The Semantics and Pro of Theory of Linear Logic Arnon Avron Department of Computer Science Scho ol of Mathematical Sciences Tel-Aviv University Tel-Aviv Israel Abstract Linear logic is a new logic which was recently develop ed by Girard in order to provide a logical basis for the study of parallelism. It is describ ed and investigated in [Gi]. Girard's presentation of his logic is not so standard. In this pap er we shall provide more standard pro of systems and semantics. We shall also extend part of Girard's results by investigating the consequence relations asso ciated with Linear Logic and by proving corresp onding strong completeness theorems. Finally, we shall investigate the relation b etween Linear Logic and previously known systems, esp ecially Relevance logics. 1 Intro duction Linear logic is a new logic which was recently develop ed by Girard in order to provide a logical basis for the study of parallelism. It is describ ed and investigated in [Gi]. As we shall see, it has strong connections with Relevance Logics. However, the terminology and notation used by Girard completely di ers from that used in the relevance logic literature. In the present pap er we shall use the terminology and notations of the latter. The main reason for this choice is that this terminology has already b een in use for many years and is well established in b o oks and pap ers. Another reason is that the symb ols used in the relevantists work are more convenient from the p oint of view of typing. The following table can b e used for translations b etween these two sys- 1 tems of names and notations: Rel ev ance l og ic Gir ar d M ul tipl icativ e I ntensional ; Rel ev ant Additiv e E xtensional E xponential M odal W ith (&) And (^) P l us () O r (_) E ntail ment ( ) E ntail ment (!) P ar ( ) P l us (+) T imes ( ) C otenabil ity () 1; ? t; f !; ? 2; 3 2 Pro of theory 2.1 Gentzen systems and consequence relations The pro of-theoretical study of linear logic in [Gi] concentrates on a Gentzen- type presentation and on the notion of a Pro of-net which is directly derivable from it. This Gentzen-type formulation is obtained from the system for classical logic by deleting the structural rules of contraction and weakening. However, there are many versions in the literature of the Gentzen rules for the conjunction and disjunction. In the presence of the structural rules all these versions are equivalent. When one of them is omitted they are not. Accordingly, two kinds of these connectives are available in Linear Logic (as well as in Relevance Logic): The intensional ones (+ and ), which can b e characterized as follows: ` ; A; B i ` ;A + B ; A; B ` i ;A B ` The extensional ones (_ and ^), which can b e characterized as follows: ` ;A ^ B i ` ;A and ` ;B A _ B; ` i A; ` and B; ` In [Av2] we show how the standard Gentzen-type rules for these connectives are easily derivable from this characterization. We characterize there the rules for the intensional connectives as pure (no side-conditions) and those 1 for the extensional ones as impure. The same rules, essentially, were used also by Girard. He preferred, however, to use a variant in which only one-side sequents are employed, and in which the negation connective can directly b e applied only to atomic formulas (the negation of other formulas b eing 2 de ned by De-Morgan rules, including double-negation). This is convenient 1 As explained in [Av2], this distinction is crucial from the implementation p oint of view. It explains, e.g. why Girard has found the intensionals (or multiplicati ves) much easier to handle than the extensionals (additives). 2 This variant is used also in [Sw] for the classical system. 2 for intro ducing the pro of-nets that he has invented as an economical to ol for developing Gentzen-type pro ofs in which only the active formulas in an application of a rule are displayed. For the purp oses of the present pap er it is b etter however to use the more usual presentation. Girard noted in [Gi] that he had given absolutely no meaning to the concept of a \linear logical theory" (or any kind of an asso ciated conse- quence relation). Hence the completeness theorem he gave in his pap er is of the weak kind. It is one of our main goals here to remedy this. For this we can employ two metho ds that are traditionally used for asso ciating a consequence relation with a Gentzen-type formalism. In classical and intu- itionistic logics the two metho ds de ne the same consequence relation. In Linear Logic they give rise to two di erent ones: LL LL B i the ) : A ;:::;A ` The internal consequence relation (` 1 n Kl Kl 3 corresp onding sequent is derivable in the linear Gentzen-type formalism. The external consequence relation (` ) : A ;:::;A ` B i the LL 1 n LL 4 sequent ) B is derivable in the Gentzen-type system which is ob- tained from the linear one by the addition of ) A ;:::; ) A as 1 n axioms (and taking cut as a primitive rule). It can easily b e seen that these two consequence relations can b e char- acterized also as follows: LL A ;:::;A ` B i A ! (A ! ( (A ! B ) :::)) is a theorem 1 n 1 2 n Kl of Linear Logic. LL A ;:::;A ` B i ` B , for some (p ossibly empty) multiset 1 n LL Kl of formulas each element of which is: Intensional (multiplicati ve) fragment: identical to one of the A 's. i The full prop ositional fragment: identical to A ^ A ^ ^ A ^ t. 1 2 n In what follows we shall use b oth consequence relations. We start by developing a natural deduction presentation for the rst and a Hilb ert-typ e presentation for the second. 2.2 Natural deduction for Linear Logic Prawitz-style rules: [A] B B A A A 3 Since linear logic has the internal disjunction +, it suces to consider only single- conclusioned consequence relations. 4 We use ) as the formal symb ol which separates the two sides of a sequent in a Gentzen typ e calculus and ` to denote (abstract) consequence relations. 3 [A] B A A ! B A ! B B [A; B ] A B C A B A B C A t t A A B A ^ B A ^ B () A ^ B A B [A] [B ] A _ B C C B A () A _ B A _ B C Most of the ab ove rules lo ok almost the same as those for classical logic. The di erence is due to the interpretation of what is written. For Linear logic we have: 1. We take the assumptions as coming in multisets. Accordingly, exactly one o ccurrence of a formula o ccurring inside [ ] is discharged in ap- plications of I nt, ! I nt, E l im and _E l im. The consequences of these rules may still dep end on other o ccurrences of the discharged formula! 2. Discharging the formulas in [ ] is not optional but compulsory. More- over: the discharged o ccurrences should actual ly be used in deriving the corresp onding premiss. (In texts of relevance logics it is custom- ary to use \relevance indices" to keep track of the (o ccurrences of ) formulas that are really used for deriving each item in a pro of.) 3. For ^I nt we have the side condition that A and B should dep end on exactly the same multi-set of assumptions (condition (*)). Moreover, the shared hyp othesis are considered as app earing once, although they seem to o ccur twice. 4. For _E l im we have the side condition that apart from the discharged A and B the two C`s should dep end on the same multiset of assumptions ((**)). Again, the shared hyp othesis are considered as app earing once. 5. The elimination rule for t might lo ok strange for one who is accustomed to usual N.D. systems. One should then realize that the premiss A and the conclusion A might di er in the multiset of assumptions on which they dep end! Notes: 4 1. Again we see that the rules for the extensional connectives are impure, while those for the intensional ones are pure (no side-conditions!) 2. the rule for I nt is di erent from the classical (or intuitionistic) one, since no o ccurrence of A on which B dep ends is discharged. In fact we have that the dual: [A] B B A is derivable, but the classical version [A] [A] B B A is not valid! 3. It is not dicult to prove a normalization theorem for the p ositive frag- ment of this system. As usual, this is more problematic when negation is included. This case might b e handled by adding the ab ove derived intro duction rule as primitive and then replacing the given elimination rule with the two rules which are obtained from the intro duction rules by interchanging the roles of A and A. It is easier to see what`s going on if the N.D. system is formulated in sequential form: A ` A ` A ;A ` B ` B 1 2 ; ` A ` A 1 2 ` A ` B ` A B ; A; B ` C 1 2 1 2 ; ` A B ; ` C 1 2 1 2 ;A ` B ` A ` A ! B 1 2 ` A ! B ; ` B 1 2 ` A ^ B ` A ^ B ` A ` B ` A ^ B ` A ` B ` A ` B A; ` C B; ` C ` A _ B ` A _ B ` A _ B ; ` C ` A ` t 1 2 ` t ; ` A 1 2 Again ; denote multisets of formulas.