LINEAR LOGIC, -AUTONOMOUS CATEGORIES and COFREE COALGEBRAS1 R.A.G. Seely Girard 1987]

LINEAR LOGIC AUTONOMOUS CATEGORIES

AND COFREE COALGEBRAS

RAG Seely

ABSTRACT A brief outline of the categorical characterisation of Girards

linear logic is given analagous to the relationship b etween cartesian closed cat

egories and typed calculus The linear structure amounts to a autonomous

category a closed symmetric monoidal category G with nite pro ducts and a

closed involution Girards exp onential op erator is a cotriple on G which

carries the canonical comonoid structure on A with resp ect to cartesian pro duct

to a comonoid structure on A with resp ect to tensor pro duct This makes the

Kleisli category for cartesian closed

INTRODUCTION In Linear logic JeanYves Girard introduced a logical

system he describ ed as a logic b ehind logic Linear logic was a consequence of his

analysis of the structure of qualitative domains Girard he noticed that the

interpretation of the usual conditional could b e decomp osed into two more primitive

notions a linear conditional and a unary op erator called of course which

is formally rather like an interior op erator

X Y X Y

The purp ose of this note is to answer two questions and p erhaps p ose some others

First if linear category means the structure making valid the prop ortion

linear logic linear category typed calculus cartesian closed category

then what is a linear category This question is quite easy and in true categorical spirit

one nds that it was answered long b efore b eing put namely by Barr Our intent

here is mainly to supply a few details to make the matter more precise though we leave

many more details to the reader to p oint out some similarities with work of Lambek

see these pro ceedings and to app eal for a change in some of the notation of

Girard

Second what is the meaning of Girards exp onential op erator Since Girard

has in fact oered several variants of in and another in Girard and Lafont

one cannot b e to o dogmatic here but some certainty as to the minimal demands

makes is p ossible in particular we show that ought to b e a cotriple and its Kleisli

category ought to b e cartesian closed in order to capture the initial motivation of the

exp onential This is already implicit in equation

Mathematics Subject Classication Revision G A

Partially supp orted by grants from Le Fonds FCAR Queb ec

RAG Seely

Acknowledgement This note should b e regarded as a gloss on Girard

providing the categorical context and terminology for that work I think the categorical

setting provides a genuine improvement and in particular indicates how the notation

may b e made clearer Others have come to similar conclusions elsewhere in this volume

De Paiva considers these matters giving a fuller discussion of the interpretation

of A as the cofree commutative comonoid over A in the context of Dialectica

categories I would like to thank Michael Barr for p ointing out that he had considered

the essence of linear categories in thus giving further evidence of the unreasonable

inuence of category theory in mathematics

LINEAR LOGIC There are several variations in the style Girard uses to present

linear logic eg one sided sequents in and traditional sequents in Girard and

Lafont I think the essence of the structure esp ecially its symmetry is clearest

when sequents in the style of Szab os and Lambeks p olycategories Szabo are

used here a sequent has the form

A A A B B B

1 2 n 1 2 m

Of course formally this is just an ordered pair of nite sequences actually sets

would do of formulas The commas on the left should b e thought of as some kind of

conjunction those on the right disjunction Better think of the A on the left as data

each to b e used exactly once and of the B on the right as p ossible alternate resp onses

Definition A prop ositional linear logic consists of formulas and sequents For

mulas are generated by the binary connectives and and by the unary

op eration from a set of constants including I and and from variables

Sequents consist of ordered pairs of nite sequences of formulas as ab ove actually

nite sets of formulas would b e b etter in view of perm b elow but let us pass over this

p oint The sequents are generated by the following rules from initial sequents ie

axioms which include the following Greek capitals represent nite sequences sets of

formulas

Axioms

id A A

IR I L

R L

d A A d A A

Rules

perm for any p ermutations

A A

cut

Linear Logic and Categories

A B

v ar

B A

R IL

A B A B

L R

A B A B

R L

A B A B

L R

A B A B

L R

A B A B A B

A B A B

L R

A B A B A B

Remarks Concerning notation In Girard a somewhat dierent

notation is used I have made changes so as to use wherever p ossible notation that is

standard from a categorical viewp oint This table summarises the changes

Girard A P or t or u

Here A I

Symbols not changed

I b elieve it is more imp ortant to pair the connectives by de Morgan duality with

with than by distributivity considerations as would justify Girards t with u

Furthermore and seem to really b e cartesian pro duct and categorical sum so those

symbols seem more appropriate than Girards particularly his I must confess to

b eing unable to nd an entirely satisfactory notation for the de Morgan dual to tensor

pro duct either in words dual tensor seems preferable to cotensor or tensor sum

or to Girards par or in symbols has b een chosen for its neutrality might have

b een b etter were it not already so widely in use elsewhere

The following sequents may b e derived

m A B A B A B A B

AB AB

e A A B B

1 1

A B A A A B

AB AB

2 2

B A B A B B

AB AB

s A B B A

a A B C A B C a A B C A B C

AB C

0 0

and similar sequents s a for

RAG Seely

A B B A

It is easy to see that m is equivalent to R w to L e

AB AB AB

to L to L to R in the presence of cut Indeed the

AB AB

rules amount to building in the required amount of cut to allow cutelimination to go

through As for symmetry and asso ciativity these follow from the rule perm and the

implicit asso ciativity of concatenation We give a as an illustration

AB C

(m) (m)

B C B C A B C A B C cut

per m A B C A B C

L C A B A B C

C A B A B C per m

L A B C A B C

(a)

A B C A B C

As for it is given by

(e)

A A B B v ar per m

R per m B A B A

A B B A

If we are to characterize the notion of a linear category we must complete the

description of linear logic as a deductive system in the sense of Lambek and Scott

First we must add the equations b etween derivations of sequents needed to get

the structure of a p olycategory Szabo these equations essentially make cut

into a p olycomp osition of p olyarrows which is asso ciative partially commutative

and has units id Analagous equations for multicategories may b e found in this

volume in Lambek for this reason I will not go into detail here for these or the

remaining equations Next we must account for the monoidal structure of I and

their duals by adding equations which make sequents A A B B

1 n 1 m

equivalent to sequents A A B B Clearly there are maps using

1 n 1 m

the evident hom notation

A A B B A A B B

1 n 1 m 1 n 1 m

given by the rules L R L R cut the p oint is that these maps b e isomorphisms

and inverse to each other

Similarly it is likely that we want the structure to b e symmetric monoidal closed

and have nite pro ducts and copro ducts each of these adds to the list of equations in

the evident way For instance a and a must b e inverse as must s and s

AB BA

This last could b e weakened if we only want a braided monoidal category as in Joyal

and Street However this would complicate the rest of the structure so we shall

not pursue this further Moreover R should give a bijection

Linear Logic and Categories

A B A B

whose inverse is

(e)

A B A A B B

cutperm

A B

Finally it seems that is a contravariant functor in view of v ar that it is

strong in view of and that it is an involution in view of d which thus must

b e inverse to d These yield further equations including the following if we are to

have a autonomous category as dened in Barr for any A B these derivations

of the sequent A B A B are equal d d Here

:B :A AB

d d is a case of the schema

(f ) (g )

C A B D

(f g )

A B C D

given by

(e) (g )

A A B B B D cut

(f )

C A A A B D cut

R C A B D

A B C D

and is a case of cut viz in general

:B :A AB

(f ) (g )

A B B C

cut

(g )(f )

A C

Since the required equations may b e easily generated from the ab ove recip e and are

in essence to b e found in the references given for the most part and since this pro cess

is familiar for instance to that of Lambek and Scott for calculus I shall

avoid the messy notational baggage needed to make all the details explicit by stating

b oldly and without discussion

Definition A linear category G is a autonomous category with nite pro ducts

Remarks For a fuller discussion of autonomous categories see Barr Here

just let me say that G is a closed symmetric monoidal category G with an involution

G G given by a dualising ob ject in our notation this means A A and

the canonical arrow A A is an isomorphism Barr uses for our

In such a category the existence of nite copro ducts follows from nite pro ducts by de Morgan duality

RAG Seely

Proposition Given any linear logic L a linear category GL may b e constructed

whose ob jects are formulas and whose morphisms A B are equivalence classes of

derivations of sequents A B given any linear category G a linear logic LG may b e

constructed whose constants are the ob jects of G and whose axioms are the morphisms

of G Furthermore G GLG and in a suitable sense L is equivalent to LGL

THE EXPONENTIAL OPERATOR In Girard these rules are given

in our notation for the op erator

der dereliction

thin thinning or weakening

A A

contr contraction

In means A A A Girard actually gives the rules for the de Morgan

1 2 n

dual we shall not discuss

It is p erhaps worth simplifying these rules

Proposition In the presence of linear logic

der is equivalent to the scheme A A

thin is equivalent to the scheme A I

contr is equivalent to the scheme A AA

if der thin contr then is equivalent to

A B

A A and f un

A B

iso AB A B iso

i iso I i iso

Remarks f un arise from the case n of iso from the n case

and i iso from the n case

Notice these rules seem to imply that we should regard as a functor by f un

indeed a cotriple or comonad by and each A seems to b e a comonoid

A A

0 0

with resp ect to the monoidal structure I in view of Furthermore this

A A

comonoid structure seems to b e the image under of the canonical comonoidal structure

A A A with resp ect to the cartesian structure in view of iso

i iso These comments will take us straight to Denition

Proof of

Linear Logic and Categories

For apply der to id For der apply cut to

A A A

0 0

For apply thin to IR For thin use IL and cut with

A A

0 0

For apply contr to m For contr use L and cut with

!A!A

A A

Given is applied to id f un is the derived rule

A !A

A B der

A B

iso is

2 1

( ) ( )

f un A B B f un A B A

R A B A A B B

A B A B AB contr

A B AB

iso is

thin B B thin per m A A

A B A A B B R

A B A B

L A B A B

AB A B

Conversely given f un and the i isos we derive as follows

First treat the n case as I A ie as a sp ecial case of the n case This is

p ossible b ecause of i iso I A may b e thought of as A Then for n

b ecomes the rule

A A

given by

A A f un

( )

A A A cut A

1 1 1

A A

iso allows the case when n to b e reduced to the n case in view of the

evident induced bijections using the hom notation of

A A A A A A A A A

1 n 1 n 1 n

RAG Seely

If we imp ose the appropriate equations on derivations it is clear that we shall arrive

at the following structure

Definition A Girard category consists of a linear category G together with a cotriple

G G satisfying the following

i for each A of G A is a comonoid with resp ect to the tensor structure

0 0

A A

I A AA

ii there are natural isomorphisms AB A B I moreover takes

the comonoid structure A A A with resp ect to the cartesian

structure to the comonoid structure in i ie these diagrams commute

0 0

A AA A I

k o k o

! !A

A A A A

Remark In fact it is easy to note that i follows from ii the diagrams dening

and However in view of the uncertainty surrounding it seems b est to keep all

the rules sep erate

As b efore we claim without further ado

Proposition Given a linear logic L with exp onential op erator a Girard category

may b e constructed on GL given a Girard category G the linear logic LG can b e

equipp ed with an exp onential op erator These constructions extend the equivalences

of Prop osition in the evident way

The essence of Girards translation of intuitionistic logic into linear logic is the fol

lowing

Proposition If G is a Girard category then the Kleisli category KG is cartesian

closed

Proof For a basic reference on the categorical notions of cotriple comonoid and

Kleisli category see Mac Lane Recall that the ob jects of KG are those of G

itself while the morphisms are given by

H om A B H om A B

K(G)

Writing A B for exp onentiation B in KG X Y for the internal hom in G it

seems likely that

A B A B

will do the trick It is an easy matter to verify that

i the terminal ob ject of KG is the terminal ob ject of G

ii the pro duct A B of KG is the same as in G

iii A B is A B

Linear Logic and Categories

Appropriate bijections are given by

C A B in KG

C A B in G

AC B in G

C A C B in G

A C B in G

C A C B in KG

A C B in KG

Remarks In general KG do es not have copro ducts Girards interpretation

of disjunction

A B AB

is mysterious from this p oint of view for the appropriate maps do not lie in KG

though we do have a glimmer of the correct copro duct structure viz the bijections

AB C in G

A C B C in G

A C B C in KG

In Girard and Lafont a stronger structure is considered for with

the intention that A I A AA What Girard and Lafont seem to require

again since they do not give a deductive system but only a logic we are left to supply

appropriate equations b etween derivations amounts to A b eing the cofree commutative

comonoid over A This condition implies and is stronger than that G is a Girard

category the question is whether it is to o strong The structure of coherent spaces and

linear maps in the next section do es not have this extra prop erty for instance nor is

the structure of De Paiva an example since it is not autonomous It seems

to me that what is really wanted is that KG b e cartesian closed so the question is

what is the minimal condition on that guarantees this ie can we axiomatise this

condition satisfactorily L Romanasked a related question what condition on makes

the EilenbergMo ore category cartesian closed

In Girard the prop ositional system is extended to a predicate linear logic by

V W

the addition of free variables of appropriate types and quantiers sub ject to the

rules

Ax t

with the usual restriction on R x not free in

W W

The rules for are given by de Morgan duality as is itself

RAG Seely

Girard is not sp ecic ab out the nature of the types here we may supp ose for

example that the ab ove amounts to the following categorical structure

An indexed linear category consists of a category S with nite pro ducts and an

indexed category G over S for each S of S the bre G is a linear category whose

structure is preserved by t t any morphism of S furthermore each has b oth adjoints

W V

a a where is a pro jection morphism of S The idea here of course is

that G consists of the linear formulas with free variable of type S and equivalence

classes of derivations of such formulas To b e certain the logic is prop erly bred

in this way we ought to add conditions to the rules of inference to ensure that in any

derivation of prop ositional linear logic the same variables app ear throughout and in the

quantier rules the only variables lost are those explicitly indicated such restrictions

are analogous to those of Seely for rst order intuitionistic logic and for

p olymorphic calculus and cause no loss of expressive p ower with a lib eral use of

dummy free variables

W V

As with the logic the adjoints are dual and so one only need assume one

exists This is analogous to the situation for cartesian pro duct and sum in autonomous