Rend. Sem. Mat. Univ. Poi. Torino Fascicolo speciale 1987 Logic and Computer Sciences, (1986)

Jean-Yves Girard

MULTIPLICATIVES

The paper discusses the general concept of a multiplicative connective in linear logie. The first lineaments of an geometrical semantics of computation are developed in the multiplicative case.

1. The multiplicative fragment ®/$

1.1. the five levels of linear logie

Linear logie, as it is now, is organized along five levels:

1 : communication (. )1 connective nil

2: cooperation «,3? connectives times and par

3 : sharing © , & connectives plus and with

4: storage/reading ! , ? connectives ofeourse and why not

5 : abstraction Ace, Va quantifiers any and some

The relevance of these five levels to computation is discussed in (Girard 1986 A). The connectives of levels 2, 3,4 are respectively called multiplicatives, additives, exponentials. The multiplicative fragment of linear logie consists of levels 1 and 2; this is by far the most satisfactory part of the syntax (ali the other levels need proof-boxes, which are a concession to more traditional kinds of syntax). Levels 1 and 2 deal with binary communication and eoo-

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peration; they are by far the best understood, and we shall try to produce an operational semantics of these two levels. In the meantime, the general concept of a w-ary multiplicative, corresponding to w-ary cooperation, will arise. Most of these connectives will not be definable in terms of ® and 3?, and for instance, there are forms of ternary cooperation which are primi­tive.

1.2. proof-nets

The operational semantics will arise from an abstract study of the notion of proof in the multiplicative fragment, so-called proof-nets-, the reference here is (Girard 1986). A pròof-structure consists of occurences of formulas, put together by means of links; there are four possible links

i) axiom link:

ii) cut-link

iii) times link:

iv) par link

A^ B

The axiom link has no premise and two conclusions, the cut link has two premises and no conclusion (the symbol CUT is not a formula, it is just here to indicate the reciprocai anihilation of A and A1), the times and par links have both two premises and only one conclusion in a proof-structure, they behave exactly in the same way. There are implicit conditions to form a proof-structure:

i) every (occurence of) formula in the structure must be the conclusion of exactly one rule.

ii) every (occurence of) formula in the structure must be the premise of at most one link

So that a proof-structure looks like a proof with several conclusions; it is not yet excluded that the proof could be disconnected. A last point; ( . )1

A AL

A A1

CUT

A B

A®B

A B

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is not taken as a connective, but defined: the atoms are of the form OL or OL1 , where a is a propositional variable; A1 is defined by

a1 is already in the syntax

(or1)1 = a

(A*>B)L = A i ^Bl

(A^B)1 =AL ®Bl

so that A11 is litterally the same as A ; in particular, the axiom links are symmetrical, i.e. we do not distinguish between A A1 and A1 A ; the same for cut-links. In order to discriminate which, among proof-structures, are noble, we intro­duce the concept of a trip: each formula is viewed as a pair (qA, aA) of a question and an answer. The prefix a is read as "move downwards", whereas q is read as "move upwards". Starting with a position (i.e. a prefixed formula aA or qA), we move to another position, following travel instructions:

i) from qA in A A1 , move to aA1

ii) from a A in A A1 move to qA1

CUT

iii)from aA move to qA when A is a conclusion

iv)the travel through a ®-link depends on a preset switcbing: attached to such a link is a switch with two positions L/R . Assume that the link is A B ; then the travel instructions are:

A®B

L: from aA , goto « (A ® 5 ) , from q (A ® B), goto #£ , from aB , goto #/1 .

R: from #A , goto qB , from tf£ goto a (A ® B) from g (A « £) goto gA .

v) the same holds for the 3? -link A B , with two positions for the switch:

A®B L: from aA , goto a (A ^ B) , from # (A3? J5) goto gA , from aB

goto ^B .

R: from «A goto gA , from a (A^ B) goto aB, from g£ goto q(A^B) .

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The travel instructions are bideterministic, i.e., when the switches are posi-tioned, we know at each position, from where we go, and where to go. In particular the global structure made of ali position, can be see as the union of disjoint cycles, or trips. When there is exactly one cycle, we cali it alongtrip-, in case of more than one trip, these trips are called shortrips. A proof-net is a proof-structure in which, for any simultaneous positioning of the switches, there is only one cycle, i.e. there is no shortrip. For instance, a proof-net must be connected. Also the structure A A1 has a shortrip, riamely aA, qAl, a A et e. . CUT

1.3. soundness of proof-nets

In (Girard 1986) we introduced linear sequent calculus, and proved the equivalence between the two approaches: to a proof P in linear sequent calculus is associated a proof-net P~ , and conversely every proof-net can be written (in a non-unique way) as P~ . This is a particular way of proving the soundness of proof-nets, by relating them to a more established and un-derstandable kind of syntax. But there is another way of proving that proof-nets are a sound concept: they enjoy cut-elimination, namely, by simple considerations of trips, it was possible to show that any cut can be eliminated in a very quick and efficient way. This form of soundness, which cuts any relation with sequential syntax, is of course the most promising for future developments.

2. Modularity and proof-nets

2.1. the problem of modularity

The general problem is the following: assume that J am given a program P , and that I cut it in two parts, arbitrarìly. I create two (very bad) mo­dules, linked together by their border. How can J1 express that my two modules are complementary, in other terms that J can branch them by identification of their common border? One would like to define the type of the modules as their plugging instructions; these plugging instructions should be such that they authorize the restoring of the originai P . We shall

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here give the solution when P is proof-net /3 , and we hope that this partial answer will be a hint towards the general solution.

2.2. the simplified problem

/3 is a proof-net, linking together a set X of occurences of formulas. A submodule fi of /3 is obtained by taking an arbitrary set L' of (occurences of) links in /3 , and to consider ali those (occurences of formulas which are either premise or conclusion of a link of L' . The complement /3" of fi consists of the submodule which corresponds to L - V , where L is the set of ali (occurences of) links in /3 . The border dfi of /3' consists of those (occurences of) formulas which are in fi and fi', i.e. dfi = X' 0 X" , with obvious notations. Obviously dfi = dfi' . If A G dfi , there are only two possibilities:

i) 4̂ is the premise of a rule of V and the conclusion of a rule of L" ; notation: aAEfi , qAE fi'

\\)A is the premise of a rule of L" , and the conclusion of a rule of L' : notation: aA Gfi', qAefi

In particular, given A E dfi , the expression "move in /3"' will be clear. it means "take the position aA (resp. qA) if aA E fi (resp. M G/3'). In a similar, exiting of /3' corresponds to the position qA (resp. aA) when ^ G j 3 " (resp. aA E fi'). Let 5 be a particular switching of V ; given A E dfi , we start to move in /3' , up to the moment we exit through B € dfi ; we set B =fs (A). It is easily seen that fs is a permutation of dfi . In a similar way, we can define, for any switching T of L" , a permutation g r of dfi' = dfi . Observe that in fi for instance, every internai position aC or </C is passed between some A and / s (A ) , for /4 G <#3' : the existence of a cycle inner to fi (not meeting dfi) is impossible, because it would be a shortrip. (This remark does not apply when fi - /3 because dfi - é) . From now on, we assume dfi ¥* (f>.

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2.3. orthogonality of permutations

A permutation o of {1, ..., »} is said to be cyclic when it has exactly one cycle, i.e. when 1, a (1), a2( l) , ..., aw_1 (1) are ali distinct. Two permutations a and r defined on the same finite set are said to be orthogonal when or is cyclic ; notation air . Orthogonality is symmetric: this follows from TO = O~1 (or) a .

Examples:

i) for n = 0 or 1 , there is only one permutation, which is self-orthogo-nal.

ii)There are two permutations of {1,2}: the identity i and the trans-position tl2 , which is cyclic. % 1 tl2 , but neither i nor t12 is self-orthogonal.

iii)There are 6 permutations of {1,2, 3}, namely i , tl2, t23, tì3 and the circular permutations s+ (k) = k + 1 (mod 3) ; s~(k) = k — l (mod 3), which are the two cycles of 1 ,2 ,3 . The orthogonal pairs are the following

2 1 S ; l 1 S ; t j2 1 ^23 » ^12 -!• ^13 > ^23 ^- ^13 > S •*-'s > 5 — 1 5

2.4. solution of the problem

For any switchings, 5 of L', T of L", fs and gT are orthogonal: the switching SU T of L induces a longtrip, from which we can extract the unique cycle of fs gT . (Starting with A on the border, we first do a bit of travel inside 0" , up to the moment we exit trough gT (A ) ; then we proceed inside fi' , up to fs gT (A), then to gT fs gT (A) etc.) In order to type our modules, we shall use the definition:

Definition

If C G a ({1, ..., »}) , then C 1 is the set of permutations which are orthogonal to ali elements of C.

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We shall not recali the usuai trivialities, e.g. C i l i =(71.. We shall type our module $ by introducing the set F consisting of the permutations / s , where S varies through the switchings of V :

Type (/?') = F 1 1

(Similarily, Type (fi") = G1 1) .

From immediate considerations, F 1 1 1 G11 , i.e. every permutation of Type (j3') is orthogonal with any permutation of Type (0").

Remark

One may ask the reason of the replacement of F by F 1 1 ; the reason is that, w.r.t. orthogonality, F and F 1 1 behave in the same way; in parti-cular, if we have two modules with the same border, and associated sets of permutations F and F' with F 1 1 = F ' 1 1 , they will be interchangeable, since the same permutations are orthogonal to F and F' .

Definition

A module is a proof-structure with hypotheses, say 0 , together with a border dfi which consists of ali hypotheses and some conclusions. We require that:

i) dp # <f>

ii) for any switching of 0, there is no cycle not meeting dfi .

The type of the module is defined as above, as the biorthogonal of the set F of the permutations of the border generated by the switchings. Two modules are orthogonal when

i) they have the same border; in order to avoid problems with occurences it is convenient to label the formulas of d$ and d& , so that to know which formula of d($ corresponds to a given formula of dfi .

ii) if A is a hypothesis in d(3 , then the corresponding occurence of A in dfi is a conclusion in d/3' , and symmetrically.

iii)Type (|5) 1 Type (|3').

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Theorem

Assume that j3 and 0' are modules, which are orthogonal, and let 7 be the proof-structure which is obtained from them by gluing their common border. Then 7 is a proof-net.

Proof: practically immediate.

Remarks

i) the theorem enable us to make locai changes in a proof-net. For instance/ can isolate a module j3 in 7 , and replace it by any module j3j such that Type (j3j ) C Type (j3) . In doing so, / have not to think about the complement of Pi in 0 , which may be quite big.

ii) the theorem enables us to undo a proof-net by starting from the conclusions, in any order. The complementary module of what remains induces permu-tations of the conclusions (conclusion switches) that enable one to speak in terms of longtrip.

iii)it is formally a bit unpleasant to have to consider hypotheses; however, a hypothesis can be transformed into a conclusion by an axiom link (link the hypothesis A with A1) . Modules with only conclusions can then be attached together by means of the cut-rule.

iv)the case of a void border is problematic. For instance, a proof-net with a void border, might be considered as a module. But then it should be orthogonal to any other proof-net with a void border, but glued together, they yield a non-connected structure. This is the reason why the case dp = 0 has been excluded. This comes from the fact that we have read the definition of "cyclic" in a certain way for n = 0 ; in fact, to say that the void permutation is cyclic means that the void set is seen as a cycle, which is a matter of definitions. Let us try to clarify this a little:

i) we change our definition in the case n = 0 : the void permutation is no longer cyclic.

ii) a module with a void border will be exactly a proof-net.

iii)nothing is orthogonal to a module with a void border.

If we adopt these conventions, the theorem will be vacuously satisfied.

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(Variant of iii): the void structure, with void border is a module. There is no switching of it, so its type is 01 1 = $.)

2.5. examples of types

i) types on one element, 1 There is only one set equal to its biorthogonal, namely {i} .

ii) types on two elements {1,2} The four subsets of <J({1, 2}) are equal to their biorthogonal. However, the void set and the full set cannot appear as types of modules.

iii)types on three elements {1, 2, 3 } The subsets of a ({1, 2, 3}) equal to their biorthogonal are: 0, {tl2}, U23 }>{fi3 }>{s+)> {s~} ,{1} and their respective orthogonals a ({1,2, 3 }), (*23, *13 >, (*12, *13>» {*12 *23>, U S+], {I, 5 " } , {S + , S~ } . '

iv)in general, let us remark that, for n > 2 , a type is never equal to its ortho-gonal. In fact, if s belongsto C = CL , then s2 is cyclic and necessarily 5 must be cyclic ; but then si is cyclic for ali s in C , so iGCL = C ; but i is not cyclic for n > 2 .

3. Generalized multiplicatives

The purpose of this section is to use the idea of module in a logicai way: a module with no hypothesis, can be seen as a propf-net with the conclusions C (dpi), D , where D are the conclusions of j3 not in dp , and C is a generalized multiplicative connective, constructed from the type of j3 . In the sequel, the arity n of the connective will al way s be at least 2 .

3.1. definitions.

A n-ary multiplicative is any subset C of a({l , . . . ,w}) such that C = C11 and C is non-trivial, i.e. C =£(f)} C=5fca({l, ..., n}). Corresponding

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to C we introduce the w-ary connective C * , together with the rule

C* C M , . . . * „

Linear negation is defined by

(C*Al ...An)i = (Cl)*A{...Ajt. •

We can define proof-structures involving generalized connectives in the usuai way; proof-nets are defined in the usuai way, from the notion of C-switch; the C-switch admits positions (s, i) for s E C , 1 < i < n

from <*i4. , with j¥=i, goto ^ S Q

from <L4. , goto aC*At ... An

from qC*Al ... An , goto qAs{{)

3.2. examples

i) there are only two multiplicatives of arity 2, namely — -Un )*; the switch has two positions (t12, 1) and (t12ì 2), in which we recognize the switchings "L" and "R" of &; this multiplicative is therefore identical to ®. — {/'}* ; the switch has two positions (i, 1) and (i, 2) , in which we recognize the positions "L" and "R" of ^ ; this multiplicative is therefore identical to ^ . •

ii) what happens if we replace C by D such that D1 1 = C ? We could have defined multiplicatives from an arbitrary set D C o ( { l , . . . , » } ) such that D, D1 are non void; if C = D11, C* is a priori distinct from D* , since the C*-switch has additional positions; but these additional positions are of no use: if the rule

Ai ... A 1

D D*Ai ... At

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is used in a proof-net, choose a particular switching for ali other links; this induces a permutation / of {1, .., « + 1 } , corresponding to the beha-viour of the trip outside the link D* , the index n + 1 being for the conclusion D* Ax ... An . f must be orthogonal to ali permutations (s, i) of {1, ..., n 4- 1} (s G D, 1 < i < n) defined by

(s, i) (/) = s (/) for / # *, w + 1

(s, i) (0 = « 4- 1

(5, i)(» + l) = s(i)

/ (n 4- 1) = i, with / =£ » + 1 is impossible, since (s, i)f (n + 1 ) = n + ' 1 ; hence / ( » + l ) = « + l , and f induces by restriction a permutation g oi {1, ...,»} . Now g 1 s , as we easily check: we arrive at the conclusion that gED1 =CX . Now, working backwards, it is easy to conclude that we can take as positions of D* ali pairs (s, i) with s E C , and stili get no shortrip. There is therefore no reason to distinguish between C* and D*. This simple remark can be used - to reduce the number of positions of the switches, replacing C by some D such that D 1 1 = C - to handle the case of compound connectives

iii)a compound connective is an expression E {ax, ..., <x ) in which each of the propositional variables ax, ..., an occurs exactly once. Compound connectives should be, if we are not mistaken, particular cases of genera-lized multiplicatives. It sufficies to treat the case of the compound connec­tive C*(D*(otì, ..., <xn), an+l, ..., ocn+m) where D is w-ary and C is m -I- 1-ary (for simplicity, we shall assume that C has been defined as a subset of a ({n, ..., n + m}). As a compound connective, our defined connective E* , has a switch which consists in combining the switch of C and the switch of D ; the positions are therefore of the form (s, i, t, j) with s G C , w < / < w + w, f G D , l < / < » . Let us denote by n + m + 1 the conclusion of the rule; we have

(s, i, t, j) (/) = n + m 4- 1 if i = n

(s, i, t, j) (i) = n + m + 1 if i =£ n

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(s, i, t, j) (n + m + 1) = t (j) if s (0 = n

(s, i, t, j) (n 4- m + 1) = 5 (0 if s (i) =£ « etc.

If we write (s, i, t, j) = (w, k) for a certain k such that 1 < k < n + w, we see that ali values & between 1 < k < w 4- m are taken, when (5, 0 and (t, j) vary. If we introduce, as we did in ii) a permutation / of {l,.. . ,w + ra + l } to describe the behaviour of a trip outside the compound link, we conclude that / ( » + w + l ) = » + m + l , and that its restriction g to {1, ...,« + w} is orthogonal to ali permutations u occuring in the pairs (u, k) of the form (s, i, t, j) ; let's cali U this set of permu­tations. Our compound connective behaves in fact like U* or U11 * .

iv)let us compute ali ternary connectives which are definable from ® and ̂ : (e*! ® a2) ® a3 ; the set U is equal to {s~, s+ }, which is equal to its biorthogonal. Ali other ways of defining a ternary connective from two ® would lead to the same answer. («i 3? a2) ^ 0:3 ; the set (/ is equal to {*'}, which is equal to its bior­thogonal. Ali other defined ternary connectives with two u5$ " lead to the same answer. (<*! ® a2) ^ a3 ; the set (/ is equal to {tl2 } , which is equal to its biorthogonal. AH ternary connectives obtained by a ® followed by a ^ lead to a connective {tA *. (at 3? a2) ^ oc3 ; the set U is equal to {£23, 1̂3 }, which is equal to its biorthogonal. Ali ternary connectives obtained by a ^ followed by a ® are of the form {t.k, t.k }* with i, j ¥= k . In particular, there are four ternary primitive connectives: {s+ }* and its orthogonal {i, s+ }* {s~ }* and its orthogonal {i,s~}* These four connectives express ternary cooperations that cannot be reduced to binary cooperation.

3.3 cut-elimination

A cut \ \ \ \ Ax ... An „ A\ ... A1 ,

C*A1...An CL*A\.:.ALn

CUT

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is replaced by

\ \ \ \

CUT CUT

We have to check that the result is stili a proof-net. Let us index the for-mulas Alf ..., An, A\, ..., Al

n as Blt ..., B2n . If ali links, but C* and C1 * in the originai proof-net j3 are switched, then this induces a permu-tation f of {1, ..., In]. It is impossible that, for some i < n , we have f (i) > n + 1 : this is because it is possible to switch C * and C1 * in such a way, that after aBf(i) one gets to #£,. (choose permutations s G C , t e C1 , switch C1 * on (f, / (f>«) and C* on (5, s~1 (i)) ) , and this would induce a shortrip. Hence / admits restrictions / ' to {1, ..., n] and /" to {n + 1, ..., 2n}; it is more convenient to see /" as a permutation of {1, ..., n} too. Now, it is easily checked that • f 1 C , /" 1 C1 ; in particular f E C1 , f'GC, so / ' 1 /" . Now, in the reduced proof-structure j3', starting with

qAx , we move to a^f"n) » t n e n t 0 4Af"n) » t n e n t o aAffu) ' ^ 6 1 1

to qAff„,x. , then to aAf„f,f,,(1) etc. . Since / ' and /" are orthogonal,

we eventually arrive at M x = a A ( w.") W(D an(^ t n e trip is a longtrip. Cuts where one of the prerhises in an axiom are reduced exactly as in (Girard 1986). Since we shall devote a lot of time to those cuts, let us ignore them now. The cut-elimination procedure shortens the proofs, so it converges very quickly.

3.4 reversed 17-conversion

We shall make an apparently minor change to our formalismi we shall restrict the axiom link A A1 to the case where A is atomic. The general pattern A AL appears therefore as a defìned link; more preci-sely, a proof-net A AL is defined by induction on A :

i) when A is atomic, this is the legai axiom link A A1

ii)when A is C* Bx ... B„ , we set

Al= I I I d B{ ...B B\ ... Bx

1 n l n C*Bl...Bn C*lBÌ....Bln

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which is easily checked to be a proof-net. This process of replacing in a proof-net general axiom links by atomic ones, is analogous to principles that are sometimes considered in typed X-calculi, namely 17-conversion and surjective pairing:

(Xx. tx)~/t (ir1 a, n2a)=/a

but the sense of the reduction is reversed. These additional principles have not yet found (reversed or not) their place in the study of computations. They seem to be cruciai for the geometrical semantics.

4. the geometrical semantics

4.1. proofs as orthogonal permutations

Let 0 be a cut-free proof-net proving A , i.e. with the sole conclusion A ; then j3 looks necessarily as the fully developed tree TA of the sub-formulas of A , whose .summits are occurences bl, ..., bn of atomic symbols af. or a. ; these atomic symbols are pairwise linked by axiom links <xf. o&.

Now, the tree TA is common to ali proofs of A , whereas the particular linkage of the summits depends on /3 . The restriction of the axiom links to the atomic case was done to obtain this common part TA . Something else is common to ali proof of A : if 5 is a given switching of the links inside TA , then / get a permutation fs of {1,... ,»} by: from position ab. , the next position as a summit is qbf ,- . The set o (A) of ali permutations fs of {1, ...., n] obtained in this way, is common to ali proofs of A . There is another permutation of {1, . . . , » } , but which depends on j3 : Sa W ~ ì w n e n b. and b. are linked by an axiom link. g0 is a symme-try (g-2 = 1), and n must be even. So gR alone defines completely our proof j3 . Moreover, the longtrip con-ditions can be stated as *g& 1 a (A ) . We propose therefore to say that a normal proof oi A is any element of a (A)1 . This deserves a discussion.

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4.2. discussion

i) a typical consequence of our definition will be that any element of a (A) (even a (/4)11) will be considered, semantically, as a normal proof of A1 : this is clear from the remark that a (/41)1 = o 04 ) 1 1 .^ This replaces the familiar duality proofs/models (i.e. proofs/refutations) by a duality proofs/counterproofs, which is much more satisfactory.

ii) in the orthogonal of a (A ) live permutations that do not correspond to axiom links, for instance, when n is odd ! It may seem strange that we do not restrict ourselves to permutations arising from axiom links; then we have to ask ourselves the reason of the syntactic restriction to axiom links: the reason is that we want to prove formulas that are stili valid when we replace their atoms by other formulas, which mean in fact propositions universally quantified w.r.t. their atoms. Now, we are interested in normal proofs of A , where the atoms are frozen, and will not be replaced by anything else, so our choice is legitimate. The treatment of the quantifiers (not included here) will restore the limitation to axiom links.

4.3 the quick elimination precedure

Take a proof-net (5 with cuts, and reduce ali symmetric cuts C*/Cl* , up to the moment each cut has one premise which is the conclusion of an axiom link. Then there are chains of cuts:

(x0 ai a! ai OL2 ocin_1 an a 1

t _ — ... _ t CUT CUT CUT

where oc0 and aj; are not premises of cut-rules. The atoms a0> •••» a„ are distinct occurences of the same atom a . Now this cut is normalized by shortening the bridge, so that to get

t t

(*) se lemma 4.5.

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Now, let us describe this procedure in terms of permutations: assume that /3 is a proof of A with cuts, on formulas Bv, ..., Bm ; if we replace the terminal symbols CUT by B. « Bf-(i= 1,..., m) and the rules CUT by («) , we stili get a proof-net, with the additional conclusions B{ « Bj-. The proof-net is easily trànsformed into a cut-free proof of the "par" A, of its conclusions. It can be therefore thought of as a permutation g~ of the atoms blf..., b of A'. Now, these atoms look as follows:

i) some of them, say blt..., bk are indeed atoms of A

ii) the other ones come two by two, i.e. n — k is even: to each atom b. of some B{ corresponds an atom, say b.+1 of B. , and to b.+ 1

corresponds b. . Define h (/") = / 4- 1 for / > k , / — k odd, b (/)= ; — 1 for j > k and j — k even, so that b. corresponds to bh^

for j > k . h (/)=/ if ; < k . We try to define the normal proof |3' of A as a permutation g„ , of {1, ..., k} . For this we consider the permutation gJ) of {1, ..., » } . If we start with i such that l < / < & , we form g*b (j),g*h(i) etc. Since g 3 is a permutation, there is a strictly positive e such that (gp b)c (0 = *" » hence there is a smallest integer f̂. > 0 such that (g~b)d*(i)<:k ; weset g8,(i) = (g«b)di(i). This semantic operation is the basis of our geometrical semantics; it computes the result by iteration of a permutation.

4.4. an example

We shall illustrate the application of a linear function of type A -o B to an argument of type A . Here both A and B will have 4 atoms ; the proof of

r — — — — I | — i A is a o e a the proof of A -o B is a b' e' d' è' f g b

We want to define the permutation of {e, f, g, h] obtained after normali-zation: for instance, start with b ; then we move to g , so we stop. If we start with / , then our link leads us to d! ; now we would.like to con­tinue by a trip inside TAi ; such a trip is not given by the data; if it were given, it would be an element of a (A*- ) . Now, our proof of A is virtually

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an element of a (AL) , since it belongs to a (A)1 = a (AL)11 , so we can consider the proof of A as a way of going inside TA± .In this proof we pass from d to a , which means that the next step after d' will be a* ; from <*' we go to b' , and again we ask the proof of A to know where to go inside TA x from b' : in this proof, we pass from b to e , hence the next step is e' . From e' we go to e which is an atom of B , so we stop. Summing up, we have obtained the proof -

i ' i *

of 5 , and we never had to investigate about the nature of B !

4.5. the soundness theorem

In cases of proofs coming from the syntax, it is clear that what we have done is just expressing what happens to the cut-elimination procedure at the level of the atoms; in particular, the usuai cut-elimination rule C*/CL* is only used to realize the opposition of the pairwise atoms of the cut-formulas. But our procedure has been defined for arbitrary semantical proofs; we have to check that the result is a normal proof.

Theorem

Let gp be a normal proof of (Bt »B\)^... ^(Bm « B ^ ) ^ A . Consider the B! S and Bj's as cut formulas, and construct a permutation g&, of the atoms of A , as we did in 4.3.; then g0, is a normal proof of A .

Proof: first let us state a lemma that we have mentioned several times:

Lemma: a (A1)1 =o (A)11 .

Proof: i) we have first to show that a (A) I o (A1) ; write the axiom link

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A A1 and progressively replace it by its definition in terms of atomic axiom links, by reverse 77-conversion. It is easily proved by induction on A that the proof we eventually find is a proof-net, in other terms that the symmetric permutation g which exchanges the atoms of A and A1

in the obvious way, is a normal proof. Let us label the atoms of A bt, ..., b and the atoms of A1 b ... ..., bnut ; an element of o (A)

will be seen as a permutation / £ a ({1, ..., n}) and an element / ' of o (A1) will be seen as a permutation / ' E a ({n + 1, ..., n + m}), whereas g exchanges i and i + n . Now, if / ® / ' denotes the permutation of {1,... ,2¾} obtained by gluing / and f\f®f'lg ; concretely this means that / l / " , w h e r e / " is the permutation of {1,. . . ,«} defined by f"{i) = f (i + n) — n . But this establishes that a (A ) 1 0 (A1 ) .

ii) conversely, we have to show that a (A1)1 Ca (A)il , in other terms, that, if g E; o (A1), g' E o (A)1 , then g 1 g' . We cannormalize the cuts C*/Cl

in the proof

g g'

*

w w CUT

following the syntactic pattern of 3.3.; the fact that the links on top are not symmetric axiom links, but arbitrary permutations does not matter at ali: / and g are just weird travel instructions from atom to atom. At the end remain only cuts between the In atoms, and we can conclude, since the structure is cyclic, as we did in case (i), that g l g ' , and the claim is proved. QED The theorem is established as follows: we takè an element / of o (A) , and we prove that g^ 1 f. For this, we use the lemma which says that / is a normal proof of ^41 ; hence

(defined) (defined)

5, B\ ... B B m m

Bx * B\ B ^ Bl A±

® (iterated) ( £ , * Bì) • ... •&* Bt)9A

m m-

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is a normal proof. This proof, if we label the atomx as we did in 4.3., looks like a permutation h' which coincides with h for arguments > k , and with / for arguments < k . By the lemma b' Lg&, hence g0h' iscyclic; start with 1: gp, f (1) = (gfi h)df(D (/-(1)) = (gfib')dfl»(l) (= u) ; for similar reason, <g0. j)

2 a) = (g0h)df(u)(f(u)) = (gph')dfn)+df(«)(l) etc .

From this it is not difficult to conclude that g„,f is cyclic, and we are done. QED

5. the geometrical semantics (continued)

We shall now develop the semantics of the multiplicative fragment (0 1 , ^ , ^ ; this semantics has to be later completed to take care of the other operations of linear logie.

5.1. vehicles

Definition

A (multiplicative) vehicle consists in the following data:

i) a permutation V of a finite set I V I

ii)a non-void subset t (V) of I V I

Definition

If (V, I Vi, t (V)) is a vehicle, its execution is the vehicle (e (V), t (V), t (V)) defined as follows: for a e t(V), let da be the smallest integer > 0 such that Vda(a)£t ( V); we set e (V) (a) = Vd*(a).

Definition

If V and V are two vehicles, then V I V' means that

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i)tV=tV'

ii)*Vl*V"

Definition

If V is a vehicle, then its (principal) type is the subset of tV defined by

Type(F) = eK±i

5.2. interpretation of formulas

To each formula of the multiplicative fragment, we shall associate a type, i.e. a set T =£ 6 together with a set P of permutations of T , such that P = P i i , and P ,&•*$-

i) arbitrary types are associated with the atoms a ; to a1 , we associate (T, Pl), if (T, P) has been associated with a .

ii) if" (T, P) and (C/, Q) have been associated with 4 and B , we first make T and t/ disjoint by making 7' = T X {0}; 17' = U X {1}, hence weget (T', P') and (£/', £?') .We interprete A ^ B as (7' U t/', P' * Q') where P' ® Q' is the set of ali permutations /® g, / G P ' , g G £)', f®g being the permutation "union" of / and g . We have to check:

Lemma

(T U if, P' ® g') is a type.

Proof: let s G P a , £ G Q'L ; select i G T\ j G [/', and define a permutation «• of T'UU' by: « (*)=*(*) when k ET', k* i, u (i)=j, u (k) = t (k) when keu', k^rHj) ,u (t~\j))= s (i) . Then ulf*g for any /GP', any g G Q' : let v = u (f®g) ; starting with / we form v (j), v2 (/) e te. up to vm~l(j) with m = card (£/'); these w points are equal to j,tg(j), (tg)2 (/), •••, (^g)™"1 (/) and are ali distinct; moreover (tg)m (;') = ; , so g ((tg)nt~1 (/)) = £ - 1 ( / ) , hence vm (;) = s (i) . For symmetric reasons, vm + n-l (y) _ j w i t h n _ c a r d (r') ^ h e n c e ^ +« (y) = y W e h a v e i n f a c t

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made a complete cycle, and we are done. This shows that (P' ® Q') is non-void. Take now an element b E(P' ® Q')11 , ànd assume that for some i E T we have h (i) = / G ( / ' ; then b cannot be orthogonal to the permutation u constructed above. This means that b can be split as b^fQg for some permutations f of T , g of U'. Now the fact that b must be ortho­gonal to a permutation like u above, is easily transfprmed into fLs,gLt. This shows that b E P' ® Q'. QED

iii)under the same hypotheses, we interprete A®B as (Tf U U', (P'1® Q'1)1) .

The interpretation \A I of a formula therefore satisfies: if \A l=(7, P), then \Al\=(T,Pl).

5.3. interpretation of linear sequent calculus

An expected, a sequent \ — Alf...,A is seen as a generalized "par" . If U . l = (7\, P.) , we form 7:' = T. X {0 etc , and we set

I \-Al,...,An\=(T[U...ur,Pl®...<sP'n).

To each proof ir of a sequent 5 , we associate a vehicle V ( ) , such that

* K ( S ) 1 | S |

c w 1 : axiom I — A, AL

If \A I = (T,P) , then | | - /1, ,411 = (T' U r", P' • P1 ") where ' and " are used to diplicate T . V (ir) is defined by I V (ir) | = tV (7r) = T' U T" , V (7r) (#') = #", V (7r) (#") = A:' . V (ir) is immediately shown to be orthogo­nal to P'vP1".

case 2 : exchange rule \-A

l - a ( 4 )

from the interpretation V (ir') of the proof of | — A , we get V (ir') by a renumbering of the indices i , according to o.

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case 3 : par rule -A, B,£

-A^ B, C

here again, the interpretation is just a matter of renumbering.

case 4: times rule I - A C \-B,D

| - A 9 B, C, D

To simplify notations, assume that, if ' and " are the proofs of the premises, that | V (n')\ and | V (ir")\ are disjoint. We define I V(7r)l= ! V (ir') I U IV (TT") I and V (TT) = V (ir') ® V (ir"). Once more the fact that eV ( ) is orthogonal to | 5 | is a simple exercise.

case 5: cut-rule , 1 - 4 , C \-A\D

CUT I - C, D

here again we do some renumbering; but the two occurences of T (such that \A\ = (T, P)) are distinguished as T and T". We define | V (ir) 1= | V (n')\ U I V (TT")I , and tV (ir) = (̂ V (TT') - T') U (tVfTr'^-T'^.Thepermutation V(7r) is defined by.

V(ir)(x)= V(ir')(x)* for ^ G I V ( T T ' ) )

V(IT)- (JO=V (»*)(*)* for ^ e IV(TT")I

where * is the operation interchanging T ' and 7" , and identical otherwise.

5.4. the execution theorem

Definition

If V is a vehicle, then the execution of V succeeds when the permuta-tion V has no cycle that wholly lies within 1 V 1 — t ( V).

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Theorem

Let 7T , ir' two proofs of the same sequent, such that the associateci proof -nets 7T~ and ir'~ have the same normalform; then

i) the execution of V (ir) and V (irf) succeed

u)eV(ir) = eV(Tr').

Proof: the result is more or less a restatement of the results of 4.5., and the proof is therefore omitted. QED

BIBLIOGRAPHY

[Girard 1986] Linear logie; to appear in T.C.S.

[Girard 1986 A] Linear logie and parallelism; to appear in the proceedings of the School on semantics of parallelism held in IAC, CNR, Roma, September 1986.

Added in print (December 1987):

i) The result of this paper have been improved by Vincent Danos and Yves Régnier, in particular by a sharper approach to orthogonality.

ii) A greater part of linear logie is now accessible through proof-nets, for instance quanti-fiers.

JEAN-YVES GIRARD - Equipe de Logique (UÀ 753 CNRS) - Département de Mathémati-ques T45-55, 5°étage - Universìté PARIS VII, 2 place Jussieu - 75251 PARIS CEDEX 05