; B ]
their internal-hom as the object,
?
[A; B ] = (V U X Y ????7?????? U Y ): The multirelation \( ? )" is given by ( ? )(f; F; u; y) = ?_ (u; Fy)+ (fu; y), where the dotted subtraction is truncated subtraction, that is ?_ + = ? if and 0 otherwise.
The truncated subtraction in the de nition above is very intuitive after reading Lawvere's \Metric Spaces, Generalised Logic and Closed Categories"[Law], where the same kind of construction is done using the positive real numbers instead of the natural numbers. Proposition 3 The construction above de nes a bifunctor [?; ?]: MSetsopMSets ! MSets. Having de ned an internal hom and a tensor product we have the obvious: Theorem 1 The category MSets is a symmetric monoidal closed category with respect to the tensor product and the internal-hom [?; ?] de ned above. The proof is simple, one has to verify the natural isomorphism
HomMSets (A B; C ) = HomMSets (A; [B; C ])
This can be done by looking at the diagrams
U V ????7?????? X V Y U
j
f jj
#
W
???? ?7????? X " j jj F jj hf; F i j # ?
W V Y Z ????7?????? V Z U
" hF ; F i jj j
?????7????? Z 1
2
2
1
and calculating the sums. If the morphism (f; hF ; F i) is in HomMSets (A
B; C ), then we know ?( ) + 0, which means ? (u; F zv) ? (v; F zu) +
(f (u; v); z ) 0. But to show that the corresponding morphism (hf; F i; F ) is in HomMSets (A; [B; C ]) we have to show ? + ( ? ) 0, which corresponds to ? (u; F (v; z )) + [? (v; F uz ) + (fuv; z )] 0, which we know, if transposing is allowed. 2 1
2
1
2
2
1
1
2
In the next section we generalise the constructions of this section to categories other than Sets and to N 's other than the set of natural numbers. 8
3 Structure on MN C
The categorically-minded reader may have noticed that we used an \adjointness situation" in N to de ne the symmetric monoidal closed structure in MSets. That is we have used the facts that in N we can say n m, also n + m 2 N and ?_ n + m 2 N - that is `+' and `?_ ' are bifunctors and there is an adjunction: ?_ (m + n) + p 0 i ?_m + (?_ n + p) 0: Thus to generalise the construction of MSets we rst de ne \a symmetric monoidal closed poset" (N; ; ; ?; e), then we show how the closed structure of N allows us to de ne a symmetric monoidal closed structure in MN C, if C is symmetric monoidal closed with products. We should mention that Flagg has, independently, the same de nition of a (symmetric monoidal) closed poset in [Fla'90], but he really considers integral closed posets, the ones where the identity for the monoidal structure `e' is also the identity for an extra `additive' structure, exactly the condition we want to avoid. We give our de nition in two easy steps.
De nition 6 An ordered monoid (N; ; ; e) is a poset (N; ) with a given compatible symmetric monoidal structure (N; ; e). The structures are compatible in the sense that, if a b, we have a c b c, for all c in N . These are called ordered monoids in Concurrency Theory, but could as well be called and posets as in [HdP'91]. Note that to be very precise we should call the monoids above, symmetric ordered monoids, see [dP'91] for a slightly more general notion.
De nition 7 Suppose (N; ; ; e) is an ordered monoid and a; b 2 N . If there exists a largest x 2 N such that a x b then this element is denoted a ? b and it is called the relative pseudocomplement of a wrt b. A closed poset is an ordered monoid (N; ; ; e) such that a ? b exists for all a and b in N . Since we de ned a closed poset to be a restriction of the notion of a symmetric monoidal closed category to the category of Posets, we have an obvious proposition:
Proposition 4 A closed poset (N; ; ; e; ?) has the folowing properties: 1. a b c i a b ? c 2. If a b, then for any c in N , c ? a c ? b and b ? c a ? c; 3. As `e' is the identity for `' a e = a a implies e a ? a for any a in N . Note that set of the natural numbers with its usual ordering and operations addition and truncated subtraction - de ned in section 2 is a closed poset hN; _ . For other interesting examples of closed posets see [Flagg]. ; +; 0; ?i Having done the rst generalisation - to consider a closed poset, instead of the set of natural numbers - we now proceed to generalise the category Sets to any symmetric monoidal closed category C with nite products. Thus, suppose that C is a symmmetric monoidal closed category with nite products and that (N; ; ; e; ?) is a closed poset as above. Write [?; ?] for the internal hom and
for (its adjoint) tensor product in C, as well as for the cartesian product. Then we can construct the category MN C as in section 1 and one of the possible symmetric monoidal structures of MN C is given by: 9
7 X ) and B = (V ( De nition 8 Given two objects A = (U ( ) ) 7 Y ) in MN C we de ne A M B their tensor product as follows: ( )M A M B = (U V ?????7??????? [V; X ] [U; Y ]) The morphism \( )M " intuitively says ( )M (u v; hf; gi) = (u fv) (v gu), where is the monoidal structure in (N; ; ; e; ?). To de ne formally the morphism ( )M consider the following map, which
we call : V 1 U V [V; X ] U?!
eval U X ?! N (U V ) ([V; X ] [U; Y ]) U ?!
Similarly we de ne (U V ) ([V; X ] [U; Y ]) ?! N . Then to get we pair and and use the monoidal structure `' of N , as follows:
N (U V ) ([V; X ] [U; Y ]) < ; > ?! N N ?! Proposition 5 The construction above induces a bifunctor,
M : MN C MN C ! MN C covariant in both coordinates, which is a tensor product. The identity IM is given by (I ( )e7 1), where the morphism I 1 = 1 !e N just picks up the identity `e' from the closed poset (N; ; ; e; ?). Associativity and commutativity of M are easy to prove. Note that we are using the tensor product and the categorical product in C, as well as the tensor in N to de ne the tensor product M in MN C. Note also that M is not in general a categorical product, for instance we have no projections, even if C is a cartesian closed category. In our previous work with the categories GC, since C was cartesian closed, it was not clear that only a tensor product was necessary in the rst coordinate, whereas a real categorical product was necessary in the second coordinate, to make the de nition above work. But as before, this tensor product M is designed to make MN C monoidal closed, if we consider the following internal-hom. 7 X ) and B = (V ) De nition 9 Given two objects A = (U ) ( ( 7 Y ) in MN C we
de ne [A; B ]M their internal hom as follows:
( ? )M [A; B ]M = ([U; V ] [Y; X ] ??????7???????? U Y )
The morphism \( ? )M " intuitively says ( ? )M (hf; F i; u y) = (u Fy) ? (fu y), where ? is the `internal-hom' in N . The formal de nition of the morphism ( ? )M is similar to the de nition of M in MN C. First consider maps and : U Y [U; V ] U Y eval ([U; V ] [Y; X ]) (U Y ) 1 ?! ?! Y V Y ?! N
U Y [Y; X ] U Y U?!
eval U X ?! N ([U; V ] [Y; X ]) (U Y ) 2 ?! Then to obtain ( ? )M we pair and and compose the result with ?, considered as a map from N N to N :
? N ?! N N ?! ([U; V ] [Y; X ]) (U Y ) < ; >
10
As an illustration of how the structure of the closed poset N relates nicely to the categorical structure of C, note that if we consider the internal hom [A; A]M that is the object ?7 U X ) ([U; U ] [X; X ] ( ) there is always a morphism from IM to it,
????e?7????? 1 " j jj jj j # ? [U; U ] [X; X ] ????7?????? U X I
as C is symmetric monoidal closed with products and e (u x) ? (u x).
Proposition 6 The construction above induces a bifunctor [?; ?]M , contravariant in its rst coordinate and covariant in its second coordinate.
Having de ned both a tensor product M and an internal hom [?; ?]M , we want to prove that they provide MN C with a symmetric monoidal closed structure.
Theorem 2 The category MN C is a symmetric monoidal closed category. The proof is very simple, to verify the natural isomorphism:
HomMN C (A M B; C ) = HomMN C (A; [B; C ]M )
we look at the diagrams
U V ????7?????? [V; X ] [U; Y ]
j f jj #
W
" hf ; f i jj j
?????7????? Z 1
2
???? ?7????? X " j jj f jj hf; f i j # ? [V; W ] [Z; Y ] ????7?????? V Z U
2
1
If the morphism (f; hf ; f i) is in Hom (A B; C ), then given u v in U V and z in Z , we know ( )M (u v; hf z; f z i) (f (u v); z ). That means, by de nition of tensor, that (u f zv) (v f zu) (f (u
v) z ). But as N is a closed poset, 1
2
1
2
1
2
(u f zv) (v f zu) (f (u v) z ) () (u f zv) (v f zv) ? (f (u v) z ) 1
2
1
2
Now to show that (hf; f i; f ) is in Hom(A; [B; C ]M ) we have to show 2
1
(u f (v; z )) ( ? )(hfu; f ui; v z ) 1
2
But ( ? )(hfu; f ui; v z ) = (v f uz ) ? (fuv z ) thus we know exactly what we need to show, if transposing is allowed. 2 This proof is basically the same as the one in section 2 for Sets, which shows that we came up with the right de nitions for the generalisation proposed. 2
2
11
3.1 Second Comparison
In this section we want compare the symmetric monoidal closed structures of AN C and MN C, but it is slightly easier to compare AN Sets and MN Sets. As we noted before the two categories have the same objects, eg N and B = V Y ?! A = U X ?! N
and for morphisms
AN Sets(A; B ) MN Sets(A; B ) as any morphism in AN Sets satis es not only (u; Fy) (fu; y) but also the
converse. The internal-hom and the tensor product are very similar in shape, but di erent enough. To make this comparison meaningful we recall the structure of AN Sets albeit in a concise way. More details can be found in [Barr79], [Barr91], [L88], [LS'91]. The de nition of the internal-hom [A; B ]A in AN Sets, for objects A and B is given by the object ( ? )A (L (A; B ) ??????7???????? U Y ) 1
to use Lafont's notation. Chu writes L (A; B ) as V (A; B ). The object L (A; B ) is a subset of the internal hom [U; V ] [Y; X ], which as Sets is cartesian closed, can be written as V U X Y . The subset L (A; B ) is de ned by the following pullback: 1
1
1
L (A; B ) ????????! V U j j jj jj U # # X Y ??????Y??! N U Y 1
which intuitively means that
L (A; B ) = fh ; i j : U ! V; : Y ! X and (u; y) = ( u; y)g 1
1
2
1
2
2
1
The diagram above is a pullback so, in particular, it commutes. To de ne the morphism ( ? )A we want a map ( ? )M : L (A; B ) U Y ! N 1
so nothing more natural than taking the transpose of either of the two composition maps in the square above. The de nition of tensor product in AN Sets is similar, for two objects A and B , A A B is given by the object ( )A (U V ?????7??????? L (A; B )) 2
where L (A; B ), in Lafont's notation is a subset of the internal hom [V; X ] [U; Y ] 2
12
given by the following pullback:
L (A; B ) ????????! X V j j jj jj V # # Y U ??????U??! N U V 2
which intuitively means that
L (A; B ) = fh ; i j : V ! X; : U ! Y and (u; v) = (v; u)g That is equivalent to saying that L (A; B ) can be `represented' by maps : U V ! N; : U ! Y; : V ! X such that 2
1
2
1
2
1
2
2
1
2
(u; v) = (v; u) = (u; v) 2
1
But note that the map U V ! N is redundant, as it can be de ned in terms of and . As before, to de ne the tensor we want a map 1
2
: U V L (A; B ) ! N 2
so we take the transpose of either of the two composition maps in the pullback square. Chu does not explicitly describe the tensor product, as it can be de ned in terms of negation, as A A B = [A; B ? ]? Since AN C is a subcategory of MN C, the two symmetric monoidal closed structures can be compared within MN C.
Proposition 7 The tensor products ( )M and ( )A can be related by the following diagram:
( )M U V ?????7??????? X V Y U
" jj i j
j
1 jj
# ( )A U V ??????7??????? L (A; B ) 2
Similarly the internal homs can be related via the diagram
( ? )A L (A; B ) ???????7???????? U Y " j jj 1 i jj j # ( ? )M V U X Y ??????7???????? U Y 1
13
In particular, the identities for the tensors can also be compared
i
1 ???7??? 1
j 1 jj #
" jj ! j
id
1 ???7??? N One very nice thing about the category AN Sets or GAMEK is its relationship to Linear Algebra, cf. [LS'91]. Seely remarks in [See] that when Chu was writing about symmetric monoidal closed categories, Linear Logic had not been invented by Girard. Hence there is nothing about additives - nor about a `!' comonad - in Chu's original construction. Also, when following Girard's and Hyland's suggestions in Boulder 87, I wrote about the categories GC as models of Linear Logic, I knew nothing about Chu's construction. But additives were very easy to construct in GC, as they are in MN C.
3.2 Additive Structure in MN C
If C has nite coproducts - as well as being symmetric monoidal closed with products - and N is a closed poset as before, products and coproducts in MN C are very easy to de ne using their counterparts in C. The method is the same used for GC and subsequently for GAMEK and AN Sets: 7 X ) and (V ) De nition 10 Given two objects (U ( ) ( 7 Y ) in MN C we de ne their
categorical product as follows:
A&B = (U V ( )7 X + Y ) ? The morphism \ & " is given intuitively by & (hu; vi; x; y; ) = (u; x) (v; y ) &
0
1
But we do no operation to (u; x) and (v; y), as we either have (x; 0) or (y; 1), but never both, by de nition of the coproduct X + Y . More precisely & is given by the morphism ` + 0
?
(U V ) (X + Y ) = (U V ) X + (U V ) Y ?????????????! U X + V Y ?????????! N 1
2
It is easy :to check that this operation de nes a bifunctor with identity given by 1M = (1 ( )7 0) - the empty multirelation - and that & is a categorical product. The projections are projections in the rst coordenate and canonical injections in the second coordinate. Similarly we have coproducts. 7 X ) and (V ) De nition 11 Given two objects (U ( ) ( 7 Y ) in MN C we de ne their
categorical coproduct
7 X Y ) A B = (U + V ( ) ?u; The morphism \ " is given by ( v; ; hx; yi) = (u; x) (v; y) 0
1
Again it is easy to check :that the bifunctor \" provides categorical coproducts and that 0M given by (0 ( )7 1) is the initial object. 14
Proposition 8 The category MN C has binary products and coproducts. It is clear that the category MN C above provides a model for Intuitionistic
Linear Logic, as described in the appendix. Theorem 3 The category MN C is a categorical model of Intuitionistic Linear Logic. The proof is trivial, as the constants IM ; 1M ; 0M and bifunctors ; ?; &; were de ned for it. 2 Observe that the additive structure of AN C is the same as that of MN C - or GC for that matter.
4 Modalities in MN C
This section should be considered as `work in progress', as we really would like to have the results for a category C symmetric monoidal closed with products, instead of for a cartesian closed category. But the calculations seem to be correct and the case of C cartesian closed is the important one for the Petri Nets applications. If we assume that C is a cartesian closed category - thus a fortiori a symmetric monoidal closed category with products - with free commutative monoids we can provide the linear logic modality `!' for MN C as a model of Intuitionistic Linear Logic. Note that, in particular the category Sets satis es all these conditions. The general idea - analogous once more to the previous work on CG - is to de ne comonads T and S in MN C and compose them to get another comonad called suggestively `!' in MN C. But comonads T and S come from monads (?)U , (?) and their composite (?) U , in C. Thus we have subsections for T , S and `!', as well as one subsection on the logical properties of the comonads. Recall that for C a cartesian closed category MN C simpli es slightly as it has as objects maps U X ?! N and as morphisms pairs of maps (f; F ) in C f : U ! V and F : Y ! X , such that in the following diagram
U F U Y ???????????! U X
j
j jj #
f Y jj
# V Y ??????????! N
we have (U F ) (f Y ) . Also we recap brie y our constructions of the last section, for C cartesian closed: the tensor product A M B in MN C is given by
7 X Y Y X ) (U V ( )
with identity I = (1 ( )7 1); the internal hom [A; B ]M is given by 0
?7 U Y ); (V U X Y ( )
15
categorical products A&B are (U V ( )7 X + Y ) &
with identity I = (1 ( ):7 0); and coproducts A B are
4.1 The comonad T
7 X Y ) (U + V ( )
The comonad T is as easy to de ne for MN C as it was for GC. Recall that any xed object U in a cartesian closed category C induces an endofunctor ( )U : C ! C
7!
X
XU
j
j jj f U #
f jj
# 7!
Y
YU
This endofunctor has a natural monad structure where the unit of the monad
1 U 2 X and the X! X is given by the transpose of the second projection U X ! monad multiplication (X U )U !1 X U is given by precomposing with the diagonal map : U ! U U , thus (X U )U = X U U X! X U . We summarize that in the
de nition below. De nition 12 For each object U in a cartesian closed category C we have a monad (( )U ; ; ) in C given by the natural transformations below: 1
1
X U U ?????????! X U
X ??????! X U
1
1
As they are monads, the endofunctors ( )U make the following diagrams commute: ( )U XU X U ??????? ????! X U U ?????????? X U 1
1
@ @@
j jj & # 1
XU
? ? .
?
X U U U ????????! X U U U 1
j jj #
j jj #
1
U X U U ????? ???! X 1
One important fact about the monads (?)U is that they also make the following diagrams commute. 16
Fact 1 The following diagrams commute. U U X U U ???????????! U X U
U U X ???????????! U X U
1
1
@ @@
j j j jj h ; evi h ; evi jj jj h ; evi # # # & U X U X U ????????????! U X h ; evi 1
1
1
1
That is a consequence of the fact that ( )U is the monad induced by the adjunction < U ; U ; ; " >: C * C[U ] also written as U a U , cf. [LSc'86]. We use the monads ( )U in C to de ne the comonad T in MN C and the fact above is used to show that T has a comonad structure. 7 X ) of De nition 13 The endofunctor T :MN C !MN C takes an object (U ( ) MN C to the object (U T )7 X U ), where intuitively the object T is given by T (u; f ) = ( (u; fu). In other words, the object T is given by the following composition:
h ; evi U X U ?????????????! U X ??????! N 1
If (f; F ): A ! B is a morphism in MN C, then T (f; F ) is given by
T U ???7????? X U
" jj F ( ) f j
j
f jj
#
V ???7???? Y V T
To show that T is an endofunctor in MN C we have to check the following diagram,
U Yf U FU U Y V ?????????????! U Y U ?????????????! U X U
j j jj h ; evi h ; evi jj = j # # U F j U Y ???????????! U X j j j jj jj f Y jj # # # V V Y ?????????????! V Y ????????????! N h ; evi j
f Y V jj
1
1
1
17
The functor T has a natural comonad structure inherited from the monoidal structure of the functors (?)U for U in C. Thus we have natural transformations : TA ! A and : TA ! T A in MN C given by, 1
2
1
T U ???7????? X U
T U ????7????? X U
" jj j
j 1 jj #
j 1 jj #
1
U ????7???? X
" jj j
1
T U ???7????? X U U 2
To show that these are morphisms in MN C we note that the following diagrams commute:
U U X U U ???????????! U X U
U U X ???????????! U X U
1
1
j jj T #
j
1 jj
#
U X ?????? ????!
j
1 jj
#
U X U U ???????????! T
N
2
j jj T #
N
Commutativity here is a consequence of the commutativity of the diagrams in fact 1.
4.2 The comonad S
Now to de ne a comonad S we assume free commutative monoids in C, analogous to what we did for the categories GC. Suppose C has strong commutative free monoids. By that we mean that there exists a functor F : C ! Monc C, which is left-adjoint to the forgetful functor U : Monc C ! C and the monad induced by this adjunction is a strong one. In more detail, recall that: Fact 2 The category MoncC consists of (commutative) monoid objects in C. That is objects in Monc C are triples (Y; Y ; Y ) where Y : 1 ! Y and Y : Y Y ! Y are morphisms in C such that the following diagrams commute: () Y 1 ???????! Y Y ???????? 1 Y
@ @@
j
jj
& # Y
? ? .
?
Y Y Y ??????! Y Y
j
jj
j jj #
# Y Y ??????! Y
Morphisms in Monc C are morphisms in C, which preserve the monoidal structure. There is an adjunction < F; U; ; " >: C * Monc C, also writen as F a U , which says that a map in C corresponds by a natural isomorphism, to a monoid homomorphism f in Monc C, as follows:
18
f X! U (Y; Y ; Y ) f (X ; X ; X ) ! (Y; Y ; Y )
We write ( ) for the composite functor U F : C ! C. Thus we have an endofunctor : C ! C given by,
X
7!
X
j
j jj f #
f jj
#
Y
7!
Y
Intuitively, X consists of commutative nite sequences of elements of X , hence we write an element of X as x, meaning x = hx ; x ; : : : ; xk i and f is just f in each coordinate. We summarize in the following de nition. 1
2
De nition 14 We denote by (( ) ; ; ) the (strong) monad in C corresponding to the adjunction F a U , which is given by the endofunctor ( ) = U F and the 2
2
natural transformations
X ?????????! X
X ??????! X
2
2
satisfying:
( ) X ??????! X ?????????? X 2
2
@ @@
j ? ? jj & # .?
X ????????! X
j jj #
j j #
j
2
2
2
X ????? ???! X
X
2
The unit of the adjunction F a U , the natural transformation : C ! C takes any object X of C to the carrier of the free commutative monoid X . Intuitively makes a singleton sequence of the element x of X and transforms a sequence of sequences into a sequence. Note that each free commutative monoid (X ; ; ) comes equipped with maps in C, 1 ! X and X X ?! X where intuitively picks the empty sequence h i of elements of X and concatenates - commutatively - sequences of elements 2
2
2
19
of X x and x . The following diagrams commute: 1
2
1 X ????????! X X ???????? X 1
@
@@ &
X X X ????????! X X
? ? ?
j jj #
j j # X X
.
X
j jj #
X j
????????! X
The co-unit of the adjunction : MoncC ! Monc C takes any free commutative monoid (X ; ; ) arising from an arbitrary commutative monoid (X; ; ) to itself. Thus : FU (M; ; ) = (M ; ; ) ! (M; ; ) where the morphism corresponds to `iteration' of the original multiplication . If the category C has a 0 object, then we have 0 = 1. In particular, if we are thinking of C as Sets and N the set of natural numbers with its additive monoidal structure (N; 0; +), as in section 2, we have (N ; ; ) = (N; 0; +). Intuitively that says (h i) = 0 and (hn ; n ; : : : ; nk i) = n + n + : : : + nk , which implies (hni) = n. More in general, if N is the closed poset (N; ; ; e; ?) we have (N ; ; ) = (N; e; ) and 2
1
2
1
2
2
(hn ; n ; : : : ; nk i) = n n : : : nk and (h i) = e: 1
2
1
2
In this stronger version the monad (( ) ; ; ) is a strong monad, so there are morphisms st Y X Y X ?!
Because we are considering free commutative monoids in C we have (X + Y ) = X Y
We also have a natural transformation m: U X ! (U X ). Intuitively m takes a sequence of u's and a sequence of x's to a sequence where each u is followed by a single x as follows:
m(u; x) = hu x; : : : ; uk xi = hu x ; : : : ; u xm ; : : : uk x ; : : : uk xm i 1
1
1
1
1
Using the natural transformations m we can de ne the endofunctor S below. 7 X ) of De nition 15 The endofunctor S :MN C !MN C takes an object (U ( ) S MN C to the object (U ( )7 X ), where, as intuitively x is hx ; x ; : : : ; xn i, S (u; x) 1
means (u; x ) and (u; x ) and : : : and (u; xn ). 1
2
2
The object S of MN C is de ned by the long morphism X U X ?! N N ?! m (U X ) ?! U X 2?!
20
If (f; F ): A ! B is a morphism in MN C, S (f; F ) is given by (f; F ) as follows
S U ???7????? X
" jj F j
j
f jj
# S V ???7????? Y
As an illustration of the conciseness of the notation used, recall that the small diagram above corresponds to the big one below:
U F ??????! U Y ?????????????! U X j j j jj jj Y jj X # # # U F U Y ?????????????! U X j j j jj jj m jj m # # # ( U F ) Y m U Y ?????????????! U Y ??????! (U Y ) ?????????????! (U X ) j j j j jj f Y jj (f Y ) jj f Y jj # # # # V Y ????????????! V Y ????m??! (V Y ) ?????????????! N Y U Y ????????????!
2
2
2
@@
2
To show that S is an endofunctor, remember that for the square most down to the right, we use that the functor ( ) preserves the order on morphisms in C, for the other squares we have equality. The endofunctor S has a natural comonad structure given by the monad structure of ( ) in C. Thus we have morphisms : SA ! A and : SA ! S A in MN C given by 2
S U ???7????? X
j 1 jj #
2
2
S U ????7????? X
" jj j
j 1 jj #
2
U ????7???? X
" jj j
2
S U ???7????? X 2
To show that the above are morphisms and in MN C we note that the 2
21
2
@&
N
diagrams below commute, which is a consequence of the following fact.
U U X ???????????! U X
U U X ???????????! U X
2
2
j j j jj S 1j # # U X ?????? ????! N
j j j jj S 1j # # U X ???????????! N S 2
Fact 3 The following diagrams commute: X U U X ???????????! U X ?????????????! U X 2
2
j
jj
#
N
S
? ? .
?
j jj m #
????? ???? N
?????? ??????? (U X )
m U X ?????????! (U X )
"
j jj #
X jj
j
U X
" @ S @ U jj @& j S U X ?????????! j X jj # U j X m jj # (S ) (U X ) ???????????! 2
N
j jj #
N
" jj j jjj j
N
Note that the rst diagram commutes because the transformation m applied to singletons is a singleton, m(hui; hxi) = hu xi; applied to a singleton is and (hni) = n. The second diagram says we can transform a sequence of sequences before applying the endofunctor S a second time.
4.3 The comonad ` ' !
Now we want to compose the two comonads T and S above, that is we want to de ne ! as S T . To give an intuitive de nition is easy: 22
7 X ) of M C De nition 16 The endofunctor !:MN C !MN C takes an object (U ( ) N to the object (U ( ) 7 X U ), where intuitively if : U ! X and u = hx ; x ; : : : ; xn i !
then ! (u; ) is given by (u; x ) and (u; x ) and : : : and (u; xn ). 1
1
2
2
But to de ne ! formally is a long process. First we de ne a composite monad ( ) U in C. Note that there is an endofunctor in C given by the composition of the monads ( )U and ( ) . Thus ( ) U : C ! C takes
X
(X )U
7!
j f jj # Y
j jj (f )U # (Y )U
7!
3 U The endofunctor ( ) U has a natural monad structure in C, its unit X ?! X is given by the composition of the units and as follows, 2
1
X ??????! X ??????! (X )U 1
2
To give a multiplication : X U U ! X U we rst note: We have the following series of natural transformations in C: 3
2 U X U ?! (U X U )
X U ?! (U X U ) U (X U ) ?! (U X U ) U a (U X U ) U (X U ) ?!
From the rst line to the second, we just take the exponential transpose. From the second to the third, we use the fact that if Y has a monoid structure, the same happens to Y U - Y is (U X U ) in this case - and the free monoids adjunction. The last step is just exponential transposition again. There is a natural transformation in C given by
X : (X U ) ! (X )U
23
De nition 17 To obtain the natural transformation it is enough to have the auxiliary map a: U (X U ) ! (U X U ) above, as we could compose it with ev : (U X U ) ! X and take the transpose. ev X a (U X U ) ?! U (X U ) ?! X X U (X U ) ?!
This is a distributive law of monads [Beck]. Thus we can use Beck's results and Finally we say that is given by the transpose of the long composition:
X
U U
3
2 U (X )U U ?! ev X ?! U hev; ?!2 i X U U ?! X
U 3 U X X U ?! Note that to de ne the multiplication we use and the distributive law , as well as evaluation on U twice, instead of . Note as well that as an endofunctor 3
2
the composition ( )U also makes sense, but it has no natural monad structure. We summarize the discussion above in the de nition, De nition 18 The endofunctor ( )U has a natural monad structure in C, with unit and multiplication given by the natural transformations 1
U 3 U X U ?! X
3 U X ?! X
We need another fact, a similar result was proved in the the work on the categories GC: Fact 4 The distributive law in C induces a distributive law of comonads in MN C, given by : TSA ! STA:
TS U ????7?????? X U
" jj j
j
1 jj
#
ST U ????7?????? X U
To check that is a morphism in MN C, we check the diagram
U U (X U ) ???????????! U X U
j 1 jj # U (X U ) ???????????! ST
j jj TS #
N
The composition monad (? )U above induces an endofunctor in MN C, which is the composition of S and T . We call this endofunctor `!', its intuitive de nition was given before. More formally, 24
De nition 19 The endofunctor ! in MN C acts on objects as 7 X ) = (U ( !(U ( ) ) 7 (X )U ) !
where the morphism ! is given by composition
;evi S N U (X )U h?! U X ?! If (f; F ): A ! B is a morphism in MN C, then !(f; F ) is given by
! U ???7??? X U
" jj F ( ) f j
j
f jj
#
V ???7??? Y V !
To show that ! is really an endofunctor we have to compose the squares we had before for T and S .
U F U U Yf U Y V ?????????????! U Y U ?????????????! U X U
j j jjj j jj h ; evi h ; evi j j # # U F j U Y ?????????????! U X j j j jj j jj S f Y j # # # V Y V ????????????! V Y ?????????????! N S h ; evi
f YV
1
1
1
The endofunctor ! in MN C has a natural comonad structure given by the monad structure of ( ) U in C. Thus : !A ! A and : !A !!!A are given by !
!
! U ????7???? X U
! U ???7??? X U
" jj j U ???7??? X
j 1 jj #
" jj j
j 1 jj #
3
3
!! U U ???7????? X U
To show that these maps are maps in MN C is just the composition of the diagrams we have shown to commute for T and S . Thus
U U U X U ???????????! U X U
U U X ???????????! U X U
3
3
j
1 jj
# U X ?????????!
j jj TS #
j jj #
U
U X U ??????????!
N
25
j jj #
N
After all the work above to de ne the comonad `!' we must show that it works. We do that in the next section, but before heading for the logic we need a last categorical proposition.
Proposition 9 The comonad `!' in MN C de ned above satis es !(A&B ) and !1 =!A !B =I Proof: By de nition of `!' we have: !1 = (1 ( )7 0 )
!B = (V ( ) 7 Y V )
!A = (U ( ) 7 X U )
!
!
!
1
By de nition of the product A&B , !(A&B ) =!(U V ( )7 X + Y ) = (U V ( )7 (X + Y ) U V ) !(
&
&
)
Taking the tensor product we have
7 X U V Y V U ) !A !B = (U V ( ) !
!
Thus to show the isomorphisms in MN C we have to show the following isomorphisms in C, 0 (X + Y ) U V =1 = X U V Y V U and that these isomorphisms induce isomorphisms in MN C. The isomorphisms in C are clear from that fact that (X + Y ) = X Y , which implies that (X + Y ) U V = X U V Y U V Actually that is the reason why we are taking commutative monoids in C.
2
4.4 Logical Properties of `!'
Now to show the logical properties of the comonad `!' we rst recall the rules for the modality of course! in Linear Logic. ?`B (weakening) ?; !A ` B
?; A ` B (dereliction) ?; !A ` B
?; !A; !A ` B !? ` A (contraction) (!) ?; !A ` B !? `!A Our next theorem show that the comonad `!' de ned in the last section really works. The details are very similar to our previous work as well as to Seely's work [See'87], to which we refer the reader. The basic idea is to show that the rules are sound by showing that, if there is a morphism in the category MN C between the objects which are the translation of the antecedent, then there is a morphism between the objects which translate the sucedent of each rule. Theorem 4 The comonad `!' in MN C satis es the rules for the modality `!' in Linear Logic. 26
It is clear that by virtue of being a comonad `!' satis es the rule (dereliction). To f wit, if there is always a morphism !A ?! A, whenever we have a map G A ?! B
G A to get G !A ?! B , which shows that we can compose it with G !A G?! the rule (dereliction) is sound. To show soundness of the rule (!) we need more. If there is always a map f f ! !!A and !G ?! !A ?! A, then we can apply the functor `!' to f , to get !!G ?! !A and if we precompose it with !G ?!!!G we get !G ?!!A, which shows the rule (!) is satis ed. But for this we are assuming that !? ` A corresponds to a morphism !G ! A, and to know that we use the previous proposition as !? =!G !G : : : !Gk =!(G &G & : : : Gk ) Next to show the soundness of (contraction) and (weakening) we show that !A is a comonoid for the tensor product M in MN C. It is easy to calculate comonoids with respect to M . They are objects, say A, equipped with a co-unit map to IM = (1 ( )7 1), A ! IM and a `diagonal' map A ! A A, satisfying some commutative diagrams. Then it is clear that !A in MN C is a comonoid with respect to M , as we have !A !A. Just check the diagrams: morphisms !A ! I and !A ! !
1
1
2
2
0
! U ???7??? X U
" jj c j
j
! jj
X U
" jj c j # ! ! U U ????7?????? X U U X U U j
jj
1
#
????! ?7?????
U
0 1 ???7??? 1
2
f !A !A to get G !A ?! B . Thus if G g!A !A ?! B we can compose it with !A ?! And if G ?! B we can compose it with !A ?! I to have G !A ?! B . 2
4.5 Comparing Modalities
Similarly to what happen with the monoidal closed structures of AN C and MN C, the exponential connectives !A and !M are comparable. In fact one could call L (A) the subset of the morphisms f j : U ! X g such that 8u 2 U; 8 (u) 2 X , if (u) = h (u) ; : : : ; (u)n i then (u; (u) ) = (u; (u) ) = : : : = (u; (u)n ). We have then the following morphisms in MN C. 3
1
1
2
(! )M U ????7?????? X U
" jj i j
j 1 jj #
(! )A U ?????7?????? L (A) 3
5 Further Work
The linear negation ( )? and the connective \par" of Linear Logic pose problems though in MN C, even if C is cartesian closed, even in Sets. 27
In our previous work, the bifunctor par was de ned in the dialectica categories using disjunction of relations, but for multirelations it is not clear how to de ne `disjunction' of natural numbers m _ n. We know from the work in GC that A2B 2 should look like (U Y V X ( ) 7 X Y ), but the problem is to de ne a well-behaved morphism 2 . One could try the following:
7 X ) and (V ( Conjecture 1 Given objects (U ( ) ) 7 Y ) de ne their\ par" as (U Y 2 VX ( )7 X Y ), where the multirelation 2 is given by 2 (f; g; x; y) = max( (fy; x); (gx; y))
The reason for the maximum of two natural numbers is that max looks a bit like logical or, if you think of of the truth table for _ and 0 means false. But then the identity for this par is (1 ( )7 1), so (the linear logic constants) I and ? would coincide. Other possibilities for the map 2 are to take multiplication of natural numbers or their minimum. But the reason none of these pars work is that they do not satisfy the weak distributive law [HdP'91] below: a2(b c) (a2b) c In the case of multiplication the weak distributive law would give us a (b + c) = a b + a c a b + c, only true if a 1 a contradiction. Also in the dialectica categories linear negation is de ned in terms of linear implication into a dualizing object \?" which is the identity for par. If one considers, as in Lawvere's paper \1" as an element of N that might induce a good choice for \?", but then linear negation is only de ned for a very small class of objects. In contrast to the situation described above, if one deals with the category AN C, one can de ne both `par' and linear negation. But then you must have models of Classical Linear Logic, as the duality A?? = A is built into the category. 0
Conclusions
Much work remains to be done. On the mathematical side we want to try to obtain the right level of generality and to prove the existence of the modality `!' when the category C is only symmetric monoidal closed. Some work, with Martin Hyland is in progress, generalizing the construction of MN C so that we model all of (full Intuitionistic) Linear Logic, see [HdP]. There are several questions as to how much of the work above can be done with non-symmetric monoidal closed categories. That would lead into non-commutative linear logic and there is some work in progress with Dominic Verity on it, as well as some other work on models of systems useful for linguistics purposes, see [dP'91]. On the applications side, there is some more work, besides [BGdP], with Carolyn Brown and Douglas Gurr on the connections between the models of Linear Logic that appear in Concurrency Theory. Finally, I would like to incorporate quanti cation into this general picture.
References [Bar] [Bar] [B&G]
M. BARR -Autonomous Categories, LNM 752, Springer-Verlag, 1979. M. BARR *-Autonomous Categories and Models of Linear Logic, to appear in JAAP, 1991. C. BROWN and D. GURR A Categorical Linear Framework for Petri Nets, LICS'90. 28
[BGdP] [Fla] [HdP] [PTJ] [LSc] [Law] [Laf] [LSt] [deP] [deP] [See] [Ver] [Wins] [Str]
C. BROWN, D. GURR and V.de PAIVA General Petri Nets and Simulation Morphisms, manuscript, Feb'91. R. C. FLAGG A Generalised Logic, manuscript, Fall'90. M. HYLAND and V.de PAIVA Lineales, manuscript Sept'90. P. Johnstone Topos Theory, Academic Press. J. Lambek and P. Scott Introduction to Higher-Order Categorical Logic, CUP, 1986. F.W. LAWVERE Metric Spaces, Generalized Logic, and Closed Categories, Rendiconti del Seminario Matematico e Fisico di Milano, 43 (1973). Y. LAFONT From Linear Algebra to Linear Logic, manuscript, November'88. Y. LAFONT and T. STREICHER Games Semantics for Linear Logic, to appear in LICS'91, Dec'90. V.C.V. de PAIVA A Dialectica-like Model of Linear Logic, LNCS 389, 1989. V.C.V. de PAIVA A Dialectica Model of the Lambek Calculus, manuscript, April'91. R. Seely Linear Logic, *-Autonomous Categories and Cofree Coalgebras, in Categories in Computer Science and Logic, AMS vol 92, eds. J. Gray and A. Scedrov, 1989. D. VERITY Enrichments, manuscript, Feb'91. G. WINSKEL A Category of Labelled Petri Nets and Compositional Proof System, LICS'88. T. STREICHER Adding modalities to GAMEK , personal communication at PSSL, March'90.
29
Appendix Intuitionistic Linear Logic We recall the axioms and rules of Intuitionistic Linear Logic, as in [Gir/L]. Axioms: A ` A (identity) ?`1
`I
?; 0 ` A
Structural Rules: ?`A (permutation) ? ` A
? ` A A; ?0 ` B (cut) ?; ?0 ` B
Logical Rules: Multiplicatives:
(unitl)
?`A ?; I ` A
?; A; B ` C ( l ) ?; A B ` C ? ` A ?0 ; B ` C (?l ) ?; ?0 ; A ? B ` C
? ` A ?0 ` B ( r ) ?; ?0 ` A B ?; A ` B (?r ) ? ` A ? B
Additives: ?`A ?`B ?; B ` C ?; A ` C (&r ) (&l ) ? ` A&B ?; A&B ` C ?; A&B ` C ?`A ?`B ?; A ` C ?; B ` C (r ) (l ) ?; A B ` C ?`AB ?`AB Note that sequents have only one formula on the right-hand side of the turnstile.
30
Full Intuitionistic Linear Logic We recall the axioms and rules of (Full Intuitionistic) Linear Logic. Axioms: A ` A (identity)
`I
? ` 1; Structural Rules: ?` (permutation) ? `
?`
?; 0 `
? ` A; A; ?0 ` 0 (cut) ?; ?0 ` 0 ;
Logical Rules: Multiplicatives: (unitl )
?` ?; I `
(unitr )
? ` A; ?0 ` B; 0 ( r ) ?; ?0 ` A B; ; 0 ? ` A; B; (2r ) ? ` A2B; ?; A ` B (?r ) () ? ` A ? B
?; A; B ` ( l ) ?; A B ` ?; A ` ?0 ; B ` 0 (2l ) ?; ?0 ; A2B ` ; 0 ? ` A; ?0 ; B ` 0 (?l ) ?; ?0 ; A ? B ` 0 ; Additives: ? ` A; ? ` B; (&r ) ? ` A&B; ?; A ` ?; B ` (l ) ?; A B `
?` ? ` ?;
?; A ` ?; A&B ` ? ` A; (r ) ? ` A B;
(&l )
?; B ` ?; A&B ` ? ` B; ? ` A B;
(*) Observe that in rule (?r ) we only deal with one formula on the right-hand side of the turnstile, according to our intuitionistic avour of Linear Logic.
31
