Semantical observations on the embedding of Intuitionistic Logic...

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Math. Struct, in Comp. Science (1995), vol. 5, pp. 41-68 Copyright © Cambridge University Press

Semantical observations on the embedding of

Intuitionistic Logic into Intuitionistic Linear Logic

SARA NEGRI

Dipartimento di Matematica Pura ed Applicata, Via Belzoni 7 - 35131 Padova, Italy

e-mail: [email protected]

Received 29 April 1993; revised 27 June 1994

1. Introduction

It is well known that Intuitionistic Logic can be faithfully embedded into (Intuitionistic)Linear Logic. A number of different solutions have been proposed for the embedding, ortranslation (cf. Girard (1987), Troelstra (1992), Danos et al. (1993)).

The purpose of this paper is to study the embedding from a semantical viewpoint,by investigating the relationship between various models for Intuitionistic Logic and forIntuitionistic Linear Logic. We will follow this pattern: given a structure that is a modelfor Intuitionistic Linear Logic we explain how to reconstruct inside it a structure that isa model for Intuitionistic Logic.

In the first section this procedure is worked out for the algebraic semantics of quantaleswith modality (cf. Rosenthal (1990), Troelstra (1992), Yetter (1990)). Here it is proved indetail that every quantale with modality gives rise to a frame included in it; furthermore,the isomorphism between the complete lattice of modalities on a quantale and the completelattice of subframes of a quantale is established. Finally, the largest subframe included ina quantale is described.

By a result due to G. Sambin, frames and quantales are representable by means offormal topologies and pretopologies, respectively (cf. Battilotti and Sambin (to appear)).In the appendix, an extension of these representation theorems to quantales with modalityis used to show that any quantale with modality is isomorphic to the class of saturatedsubsets of a suitable pretopology endowed with an operator, which was introduced inSambin (1989), and called the stable interior operator. As an application, we show how,given a formal pretopology with a stable interior operator representing a quantale, wecan obtain a formal topology representing the subframe determined by the modality.

All these semantical considerations lead us to consider, in the second section, 'natural'requirements on translations from Intuitionistic Logic into Intuitionistic Linear Logic.More explicitly, we define a faithful translation corresponding to the insertion of asubframe into its matching quantale with modality, and such that translated formulas areequivalent to their exclamation. Furthermore, we define a faithful translation which isalso a translation of schemes, that is which commutes with substitution.

In the final section we deal with categorical semantics, which has been extensivelystudied in Linear Logic literature (cf Barr (1991), Marti-Oliet and Meseguer (1991), de


Sara Negri 42

Paiva (1989), Seely(1989), etc.). We study sufficient (and in a way necessary) conditions

such that the co-Kleisli construction, applied to a category that is a model for Linear

Logic, gives rise to a categorical model for Intuitionistic Logic. In particular, we propose

an answer to a question, raised in Seely (1989), concerning the existence of coproducts in

the co-Kleisli category. Finally, we show that the algebraic construction developed in the

first part is just a particular case of this categorical one, namely its reduction to partial

orders.

2. Frames and quantales with modality

In what follows we will show how a quantale endowed with a modality gives rise to a

frame and that every frame included in a quantale can be obtained in this way.

We first review some definitions.

Definition 2.1. A quantale Q is a quadruple (Q, •, 1, \/) such that:

(1) (Q, •, 1) is a commutative monoid with unity 1;

(2) (8,V) i s a complete lattice;

(3) for any family (b;);e/ of elements of Q and any a G Q, infinite distributivity a • \JieI bt —

Vig/ cit • b holds.

One of the basic properties of quantales, which will be needed in the following, is stability:

(S) a<a' &b<b' -+a-a' <bb'.

Stability can be easily proved, in fact, by distributivity applied to the right-hand side,

we have (a • a') V (b • V) < (a V b) • {a! V b') ; by hypothesis a V b = b, a' V b' = b', thus

(a • a') y(b-b')<b- b', which is the claim.

In a more general framework, e.g. in Rosenthal (1990), commutativity and the presence

of unity are special features and are therefore explicitly stated. Since we deal with

commutative linear logic, our quantales will be assumed to be commutative, even if many

of the following results continue to hold in the non-commutative case.

Definition 2.2. Let S be a subset of a quantale Q. We say that S satisfies weakening if

a < 1 for all a G S and that S satisfies contraction if a • a = a for all a G S.

A complete lattice where meet is distributive over arbitrary join is a special kind of

quantale, in which the monoid and the meet operation coincide; it is known in the

literature as locale, frame or complete Heyting algebra, according to the notion of morphism

(cf. Mac Lane and Moerdijk (1992)). Here we consider morphisms preserving arbitrary

join and finite meet, namely frame morphisms.

The following proposition provides a useful characterization of frames.

Proposition 2.3. A quantale Q satisfies weakening and contraction iff a A b = a • b for all

a, b G Q, i.e. iff Q is a frame.

Proof. By stability applied to a < a and b < 1, we get a • b < a, and in the same way

a • b < b. Thus a • b < a Ab. By stability applied to a A b < a and a A b < b we have

(a A b) • (a A b) < a • b, thus, by contraction, a Ab < ab. The converse is obvious. •


Semantical observations on the embedding of Intuitionistic Logic 43

Definition 2.4. (cf. Rosenthal (1990)) In any poset P, a map fi : P —> P is called a coclosure

operator if it satisfies:

(a) fia < a;

(b) a < b —> fia < fib;

(c) \ia < fifia.

A coclosure operator on a quantale g such that fia • fib < fi(a • b) for all a, b G g is called

a quantic conucleus.

A coclosure operator on a frame L such that //a A fib < fi(a A b) (hence by monotonicity

fi(a Ab) = fia A //£>) for all a, b G L is called a conucleus.

A coclosure operator such that

(d) fi(a Ab) = fia- fib,

(e) fiT = 1, where T = Vg

is called a modality, (cf. Troelstra (1992)).

It is easy to prove that fi is a coclosure operator iff it satisfies the following:

(*) fia <b <-> fia < fib.

In the language of categories conditions (a), (b), (c) define a comonad in the partial order

Q-

Troelstra (1992) calls a quantale g with such a modality fi a complete IL-algebra with

storage. If, moreover, g is provided with an involution, that is a unary operation ~ such

that for all a,b G Q, a = ~ ~ a and a < b —»~ fo < ~ a, we obtain a structure that is easily

seen to be equivalent to a modal Girard structure as defined in Avron (1988). We say that

a G Q is a fi-element if a = /ic, for some c G g ; by idempotency of /z, a is a //-element iff

fia = a. We shall denote by Q^ the set of /i-elements of g.

We can give an alternative definition of the modality fi involving only the monoid

operation.

Proposition 2.5. A coclosure operator fi on g is a modality iff it satisfies the following

conditions:

(1) fia • fia = fia;

(2) fia- fib < fi(fia • fib) (hence fia • fib = fi(fia • fib));

(3) fiT = 1.

Proof. If fi is a modality on g, then (1) is an instance of (d). Moreover, since fia • fib is

a //-element by (d), then fia- fib = fi(fia • fib), thus (2) holds.

Conversely, from monotonicity and stability applied to a A b < a and a A b < b, we get

fi(a A b) • fi{a Ab) < fia • fib, which gives fi(a A b) < fia • fib, since by (1) the product of

//-elements is idempotent. As for the opposite inequality, first observe that monotonicity

applied to b < T gives, by (3), fib < 1. Thus, from fia < a and stability, we obtain

fia • fib < a. In the same way we get fia- fib < b, and therefore fia- fib < a Ab. Thus by

monotonicity and assumption (2) we have fia- fib < fi(a A b). •

We have a useful presentation of //-elements as suprema:


Sara Negri 44

Lemma 2.6. If /x is a coclosure operator on Q, then for all a e Q, fia = V{/*c : /xc < a}.

Proof. As /xa < a, we have /ia < V{/xc : /xc < a}; conversely, if /*c < a, then /ic < fia and

therefore V{/xc : fie < a} < fia. •

The following lemma lists some technical results we shall need later, and shows, in

particular, that any modality is a quantic conucleus.

Lemma 2.7. For any coclosure operator the following four conditions are equivalent:

(1) 2/j is closed under •

(2) fia • fib = jic for some c

(3) fia- fib = fi(fia • fib)

(4) fia • fib < fi(a • b)

Moreover, each of these together with

(5) fie < fie • fie

is equivalent to

(6) fi(a A b) = fia • \ib.

Proof. (1)=>(2),(1)=>(3) As fia, fib ! Q^, we have fia- fib ! Q, that is, fi(fia • fib) = fia- fib

with c = fia- fib.

(2)=>(1) and (3)=>(1) are obvious.

(3)=>(4) fia- fib = fi(fia • fib) < fi(a • b).

(4)=>(1) For all a,b e Q^, a • b = fia • fib, thus by hypothesis a- b < fi(a • b); since the

opposite inequality always holds, we get a • b = fi(a • b), that is, a • b G Q^.

As for the second statement, it is obvious that (6) implies (2) and (5); we are now going

to show that (3) and (5) imply (6): by stability applied to fia < a and fib < 1, we get

fia • fib < a and similarly fia • fib < b, hence fia • fib < a A b, therefore by monotonicity

fi(fia-fib) < fi(a/\b), which together with the hypothesis gives fia-fib < fi(aAb); conversely,

fi(a A b) = \/{fic : fie < a A b}= V{fic : fie < a & fie < b} < V{/xc : fie • fie < fia • fib}=

y{fic : fie < fia • fib} = fi(fia • fib), where inequality follows from stability, and the last

equality from the hypothesis. •

For any modality fi on Q, the relation

a <pb = fia <b

is a preorder on Q. Indeed, by (*), a <^ b iff fia < fib, and thus <f, is reflexive and

transitive because is also <.

The preorder <^ is not antisymmetric in general, and actually antisymmetry of <^ is

clearly equivalent to: fia = fib implies a = b, i.e. injectivity of fi. Thus, in order to get

antisymmetry, we consider, as usual, the equivalence relation =,, induced by the preorder,

that is, a =^ b = a <^b & b <M a, which then becomes

a =ftb = fia = fib.

Let Q/=ii be the quotient set of Q with respect to =ll and [a]^ = {b e Q : fia = fib}. Then

it is obvious that

[a] <» [b] = a<^b



is a (well-defined) partial order on Q/= , which is also denoted by <^, since no confusion

is likely to arise'.

The following lemma shows that the supremum of /i-elements is a /z-element. This fact

will be used to prove that Q/=ll can be given the structure of a frame.

Lemma 2.8. If fi is a coclosure operator, n{\Jieifiai) = Vje//ia,, thus the set of /^-elements

of g is a complete sub-lattice of g.

Proof. One inequality is an instance of (a); conversely, /*a, < V,e//xa,- gives, by (*),

[iat < /i(Vie/yua,) for any index i £ I, hence V,e;//a, < /i(V,e//zaj). •

Proposition 2.9. For every quantale g with modality fi, the quotient set Q/=ll is a frame

with the following operations:

[a] A" [6] = [a A ft].

Actually g/=( i is a quantale where the product

[a] •" [ft] = bia • fib]

and the meet coincide.

Proof. It is straightforward that V and AM are, respectively, the join and the meet with

respect to the partial order </1. We show that Q/=lt is a quantale where the monoid and

the meet operation coincide. The operation -̂ is associative, since fi(fia • fib) — fia • \ib,

and makes Q/=ll a commutative monoid with unity [T] = [1]. By the definitions and

the previous lemma, the distributive law [a] -̂ vf6/ [/>;] = vf6/ [a] * [bi\ holds, so Q/=ll

is a quantale. Finally, since fi(a Ab) = fia • fib, we have [a] A** [b] = [a] -^ [b], for all

W,We2/v •

In the frame Q/=/1, an operation of implication is defined as usual by:

[a] -> [b] = V{[c] ! 6/=, : [a] A" [c] < , [ft]}.

There exists an interesting connection with the implication a —o b = V{c e g : a-c <b)

of the quantale Q :

Proposition 2.10. For all [a], [b] e 6/=/1, [a] -»• [b] = \jia —o ft].

Proo/ We have the following chain of equalities: V{[c] e g/=(1 : [a] A'1 [c] <,, [ft]} =

V{[c] e 6/=, : W -̂ W <M [ft]} = [V{/ic e g : /zc < \ia —o ft}], by definition of the frame

structure of Q/=ll, (2) of proposition 2.5 and the adjunction existing in g between the

functors a • — and a —o —. Finally, since n(fia —o ft) e {fie e g : \ic < fia —o ft}, we have

fi(fia —o ft) < \/{fic G g : fie < fia —o ft}, that is \pa —o ft] <^ [a] —> [ft]. Conversely, from

' We could say that Q/= is the abstraction obtained from (Q, n) disregarding the difference between a and /xa,

that is, via the interpretation, between a formula A and its exclamation \A.


Sara Negri 46

v{/ic ! Q : IJLC < fia —o b} < fia —o b, one gets fi(\/{nc ! Q : fie < fia —o b}) < \ia —o b,

that is, [a] -> [b] <„ [p.a —o b]. t D

An equivalent, and somehow simpler, approach is to consider instead of Q/=ll, the set Q^

of ^-elements of Q. We will show that Q^ can be provided with the structure of a frame

isomorphic to Q/=ll. We first need a preliminary lemma.

Lemma 2.11. Let [i be a coclosure operator on g. Then Q^ satisfies weakening iff fil < 1;

Qp satisfies contraction iff \ia- fia = fia, for all a G Q.

Proof. If Qn satisfies weakening, then, in particular, fiT < 1. The converse follows by

monotonicity, since a < T for all a £ Q. The second part holds by definition. •

We are now in a position to prove that Q^ is a frame.

Proposition 2.12. Let Q be a quantale and ^ a unary operation on Q, then:

(1) If \i is a coclosure operator, Q^ is a complete sublattice of Q;

(2) If ^ is a quantic conucleus, Q^ is a subquantale of Q;

(3) If fi is a modality, g^ is a frame.

If p is a conucleus on a frame L, LM is a subframe of L.

Proof. The first two statements and the last are also in [R], proposition 1.1.3 and

theorem 3.1.3, but we prove them here anyway, for the sake of completeness.

If fx is a coclosure operator, by lemma 2.8, Q^ is closed under arbitrary joins, i.e. Q^ is a

sublattice of Q. Moreover, if fi is a quantic conucleus, by lemma 2.7, Q^ is closed under

binary products. If n is a modality, by lemma 2.11, Q^ satisfies weakening and contraction,

thus, by proposition 2.3, Q^ is a frame.

Finally, let / j b e a conucleus on L. Then n is also a quantic conucleus on L, hence L^ is a

subquantale, therefore a subframe, of L. •

As already announced we have:

Proposition 2.13. For every quantale Q and modality n, the map

<t>-Q/=» —» e ,[a] i—• /xa

is a frame isomorphism.

Proof. The map >̂ is well defined and injective by definition of =/J and surjective by

definition of Q^. Moreover, (f> preserves arbitrary sups because <̂ (Vf6/ [a,]) = <£([V,e///a,]) =

' Proposition 2.10 can be proved in a more general, but not shorter, way, using categories. Via the equality

holding for quantic conuclei /i(/ia —o c) = /X(/«J —o /it), which can be proved directly or as a consequence of

Cor. 1 to prop. 3.1.1 in Rosenthal (1990). One shows that the map

e/=, —+ e/=,

[c] i—> \pa —o c]

is well defined and monotone, i.e., is an endofunctor of the partial order Q/=fl. Then, by straightforward

application of definitions, one shows that for all a,b,c e Q, [a] N1 [b] <h [c] iff [b] <^ [pa —oc], which means

that the functor [pa —o • ] is right adjoint to [a] AM • . Thus, by the uniqueness of the adjoint functor, we have

[a] —> [b] = [pa —o b], for all [a], [b] ! ()/=„, since, by its definition, [a] —> • is right adjoint to [a] A11 • also.



= ViejiMf = V,e/</>([aj]) and preserves finite meets because </>([a] -̂ [ft]) =

• fib]) = »(tia • nb) = na-fib = <£([a]) • <j>([b]) and 4>([T]) = /zT = 1. •

The following lemma, which will be useful in subsequent sections, defines the operations

of the frame Q^ in terms of the operations of the quantale Q.

Lemma 2.14. Let / i b e a modality on Q and L = Q^. Then for all a,ft,a, £ L,i G I:

(a)

(b)

(c)

(d)

(e)

(f)

aALi

aVLi

Akia

Vfe/aa^L

0L =

U = =

u ^

,- =

i =

b =

On.

Prao/

(a) Since a Ae ft < a and /j(aAefc) < aA 2 b , we have n(aAQft) < a; similarly n(a AQb) < b.

If c is an element of Q^ such that c <a and c <b, then c < aAb, hence c = fie < n(aA°-b).

(b) It is enough to observe that if a, ft e g^, then by Lemma 2.8, a Ve ft e g^.

(c) and (d) can be proved generalizing (a) and (b).

(e) By uniqueness of the adjoint functor, it is enough to prove that, for all a,b,c G L,

c < n(a —o ft) iff a AL c < b

in fact c < [i(a —o ft) iff fie < a —o ft iff, by adjointness, \xc • a < ft iff, since a = fia,

fie- j*a < ft iff n{c AQa)<b iff, by (a), a AL c < b.

(f) is obvious since /xO = 0. •

Given a modality / i o n a quantale Q, we have constructed the frame Q^ £ Q. Conversely,

given a frame L c g, we obtain an operator fiL on Q, denned by

/xLa = \/{b e L : ft < a}.

As we shall prove below, this operator is in fact a modality. Moreover this procedure

defines a biunivocal correspondence between frames included in quantales and modalities

over quantales.

We have already proved (cf. Lemma 2.8) that if /i is a coclosure operator, Q^ is closed

under arbitrary joins. In fact also the converse holds (cf. Rosenthal (1990), Proposition

1.1.3):

Proposition 2.15. If L is a subset closed under arbitrary joins of the quantale Q, HL is a

coclosure operator.

Proof. The inequality /x^a < a holds by definition of supremum. If a < ft, then

{c G L : c < a} c [c G L : c < ft}, therefore fiLa < \iib, so HL is monotone. Since L is

closed under joins, HL{O) G {ft G L : ft < HLO], therefore IIL(O) < v{ft ! L : ft < /ILO},

which by definition is HLIAL{<Z), and idempotence is proved as well. D


Sara Negri 48

The following proposition show that the correspondence thus obtained is biunivocal.

Proposition 2.16. Let ^ be a coclosure operator on Q and L a subset of Q closed under

joins. Then:

(i) HQ>=P,

(ii) Q^ = L.

Proof.

(i) For all a e Q, ^ ( a ) = V{b ! Q^ : b < a} = V{b G Q : fib = b & b < a} = fia, where

the last equality holds by definition of coclosure operator.

(ii) Q^ = {a e Q : fJ.^ = a} = {a e Q : V{b ! L :b < a} = a} = L, since L is a subset of

Q closed under arbitrary joins. •

This biunivocal correspondence extends to subquantales-quantic conuclei and subframes-

conuclei (cf. Rosenthal (1990)). If fi is a quantic conucleus, then, by Lemmas 2.7 and 2.8,

Qn is a submonoid of Q closed under arbitrary joins, i.e. a subquantale of Q. Moreover

it follows from the definitions that if fi is a conucleus on L, then LM is a subframe of L.

Conversely:

Proposition 2.17. If S is a subquantale of Q, then fis is a quantic conucleus. If S is a

subframe of a frame L , then fis is a conucleus.

Proof. By proposition 2.15, /^ is a coclosure operator, thus proving the claim amounts

to proving the inequality n(a) • fi(b) < fi(a • b) for arbitrary a,b G Q. By Lemma 2.7 this

is equivalent to proving that Q^s is closed under •, which holds by proposition 2.16 from

the hypotheses on S. As for the second statement, just observe that the first part gives

Hs{a) A ns(b) < ns(<* A b). Since the converse inequality holds by monotonicity of the

coclosure operator, us is a conucleus. •

At last we can prove:

Proposition 2.18. If L is a subframe of the quantale Q, [II is a modality.

Proof. By Proposition 2.15, to reach the conclusion there remain (d) and (e) to be

proved. As for (d), by Lemma 2.7 we have to show that Q^ is closed under • and that

Hi.a < PL^ ' A*Lfl- The first statement holds, since we have already seen that Q^L = L. To

prove the second amounts to proving that

\/{d e L :d<a] < V{b G L : b < a} • V{c e L : c < a],

i.e., using the distributive property twice,

V{d G L :d<a] < V{b • c : b G L,c G L,b < a,c < a},

which is now evident, since product in the frame L is idempotent.

As for (e), /iLT = \/{b G L : b < T} = 1 holds since for all a G L, a < 1. •

We thus may summarize the above results with the following:

Theorem 2.19. Let Q be a quantale. Then any modality (respectively quantic conucleus,

coclosure operator) // gives rise to a frame (respectively subquantale, complete sublattice)

Qn £ Q, given by Q^ = {a G Q : \ia = a}. Conversely, any frame (respectively subquantale,

complete sublattice) L £ Q gives rise to a modality (respectively quantic conucleus,



coclosure operator) HL, defined by n^a = V{b ! L : b < a}. Such a correspondence is

biunivocal. In the case of frames, it gives a biunivocal correspondence between conuclei

f iona frame L and subframes of L.

A partial result in the same direction was announced in Avron (1988).

As an application we can show how this construction works in a special case.

If we consider the pointwise partial order on the set of modalities on Q, that is,

Hi < fi2 = ixi(a) < jU2(tf) for all a ! Q, then the biunivocal correspondence stated above

is easily seen to be order preserving, thus becoming an isomorphism of partially ordered

sets. It is known (cf. Rosenthal (1990), Proposition 3.1.3) that the set of subframes of a

given quantale Q is a complete lattice, where the meet of a family (L,);e/ of subframes of

L is their intersection, and the join is the meet of the family of subframes of Q containing

U,e/L,-, which we shall denote by V;6/Lj. By the isomorphism we get:

Corollary 2.20. The set of modalities on a quantale Q is a complete lattice. '

Through the top element in the lattice of subframes of Q, we can now give an explicit

description of the top element in the lattice of modalities on Q.

Definition 2.21. Let Q be a commutative quantale; we define:

C(Q) = {aeQ:a<l&a<a-a}

and call it the centre of Q.

We have:

Proposition 2.22. For any quantale Q , C(Q) is a frame and is the greatest frame included

ing-

Proof. C(Q) is closed under •, since if a < 1, b < 1, then by stability a-b < 1; if a < a- a

and b < b • b, then by stability and commutativity we have a • b < (a • b)(a • b). C(Q) is

closed under V: if for all i e / , bt < 1, then Vi6/b; < 1; moreover, if for all i £ I,bt < bfbj,

we have V,!/fr, < Vie/b; • bt < Vie/je/ft,- • bj = (Vie/ft,) • (V,-6/fc,), by the distributive property.

The subquantale C(Q) is a frame because a • b = a A b in C(Q), by Proposition 2.3. C(Q)

is the greatest frame included in Q because, if L is a frame included in Q, for all a e L,

we have a < 1 and a < a- a. •

The previous construction allows us to describe explicitly the modality corresponding to

C(Q). By the definition of induced modality, nc(Q)(a) = \/{b e C(Q) : b < a}, hence, by

the definition of C(Q), Hc(Q)(a) = V{b e Q : b < \,b < b • b,b < a}. Moreover, by the

biunivocal correspondence stated by Theorem 2.19, we can characterize elements of C(Q)

as fol lows: a e C(Q) iff nC(Q){a) = a, t h u s a e C(Q) iff a = V{b ! Q : b < \,b < bb,b < a}.

We can now give a simpler characterization of modalities on Q. By Theorem 2.19 the

set Mod{Q) of modalities on Q is in biunivocal correspondence with the set of frames

included in Q, which is the same as the set of frames included in C(Q), by Proposition

' We could ask whether it is also a frame, i.e. whether distributivity holds. But the following example shows that

the inclusion (Uje;(Lo n L,)) £ LQ n (Ujg/L,-) may be proper. Let L be the frame {0,a,i ,T}, with 0 < a < T

and 0 < b < T; consider the subframes Lo = {0, T}, L\ = {0,a}, L2 = {0, b}. Then Lo n {Lx UL2> = Lo, while


Sara Negri 50

2.22. Since C(Q) is a frame, the set of frames included in C(Q) is, by Theorem 2.19, in

biunivocal correspondence with the set of conuclei on C(Q), CN(C(Q)). We therefore have

the following chain of biunivocal correspondences:

Mod(Q) <—> subframes of Q <—> subframes of C{Q) <—> CN(C(Q)).

Thus, since the composition of left to right arrows is the identity map, any modality on

Q is a conucleus g, when restricted to C(Q). Conversely, any conucleus g on C(Q) gives a

modality on Q, denned by composing from right to left:

Aig(fl) = V{b ! C(Q) : g(b) = b&b<a}.

We may thus conclude with the following

Theorem 2.23. If fi is a modality on Q, its restriction to C(Q) is a conucleus on C(Q). If

g is a conucleus on C(Q), ng, as denned above, is a modality on Q. The correspondence

between modalities on Q and conuclei on C(Q) is biunivocal.

Since /x = /IQ^, for any \i, if a is an element of Q, by definition of modality induced by a

subframe, n(a) e gM, hence, a fortiori, //(a) ! C(Q). We thus obtain:

Corollary 2.24. For any modality n on Q, fi(Q) S C(Q), hence also n(Q) £ /*(C(g)).

3. Semantics of the translation from IL into ILL'

In Lemma 2.14 (e), we have seen how the implication of the locale Q^ is related to the

implication of the quantale Q via the modality fi: if a, b G Q^,

a —> b = n(/xa —o b) = fi{a —o b).

Since the valuation for ! in a quantale with modality is given by V(\A) = f-tV(A), this

suggests a comparison with Girard's translation of the intuitionistic implication into the

linear one, given by

(A->B)g =\Ag -^>BS.

In general, as it is known that frames and quantales with modality provide us with

a complete semantics for intuitionistic and linear logic, respectively, we would like to

investigate more deeply the analogy between the insertion of the frame L = Q^ in the

given quantale Q with modality fi and the translation from IL into ILL. We can draw

the following diagram:

VL

IL > ILL

(1)

'See appendix B for a sequent calculus definition of IL and ILL.



where VL and VQ are the valuations of intuitionistic and linear logic, respectively, into

the frame L and into the quantale Q, ()° is a translation from IL into ILL and ( ) o is the

embedding of the frame into the quantale.

Our purpose now is to find a suitable faithful translation that makes the diagram

commute. Let us recall that Girard's translation (cf. Girard (1987), Schellinx (1991)) is

denned by induction on formulas via the following clauses:

P« = P, for any atomic formula P

(A A BY =

(AW Bf =

{3xAY =

This translation satisfies faithfulness in the form:

r \-1L A iff i n

One can easily realize that this does not make the diagram commute, since, for instance,

VQ((AABY) = VQ(AS&BS) = VQ(A*)A<i VQ(B*), while (VL{AAB))O = (VL(A)AL VL{B))O =

H(VL(A) Ae VL(B)), which in general are not the same, assuming as inductive hypotheses

that VQ(A*) = VL(A) and VQ(B«) = VL(B).

There is another requirement on translations to point out. In the Godel-Gentzen

double negation translation from Classical into Intuitionistic Logic, translated formulas

are stable, that is, they are equivalent to their double negation, and this is perfectly

sound, since a translation should not distinguish more than its source does. In the

same way, since intuitionistic formulas can be weakened and contracted, that is, they

are the same as their exclamation, a translation of an intuitionistic formula should be

a formula that is exclamative, that is, a formula A such that A =ILLIA. Since \A h

A always holds by !L, A is exclamative iff A \-\A. To find such a translation, note

that:

Proposition 3.1.

(i) The constant 0 is an exclamative formula.

(ii) For any formula A, \A is an exclamative formula.

(iii) If B and C are exclamative formulas, then B®C,B®C, 3xB are exclamative formulas.

Proof, (i) holds since 0 h A for all A, hence 0 h !0.

(ii) is an immediate consequence of the IR rule.

(iii) Suppose that B \- \B, C h !C; by ®R we get B, C h !B® !C. Since \B® !C h \(B ® C) al-

ways holds, by cut and ®L we obtain B®C \- !(J3®C). As for the connective @, by applying

®R to the hypotheses, we have B H B e ! C and C h!Be!C, hence, by ®L,B®C HJ3©!C.

Moreover, we can easily prove !B©!C \-\(B ® C), thus, by cut, B © C h ! ( B ® C). Finally,

since B h ElxB, by !L and \R, we have \B \-\3xB. By hypothesis and cut, B \-\3xB, thus

•


Sara Negri 52

These considerations lead us to propose the following translation, which satisfies the above

requirement but is more economical than prefixing every subformula with ! :

P' = !P, for any atomic formula P

_L*=0

(A A B)' s\(A'&B")

(Ay B)' =Am@B*

(A^BY =\{A' —oB')

(VxA)' = ̂ M

(3xAY =

It is clear that for any formula A, its translation A* is exclamative. Moreover, observe

that since \(A&B) =,LL\A®\B, we could equivalently define {A A BY =\A'®\B', or, by

Proposition 3.1, (A A BY = A* <g> B*, but we prefer the former since it allows a simpler

translation of proofs.

We now show that (•)* is a faithful translation.

Proposition 3.2. For any set of formulas F and any formula A of IL,

F \-iL A iff T* h/LL A*.

Proof. Suppose that F h/£, A. We prove that F* \~ILL A' by induction on the length of

the derivation of F \-IL A.

Let F h A be an axiom: if it is A \- A, then A' \- A* is an axiom of ILL; F, JJ- A becomes

F*,0 h- A", which is an axiom of ILL as well.

Let us now examine the last rule applied in a sequent calculus derivation of F h/i. A.

If A = B A C and the sequent f h / t - 4 is obtained by means of AR, let y and 5 be the

derivations in IL of F I- B and F h C, respectively; by inductive hypothesis we obtain

the derivations in ILL y* and S* of F* h B' and F* h C that by &R give F* h B'&C

and therefore, since all formulas in F* are exclamative, by IR, F* \-\(B*&C). For AL\ we

have:

r,T',A

yA'\-

'&B'

C

h

*

C*

T,AAB\-C

The rule AL2 has a similar translation.

The V-rules are translated into the corresponding ©-rules. As we have done for AR, also

in the translation of —» R and V.R, we use the fact that all formulas in F* are exclamative,


Semantical observations on the embedding of Intuitionistic Logic

and the dots denote repeated applications of \L, \R and cuts.

53

r,B\-cri-B->c

R

y

r\-A6

T,B

r,A->B\-

y

r \-Bx

\-cc

wr h VxBx

VL

' h 3xBx

y

i , 3XAX I x5

The translation of structural rules is:

y

T,B\-A

T',B* hC*

r >c*r

7 5'

T*\-A* T*,B*\-C*

B*)\-C*

r* h VxBx*

fr,At* \- B'

T\VxAx'\-~B*~

r, \(VxAx*) h B*

y'

T* I- 3xBx*

y '

T*,At' \-B*

T',3xAx'\-B'

B*\-\B' T',\B'\-A'

T',B'\-A'

T,A,A\-B A' HA*

\A*\-A* r,A*,A*Y-B*

\A*\-A* T*,\A',A*\-B*

r, \A\ \A' v- IT

r, \A* v- B*

T\A'\-B*

Finally, translation of cut is unproblematic.


Sara Negri 54

Conversely, assume a deduction in ILL of F* (- A* is given. Then, by replacing \A with

A, for any formula /I, © with V, ® and & with A, —e with ->, 0 with _L, we obtain a

deduction in IL of F h- A. To convince yourself of this, just observe that the axioms and

the rules for the constants and the connectives of ILL that are translations of constants

and connectives of IL, after these substitutions, turn into valid inferences of IL. Let us

give full details:

r,Of-/i becomes r ,±h^

which is an axiom of IL.

r'A'BhC gives r>A'BhC

T,A®B\-C ° T,AAB\-C

which is easily proved in IL.

ri\-A r2 h B „ . . r,\- A r 2 H B

U A turns int0 r^yAABwhich is obtained from AR with several applications of weakening.

The rules for the connectives &, ©, —e become the rules for A, V, —> respectively.

r becomes weakening,

' and ' I r after substitutions are tautologies.

'" ' ' yields contraction.

The rules for quantifiers, cut and exchange remain the same. •

We are now in a position to show that the above translation from IL into ILL is, by

means of semantics, the syntactical counterpart to the algebraic construction of a frame

inside a quantale provided with a modality, i.e. that diagram (1) commutes. Observe that

for all A, VQ(A') e %, since A* =ILL\A' implies VQ(A') = fiVQ(A').

Theorem 3.3. Let VL and VQ be valuations of IL and ILL into the frame L and the

quantale Q, respectively. If VL(P) = VQ(P) for all atomic formula P, then for any formula

A,

VQ(A') = VL(A)0

i.e. if diagram (1) commutes on atomic formulas, then it commutes on all formulas.

Proof. In what follows we will not write (•)„ explicitly. The proof is by induction on

formulas.

If A is the atomic formula P, VQ(P') = VQ{\P) = nVQ(P) = nVL{P) = VL(P), by

hypothesis and definition of L = Q^. If A =_L, VQ(±') = VQ(0) = 0e = 0L = VL(±). If

A = B A C, VQ(A') = VQ(](B'&C*)) = n(VQ(B') A VQ(C*)), thus by inductive hypothesis

VQ(A') = v(VL(B)AQ VL(C)) and, by Lemma 2.14, VQ(A') = VL(B)AL VL{C) = VL(AAB).

For all other connectives the proof is completely similar. •

We say that a translation (•)* is a translation of schemes if it commutes with substitution,


Semantical observations on the embedding of Intuitionistic Logic

that is it satisfies the following diagram

55

A(P) A(P)*

(2)

If P* = !P, for atomic P, the previous condition is not satisfied, e.g. taking for A and

B the formula P itself, since, in this case, A(B)* =\P, while A(B*)* = !!P.

This counterexample also shows a general fact: if (•)* has to be a translation of schemes

it has to be the identity on atomic formulas. We see that this condition is also sufficient

for a translation to be a translation of schemes. This amounts to proving that

A(B)* = A*(B*) (3)

for all formulas A and B. In fact, if the previous equality holds for all B, in particular, for

P atomic, it gives A(P)* = A*(P*) = A*(P) and the commutativity of (•)* with substitution

becomes:

A(P)(•r

A*(P)

P/B

A(B)(•)*

P/B*

A(B)* = A*(B*)

(4)

Thus, to show that (•)* is a translation of schemes, we just have to prove (3). If A is

the atomic formula P, it collapses to the identity P* = P*. If A is the constant _L, there

is nothing to prove. Otherwise (3) is satisfied by any translation defined inductively, thus,

in particular, it holds for Girard's translation.

Let (•)' be the translation that is defined like (•)«, except for (A A Bf ^.(A'&B'),

(A -> B)' = ! (W -^B'), (VxA)' =\^xAK By the previous remark, (•)' is a translation of

schemes. It is easy to prove that Theorem 3.3 holds also for the translation (•)'. Moreover

(•)' satisfies the same faithfulness property of Girard's translation:

Proposition 3.4. For any set of formulas F and any formula A of IL,

F h/ L A iff !F' HLL A'.

Proof. The proof follows the same pattern as in Proposition 3.2, and is straightforward


Sara Negri 56

except for the three following cases:

y'

in, \AI \- c

in, \A\ \BI \- cy \{A'&B')\-\At®\Bt) !r ',!i '«!5'hC

T,A\-C irWiA'&B^hO

T,AAB\-C

\A'\-\A' B'\-B\A' —o B', \A' h B' yt

!(L4' —o B'), \A' h B' | r ( h A1

i( \Al —o B'), \Al V- IB' in h \A' in, IB' h C

y s \(\A' -oB')HA' -^\B' i r . U ' - e l B ' h C

r\-A r,B\-c in,in,

WxA'

WxA* h

!(L4f-

!!(L4(-

\-A'

HA'-VxL4<

•oB")

-oB)

inin,

h C

/!, \A' h B'VxM'hB1

r,/iihB in, ivx^f-B1

VLB ViJ -^ m,!!\

•

Since (•)' is the identity on atomic formulas, in general A' is not exclamative. However,

we can prove:

Proposition 3.5. Let A be a formula of IL. If P =ILL[-P for any atomic formula P

occurring in A, then L4' =ILL A'.

Proof Immediate by induction on formulas (cf. Proposition 3.1). •

Observe that the previous result does not hold for Girard's translation: for instance, if

A is B -»• C, we have A« =!Bg -^> C«, Ug =!(!Bg ^ > C«). In this case the assumption

on atomic formulas does not imply that \Ag =ILL A%, since there is no cut-free proof of

!B« _ o c« h/LL!(!B« —o Cg) under the inductive hypotheses B« HB«, Cs HC«.

Before giving an 'internal' formulation of the above Proposition, a remark on notation

is in order: if P denotes the multiset of formulas (P,),=i,...>n, then P —©IP denotes the

multiset of formulas (P, —©!P,)i=i,...,n-

Proposition 3.6. If P are the atomic formulas occurring in A, then !(P —e IP) \-lLL A' —e \A'.

Proof. It follows from 3.5 and the proposition on p.70 in Troelstra (1992). •



Corollary 3.7. Let P be the multiset of atomic formulas in T. Then

T \-,L A iff !(P - e ! P ) , r ' hLL A'l

Proof. If F \~n A, then by Proposition, 3.4 !P H/LL A'. By the above result, for all

formulas B in T, we have !(P - e ! P ) h/ L L B< -e !B ' , that is !(P - e ! P ) , B ' h L L ! B ' . Hence,

after several applications of cut, we get the claim.

Conversely, with the same replacements as in Proposition 3.2, we obtain P —> P, T \-jL A

and thus, by axioms and cuts, F h//, X. •

4. The co-Kleisli category associated to a comonad

We are now going to see how the construction of the first part is a particular case of a

categorical one. Let us first recall some definitions concerning the categorical semantics

of linear logic (cf Mac Lane (1971), Troelstra (1992)).

Definition 4.1. A symmetric monoidal category (SMC) (#, ®,/,<x,y, X) consists of

— a category #

— a bifunctor • ® • : <! x <! —> <!

— an object / ! Obl<!)

such that for all A, B, C G Ob(^) there are the following natural isomorphisms

<XA,B,C :A®(B®C) —y (A ® B) ® C

:A®B —> B®A

kA:l®A —> A

satisfying the Kelly-Mac Lane coherence conditions.

A closed symmetric monoidal category (CSMC) is a symmetric monoidal category <$ such

that for all A eW, the functor • <g> A has a right adjoint A —o •.

A closed symmetric monoidal category with products, coproducts, terminal (T) and initial

(0) objects e&,®,I,0L,y,l,<t>,Yl,]J,T,O) is called an intuitionistic linear category*.

A comonad on # is a triple (T,r,s), where T : <6 —> *! is a functor and s : T —• T2,

r : T —> id are natural transformations making the following diagrams commute

T This result is, with the same conventions of notation, the analogue of

r \-CL A iff -.-.p -> 7, r G h / L AG,

holding for Godel-Gentzen double negation translation, with PG = P in order to get a translation of schemes.

* As is done elsewhere (cf. Marti-Oliet and Meseguer (1991)), instead of J~J and TJ, the more suggestive

symbols & and © are used for binary product and coproduct, respectively.


Sara Negri 58

T2A

(5)

(6)

TA <—— T2A ' > TATo get a suitable interpretation of the modality ! of linear logic, we need a furtherassumption on T, namely we suppose that

T(T) s /

and that for all A, B there exists a natural isomorphism

WA,B :TA®TB ^ > T{A&B).

(7)

(8)

It is a standard construction (cf. Mac Lane (1971)) to associate to a comonad (T,r,s) on^ the 'co-Kleisli' category K(^), whose objects are those of ^ itself, while the arrows aregiven by

If / ! K(<g)(A,B),g e K{<#)(B,C), then their composition is denned by

g °K(<g) f = g o T / osA.

As usual, let nA and %B be the canonical projections and let (•, •) be the universal morphism

of coproduct. Then if we add the following coherence condition (cf. Hoofman (1992))

T(A&B) -^ TA <g> TB

WTAJB

T2(A&B)T(TnA,TnB)

(9)

T(TA&TB)

we get a a basic property of this construction:

Proposition 4.2. Let ^ be a CSMC with terminal object and binary products, (T,r,s)a comonad on (

!, satisfying the above conditions (5)-(9), and K(f) the correspondingco-Kleisli category. Then K(%!) is a cartesian closed category, with terminal object T,binary product A x B = A&B, projections pA = rAo TnA, pB = rBo TnB, universal arrowof product as in e

!, and exponentiation A => B = TA —o B.If, moreover, ^ has arbitrary products, then K(^) has arbitrary products also.



Proof. See Hoofman (1992) for the first part; the second is a straightforward general-

ization of the proof of existence of binary products. •

We can require additional structure on the CSMC %>, for example, the presence of finite

and arbitrary coproducts. To get this structure inherited by the co-Kleisli category K(^)

we need further hypotheses on the comonad T. The following fact is useful for finding

necessary conditions.

By a general property of the co-Kleisli construction (cf. Mac Lane (1971)), there exists

a couple of adjoint functors

G

where G(A) == A, G(f) = f o rA, if / e <$(A,B), F(A) = TA, and F(f) = Tf o sA, if

/ ! KC$)(A,B). Thus F preserves coproducts, as every left adjoint does.

We are now ready to characterize the existence of an initial object in K(<t>).

Proposition 4.3. Let # and K(^) be as in the hypotheses of Propostition 4.2. Then K(^)

has initial object A iff TA ^ 0.

Proof. Suppose that A is initial in K(^), then FA = TA = 0, since the functor F

preserves coproducts. Conversely, for each B in %> there is a unique arrow TA = 0 —> B,

thus A is initial in K{<g). D

Now, let us suppose that ^ has binary coproduct A ® B with injections eA, eB, and the

universal morphism of / : A —* C and g : B —> C, [f,g] : A® B —• C. We can define in

a natural way a coproduct A @K B in K(^), by imposing suitable conditions on T, to be

specified below. Indeed these conditions are also necessary if we require that this position

defines a coproduct, since they are equivalent to conditions arising from the fact that the

functor F preserves coproducts. These facts are stated by the following two propositions.

Proposition 4.4. Let ^ be a CSMC with comonad, as in the hypotheses of 4.2, with binary

coproducts. Suppose that for all A, B e # there exists an isomorphism

4> : T(TA ® TB) -+TA®TB

such that the following diagrams are commutative

T2A 6TA> T(TA ® TB) T2B !TB> T(TA ® TB)

TA TA®TB TB

(10)

v^> TA ® TB

and, moreover,

<f> 1 = [TeTA o sA, TeTB o sB].

Then K^!) has binary coproduct A ®K B = TA ® TB, with injections e$ = CTA, <

and universal morphism [•, -]K = [•, •] o (j>.

(11)

= !TB


Sara Negri 60

Proof. Routine checking of definitions. •

The existence of the isomorphism $ and the coherence conditions to be satisfied are also

necessary, and, moreover, <j> is in a way determined by the coproduct. Indeed, the previous

proposition admits the following converse.

Proposition 4.5. Suppose that K(^) has binary coproduct given by A®KB = TA®TB, with

canonical injections eTA and eTB. Then there exists an isomorphism <f> : T(TA ® TB) ->

TA@TB satisfying conditions (10) and (11) of Proposition 4.4 and such that the universal

morphism of coproduct in K(^) is given by [•, ]K = [•, •] o <j).

Proof. The isomorphism <f> exists because the above defined functor F preserves

coproducts. Thus F{A ®K B)^FA® FB, that is T(TA © TB) ^ TA e TB. Conditions

(10) and (11) are obtained by writing up what it means for the functor F to preserve

coproducts. In particular, (11) easily yields the last statement. •

These propositions can be generalized to arbitrary coproducts in the obvious way:

Proposition 4.6. Let ? be a CSMC with comonad, as in the hypotheses of 4.2, with

arbitrary coproducts \\x Ax- Suppose that there exists an isomorphism (j> : T(]JX TAx) -*

Ux TAi, such that for all X G A the following diagram is commutative

TA,)

(12)

and, moreover,

<j>-l = [(TeTAxosAl)x]. (13)

Then K{^) has arbitrary coproduct JJ^ 4̂̂ = ]\k TAx, with injections e^ = eTAi, and

universal morphism []K = [•] o <f>.

Conversely:

Proposition 4.7. Suppose that K{^) has arbitrary coproduct given by Jjf Ax = U^ TAx,

with canonical injections eTAx. Then there exists an isomorphism <f> : Tl(]J/l TAx) —>

UA TAx satisfying conditions (12) and (13) of proposition 4.6 and such that the universal

morphism of coproduct in K(^) is given by []K = [•] o <j>.

It is well known that a CMSC with arbitrary coproducts that is a partial order is in fact

a quantale and that a CCC with arbitrary coproducts is a frame. The natural question

arising now is whether the construction of the first part is a particular case of the co-Kleisli

construction applied to partial order categories.

To make these ideas precise we need a 'partial order' functor between the given category

and the corresponding partial order. First of all, to get a preorder, we have to identify all

the morphisms having the same domain and codomain. Then, to force antisymmetry, we

have to identify isomorphic objects, that is to consider the skeleton of the category thus



obtained (this is essentially a quotient on morphisms and on objects). The first step is a

particular case of the construction in Mac Lane (1971, p. 51), which we recall below:

Proposition 4.8. For a given category c!, let R be a function that assigns to each pair of

objects A, B of # a binary relation RAB on the set <&(A,B) x ^{A,B). Then there exists a

category <$/R and a functor Q = QR : <$ -> <g/R such that:

(ii) If H : <! -> ® is any functor from # for which fRAjg implies Hf = Hg, then there

exists a unique functor G : ^ / R —> 2 with G o QR = H. Moreover the functor QR is a

bijection on objects.

This proposition yields the desired result taking for RA^ the whole set ^{A, B) x ^{A, B),

for every pair of objects of c!.

Let S be the projection functor of the preorder ^/R on its skeleton and let P be the

composition of this functor with Q. Then P{%>) becomes a partial order with the position

PA < PB = there exists a morphism f ! <£(A,B).

Actually, P(^) is a quantale with product

PAPB = P{A®B) (14)

and lattice operations

PA A PB = P(A&B) (15)

x). (16)

In fact these are good definitions by the meaning of skeleton. The product is associative,

commutative and with unity P(T), since the bifunctor ® is associative, commutative and

with unity up to isomorphism. Furthermore, it is easy to prove that (15) and (16) give the

meet and the join with respect to the above defined partial order. Distributivity holds,

since, for all A, the functor A ® • is a left adjoint and therefore preserves coproducts. Of

course P(T) is the top element of the quantale.

The comonad T induces in P(#) the modality defined by

HT{PA) = P(TA).

Indeed HT is well defined, since if A and B are isomorphic in '£, so are TA and TB,

because T is a functor. The comonad natural trasformations give \ijPA < PA and

HTPA < HTHTPA, while monotonicity follows from T being a functor. Equality between

HT(PA A PB) and \iTPA • \iTPB holds by the isomorphism T(A&B) = TA® TB, while

T(T) = / yields fiT{P(J)) = PI.

Therefore, as we have seen in the first part, we can get - with suitable smallness

conditions - the frame of the /zr-stable objects of P(^) . On the other-hand, the partial

order obtained from K{<g), P{K{%)), is a frame, since K(#) is a CCC with arbitrary

coproducts. As one might expect, these frames are isomorphic. More explicitly:


Sara Negri 62

Proposition 4.9. In the above hypotheses, the map

is a frame isomorphism.

Before proceeding to the proof, the following consequence of coherence conditions is useful.

Lemma 4.10. Let %> be a category with comonad T and K{^) the associated co-Kleisly

category. Then for any pair of objects A, B !<£

A 3 f c w B iff T ^ ^ TB,

where =K(*) and =<g denote isomorphism in K(^) and in <!, respectively.

Proof, of Proposition 4.9. By Lemma 4.10, 4> is well defined and injective, and it is

surjective by definition of (P(#))M7.. The map 4> preserves infima, in fact cf>(PA A PB) =

4>P(A&B) = PTG4&B); on the other hand, <pPA N1* <j)PB = nT{<l>PA A <£PB), and by

the definitions iiT{<j>PA/\ <j>PB) = PT(A&B). As for arbitrary suprema, observe that

WxPAx) = 0(P(I]*W4i)) and </.(P(Ljf *'^)) = Wfltf T^)) = P(T(U* T^)) bydefinition of coproduct in K^!). By the hypothesis on T ensuring the existence of coprod-

ucts in K(V), P(T([fx TAX)) = P(LL? TAX) and, moreover, ^T<I)P(AX) = V^TnT(PAx) =

V ^ r ( P ^ ) by Lemma 2.14 (d). Furthermore, yxnT(PAx) = TJA iiT(PAx) = Uxp TA^ t h u s

4>{\/xPAk) = Vf 4>{PAx). Finally, 4>PI = nTPI = P / , since \iT is a modality on P(#). D

We thus obtain the diagram

(17)

Appendix A. Formal topologies and formal pretopologies with stable interior operator

The relationship between frames and quantales with modality described in the first section

can be expressed equivalently by means of formal topologies and pretopologies. For this

purpose we first extend the representation theorems of quantales and frames (cf. Battilotti

and Sambin (to appear)) to quantales with modalities. We show that if #" is the formal

pretopology representing the quantale Q, that is, Q = Sat(^\ then modalities fi on Q

correspond to stable interior operators R on #". Furthermore, any formal pretopology 3P

with R gives rise in a natural way to a formal topology si such that the frame of its

saturated subsets is isomorphic to the frame Q^.

We briefly recall some definitions from Sambin (1989).

Definition Appendix A.l. Let J* = (5, •, l ,<jr) be a formal pretopology. A map R :

0>S —> SPS is called an interior operator on J5" if the following conditions (a)-(d) hold, it



is a stable interior operator if it also satisfies (e)-(g), where U, V are arbitrary subsets of

0>S:

(a) RU e Sat{3F);

(b) RU<U;

(c) U< V^>RU<\ RV;

(d) RU<RRU;

(e) RURU=^RU;

(f) RU • RV < R(RU • RV);

(g) RS = &\.

Observe that an interior operator on 3F is just a coclosure operator on Sat(^). Moreover,

it is determined completely by its restriction to Sat(!F), since it follows directly from the

definitions that for all U £ s, RU = RÛ. Indeed, when R is a stable interior operator,

this restriction is a modality on the quantale Sat(^), since its definition is equivalent to

the definition of modality given in the previous section.

We can make the analogy between modalities and stable interior operators more explicit

through an extension of the representation theorem for quantales, which we recall now.

For further details the reader is referred to Battilotti and Sambin (to appear).

Let S be a base for a quantale Q, that is, S is a submonoid of Q such that for all

a e Q, a = V{b e S : b < a}. We can define two maps </> : 0>S —• Q, <j){U) = Vl/ and

\p : Q —y 3?S, xp(a) =[s a = {b ! S : b < a). It is straightforward, by the definition of

a base, that <j>(\p(a)) = a, so <$> is surjective. In general, <j> is not injective, and imposing

injectivity actually amounts to considering the quotient set determined on 3PS by the

equivalence relation U =& V =[$ Vl/ =[s VF. It is not difficult to prove that =& is

indeed a congruence relation with respect to the quantale structure of 0>S defined by

the set theoretic operations of (arbitrary) union and pointwise multiplication, and the

quotient ^ S / = j r can be viewed equivalently as the quantale Sat(Jir), where the base of

the pretopology & is S itself and for all U £ S, &U =is Vl/. The (well-defined !) map

induced by <j) on the quotient is an isomorphism of quantales (by abuse of notation

we call it 4> again). We can extend this isomorphism to modalities and stable interior

operators considered as unary operations on Q and Sat(^), respectively. In order to have

a morphism with respect to the additional structure, the following conditions must be

satisfied:

(a) xp(na) = R^xpa

(b) 4>(RU) = HR4>U,

where R^ and nR must be suitably defined in such a way that if n is a modality, R^

is a stable interior operator, and conversely if £ is a stable interior operator, fiR is a

modality, and the correspondence is biunivocal, that is, R^R = R and fiRit = /i. Observe

that RÛ = Rn is Vl/ = ^tp(Vl/) = xpn{\/U) =[s n(VU), where the first equality holds

because RU = R!FU and the second because xp is a morphism from {Q,n) to (Sat(J^), R).

On the other hand, nRa = /JRV Is a = /iR4> [S a = (f)(R J.s a) = Vi? [s a. This justifies the

following definition.

Definition Appendix A.2. With the above notation we put RÛ =is A*(Vl/) and nRa =

VRlsa.


Sara Negri 64

By routine checking of definitions we can prove the following result:

Proposition Appendix A.3. If /z is a modality on Q, then Rf, is a stable interior operator onJ*\ Conversely, if R is a stable interior operator on #", then pR is a modality on Sat(^).

This correspondence between modalities and stable interior operator is biunivocal, thatis, RM = R and /iR)i = fi.

Proof. Let n be a modality on Q; we prove that R^ satisfies conditions (a)-(g) ofdefinition Appendix A.l. For all U e SPS, PRÛ =is V|sH V U =|sM V C/ = ^ [ / , thus(a) is satisfied. As for (b), observe that fiVU <VU gives J,s //V L/ c | s vl/, that is RÛ< [/.Suppose [/ <l V, that is 1/ £j,s W. Then Vt/ < VK. Thus by monotonicity, n V U < fi V 7,and, therefore, |s M v ^ S-U ̂ VK. Hence, by reflexivity, RyU < R/iV, thus (c) is proved.Condition (d) holds iff | s // V [/ s | s V [s n(V | s M V I/). By the definition of a base, theright-hand side is equal to j , s V | s /z(/z V (7) and therefore, by idempotence of /*, it is equalto |s V Is V- V U and again, by the definition of a base equality, follows. Condition (e)amounts to Is n V U = | s V(|s fi V U- Is n V [/), the two members are equal by distributivityof joins with respect to product, the definition of a base and idempotence of the productof//-elements. By (a), condition (f) holds iff |s^Vt/-|s/zVK c| s / iv(l s / iVl/- | s / iVK). Bydistributivity and the definition of a base, the right-hand side is equal to J,s/i(/zVL//xVF),so inclusion follows from (2) of Proposition 2.5 and stability. As for (g), observe thatR»S =isfiT, thus by Lemma 2.4 (e), R,,S = JH.

Now, let R be a stable interior operator. We prove that HR is a modality via thecharacterization given in Proposition 2.5. Since Rjsa<] J,sa, by definition of precover andbase, Rlsa s j sa , hence VRlsa < a, that is, fiRa < a. Suppose a < b, then Is a îsb. Byreflexivity of the precover and monotonicity of R, we get R Is a < R is b, i.e. R[sa £ R[sb,

and therefore VR Is a < VR [s b, that is fiRa < fiRb. By idempotence of R and the factthat RU = RÛ, we get Risa =? RIs VRlsa, and therefore VR[sa=^ VR[S VR[sa,

which by definition means fiRa = nRnRa. We have thus proved that nR is a coclosureoperator. The product of /i/j-elements is idempotent, since by (e) of Definition AppendixA.1, we have | s V(R [s a • R [s a) = R J,s a, and therefore V | s V(R [s a • R J.s a) = VR J,s a,

which by distributivity and the definition of a base gives the claim. To prove thatURa • fiRb < ^ R ( n R a • n R b ) , t h a t i s , VR \,s a • VR I s b < VR | s (VR [ S a - V R [s b), o b s e r v e

that, by distributivity and the property of R of depending only on the saturation of itsargument, the claim amounts to V(R is a • R is b) < VR(R is a • R is b), which followsdirectly from the definition of R. The last condition /iRT = 1 or, equivalently, VR | S T = 1holds since | S T = S and RS = JH = | s Vl.

The correspondence is biunivocal, in fact RÛ = | s VRis VU — RU, since RU = RS'U

and RU is ^-saturated and fiR)ia = V is fiV is a = pa by the definition of a base. •

Therefore we have proved that any quantale with modality is isomorphic to the class ofsaturated subsets of a suitable pretopology, endowed with a stable interior operator. Ournext problem is to define the formal topology si representing the frame Q^ directly interms of & and R. The idea is to use the representation theorem for frames applied toQp, where n is defined in terms of R. By now, we can take the whole frame as a base ofitself, that is we define S^ = {a e Q : VR J, a = a}. This set inherits from Q the structureof a commutative monoid with unity. The cover relation is, as usual, a < ^ U = a <VU,



and this shows, in particular, that the restriction of the precover <jr to S^ x ^(S^) is

indeed a cover.

We obtain the following diagram, in which the vertical arrows are the isomorphisms

given by the representation theorems:

, R) Satstf

(18)

The bottom arrow is absent because the codomain of R is not exactly Sats*/, but the

frame associated to the quantale Sat(^) and the modality R, L = Sat(tF)n = {U G

Sat(^) : RU =& U} ' ; this set is a frame with the following operations (cf. Lemma 2.14,

and the definition of the quantale structure of Sat(^) in Sambin (1989)):

u ALv

It is easy to prove that these operations indeed define infima and suprema for the

partial order induced by the precover (set theoretic inclusion in fact) on L. Moreover, the

supremum can be defined without involving !F, since one can prove that if the [/* are

K-stable, then ^(Ui!l Ut) = R(Ui!j I/;).

The following proposition shows that this frame is in fact isomorphic to Q^:

Proposition Appendix A.4. The map

W-% —• L = {U e Sat(&) : U = RÛ}

a i—> is a

defines an isomorphism of frames with inverse (f> : U i—> Vt/.

Proof. Let a G Q^; by the representation theorem we already know that is a G SatiF.

Moreover, R is a = | s (URG) =is a, because a is /i-stable, hence is a e L. Conversely, if

U G L, then //(Vl/) = \/{RÛ) = V[7, so the given maps are well defined. We now prove

that ip is a morphism of frames. Let a,b ! Q^. Then \p(a A^ b) = xp(fi(a A b)) = R^y)(a A b),

since \p is a morphism (cf. (a) above). Moreover, xp(a A b) = \pa n tpfc, thus by definition

of the frame structure of L, \p(a A^ b) = Rîs a AL R^ is b. Then by (a) again and the

fact that a and b are //-stable, we obtain xp(a A^ b) = xpa AL xpb. The map \p also preserves

arbitrary sups. Let {a,},e/ be an indexed family of elements of Qp. Then tp(vfe/aj) =

xp{n(yielai)) = ^v(Vte/a,). On the other hand v j^yfa) = ^(U,-6/tp(aO) = R^(UieIxpfa)),

and it is easy to prove that R^(Uieixp(ai)) = K(|s VU,e/ i/)(a,)) = K(|s VieI(Vip(ai)). Finally,

R(ls V,e/(Vi/)(aj)) = i?(|s Vig/flj) = /?(i/> Vje/ a,-). The verification that (p is the inverse of tp

is immediate. •

' Observe that by condition (a) we can equivalently write L = {U e &S : RU = U}.


Sara Negri 66

We thus obtain the following commutative diagram:

(Q,H) >

(19)

^ Satstt

Appendix B. Table of sequent calculus rules for IL and ILL

Here follows a sequent calculus version of Intuitionistic Logic and Intuitionistic Linear

Logic. We use greek upper case T, T', A, A' ... for finite sequences of formulas, A, B,

C, ... for arbitary formulas. As for linear connectives we conform to Girard's notation,

but use two-sided sequents. For the rules of V.R and 3L, the usual side conditions on the

variable are assumed.

Intuitionistic Logic IL

Axioms:

Ah A

Rules:

ThA ThBThAAB

ThA w n

AR

T,AhB

ThA-+B

ThAx v

ThVxAx V

ThAt .,

w

R

ThAT,BhA

r2,AhB

r,±hA

r,Ahc

T,AABhC

ThB

ThA T,BhC

T,A-+BhC

T,AthB

T,VxAxhB

T,AxhB

T, 3xAx h B

T,A,AhB

VL

r,,r2i-Bcut

T,BhCAL2

T,AABhC

T,AhC T,BhC

T,AVBhCVL

T,A,B,AhC

T,B,A,Ah C



Intuitionistic Linear Logic ILL

Axioms:

Ah A

T,0\-A

Rules :

TTFB 1 L

T,A,B\-C

T,A®B\-C

r,A\-cT,A&B h C

T\-A

T\-A®B

r,A\-B

r h~ A. —o B

T\-B , ,T,\A\-B

Y\-Ax w

T\-VxAx V

&Li

>R

T2,A\-Bcut

h 1

r i - T

n i- A r21- B

rur2\-A®B

r,B\-cT,A&B h c

&L7

TxhA T2,B\-C

Y\, PT, A —O B \~ C

T,A\-B

T, \A h B

T,At\-B

\L2

VL

T,A,B,A\-C

T,B,A,A\-C

ThA ThB&R

r\-A&B

T,A\-C T,B\-C

T,A@B\-C

n J

\r\-c!D-!C

T\-AtT\-3xAx

\Rr, \A, \A\-B

r, \A v- B\c

3Rr,3xAx\-B

3L

Appendix. Acknowledgement

The author wishes to thank Prof. Giovanni Sambin for discussions and encouragement

during the preparation of her work. Thanks are due also to Raymond Hoofman for some

helpful suggestions on the existence of coproducts in the co-Kleisli category and to Giulia

Battilotti for useful comments on the representation theorem for quantales and frames.

Finally, she is grateful to the referee for the careful reading and valuable remarks.

Appendix. References

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161-184.

Barr, M. (1991) ""-Autonomous categories and linear logic. Mathematical Structures in Computer

Science 1 159-178.


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Battilotti, G. and Sambin, G. (to appear) Pretopologies and a uniform presentation of sup-lattices,

quantales and frames.

Danos, V., Joinet, J.-B. and Schellinx, H. (1993) On the linear decoration of intuitionistic derivations,

Equipe de Logique Mathematique prepublications, Universite Paris VII 41.

Girard, J. Y. (1987) Linear Logic, Theoretical Computer Science 50 1-101.

Hoofman, R. (1992) Non Stable Models of Linear Logic, Ph.D. Thesis, University of Utrecht.

Lafont, Y. (1988) Introduction to Linear Logic, Lecture notes for the Summer School on Constructive

Logic and Category Theory, Isle of Thorns.

Mac Lane, S. (1971) Categories for the Working Mathematician, Graduate Texts in Mathematics,

Springer Verlag.

Mac Lane, S. and Moerdijk, I. (1992) Sheaves in Geometry and Logic, a first introduction to Topos

Theory, Universitext, Springer Verlag.

Marti-Oliet, N. and Meseguer J. (1989) An algebraic axiomatization of linear logic models, Report

SRI-CSL-89-11, Menlo Park, California.

Marti-Oliet, N. and Meseguer J. (1990) Duality in closed and linear categories, Report SRI-CSL-90-

01, Menlo Park, California.

Marti-Oliet, N. and Meseguer J. (1991) From Petri nets to linear logic. Mathematical Structures in

Computer Science 1 69-101.

Paiva, V. C. V. de (1989) A Dialectica-like Model of Linear Logic. Springer-Verlag Lecture Notes in

Computer Science 389, 341-356.

Rosenthal, K. (1990) Quantales and their applications, Longman Scientific and Technical, Longman.

Sambin, G. (1989) Intuitionistic formal spaces and their neighbourhood, In: Ferro, Bonotto, Valen-

tini, Zanardo (ed.) Logic Colloquium 1988, North-Holland Amsterdam 261-285.

Sambin, G. (to appear) Pretopologies and completeness proofs, The Journal of Symbolic Logic.

Sambin, G. (in print)The semantics of pretopologies. In: Dosen, K. and Schroeder-Heister, P. (eds.)

Substructural logics, Oxford University Press.

Schellinx, H. (1991) Some syntactical observations on linear logic. Journal of Logic and Computation

4, 537-559.

Seely, R. A. G. (1989) Linear logic, *-autonomous categories and cofree algebras. Contemporary

mathematics 92, 371-382.

Troelstra, A. S. (1992) Lectures on Linear Logic. CSLI Lecture Notes 29.

Yetter, D. N. (1990) Quantales and (noncommutative) linear logic, The Journal of Symbolic Logic

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