I. Quantales Definition 1.1 a Quantale Is a Complete Lattice Q With

Home , Complete Heyting algebra, Quantale

Logique cfe Analyse 153-154 (1996), 113-151

EQUALITY IN LINEAR LOGIC

Marcelo E. CONIGLIO and Francisco MIRAGLIA

In this section we introduce the basic definitions and results of the theory of quantales (a good reference is [Rosi). Quantales were introduced by Mulvey ([Mu ll) as an algebraic tool for studying representations of noncommutative C*-algebras. Informally, a quantale is a complete lattice Q equipped with a product distributive over arbitrary sup's. The importance of quantales for Linear Logic is revealed in Yetter's work ([Yet]), who proved that semantics o f classical linear logic is given by a class o f quantales, named Girard quantales, which coincides with Girard's phase semantics. An analogous result is obtained for a sort of non-commutative linear logic, as well as intuitionistic linear logic without negation, which suggest that the utilisation o f the theory o f quantales (or even weaker structures, such that *-autonomous posets) might be fruitful in studying the semantic of several variants of linear logic. As usual, we denote the order in a lattice by w h i l e V and A denote the operations of sup and inf, respectively. We write T for the largest element in a lattice and 0 for its smallest element.

Definition 1.1 A quantale is a complete lattice Q with an associative binary operation 0: Q X Q Q , which distributes on the right and on the left of arbitrary sup's, i.e.:

[Q1] a 0 (b c ) ( a b ) c , for every a,b,c E Q [Q2] a O(V,Efa t ) = V , E / A qu(anatal e Qa is, unital) if it has an element 1 E Q such that a 0 1 . 1 0 a = a, fo,r every a E Q. A quantale Q is commutative if a 0 b = b O a, for every a, b E Q. ( V i E / a A morphism of quantales is an operator between quan tales which pre- servets 0 and a) rbitrary sup's. 0 It's easilya seen that the above axioms imply that (g) is increasing in both coord=inates, that is t o ( a , 0 114 MARCELO E. CONIGLIO AND FRANCISCO MIRAGLIA

If a b then, tic Q , a 0 c 1 , 0 c and c 0 a 5 - c 0 b

We register a classic result:

Proposition 1.2 The endomorfisms a 0 • , • 0 a : Q Q have right adjoints, denoted by a —› • and a —4 1 respectively. Thus,

a0c--<-b f f c c 0 a - - and cao-ns-e>quently b r a—>,b=V ab — ) 1 Definbit=ionV 1 .3 Let Q be a quantale. A map j : Q Q is said to be a a) quantic nucleus if it satisfies: I c E[NQ1] a b implies j(a) j ( b ) [ N Q 2 ] a j ( a ) Q[NQ: c3 0 b) qua] anbtic conucleus if it satisfies: )A l. [CWQ1] a b implies g(a) g ( b ) [ C WQ 2] g(a) a [(CWa Q3] g(g(a))= g(a) [ C N Q 4 ] g(a) g ( b ) g ( a b ) ) ) Qua=ntic n uclei and conuclei are important, because they determine the quot[ients and subobjects in the category of quantales. N DefinOition 1.4 Let Q be a quantale. A subset S C Q is a subquantale of Q if it is closed under 0 and arbitrary sup's. ] Propjosition 1.5 [Rosl: ( a) If Q _ > ) Q is a quantic nucleus, then Q. =((..t E Q : j(x) = i s a quantale where the operations Of ,\/ and A ' ianQ .are given by: ) ja 0 () Morebov er, the map j : Q Q j given by a j ( a ) , is a surjective mor- phismb of quantales. Further, every surjective morphism of quantales can be repre)=sen te d in this form. j ( a 0b b) )V i e , a i = j ( V i e , a i ) N E / a . = A i e , a i . EQUALITY IN LINEAR LOGIC 115

(b) I f Q Q is a quantic conucleus, then Q, = [ x E Q : g (x)= x ] is a subquantale where, with natation as in (a), N : EMo1re oaver , every= su bquangtale is of this form, i.e., ( if SA C Q is a, subquantale, then there exists a quantic con ucleus g in Q such that S Q E 1 g a Definition 1.6 An element I E Q is dualizing if

(a —> r ) ---> I I = a = (a - 3 1 I , An eflemenot s Er Q is cyclic if e v e r y aa --> r s = a —› 1 s, for every a E Q. E Proposition 1.7 [Yet]: Let Q be a quantale and s, I be elements of Q. a) sQ E Q is cyc.lic iff for all a l , a n i n Q flo ,r all cyclic permutations ir of ( I•, n ) . b) I •f I E Q is dualizing, then Q is unital and we have: • 1 = r = O a Definition 1.8 A Girard quantale is a quantale which has a cyclic dualizing n element I . The Operator - —) s 1ne Aa=tion, and• we wr-ite- a- > a ) .i rl = da e- - -f m •> I Nextp proposition is of frequent use when computing in a Girard quantale. —( n o › t e l 1t h a t ! Propiosition 1.9 [Ros]: Let Q be a Girard quantale with a cyclic dualizing i= s elemeent I and let a,b E Q. Then: c= a l l s e1 d a(1) a ----> 1 b = (a b l, (3)i a bn = (a e—› 1 b (4) b a = (a —› b ,1 1 a 1(5) r a r b =1 , 1 () n )1 - - - > 1 ) 1 ( 2 ) dProp_ao1sit_ion 1 .10 [Yet]: Let Q be an uni1tal quantale and s E Q cyclic. Then, )a ja : Q l Q , given by j( a)= (a —› s) —› s is a quantic nucleus, and Q. = 0r =l a E( Q :6 j(a)=) a i s a Girard quantale, where I = s E Q ab a ,= — ( 1 3 › 21 l )0 b •a = •) b • - - O - > a r , a ( , ) - - 5 . - s , 1 16 MARCELO E. CONIGLIO AND FRANCISCO MIRAGLIA

Example 1.11: The Phase Quanta/es (Girard) Let (M , • , ! ) be a monoid. We define A• B = a b : a E A , b E B ] V A,B C M. Let I C M be such that a• b E I implies b•a E I (for example, I can be a semiprime ideal or the complement of a prime ideal of M). Thus, to (M) is a quantale with the product defined above, and with sup's e inf's calculated as unions (U) and intersections (n). In fact, it's an unital quatztale, where 1 -= a n d I i s cyclic. Then, by proposition 1.10, O M ) i s a Girard's quantale containing 1. Their elements are called facts, and we have:

A 0 /3= (A • B r Va , E 0 , = (Li ,e/A,) 11 ;A iind pA ai r t i c u l a r , The Anext resul t tells us that every Girard quantale is of this form. yi Proposition 1.12 [Ras]: If Q is a Girard's quanktle, then Q is isomorphic to E a phaBse quantale . (/ A We auAre now going to relate the exponentials treated by Yetter ([Yet]) and Avron8i ([Av)rj) with certain concepts in the theory of quantales. 1; 1 Defidinition 1.13e [Yet]: An open modality in a quantale Q is a map Qfn Q satisfying: Ap ia[ Mil p„ (a) a [ M 2 ] a b implies p, (a) p . ( b ) Er[M3] ( ( a ) ) = ( a ) [ A 4 4 ] ( ( a ) ( b ) ) = ( a ) P Dt, ( b ) . An open modality p. is said to be Bi c ,— central i f b 0 p„ (a) = p, (a) 0 b for every a,b E Q. u— idempotent i f p, (a) 0 p„ (a) -= p, (a) for every a E Q. l— weak i f Q is unital, p, (1) = 1, and 11 (a).„ a- Let Mrç•( Q)1 = { fp,: op, ris ane opven emodr alyity in Q}, partially ordered by point- wise order. ,a E AQ . PropAosition 1.14 [Yet]: Let Q be a quantale (resp. unital quantale). Then, thereB exists a unique maximal open modality in Q central (resp. central, idem=potent and weak) denoted by c (resp. m ) gAi v e n b y : nc !,n(x)=V[aGQ:axAl,a=a0a,aEZ(Q)1, Bm -( fx A) &= BV .t a E Q : a x , a E Z ( Q ) ) EQUALITY IN LINEAR LOGIC 117

where Z(Q)= I a E Q : a 0 b = b 0 a, for every b E Q].

Definition 1.15 [Avr]: Let Q be an unital quantale. A map B: Q Q is a modal operation if it satisfies, for all x, yE Q:

[BI] B(1) I [B2] B(x) x [B3] B(B(x))= B(x) [ B 4 ] B(x) 0 B (y ) , B ( x A y ) . Proposition 1.16 Let Q be a quantale, and Q Q a map. (i) 1.1, is an open modality i f p, is a quantic conucleus. (ii) I f Q is unital, then [1, is an open, idempotent and weak modality iff is a modal operation.

Proof. ( i ) W e must prove that [M4] is equivalent to [CNQ4]. Let p, be an open modality. Because p (a) 0 p. (b) 5- a 0 b, we get p, (a) O p (b) ILL (p, (a) O p. (b)) 1 1 . (a 0 b). Conversely, if p, is a quantic conucleus, then

p . ( a 1 ) 1 (ii) A®ssume that p, is0 an open, weak and idempotent modality. Then, it clearlyp .satisfies condit1ions [BI ],[B2],[B3]. Because p, (a) 5_ a and p,(b) 1, we( bget p,(a) O p, (b( ) a 0 1 a . Similarly, pi, (a) O p. (b) b and there)fo re, p, (a) 0 p, (b)a S a A b. Thus, FL (a) 0 p, (b) = p. tp, (a) 0 p. (b)) p„ (a A b). ( ) Now, since p, is increasing and a A b S a, b, then p, (a A b) 5_ p, (a), p,(12)l. Thus, 0 - ( p(czAb).11(aAb)0p(aAb)p(a)0p(b),t b , ) and s(o p, satisfies [B4]). Coanversely, if p, is (a modal operation, then [ MI ] is just [B2]; [M3] is [B3],) while [M4] is eqauivalent to [CNQ4], in the presence of [MI], [M2], [M3],) by item (i) above) . CoPndition [M2] is itemI (5) of Lemma 4.2 in [Avr] and p, satisfies [CNQ4] by it,em 8 of Lemma L4.2 in [Avr]. It follows from items 4 and l in that same Lemma, that IX ( X ) 1 for every x E Q. Thus, [B I] yields that p, is weak(. D• L p ( Defin. ition 1.17A frameb (or Complete Heyting algebra) is a quantale where (g) = (A. ) I - ) ) ) 118 MARCELO E. CONIGLIO AND FRANCISCO MIRAGLIA

Definition 1.18 Let Q be a quantale and let T be the maximum o f Q. We say that x Q is

a) right-sided (in Q) i f x 0 T b) left-sided (in Q) i f T x . r , c) two-sided (in Q) i f it is right and left sided in Q. d) idempotent ifx0x=x.

Proposition 1.19 [Ros]: Let Q be a quantale, and g a quantic conucleus in Q. Are equivalent:

(a) Q 8 is a frame. ( b ) g(a) g ( b ) g ( a A b). (c) every xE Q g i s Defini itdione 1m.20p A omap g satisfying the conditions above is called a localic conutcleeusn ant d Q g is called a localic subquantale. Since open, weak idempotent modalities are localic conuclei p. such that p, (1)a = 1, wen have d Corot llarwy 1.2o1 Let- Q be an unirai quantale, and let p, be an open, weak and idemspoteint md odaelity. Then, Q p is a frame. Thdus, th e fixed points of the interpretation of the modality ! (of course) lie in( a localic subquantale (complete Heyting algebra) of any quantale in whichi linear logic is interpreted. n Q 2. L ign e a r Calculus with Equality ) . In this section we discuss the laws for a binary predicate representing equality. The goal is to define a reflexive, symmetric and transitive predicate satisfying the substitution (Leibnitz's) rule for the class of all formulas. We may assume, just as in Classical Logic (CL), that we have substitution for atomic formulas. With this model in mind, we shall define a prototype of a linear calculus with equality, called (L L E 1sequents in Linear Logic (LL). Analogously, we will set down a calculus )w.i thO equalityu r fforo ther (mMAuLLl ) afragment,t i o n i.e., the fragment without expo- wnentialis, indlicateld by (LLE 0 u)e.l emSetntaasryr formulas,t i neg we prove that (• = •) must be 1 and idempotent in f(MArLLo), anmd op en in the general case; in other words, (• = • ) must be intuitionistic. t h e p r o p e r t y o f s u b s t i t u t i o n f o r EQUALITY IN LINEAR LOGIC 119

Definition 2.1 A first order linear language with equality L consists in a countable set of predicate symbols = ( P : n E w ) U f = } (where "=" is binary), a countable set of variables V =f v n : n e w ) , together with the symbols:

1, T , 0, 1 , V , DefinAitio,n 2 .2 The formulas of L, FOR(L), are defined recursively: e , S[Fl] E1 , 1, T, 0 E FOR(L). ,[F2] I f P Ep is a predicate of arity n and x 1 0, . . . , xt hen P a ( r xe 1 v[F3]a ir,f. i.F.,a, Gx bEn l FeORs(L,) then G , F u G , F & G, F IED G E FORM., , ) a n d [F4] i Pf F E( FOxWL ), th1en !F ,?F E FOR (IL). ,[F5] i f F E FOR(L) and x E V, then A x.F and V x•F E FOR(L). 7 , . . . , x n Ever.y occurrence of a variable x in a formula F is free except in a subfor- mula of the) type A 1x.G o r v x.G (which are bounded occurrences). We shall write E F O R ( L ) . ELF(L) =f 1 3tary formulas (x 1 and , . . . , x n E L F ) (taDry) formulas. : = P DefiEnition 2 . 3 T h e syn t a ct ic l i n e a r n e g a tio n i s a m a p E 1 : FLO RF(L )—> FO R(L ) given by the usual rules: a ( L —[Nil.] 1 ( 1 ) = 1 , 1 (1 ) = 1, 1(0) -= T , I (T) = 0 ) f[NL2] l ( P ( x U =l p i ( }, . . . ,,x (henre, P(x 1) x x)[NL3]) ,_ L, .(.F. , XG ) =l l ( F ) u 1 ( G ) , _L(Fu G ) = _L(F) I ( G ) , = E= nl ( F &) G )=) _ L(F) l ( G ) , _L(F Et) G) = 1.(F) & l (G ) y[NIA] 1 ( F ) ? l(F), 1(?F) = I ( F ) VP E = )[N1,5] 1 ( A x.F) = x• -1 -(F) , x . F ) = A x. ]( x E L F P : ,1 ( L ) ) ( We wx rite l ( F ) = F tn x l ; , h) 1 c l ye a r l y e1 , F E s( ( . = V F ex . l ) t= . ' , f o oy , r t f) x e h v e e) n r e y l( ) F e ex , E x m= 1 F t O ey ( R e M n) ( . n -1 x d = e y d ) s ) e = t ( o x f = e y l ) e m e n - 120 MARCELO E. CONIGLIO AND FRANCISCO MIRAGLIA

Definition 2.4 Let A a formula; we define A[ylx] to be the formula obtained from A by replacing all bound occurrences of y by z, where z is the first variable (in the natural order of V) not occurring in A, and then replacing all free occurences of variable x by y.

Definition 2.5 (Girard) The calculus (LL) for first order commutative linear logic is defined by the axioms and rules below (here, A, B denote formulas, and F,A denote multisets of formulas):

[AX1] I - A l ,[C UAT] I - I [- A F, A X, A 2 ]I[EX -CII] I f A is a permutation ° f r FA. , A, t h e n [T&] 1 - A, F B , , I -A & B , F [ e lA] F-A GB ,F FX A3[ I f F is not empty, then I- .] I- F F [1[(g)] FA,F[u] I-A,B,F I-A OB, F, A I-A u B,F G 2 [dereliction] F A IF [ w e a k e n i n g ] ] ?A, I- ?A, F [Bcontraction] I - ?A. ?A. F r j F A ?Ir I I- ?A, 1" !A, ?I'

[V] 1 Ary/xl. IF [ A ] I f x is not free in F, then I - A I- x.A,F h A x.A, F

We can consider as defined connectives the linear implication and the linear equivalence, given by:

(A B ) --- def u B) = (A ) .( A( 0A-0 B) clef (A —0 B) & (B —a A). l EQUALITY IN LINEAR LOGIC 121

Definition 2.6 The linear calculus with equality (LLE 0 fragment) f o r otf (LhL) eis d efin(edM by aAddinLg thLe a)xioms below to those in the preceding definition:

[R.] : ( x = x); [ S = ] : ( x = y) , (y = x); [T=] : F (x = y) -1 - , ( y =[SUBST]: z )( x 1 = y 1 1) F(x 1 ,1wh er,e ,(F(xx 1, . . . , x(=, .1. . , x ) x yn ,=ndb.y. replacinge n o ) some occurrences o f x (wh ich are not in the scope of a y i .z)t, xe s) 1 - f.-a 1 - , [1 rq,n F . eu l e ( Defin=it]i on 2.7 The linear calculus with equality (LLE 1 eam e n x )b yf aod:dr in g, (beLsidLes )the axioi ms s[R=] ,[S=],[T=],[SUBST] the following rule ant t a r l d ei(f f i n e d n[!=]y : ( x = ye) 1 , !(x y ) x dif e o y = Obviously,Frr ) m all the1 rules (with exception of [R=]) could be formulated as , linea(bur implications,l , for example [T=] could be stated as ( xya — x 1ywF = =, = = 2i x y We haveyt as well„ that (LLE 1 (LLE 1 ) i sxh) ) b y e q y(0u i ve a l e n t r e [Sl=] ( x = y) t ,v( o e p l I , = t ,nxh t e a c xi ) 3c. Se)au=m a nl aticcs u l n g u ilys l [ Ion thissy) b s e c)tiot n wea deveilop interpretations for the calculi described above. It S wn ill ob1e necdessary to extend the definitions in [Yet] so that the axioms = ifnvobl]vin g requality aore verified. ] m t: a a( n ix d n= [ ey ! d) = f- ] r1 b o- y m, t F1 h ( e x r , u , l . e . : . , x , i ) 122 MARCELO E. CONIGLIO AND FRANCISCO MIRAGLIA

Definition 3.1 An interpretation of a language L is a triple < st ir, • la> where

- s i is an algebra for the theory with constants 1, 1 , T, 0; - a set of unary operations , , ? ) (the interpretation o f the modalities); - b in a ry operations 0, u , &, ED, and infinitary operations V and A; - C sti, the set of valids elements; and - 1 • 1 : ELF '(llL) - › 51 is a map.

It is straightforward to see that there exists a (unique) extension of 1 • 1, 1FOR(L), t o also denoted by • 1, 1l• 1.for1•154 • . W h e n tDehfiniteionr 3.e2 A semantics for L is a class of interpretations. A semantics is isounds (with respect to a linear calculus P) if: n o r i A s k for every < si,V,1 • 1 o l f 3 1 c o , n f u s i A se>m aniticns f or tL ish coemp lete (with respect to P) o n . , s .e m a n t i w IFI e c .s . w dimpliesi r 1- F isi provable in P. , t E e A Defin1ition 3.3 An interpretation of quantales for L is a Girard quantale Q, n togetherr with an assignment ! E M(Q) and a map i • 1 Q, • E L F(I I ) Q , { c t e Q : a 1 s} s u c h t h a t f L p o- V a, b E Q, a u b d e f (a 0 b ) a n d ?a d e l ( r r[- W e interpret A x.A and V x•A as A r o and r E v lAbilx11 Q e1E1V I A [ y / 4 e v , v• ( e l ) ) Recalla thatr e Ms(Qp )e isc thtei latticev e of open modalities in Q (Definition 1.13). e b l y . Defirnition 3.4 The semantics of quantales for (commutative) Linear Logic l with yequality is the class of interpretation of quantales for L such that: e < i s[SI] Q is commutative. [ S 2 ] 11 d e f ! is idempotent and weak. n t P i[S : E L F i , * 3m V(a) Fo r the Calculus( 1 L (LIE 0): p ,] ) l 1 Q Hi • s e 1I a t s 4 i s QI > f i A i e 1 n s : u t . h . e . s 0 e A m n a 1 n , t 4 i E c s , EQUALITY IN LINEAR LOGIC 123

[=11: x=x 1; [=2] : = YI 0 IY = z1 [=3] : y = ly = • [=4]: x tx i n IF(x , = (same notatio=n as {Sl UBSTD; [ = y y e 5 1 , y (1b ) FoIr the Calculus (L L E i the stronger: I 1: • , . )I, c o• n d i t i o n s . . [x= = 5 ] 7 , x a= n d , The ]next result shows that it is always possible to define a map I • I [Y = 6 ] e Qfying s: tah et aibsov-e conditions. aI r e y rII e p l a Propxos ition 3.5 If • I n to ELcxF + e d Q := E L F ) (thleL )gb= en eral case, conditions [=1],...4=41,[=7]. ( L Y) I s a yt i s Q I fPryooilfn: L e t 's begin by the general case, in which ! E M(Q) is idempotent ai nd w! eask. g , 0 a I i LeIt I • I Q •• ELF(L) Q be a map, and let x,y E V, x y . Define Am ,,,, xan da B n x rp = , t = ,t, aAY sh: h Y e xBl =n E ELF(L): x or y occur in F} e I i ,• (In B ;co n sid e r the relation: t = M [ c I A = a F — G i f f G is obtained from F by replacing some occurrences of L 6 x by y and/or some occurrences of y by x n E L ] b E )For example,: P(x,z,y) — P(x,z4) — KY,z,Y) — Ikv,z,x). Clearly, — is an e L ecquiv=ale nce relation. Now define, e F x t a Y e ( n d s I e L d e Fact) 1: I f I • I satisfies [=7] and [=31, then it satisfies [=4] iff Jx = T , for x: y . c Ton see this, assume that F, G E B , , , F — G . Note that there is o H Ee ELF(L) such that H = F(x y ) , G = x ) . n Sini ce ! is idempotent, [=7] implies [=5] and so, by [=3],]=4] and [=5] we hdave that Ix = yl 0 IFI H I ; thus, t i t h i e o r n x s n [ o = r 1 y ] , o . . c . , c [ u = r 6 i ] n a F n I d , i n 124 MARCELO E. CONIGLIO AND FRANCISCO MIRAGLIA

= 0 I F i TherefoI rex, Ix = yl - 4 , proving that Ix = y T x • Con=verse ly, suppose that Ix y f o r x y . Then, i f F E UY ; 1 I =1 ( ,1 1 F ( x 1 B I , . . . , x n and therefore,) 1 Ix, = Yn1 0 )1 n_1,X Y 01. Analxogously, = — › y Y = 11 n , - ® = , 1 0 1F(x n I x n-11° ) ) 1 I ) 1F(x ,Y,)1 0 y I n We mayF proceed by induction to get: - I o Ix]) = Y 1 1 x 0 • as desiIred. n • x • y If F E n i 0 = n n I x Ix Yi = Y 1 1 ) = „1 0 I • I A = show•in0g th•at[= 4] is valid and proving Fact . x Y, By0 inI du ction, define a map I • I as follows: for x , y E V, x y , set t nh e I Hx n 1, nx I= Y 10 T , s 0 i =x '= = rxo (Ix z1 z1 n I c (a <---> b means (a -4 b ) A (b - › a)); Y G e FIx = ) n I [ (7 I . Fa=ctx1 2-: 1 • I 0• 0 ,, 7In l=fact : condition [=1] is clear, while [=3] is verified because it's true for Ix = YI in , for all n E s a1 t?Vi s f i F e is Y\ m Sin' ce Ix = yL = !Ix = [ = 7 ] is satisfied (the case x = y is valid too, be- t p hi' ,le cause( !1 = 1). Observe that, for every n 0 and z x , y : p i r .Eoe p e sr 1 t i .l y ,+1 0 ix = z1, I ) ) = zIn and therefore, e [ s• .y y - ( r = e• .= g q 6 •u i .Y ) r ] e) xi I d , = n x f w ?) = o e Y; z r h nf l t a )i - h v In I e e ,a x f : l = u l Y l y I l , n l s + o e 1 g t 0 i I I c x x . = = x z l l _ n = = 1 z . l n f o r e v e r y n O . EQUALITY IN LINEAR LOGIC 125

Thus,

Ix = yl .0 0 I x Since= Ix = yl I x = yl = T n FoFzr thale c( MtAL L) fragment,1 it's enough to take ! = ( s e e Proposition 1.14) in the above computations. [11 g uA a r a n t e e s n Discussion 3.6 Are requirements [=5],[=6] (resp. [=71) too strong ? t eh a t • The motivatio n for them is, starting with substitution for elementary for- mulaNs, to h ave substitution for every formula. Obviously, they are suffi- cientI, but inodeed, they are also necessary. To see this, note that, if a = s = vYa t i s f ai , c ;e b s , (i.e . a ( b c ) ) must imply a ! b ! c (and, of course, that a[ 0b nl c !=b ) . Sin4ce 1 is a formula, we must also have a = a 0 1 s 1. The c] ri=ticoa l ca. ses are 0 and !. We have I w LePmm[ a 3.7 Let Q be a Girard Quantale and let a be an element of Q such that a 5_ 1. (a ) Are equivalent: ( = 7 x (i) V b, c, d E Q, a 5 (b c ) , a 5 (d e ) ] ) implies a 5 ((b 0 d) <---> (c O e)). a (aii) a 5 a 0 a n n (b)d Assumed th at I E M(Q) is idempotent and weak. Are equivalent: c [ 1 (Mi) V b, c E Q, a :5 (b c ) imp lie s a 5 (!b ! c ) . P (4ii) a = !a ( ] (c)x Vy b, c E Q, a 5 (b c ) i m p l i e s a 5 0 1 y i Pro) oef: (a ) ( i ) ( i i ) : since a .5 1 , then a :5 (1 a ) and so a 5 ((1 0l 1) <---> (a 0 a)); thus, a = a 0 (1 0 1) 5 a 0 a. (i) 1( i i ) : I f a Ls (b c ) , a 5 (d e ) , then we have a b s c and , d a d 5 e , and so a 0 (b 0 d) ( a a ) 0 (b 0 d )= (a 0 b )(g )(a 0 d) tc 0[ e. Analogously, a 0 (c 0 e) S b O d. (b)h =(i) ( i i ) : sin ce a 5 1 and !1: = 1, then a :5 (1 <---> a) and therefore a 5e (12 <---> !a); thus, a = a 0 1 5 !a S a . (ni) ] ( i i ) : suppose that a S (b c ) ; thus, a 0 b S c and then a O !b wa O. b s c. Thus, by [M4] , we have a O !b = !a 0 !b = !(!a O !b) = !(ae 0 !h) 5 !c. Analogously, we can prove that a 0 !c S !b. (c)m Since Q is commutative, then (x —› y) = (y 1.9. El 1u ) , b y Ps r o p o s i t i o n t h a v e t h a t a b c a n d 126 M A R C E L O E. CONIGLIO AND FRANCISCO MIRAGLIA

We can now show that we have substitution for every formula:

Proposition 3.8 Let I • I: ELF(L) Q satisfying [=1] — [=6] ([=1] — [=4], [=7] resp.). Then, the extension to FOR(L) verifies:

Ix = 1 1 0 • wher•e 4 )(x xl • l,re. p. .la,0cxi ng some occurrences of x , (not in the scope of a y t ye. ,E- q uI a nxt i f i e r ) b y .F O R Proo=f: B y in duction in the complexity of (I). As a first step, we have two .c(aseLsn to) co nsider. .h 1 ,a i)0 e E LF + xs a1 ) . It's true by [=2],[=3],[=4] and [=6] (or [=7] and ! weak). a( e4ii) v 1 , 1 ne n0, Tt , Tduo par10ocele d wI ith. the induction, assume that substitution holds true for every s(l1) wyi,Ith )ctom ' p lesxity k and let 4) be a formula with complexity k + 1. We zfhave,t the rfollouwing cases: lr xe a) = ci 0 13. )e „b It's immediate from [=5] (common to both systems) (e )y a nd lemma 3.7. xo c 1 bI[ ) (i)== a • yc 46u Sin]ce lu 1r c)(r) l(I). A x.ol G i v e n z E V, we have: ,e (on l .c e xr = 7.,1l a Y i I 0 A VE V l .s a ( X 1[ 1 1 ' Ix e l .o • . . la (x ?= , ' 0 ,f y7 X n • . 0 i l a t [ hY xv 1] 0e 1 e ( x x 1 1 Avevia (x ya ,a ;c o a 1 i ,r .n 1n c n , . . d) 4 = a & p. Similar to c). o ,i .d -l u d . , x e) (f u l l logic) 4) = !a. It follows cfrom [=7] and Lemma 3.7. )a .! 1s i t , x K ib ,w Ao n h ) Since the other connectives are defineyd by duality, the proof is complete. E! sl xe vf e [ z L Now,oe ,a weEo shall extend the resunlts in 1[Yet]4 to the calculus with equality. v bs ik vl ) 1 i t 2. ll l a y ao . i (w n xs e 1f d ,r f .o r .m o .L m ,e . xm 1 ,m ) ,a ( x3 x ). 1 [7 , y. . / . x . ] , x n ) b y EQUALITY IN LINEAR LOGIC 127

Theorem 3.9 (Soundness) The semantics of quantities for commutative Lin- ear Logic with equality is sound with respect to the given calculus.

Proof: W e prove the validity of the new axioms (for the other rules, con- sult [Yet]). Define IF A 1 ,e.v.e.ry, Aa, b En Q , aI • b iff a =lte eud by [=1]bI — A [=61 ] (rest, p. ] —[=4 1 ,[=7 ]). ut h e •vTheao•reml 3•i.10d (uCoi mpt letyene ss) The semantic of quantales for commutative Linear Logic with equality is complete with respect to the given calculus. Ao f e a c ,Proof Th e proof is an extension of the proofs by [Yet] and [Gir]. Let M h .be the set of finite sequence of formulas in L; M a x i eBI raitiosn o fa cyon camtenaotio nn (ando i idden tity the null sequence). Let M be the o m ow(commbi utattsive)h meon oid obttainedh by ideentifying sequences which are distinct i ronly vby ap pi ermnu-tatiogn of t heir elements. Just as in example 1.11, 1 (M) is a tscommutahtive q uantale and we set I = F : i s provable ) E ( m ) . ag Sinceu Mt is commutative,a I is cyclic and so we can consider the phase ,rquantaale Q = n( M ) I f-,(A w—›h I e) —)r e oj Let P r : r0 : ( M ) Pr(A)= { F: I- A, F is provable }. —F O ) R Byp theorem 3.4 in [Yet], Pr factores through Q, i.e., Pr(FOR(L))= Q. ( M M —Let • I = Pr ELF+ (L); clearly, the unique extension of l• to FORM is Pr, o) nce we have defined in Q the open, weak and idempotent modality ! as: i › s ( !(x) = V P r(! A ): Pr(!A) x 1 g m i vFa)ct: ! e1 = 1 (where, by definition, I -= A { Pr(A) : A is provable}). n Tdo see this, Let F E P R ban(ed! 1let A E FOR(L) be such that I- A is provable. Then: y )f ; t h u s , j Ii - ( Fn A ) (e i . e . , = Id - (b ? I ) 1y FA , ,: F ) i s p r o v a b l e 128 MARCELO E. CON IGLIO AND FRANCISCO MIRAGLIA

i.e., F E Pr(A), and so Pr(!1) A I Pr(A) :1- A is provable = 1. Thus, Pr(q ) v P r(! A ): Pr(lA) 1 1 = !1. In fact, equality holds because I- !IL is provable, and so 1 .5 Pr(!1) ! 1 5- 1, i.e. 1 •= !1. We shall now show that I• I satisfies the required properties. Since I-(x =x) is provable, 1 :5 Pr((x = x)) = Ix = x Th is verifies I= 11. To prove [=21, let FA E = yI • y = zl. Then we have:

(x y ) , ( x = y ) 1 (y = z)", (x = z) , (y = z), A , ( y = z ) ( x (x = z) , F, A z ) i.e., F A E x = zl. Thus, Ix = YI • ly = zi I x = zI and so Ix = yl® = zl = (Ix =Y1 • IY = zi)

To p1ro ve I[=3],x let F E =x = yl. We have: z l • (x y ) , ( x •-=•• y)", (y = x) I- (y = x) , F i.e., F E ly = xl and thus Ix =y1 y =x1 Properties [=4], [=5], [=6], [=7] are verified in a similar way. This shows that we have an interpretation of quantales such that IA I 1 iff h A is provable, completing the proof. EI

4. Generalisations of the Calculus

There are alternatives to the treatment of equality given above, using the exponentials of Linear Logic. For example, instead of requiring (• • ) to be open, we could establish that its characteristic properties be valid in the interior of (• = -). Thus, we define the calculus (I' t ) b y t h e a x i o m s : [1?!.] 1 - !.(x = x) [ S 1[7 1- ! ( x =!] yF ) —t 0. !( ( y =x x= ) y ) 0 1 . ( y Z ) ! ( r = Z ) EQUALITY IN LINEAR LOGIC 129

For every predicate symbol P,

[SUBS!]! ( x l = ----0 P (x y 1 1 Clea)rly , we have substitution fo r ev2e ryy formula, and we can say that !(x = y) has the behaviour of a lineari equality. An even weaker form of subs0titution can be defined as follows: • , . . . , • x = y • „ i IP(xly y 0 e ) n or, e!ven y 0 ) ( x n • , !(x ) • = l —0 ?P(x 1y y • y = , n 0 We ctany modify each axiom combining) th.e modal operators in all possible ! ways). Iit is straightforward to verify that (incorporating id, the identity oper- ( ationP o)n formulas), there are only seven modal operations obtained by suc- cessive appxlications of ! , ? }, namely, the set ( • , x • i • 2 = 0 y ! „ ( ) x 0 , ! = P y ( t dtit, = !x, !?!, ?!, !?,?!?, ?, id i d !? r i ) , ! . P . ( . x , 1 x , , . ) fig. I . . , x 130 M A R C E L O E. CONIGLIO AND FRANCISCO M1RAGLIA

with its natural lattice structure (see figure 1). This is in complete analogy with Kuratowski's problem, in which we can prove that, given a topologi- cal space X and A C X, there are only 14 distinct sets that can be obtained from A by combinations of taking complements and closure (in our case, we do not consider modalities of the form m'(x) = m(x ) 1St arwtinig twhith this,m we ca n definEe a general scheme to create a linear calculus with equality.

Definition 4.1 Let ( r a )equality P((m ,)) is defined by the following axioms: i [R] m i iM F ra 4 ° ( rxa 5 ,F or be(=axceh pr edicaate sym bol P, the axiom s e=x q u e n [SUBS] m c ey) ) 7 1 i 0[ n A S( ( (t ) , . 0 Remmarks 4.2 ( a) I f m 3 M ( Ar(x =]n y), s(x = y), r(y = x) and s(y = x) are all equivalent. 2 5 1 l (b)m_ iI f mn , t( h ye n 3 e i(c)i2 Iaf m 7 =r id, m 8 E ! , !?, !?! } and m [ S= ] ( fcor=9 e s(vmeiaryd fo rml ula (I) built up from formulas of the form m 9 (P), where P E Ei LFm(xzL).p l i e s x c o1 (0rux l t (d)m) hFor aeacht m , ! , !?, !?! 1, w e1 h ave m ( 1 0 u m, = st h e n f1711 3) mo , (r 1 1 , ) ) = Y w is yu b is t i t u t i o n m!( 0:0r n tEx=ham, pleso 4h.3 (la) Thde systsem (LLE r1 a7 , v 1as!sig)nment s=1 6= a ) , o) f s e c t i o n m M( ) 2 t 0 i s 2, ix ) o h mb t a i n e d ,)(b) I=f we set . b e 9 y ) ! t n ( h e . aM I M 1 P2 n- we- get( the system (P ,), in which "equality" is the interior of (• = •). This systedm- satisfies substitution for every formula. mxM 1m5 ( - = x ,M . i6 = .- . d x ,M x ,8 ) )- f! i ) oa s )n rd p ei r n v4 o = e v m r7 a = y b m i9 l = 3m e .1 f 0 = o i r d , e v e r y m E h t . EQUALITY IN LINEAR LOGIC 131

M - M • 2 and M 5 - m 1 ? M 3 • • 6 - then Mwe have a system in which "equality" corresponds to the closure of (• = 8•); from [S] will follow that that ?(• • ) is "clopen". This system satis-fies the substitution rule for every formula. 9 , (d) BMy considering 4 - M M =M - m - m M I= 3 =M 6 t O • •, 2 — 5 = M 8 = M 9 = I ? ! 7 and - 9 - Mm = m = !?or 19 !I 0 we g-et a system containing a translation of the classical theory of equality insidie (11), to be studied in next section. d , (e) Another translation o f classical equality in (LL) can be obtained by considering the assignment

in 1 = and i d , m m2 3 = m 6 = I 'll 10 = = M Now,4 we would like to identify the classes of equivalent calculi. Instead of doing= a complete classification of the possible calculi, we present quantale = theo=retic techniques that are useful in deciding this problem. We start with refleixivity ( [ M ) :n 8 Prop=osit ion 4.4 Let Q be a Girard's quantale,and let x E Q. Then: m (a) A9re equivalent: = ? (i) ? x 1 ( i i ) ! ? x >- 1 ( i i i ) ' M x 1 132 MARCELO E. CONIGLIO AND FRANCISCO M1RAGLIA

(b) Are equivalent:

(i) 1 (ii)!.x 1

(i) ?!.x 1 (ii)! ? ! . , , Proocf (a1): I f ?..r •• 1, then !?.x ! 1 = 1. If !?..v 1 , then ?!?x ? 1 1. Finally, ?!?x 1 implies ?x 1 , because ? ? ! ? . (b) comes directly from !1 = 1, while (c) is consequence of (b).

Thus, introducing the notation r = s to denote that calculi obtained with 1 = r and in 1 = s are equivalent, we have:

(a) ? ! ? ? ! ? ( b ) id = ( c ) ? ! ! ? ! and so there are only three non equivalent possibilities for [R]. With respect to axiom [S], consider At 1 = I?, ?!, ?!? --, I ?m: in E Ait 1, together with At 2 = ! , !?, !?! = { !m: in E itt }. Write (r,$) = (r',s') to denote that calculi obtained with m 2 = r, m 3eq u=iva lenst. Th ena wen hadve: m 2 = rPropo' sition, 4.5 (a ) I f mm 3 E A t = Si (i) ( ! ', m 3 (iii) (id , in 3 a, )t -h=r e n e: () ?= ! , (b) im(f ? , m3 m , )- 3 E =)(i) ( m 2 (rn 2 . 4 (, ? )! ( ?m , i d ) 2 !2 , m Therefore, t(hemre are only 26 non equivalent cases for 1S , 3, ! ? ) - ( m 2 t ()1): ( 1: 2 , ! h !( , ! —= (!?!,!?) ( ! ? ! , ? ) ( ? ! , ? ) , ? )! e ?i(2): !?, !?) ( !?,?!?) ! ? , ? ) —= ( ?!?,?!?) (? ! ? , ? ) ? ) n (i 3 )! : ( ! , ? ! ) ( ! ? ! , ? ! ) ( ? ! , ? ! ) ( ! , ! ? ! ) ( ! ? ! , ! ? ! ) ( : ,)(4)?: ( id,?) (?,?) ( 5 ) : (!? ,? !) (? ! ? , ? ! ) (! ? , ! ? ! ) i(6): !( (id,?!) (?,?!) ( 7 ) : (id , ? ! ? )= (?,?!?) ( 8 ) : ( L i d ) =- (L!) i(9): (!?,!) ?! ) (!?,id) ( 1 0 ) : (!?!,!) ( ! ? L i d ) ( 1 1 ) : (id,id) )(12): ( id,!) (13): (id , ! ? ) ( 1 4 ) : (id,!?!) ( 1 5 ) : (?,id) (? ( m !, 2 ,m , ? ?3 ! ) ) ( (= m !( 2 ? , ! ! ? ! ,? ) ?, !m ?3 ) ( ? ! , ? ! ? ) EQUALITY IN LINEAR LOGIC 133

(16): (?,!?) (17): (?,!?!) (18): (?,!) (19): (?!?,id) (20): (?!?,!?) (21): (?!?,!?!) (22): (?!?,!) (23): (?!,id) (24): (?!,!?) (25): (?!,!?!) (26): (?!,!)

For transitivity ([7]), we shall write (r,s,t)-- (r',s',f) when calculi obtained with m 4 = r, m 5 S , M T6h ent we haave:n d m 4Prop osition 4.6 If m 5 E .Alt 2 , then: r (i) ' I f , m m6(ii ) E( r , m 5 5 5 i, ?t )t (iii) ,( r , m 5 = i( s , m (!?!,m, ? ! )! 5 s ,5 ' , , !(? s! ) , ? m mt, ?h ) fTh5oe rer s) u lts above imply that, for m 5 E At 2 fixed, we have only 8 possibil- 6 e( ni eitied,sv? fo!(r, p) a irs (m 4 ,M 6) (in contrast with 26 possibles ones), where the num- bt e(rs referr to the pairs described above. e ( r ti ' ,m m y d ,, a 5s Eli: (r1) (2) (4) ( 7 ) -=- (13) [2]: (3) (5) (6) -=- (14) r m m e , [3]!: ( 8) (11) (12) [41: (9) (10) , 5 5 e M? [5q]:) (u1 5) i (19) -••• (23) [6]: (16) (20) (24) s , !, v 6( [a (l17) e (21) (25) [8]: (18) (22) (26) E ? ?! n )! 7t . AW(,)ith] re ! spect to substitution, if we fix m 8 t,for sm(1 p7:a!1i?rs (,m9 7 e A / 1 t2,amlei5mn t1 c)a0l culi that can be constructed with the exponentials, satisfying the .,)us,n,5 u al rulest( h of eequalnity. W e register that this corresponds to only 8% of the original universe of possible calculi. to 5!, nh!ul ye? r e at ,?! h ), re e 8 ( t! 5a. m)Integrp re taing Classicali n Equality, in Linear Logic with Equality c 6(? a n, s Ite )!mis wse5ll k n ow that the exponentials of linear logic (LL) are important in iantf?5erpbr,eting intuitionistic logic (IL) and classical logic (CL) inside LL. For eoao,ch ov?f! these logics, we have two translations, one of them based in the fact tehrm?at e.v!e! ry (c ommutative) Girard's quantale contains a frame (complete THe5) ytin? g algebra) (corollary 1.21) and a complete Boolean algebra: h v,- ) u e! It=(!x:xEQ}fs and g3={?!x:xEQ), , r? o respectively. Operations and constants in each of these algebras are given t y) r by: h r e e , v s s e e E r a A y r t r e , t s h , e t n , o u n E e q u i v - 134 MARCELO E. CONIGLIO AND FRANCISCO MIRAGLIA

(H). F o r the frame X:

= 0, T . 1 , ( x AxY)=( x ® y )y) == ( x —! › (y),x x A ( x O A y x ) , g ) = ( x (B). tF o(r -the c-xomp>lete Bool0ean a)lgeb,ra: S \ h e .05 y ) !%(x vg lY )= (xu y ) = ( x 0 y = y) = — › y ),-1 x = (x 0 _ A=1 (S 9?1 S,x Vv3 S = I S 1 . SI) v ) = ( ! x ,I ThusT;, w ey ca=n inter)pre t A 7e ;F(ORx(IL) as A' E FOR(L) by the rules: A VI ? 1 op=e rxations in (H) to define A f o r all intuitionistic formulas (here, FOR(IL) 1 ) ; de!noSte the set of intuitionistic formulas). , AFo=r classical logic, we have our first translation: ( i V x (*)f S A ? ! A if A is atomic; then proceed by induction on complex- y ity, using the rules in (B). A . ) i Thus=, i f we assume two-hand sequents for (LL), we have ([Girl): s ? a ( A 1 t ! ,if.f. (A 1 o x .) , anmd ! A1 i y „, . . . , (1c F( A ) !(A 1 8 ; i) t . ) , 1 p ,I Anot?her int.er.p.retation for classical logic is constructed from polarities for T r A- form!ulas. Thus,, ! given a sequent for classical logic F Fa A, we'll say that ) o iA occu(rrences( Aof formulas A E F are positive (denoted by pA) and occur- : c s' rencexs of formulas) B E A are negative (denoted by nA). We have the fol- lowingA e rulepis of a second translation (" ): A c C e rs y A L d pA = Aop = nAr if A is atomic. ) C A b p(-1 Avo) = (n A) , i i y IPO , Bnav)( =- 1P s s i A(pA(A A) baB) = ?p(A) 0 ?p(B), n(A A B)= n(A) & n(B), p p n ()p (ApE BADlb ) ) l= n (A) r r d -p1 1p -e( B ) , o o u ,(n B ( )i A v v c ,B n ) a a t n= iL b b i (? ApnL ( Al ) l o v t, e e n B! nu) i i o =( Bi n n n !) n,t L c c ( Ai L l o ) o . a m u n s p ! ni s l ( Bs i e ) ,t c x i a i c l t l l y o o u g g s i i i c c n i g f t f h e EQUALITY IN LINEAR LOGIC 135

p((V x)A) A x.?p(A), n((V x)A) A .v.ii(A), P((3 x)A) = V x.p(A), n((3 x)A) = V xin(A).

With this definition, we have ([Gir]):

A 1 ,is. .provable in (LL). . , A We now tuarn to the question of determining a linear theory of equality such that, given Aa cla ssical theory of, its translation into linear logic is defines a linear theory contained in the original linear theory of equality. i For this, we need to relate the deduction of a sequent in (LL) from a set of sequent-axsioms and the deduction o f a formula from a multiset o f hypothesis (inp the left-hand side of the sequent), called external and internal relations ofr consequence (respectively) by [Ayr]. o Definition 5v.1 Let A E FOR(L) with exactly p i as free variables. The universal calosure of A, denoted by A A is the formula obtained from A by quantifyinbg universally a l l the free variables o f A, i.e., A A A x l i It is straightforwarde to prove next result: . i •Pro position 5.2 Let A 1 A a multiset of formulas. Are equivalent: •, . . . , A (b e (i) 1 - A is provable from the sequent-axioms F A n •f o r m Cu l a s ;(ii) t h e r e exists k 0 such that 13, B FA is provable; A, L k time.v xB ) ! (iii) ! A n A 1 i .ANow, assufme th at ,9i is the set of axioms (without free variables) of a theory 1 , . . . , Aof! (ACL ) inf a la nguage without functional symbols. It follows from Prop- .&os ition 5.2! that if A • A Aleqnuivalen t,n for the first translation (*) of CL into LL: • (i) A 1,...,A C L A; n, .I. . , A ( • (ii) K A 1 an - E A & )(iii) Th e re is a proof of F A ls A i 1 !thacCt thfen rA ot m t h e oa i n )d A' s)a x i o m s gA s , AhThAIuas,v toe e-a ch ax iom A E ,91 of a classical theory, corresponds a sequent- oiaxpiomr Is- A. c in ( LL). Similarly, for the second translation (**), it can be seen n e( o A 1 uathaot tov eac.h axiom A E si, there corresponds a sequent-axiom F n(A). ,f i) r e sf aNowo b asrs.umme that .9i defines the classical theory of equality. Thus, si con- ae sc lusislts line thae, fol,lowi ng axioms: nv p, a . rr . . , I - yt . h ! ,dei o(a bv nAn wtl ane sb ( eh) l). e A de ic k ef n( o r ) l e c fl (a o l F w l ii L n ? g na L p er ) ( Ve ; A A ) , t h e e x i s t e n t i a l c l o s u r e o f A . 136 MARCELO E. CONIGLIO AND FRANCISCO MIRAGLIA

Now assume that .94 defines the classical theory of equality. Thus, .94 consists in the following axioms:

(V x)(x = x); (V x)(V y)((x = y) ( y = x)); (V x)(V y)(V z)((x = y) A (y= z) ( x = z)); ( V x 1 P(x )(where P varies over every predicate sy1m bol of arity n o f the lan- (guVagye). ) 1 7 For the first translation (*), we get a calculus (E 1 oms:) „ . 1 ) w (i Vt hx t h e , f o n l l o w i n g - - a [Rx1] ? i! A -? ! (x = x) [ S i ] F ?!A ? ! A y (!?!(x = y) ? ! ( y = x)) )[Ti] ? ! A x ?!A y ?!A z (!?(!?!(x = 0 - r n(y = z)) - - (0[SV UByS=T11 I f ZP is )a n)-ary predicate symbol„ o f the language n ? ! A x „ ) !?l! (x?n =y0 0 !? !P (xl,...,xn )))-0 ? !P (xl? y n ( ( x ) ! ) ) For t1he second translation (**), we get a calculus (E axioms: A y 2 = i ) wy i t h t h• • e f o1[Rl2] lF o A wx (xi= nx) [gS 2 ] F A x A y (?(x y ) —0 !(y = x)) [12] F A x A y• A z CM(x= y) ' k y = z)) — ) 0[S U!B(S7y2 ] I =f ?P is! az n-a)ry )predicate symbol in the language A Ax • Ax • i n n Cons•ider th e linear A?the!ory (P 1 = ) d Ae t e r m i nyAe d b ([LyR1] = y iy A t x-[LhT1] eF !?!) (x•n = y) r = z) -- 0 = z ) a n)[xL S[UiBLSTSo1]1 !0m]?•( ! !( sx 1 = y) ® • • • 0 !?!(x = y n 1) 0 ! ? ! 'P? !( Px( x l 1 ? Y 1 : ?•? -, . . . 7, x! (, ! „ ) ) Clearly, (P 1 PA ( x ?• • •! )( P ( = x(, x cilinoeanr ttheaoriy of equality in the sense of section 4. y ) x „tx , , 2 y n), Ss im ilarly, define the theory (P 2 — l A=n (u )E b y t h e 0 , y) . 1s a x i o m sy : = . 1 )i n ) (n b e . ( 0 cg a u . ?• st e , (• eh v x A• e r n 0 e x y ) aq i xu ) = i - y ov - , ma - ) l i - 0 ne 0 (n • ! Ec • 1e P• )s ( 0 io x sf 1 dl 2 a e y ds ut , cs y ie ) bc ) lt . ei io n ) a n d w e h a v e t h a t ( P 1 ) d e f i n e s a EQUALITY IN LINEAR LOGIC 137

[LR2] ( x = x) I L S 2 1 ? ( x = y) —0 !(y = x) [L121 ? ( ? ( x = y) 0 ?(), z ) ) [LSUBST2] ? ( ? ( x 1 = v 1 ) 0 • • ---0• !P (x 1 2 y y 0 ? (n x We c=an reform ulate) (P. y 2 ) n a s : [LR21 ( x = x) [ I . S 2 1 ( x = y) —0 (y = x) [)L121 ( x = y) 0 (y = z) ( x = z) O ? P ( x [LSUBST21 ? ( ( x 1 = y 1 l ) • • • !P( x 1 y , . . . , x 0[? ! ] ? (lc ==y) —i 0 !(., v = y). n y , . . . , ) ) It is sntraightforw ard to xch,e ck that (P 2 (P ) a )n d ( E y 2concerned) of the calculi already defined. 2 0 ) a?n r e ) e qP u (i v xa l e) n t s . i I l n . . s6. Eq u a lity and intuitionistic Linear Logic f . a . c . t , t In thxis section we shall study how to recover classical logic and intuitio- h nisticn logic from a linear calculus without exponentials, as well as analyse e anoth) er exten) sions of the relationship between linear logic and quantales. Os ur basic system is (commutative) first order intuitionistic linear logic, fromt now on denoted by (I I I ). The appropriate semantics will turn out to br e furnished by commutative unital quantales. Similarly, the semantics for noon-commutative first order intuitionistic linear shall be proven to be given nby unital quantales. From this, it will be seen that, adding appropiate axi- goms, we can recover intuitionistic logic, classical logic and classical linear elogic. Thus, an intuitionistic linear theory of equality provides an intuitio- snistic theory of equality, a classical theory of equality, and a classical linear the ory of equality, simultaneously. The most natural candidate is system ((LLE 0 We define a sequent calculus for (commutative) first-order intuitionistic a), tlinheaer logic without negation, simply by extending the system in [GiLa] and tnhen proving soundness and completeness for this system. ds i m p Dt efinition 6.1 The language 0 f o r commutative first-order intuitionistic hl e s tlin ear logic consist of a countable set of predicate symbols, to = { P (eI) }, a countable set of variables V = v n an l :rE D,n v , AE an d thwe sam e rules) , as in Definition 2.2 for the formation of the set r: n eE teof formulash e F OR(0_ i a sf)D.e fiyFnitioomnr 2 .b4. o l s d 0oA , E y ,rF O R d &e( L , m,e ) oaf n rdi exn , rye eEd sV. t, rA i[ y l cx ] ti es da as si fn a r a s s e m a n t i c s i s 138 M A R C E L O E. CONIGLIO AND FRANCISCO MIRAGLIA

Definition 6.2 The calculus (H i ) for commutative first order intuitionistic linear logic consist in the following rules and axioms (A,B,C denote formulas, and F, A denote multisets of formulas):

[AX1] A I- A [AX2] T [AX3] 1 [ A X 4 ] F , O F A [CUT] A A, A B [EXCH] , A, B A C F, A B , B, A, A C

I- A [ 0 R] r 1- A A B [ 0 L] F . A . B I - C IF, AI- A (g) B F,A0BI-C

[SE R] FA FHB [8c1] F•AFC [84.2] ,BI-C FI-A&B F,A &BI- C F,A&BF C

[ED Li FAI-C C [El) 1] FI-A [(f)2] FI-B F,AeSt-C FFAEDB FI-A1EDB

[-0 R] FAFB [—o I- A A, B C F,A,A—DBI-C [v Ri r A w l [v r A 1- B I- V x.A F , V x.A B i f x not occurs free in F , B

[ A [AR] A L ] , A x•A F B F I- A x.A i f x not occurs free in A DefinLition 6.3 An interpretation of quantales for (ILO is a commutative unital qvuantale (Q , V , * ,1) and a map I•I 1. 0 1 = 0 , 1 1 1 Q :/ F O R ( L s 2,. A B I = IAI I B I , B I = IAI B . ) x - - ) 31. , A STt BI =. I Q 1 A4. IA xA. A 1 . A y E VAR(L,) [y /x ] I, s aB t i s f y i n g : I B I ,L AI= v IWe sAay tyh a t Ae E F O R ( L ) i s EAv saeqllui ednV)it nFA 1 Q- A1 isf lvalidA I in Q if n Q , S 1RA( 1L Q BQ I 1 1 . if =0 =, 1 w i h e r e I F ) ( 8 ) l A A I I I b i Beyfore stating a n1dA proving soundness we establish the following simple Q l - B , 1 I = •t i l n • Q 1 EQUALITY IN LINEAR LOGIC 139

Lemma 6.4 Let Q be a quantale. Then, a 0 0 0 a =0 for every a E Q.

Proof: 0 a —› 0 implies a 0 0 s 0; 0 s — > 1 0 i m p l i e s 0 aTheorem 6 .5 (Sousndness)- I f r I- A is provable in (1,1,1), then it is valid in 0every in.terpretEation of quantales, i.e., Q e for every unital quantale Q.

Proof B y induction on least lenght n of a proof of I - LAet Q. be a quantale. I f n = I , [AXIMAX21 and [AX3] are immediate, while [AX4] is a consequence of Lemma 6.4. Assume the thesis holds for all sequents with a proof of length s n (n l fixed) and let A D be a sequent admitting a proof of minimum length n + I. We discuss the last rule applied in the proof of A I- D: [CUT]: We have IFIs AI and IAI I B I by the induction hypothesis; thus, Ill* II IAI * IA 5- I Bl. The passage through the rules [ UCH] and [1L] follow from the fact that Q is commutative and that 1 is the unit of Q. [ 0 f a 111 I A I and IAI I B I (induction hypothesis) yield, then IF * IAI BI. [ 0 Li: T h is works by definition of interpretation. [8z, lib]: B y induction, IFI I A I and I B I , and so IFI I A[8E II ] anAd [8E 2I]: BW eI h•ave IF * Al ( induction hypothesis); thus (IAI A IB

The other rule is similar. [ED L],PED IMCD 2]: Same argument as in 8E, using v in place of A. [--o R]: Induction yields IF * I AI AI I —I B I ; tI BhI L]:. uThes in,duc tion hypothesis yields A l and IAI I B I 5- ICI. Thus, b y a d j o i n t n e s (1s [ V R,]4: B y induction, there is a variable y such that IFI A [ y l1x ] 1 I ; —b u t t h e n 1 We now>I state a Fact whose proof is routine: - B 1 I) 5 I - r - l I I A I L ( Y I / A x I 1 H 1 B y O I * A I [ A Y * 1 I X I ] I l B • I I A I - 5 - I C I . 140 M A R C E L O E. CONIGLIO AND FRANCISCO MIRAGLIA

Fact: I f F ,A I- B is provable in n steps and x is not free in either F or B, then f , A [y/x] B is provable in n steps for every y E V. [V L]: Suppose that last rule applied was [V L]. Since x is not free in F, B, by induction and the Fact above, we may assume that 111 IA[y/x[l I Bfori e, very y E V. Thus, IA[y/41 r F —> I B i f o113r1, i .e. I rl * IV LAI I B I • e[ A vL]: eT hr isy is treated just as [V R], above. y[A R]: S uppose that last rule applied was [A R]. Since x is not free in F, indEuction an d the Fact above allow us to assume that F i 5-- A[y/x]I, for every y E V. It is then clear that 11 V - a n d 1B eLfoçre_ w e proveA cIomApl[eteyne/ ss, we need a general result about quantales s o x(ge]nel ra, liz ing an analogous result about Heyting algebras in [Mir]). I V e n d i n g x tDefinitihon 6.6 Ane auton omous poset is a partially ordered set P with a bi- naryA associativle operation 0 such that the endomorfisms a • and • 0 a p r o o f . ha=ve right adjoints, d enoted by a --> r • and a 1 , respectively. E l VA [(g) , VI-autonomous lattice L is an autonomous poset where L is a lat- tic,e and such that 0 is distributive for all V' s which exists in L, i.e., if S C L sI uch that exists V S in L, then, for every a E L , V s E S (a 0 s) and V As E S (s O a) e xists in L, and we have that V ,S• E S (a O s) = a O (V S), V Ns E S Cs O a) = (V S) O a. I f S,T are subsets of L, define M I S • T = (a 0 b : a E S and b E T).

Definition 6.7 Let L be a lattice and C L an ideal (i.e., if x E I and y x then y E I; if x,y E J then x y y E I). We say that l is complete if it satisfies:

if S C I such that V L S exists, then V L SinceS aEn aLrbitrary meet of complete ideals is again a complete ideal, Cl(L) = [ I C L : l i s a c o m p l e t e ideal in L ), is a complete lattice ordered by inclusion, and containing, for each a e L,

a = fx E L : x If a Ea L a nd S) C L is a subset of L, define . a , S e l e f U c E S ( a r C ) a n d a I S d e f t — ) c E S ( a ' 6 ) 4 - • EQUALITY IN LINEAR LOGIC 141

Lemma 6.8 Let L be a [0) , VI-autonomous lattice. Then (a) I f a E L and I E CI(L), then a r I and a ---> 1 la re in CI(L). (b) Fo r every S,T C L and K E CI(L), we have

S•TCKiffSCr)T(a—>IK)iffTC aEs r

Proof ( a ) I f x E a —› / , then there is c E / such that x a ---> r c; thus, if y x then y a -4 c and therefore y E a —› r /. If x,y E a —› r /, then there are c,d E / such that x a —› c , and v a —› r d; therefore

„vvy(a r V (a r cl) a r(cVd),

yields x y y E a —› r I, because c y d E I. Let S Ç E a/ such that s a c fors r all s E S. Then, , / i . e . , a b a O c a ® VSES s ' V sesaOsE s e c bsecause / is complete. But this means that c E a —› r V A similar computa- tiosn will prove that a 1 I E CI(L). . (b)u Let x E S and a E T and assume that S • 7 tShe.c n Ci a nK —›; cI c asnd ithuns xc E ea — > 1 K, for every a E T. e xNow,h / s®ui pposse th at S Ca n aeT (a —› 1 K), x E S and a E T; since x E a =t-4 1 K, thcere is c E K such thaEt x •• a —) I c , i.e. x 0 a c . Since K is an indeKhal, we get, x O a E K. The other equivalence can be handled similarly. D i a d e aThteore ml 6.9 (,The co mpletion of a [ 0 , VI-autonomous lattice) a cLet L be a [(g) , v]-a utonomous lattice with 0, T . Then, there exists a qsu=antale Q and an injective map 41: I, Q such that E V /. 4 )(0 ) = 0, 4)(T) -= T. Moreover, 4) preserves all V' s and A's existing I L ,in SL, i.e., if V , E e1 a , 4)(V , a x n d E A i 2. F1o r all a, b L , j s E j a I t (i) 4) (a 0 b) = 4)(a) 0 4)(b) , ) 3 s ii) 4)(a —› r b) = 4)(a) —› r 4)(b) and 4 )(a —› 1 b) = 4)(a) —› 1 4)(b). = I . V , 3e. FI fx a 0i b = b 0 a for every a,b E L, then Q is commutative. E s o t s i i r ( 1 n e ) L v ( 0 , e t 1 r 1 h y ) e s a n E n S d t ( h V e A r b e j i ) s = c A s j 4 ) ( b 142 M A R C E L O E. CONIGLIO AND FRANCISCO MIRAGLIA

4. I f L is unital, i.e., there is 1 E L such that a 0 1 = a =1 ® a, V a E L, then Q is unital and 4)(1)=1 Q . 5. I f in addition, L has a cyclic dualizing element I , then Q is a Girard quantale and 0)(1) =

Proof S e t Q = CI(L) and define in Q the operation

( def. r l i K E Q : K D I • J I , 0 w h e r e i • J = t a g b : a

Clearly * is increasing in both variables. By Lemma 6.8, • * I and / h a v e right adjoints / - - - - > r J a nnd ald- (-a> --> r J) and n aEl (a 1 J), I Jre,sp ectively. Thus, V I,J E Q, g i(II) v /*JCKiff/Cfe n K b y Formulas (I) and (II) will be of constant use. To show that * is associative, we need

Fact /: Let I, J, K E Q. Then

, 9 1 = { Proo1f of Fact I: W e have, using Lemma 6.8, the following sequence of implications1 E Q : HD(1*.1)•KHD(I•J)•K=1•Cl•K) H D ( 1 * . 1 ) The last term implies H D / • (J K ) and A C S i m i l a r reasoning yields g3 C• K } It f=ollIows directly from the Fact that * is associative. We now must show that H* diEstributes over suprema. LeQt J: ObEse HrveD that V i C / *I V• iC caa Qtion=s n Ia a /li *s K Ea nl )Qw) : Ha dy=D sM Lt I r. . Ju / e a. a 'N1 ao a }wE ,, A awC se Q wh ; ea w lv e l ae s st h th a he l af l to p Vl r (l o /o v /we ai t )n h g a s t e / q * u V e I n a c = e V o ( f I i * mI p a l ) i . - EQUALITY IN LINEAR LOGIC 143

KDU(/*/a)KD/*/,„VotEA 'a C/—>,K,VoLEA U / a and so / * V / C ade edC a q uantaleI. V Be fore defini—ng the map 4,, we state ( / › Fact 2: Fo r a, b E Q * / K a (i) a ' (ii) I f S C L is such that a = V S exists in L, then V s ) -. „(iii) I f S C L such that a = A Sexists in L, then a R b i g s4(iv -) sa=" =A as E" s s h ' t =-f - S E S dProiof sof Ftact 2: ( i ) L e t K E Q be such that a" (sb " • r-w e *iha bve Kau 0 x bC an d so, x ; a — >, b. Thus, K C (a —) r b ) a- tf-thait aov" i r x —= tE,- (Foryp rt hoe revv ei rsne ginclusion,K co,nsider x a and y a —) b ; then x 0 y i—a )(›a — › r b) b and so a" * ( a ba sb-C b•b" ( a — › b Y Fo)r the operation —> s-b4 -) r i i (' ii) I f V S a , then s C a " mC-T - i C l(b,Vh -at hf aoe r e v e r y If H D U ra4,i snr g u m e nE s(iii) I s similar to (W. a—,tEs Sd i , s s a,(ivi ) mI fnx i 15- la,d v b t h e n x0 y a b imp lie sa nT)il hus., * e b C. ' (a, b ) " a„ sf*- r . •s. bo" d-ba Now, if H D a 4.- • b v i-<- - C ( ta 0 b ) ' s)4definitions - of * (formula (I)), this yields (a 0 b ) 4 —,-4 - t h e n o"Thist i ends thes p roof of F act 2. 4 -s.a iC n a "b Q-r Wc) e nowl defi nee a - cE e * b " i-e ri - H a , n a s sCl tb 4): L -h- > Q bya (1)(a) = a (- d id e s i r e d . ia t' sw e n"tClearly= 4 ) is injective and, by Fact 2.(ii) and (iii), preserves all existing V's -an(cgd Ao's in eL. Furthermore, 41(0) 0 ) (the smallest complete ideal in L) -i t anamd 4)p(T ) = L = T Q o ( .b y—l TFahect e2. (i) and (iv). This shows that 4) has the properties in items 1 and 2 nofa th e statemen t. p ›tr ee s e r v a i b t 1,iIt ois nquite clear from the definition of * (see (I), above) that Q will be cmo)mmutative if * is commutative in L. Moreover, if L is unital, straightfor- o bi f pwaCrd computa tion will show that (1)(1.) •= 1(- is the unit of Q. t )t h l H(d) Let I E L be a cyclic dualizing element in L. Clearly I E Q is e 4f icy.clic and therefo re Q o o t h ei B e l r s= y1 / o l p e r (E t Q : j a o t i o a( h / ) n w s -= e i s 4/ s t r) g h u a bi s r a a n )a t t e e 4G i r a -d a r E d Cq u a n a H t . a l e 4, -w h re r 1e 2 4 - , a s n e e d e d . 144 M A R C E L O E. CONIGLIO AND FRANCISCO MIRAGLIA

: Q - - - > Q given by j(/) = (/ —› J.. ) —) 1 Mo4re- o v(erP, cri o p o s i t i o n *an-1- eEmb.e dd1ing of0 L in) a .Girard quantale, with all the required properties. Q We can now prove 1 . f o r Theorem 6.10 (Completeness) I f 1 e v e q1uantales Q, then F I- A is provable in (LU). r y 1 2 I A I Q a Pf rooof O ur r m ethoad willl be tlo sh ow that the Lindenbaum algebra L of E (iL Un) ctane be rempbedrdeed int a acomtmutative unital quantale Q, such that L 1AI i f f F I- A is provable. i o n s . Define in FOR(L,) the relation: i n T h A - B i f f A B and BI- A are provable. u s ,It is e asily seen that - is an equivalence relation. Its equivalence classes shall be denoted by AL. t Let L be the set of equivalence classes, i.e., L = (A L : A E FOR(0_ i h L. )d e)fin. e thIe rnelation: e m AL B L i f f A I- B is provable in (113). a Bpy [CUT], i s a partial order in L. Moreover, for formulas A, B in 1±1, we jhave 0 4Fact I: I f A, B are formulas in (111), then ) I. ( A & = (AL) A (BL) and ( A El) B)L = (AL) v (BL). : 2. A , v (A [y/x]L) , (A _LA )3/. 0a ,n d L - - V , PrEoof of Facty I: A l l these equalities can be read off the corresponding r-ules of the calculus. We do the first one in each of items 1. and 2. in some (O L A [ y / d- etail, just naming the rules that should be used for the other cases. xa n] L ) > 1. F r o m [AX1], [&1] and [(k (d v AL,QT] (Aw &e B )I_g Be L t. On the other hand, if C I- A and C B , then [&1i 1yJieAlds C I- A & B&. This m eans that ]i B I - s A CL A ,L , CL B l _ C L ( A & B)I_, proBving that (A & B)1_. (AL) A (BL) in L. a n d Similarly, [AX1], [ED [ e 2 ] and [ e l L ] will yield (A ED B )1_ = (AL)s y (BL). o ( A & B ) I _ • • EQUALITY IN LINEAR LOGIC 145

2. B y [AXI] and [A Ll, (A x. A)I_ A [y/x]/_ , for every y E V. If BI _ A[y1x11_, for every y, let z be a variable not occurring in B. Then, BL A [ d x 1 1 _ and so [ A R] yields ( A x A)I_. This proves that A , (A [ y lx ] I _ )= (A L A ) For the existential quantifier, the reasoning is the same, using [A X1], _[ V_ R] and [V L]. 3. T h i s follows directly from [AX2] and [AX4], ending the proof of Fact I. Define in L a binary operation * by:

( A O * ( B * is well defined, is associative, commutative, increasing in both variables and i s the unit. We have I - ) — Fact 2: With the operation * defined above, L is a [0, V] autonomous lattice. = c Prool f of Fact 2: I t remains to be shown that * has right adjoints and dis- tribuetes over the suprema existing in L. If fA, B, C are formulas in (1,1,1), then 0 (-1- A 0 BI- C is provable i f f A , B f- C is provable ) iff A I- B --o C is provable.

To see this, note that [ 0 R] yields A I- A B 1- B A,BHAOB

Thus if A 0 B C is provable, [CUT] implies that A, B C is provable.

The converse comes directly from [ 0 L] as A . B C AOBI-C

From [--0 R] we get A, B C A - B —0 C

To show that the last clause in (+) implies the second, first note that B , B — ° C F CThus, the provability of A I- B —0 C and B, B —0 C C and [CUT] yield ithat of A, B !- C. s p r o v a b l e , b e c a u s e w e c a n u s e [ — o R ] a B F B C I - C . 146 M A R C E L O E. CONIGLIO AND FRANCISCO M1RAGLIA

It is clear from (+) that (A —0 •)/_ is the rigth adjoint to AL * • = • * AL in L. To show that * distributes over the sup's in L, let S C L be such that there is AL E L satisfying AL = V L S. Fix Bl_ E L. Since * is increasing, (B O * (XL) ( B L ) * (A/_), for every XL E S. Let FL e L such that (BL) * (XL) F L , for every XL E S. By (+) we have that, for XL E S,

B 0 X I- F is provable i f f X F (B —0 F) is provable,

V XL E S; taking sup's, we get A I- (B —0 F) is provable. Thus, B 0 A I- F is provable, a n d s o (B L ) * (A L) L . B u t t h is means th a t xl-Es((BI-)*(XI-)) exists in L, and it's equal to (BL) * V 1 pro, oSf of, F aect 2n. d i n g t h e Therefore, L is a commutative [ 0 , V]-autonomous lattice and by theorem 6.9 it has a completion 4) : L Q , where Q is an unital commutative quantale. Moreover, (I) preserves 0 , —0, 0, T , 1 and all existing V's and A's in L. Consequently, if X, Y are formulas in (L i]) we have

I-1)(XL) (I )(Y L ) i f f X L Is YL.

It f o l l o w s f r o m th e se preservation p ro p e rtie s t h a t t h e m a p 1.1 Q • qTuOanRtales such that IFI I At(hLeI p roiof off th e ThFeorem . D 1) - A Before we give logical applications of the preceding results, we need to in- i> s vestigate when is it that a quantale becomes a frame. From Lemma 6.4 pcQomres:o v a b l eg , aCi orollary 6s.11 Let Q be a quantale. I f Q has a largest localic subquantale rLv, theen q u ie r e d tn L = 1 ( Q ) = ( a E Q : a ob 2 cyProo=f ao W1e. mustm have L C 1(Q) because every x e L is idempotent (prop- p!osition l1.19).e tA For ea E 1(Q), set L = ( 0 , a ). By Lemma 6.4, L„ is closed under 0 ; it lis clearly closed under arbitrary V's, and so L =a is iidsem poaten t, sit fuollbowqs thuata L n t a l e therefore 1(Q) C L. 0 (ao i s fl o c a l i c .Q . ITS h i u ns , c e )L (C AL , Li . e . , )a iE sL , a n nd i n t e r p r e t a t i o n o f EQUALITY IN LINEAR LOGIC 147

The next result is a correction of Theorem 3.4. I in [Ros], which is false as it stands. In fact, the usual quantale of phases of Linear Logic is a counter- example to that result.

Proposition 6.12 Let Q be a quantale. Are equivalent: (1) Q has a largest localic subquantale. (2) 1(Q) is a localic subquantale of Q. (3) F o r all every a,b 1 ( Q ) , a 0 b = b 0 a 5. a A b (A in Q or in 1(Q)).

Proof. ( 1 ) ( 2 ) comes from Corollary 6.11, while (2) ( 3 ) is immediate. (3) ( 1 ) : Because 1(Q) is commutative, it is closed under O. To show that 1(Q) is closed under sup's, let 1 a , 1 ,, 1 C 1 ( Q ) . T h(*) eV ina :a , = V , E 1 ( a , = (V ;0 ® (V jel a d; a(**) )( V ia a i)® ( V Vi J becaEaus e a,a 0 I ad V a i is idempotent and so 1(Q) is a subquantale of Q. j a(= V a i To sf hoow thrat 1 (Q) is localic, observe that it follows from 3. that a 0 b :5 a, fo,ir eveary b E 1(Q). Thus, V a E 1(Q), e va( e Vr J a y aj( aV 1(Q) = \/v bE/(Q)5 a and (V1(0) O a a, j )t E (a whicOh shaows that allb ele) ments of AO are idempotent and two-sided in 1(Q). Now,/ ) Proposition 1.19 guarantees that 1(Q) is localic. If L C Q is a localic s. ubq)ua5nt a_le tVhen every x E L is idempotent (Proposition 1.19) and so L C F1(Q).I DEr I o a ,m , (Thus, in a commutative Girard quantale Q, 1(Q) n 1 i s the largest frame *contained in 1 ' . From the point of view of Logic, we have a privileged )localic su bquantale, that corresponds to intuitionistic Logic. However, the asituation is quite different in the non commutative case: in general, a Girard nquantale will not have a largest localic subquantale. If one considers Propo- dsition 1.14, one realizes the importance of commutativity: there is a largest interpretation for ! in a Girard's quantale Q, corresponding to the largest a( mong frames L satisfying * * (i) L C Z(Q) ( i i ) C - ) , Iin fa{ct,x Q,E = 1Q(Q): nx Z(1Q) }n 1' , where Z(Q) is the center of Q. t f o l l o w s t h a t 148 MARCELO E. CON IGLIO AND FRANCISCO MIRAGLIA

The quantale theoretic setting will be helpful in finding axioms that, when added to various systems of Linear Logic, produce classical intuitio- nism and classical logic. We have the simple (compare with proposition 1.19),

Lemma 6.13 Let L be a complete lattice and * : L x L a binary operation on L. Are equivalent:

(i) = A. (ii) (a) x * x = x for every x E L. (b) T x , T x for every x E L. (c) * is increasing in both variables.

Proof: ( i ) ( i i ) is clear. (ii) ( i ) : W e have a * b a * T a ; a * b < T * b b and therefore, a *1 ):Ça A b . On the other hand, since * is increasing a A b = (a A b)* (a A b) 5_ a * b.

An analogous (and dual) result holds for the operation y. From the preceding results we get

Corollary 6.14 Let Q be a quantale. Are equivalent: (i) Q is a frame, i.e., = A. (ii) every x E Q is idempotent and two-sided.

This Corollary and the completeness Theorem 6.10 yield

Proposition 6.15 ( a ) Th e system (Li) obtained from (LLI) by adding the axioms

[1D1] 1-A0A-0A [1D21 1-A-0A0A [2S1 1 - A 0 T —0 A

determines intuitionistic logic, and therefore if we add to (LI) the axiom

[ — 1 ] (A —0 0) —0 0 A

we get classical logic, where 1 is equivalent to T and 1 - 0 0 is equivalent to O. Thus, adding the axioms [1D1],[1D2],[25] and [R=], [S=j, [T=], [SUBST], [5- 1] (see section 2) to (Lid) we obtain a intuitionistic theory of equality, EQUALITY IN LINEAR LOGIC 149 which becomes first order classical logic with equality upon the addition of

(b) The system (MALL1) obtained from (LU) by adding the constant formula I to the language and the axiom

[ —11 (A —0 I ) —0 I - A , is equivalent with (MALL) in the following sense:

A is provable in (MALL) ifft- A is provable in (MALL1), where A I(( A 0 1 )0 (B -0 _ 0 ) - 0 I , respectively. Thus, adding I to the language we ahaven that d A (LL1 ) + the axioms [—I --11,[R=],[S=],[T=1,[SUBS1],[1=1,[ 1 ] u is equivalent to (L L E B 0 where (L L E a ), 0 (c) Adding [1D1],[1D2] and [25] to (MALL I) produces classical logic, r )w hierse 1 is equivalent to T, and Ois equivalent to I . D e d e s c r i n Ri ebmaerkd 6.1 6 The possibility of obtaining (MALL) from (LL1) in part (b) ap- t e pi ears nin [Dos ] proved by a different method: one interprets A —0 B, A & B arnd pA x.A as (A 0 (B —0 1 )) —0 1, ((A —0 1 ) e (B —0 ) ) —0 1 and D e f r(V xe.(A I ) ) —0 respectively, together with the interpretation for u i n i t mt adee in (b). i o n d Sin ce (a 0 b) c a ---> (b --> c) and b = 1 ----> b, in every unital com- 2 mautative quantale, it can be shown that his interpretation of linear implica- . stion is equ ivalent to adding [--1 —1] to (I I I ) to obtain (MALL) (translations f6o r & and A are derived and a consequence of the ones given). A . As a matter of fact, we can extend this result to non-commutative intu- I itionistic linear logic, simply by observing that each rule of (LL1) deter- ma ines an algebraic property of the Lindenbaum algebra of the calculus. For an general unital quantale, we need two implications, —0/ and — o crdard the exchange rule [EXCH], as well as to modify the rules in order to d, estcroibe thde pirospe-rties of each operation. We can define a linear calculus which describes unital quantales, essentially the same as in [Abri.

Definition 6.17 The calculus (NCLL1) for noncommutative first-order linear intuitionistic logic consist in the following rules and axioms: 150 MARCELO E. CONIGLIO AND FRANCISCO MIRAGLIA

[AX11 A I- A [AX2] T [AX3] 1 [AX4] F , 0, A f- A

[CUT] FI-A /,A Al-B [IL]

[0 R] I- A A B [0 L] F. A. B, A C F,A1-AOR F,A0B,AI-C

[8 R] I-A FI-B [E[L] F.A,AI-C F,B,A I-C F[-A&B F,AEDB,Al-C

[8e, L l] F . A, A C [ 8 c . /21 F . B. A C F, A & B, A C F, A & B, A C

[ S RI ] F I- A [ED R2] I " B ri-AEDB FFAEIB

[—D i L ] F[—o C Ir -L ] [V RI A l y / x 1 AF x.A ,I [B- V UA F , LA, B IF. fC[AL]1 A B x[3 I', A LA, A B i—A oF[ A R] I f x is not free in F then F I- A s 1- A x.A nrC R[ Witho the same method employed in Theorems 6.5 and 6.10, Theorem 6.9 t]- yields0 that unital quantales are a complete and sound class of models for (NCLfFIM: r., eAR eI] i- nB F , B t h e n F . A . A I - B EQUALITY IN LINEAR LOGIC 151

Theorem 6.18 (Completeness and Soundness for (NCLL1)) A sequent F I pre-t atAion oif qsua ntales. p r o v a b l eIns tituto de Matemdtica e Estatfstica, Universidade de Sao Paulo i n Caixa Postal 66.281 - Cep 05315-970, S.Paulo, S.P., Brasil e-mail: M . E . Coniglio: [email protected] P I F. Miraglia: [email protected] or [email protected] C L L I ] REFERENCES t [Aibr . f Zeitschr. f math. Logik und Grundlagen d. Math. 36 (1990), 297- ] f 3 18. A[AIyr] A v r o n A., 'The Semantics and Proof Theory o f Linear Logic', b - Theo retical Computer Science 57 (1988), 161-184. r[BrGu]A Brown C. and Gurr D., 'A Representation Theorem for Quantales', Journal of Pure and Applied Algebra 85 (1993), 27-42. u i [Dsos] Do g e n K., 'Nonmodal Classical Linear Predicate Logic is a Frag- s ment of Intuitionistic Linear Logic', Theoretical Computer Science v c i 102 (1992), 207-214. [VGairl G i r a r d J.Y., 'Linear Logic', Theoretical Computer Science 50 . l (1987), 1-102. [GiLa]Mi Gira rd J.Y. and Lafont Y., 'Linear Logic and Lazy Computation', . d Le ctures Notes in Computer Science 250 (Springer, 1987), 52-66. ,[Mi ir ] M i r a g l i a F., Sheaves and Logic, preprint. '[Rnosi Rosenth al K., 'Quantales and their Applications', Pitman Research N e Notes in Mathematics Series, vol. 234 (Longman, Harlow, 1990). o[Yvet] Y e t t e r D., 'Quantales and (Noncommutative) Linear Logic', Jour- n e nal of Symbolic Logic 55 (1990), 41-64. - r c y o i mn mt u e t r a - t i v e I n t u i t i o n i s t i c L i n e a r L o g i c ' ,