A Full Completeness Theorem for Multiplicative-. Additive Linear Logic is proved for this semantics. Contents. 1 Introduction. 1. 2 The concurrent games model. 2.
Concurrent Games and Full Completeness Paul-Andr´e Melli`es LFCS Univ. of Edinburgh
Samson Abramsky LFCS Univ. of Edinburgh
1 Introduction
Abstract A new concurrent form of game semantics is introduced. This overcomes the problems which had arisen with previous, sequential forms of game semantics in modelling Linear Logic. It also admits an elegant and robust formalization. A Full Completeness Theorem for MultiplicativeAdditive Linear Logic is proved for this semantics.
This paper contains two main contributions:
the introduction of a new form of game semantics, which we call concurrent games. a proof of full completeness of this semantics for Multiplicative-Additive Linear Logic.
We explain the significance of each of these in turn.
Contents 1 Introduction
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2 The concurrent games model
2
3 Proof-structures from strategies 3.1 The multiplicatives . . . . . 3.2 The MLL fragment . . . . 3.3 MALL proof-structures . . . 3.4 Main construction . . . . . . 3.5 Symmetry . . . . . . . . . . 4 Correctness criteria 4.1 MALL proof-nets 4.2 Oriented cycles . 4.3 Acyclicity . . . . 4.4 Connectedness . 5 Main result
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Concurrent games Traditional forms of game semantics which have appeared in logic and computer science have been sequential in format: a play of the game is formalized as a sequence of moves. The key feature of this sequential format is the existence of a global schedule (or polarization): in each (finite) position, it is (exactly) one player’s turn to move1 . This sequential format turns out to have important limitative consequences when we wish to use game semantics to model programs or proofs:
12
There is a modelling limitation. Sequential games can be used to model sequential computation, but do not yield models of parallel computation in a natural way. There is also a mathematical limitation. Despite the evident inherent duality in games (interchange the rˆoles of the two players), sequentiality is an obstacle to modelling logics in a classical format, such as (Classical) Linear Logic. In fact, sequential games have only yielded satisfactory models of fragments of Linear Logic: the Multiplicative fragment in [AJ92b], the Multiplicative-Exponential fragment in [BDER97], and the negative fragment in most other work. This has been sufficient to model -calculusbased programming languages, but is inadequate as a general account. We will illustrate this problem with respect to Andreas Blass’s pioneering work on game semantics below.
We will solve these problems by making a radical departure from all the formal versions of “games in extensive form” used to date in Logic and Computer Science of which we are aware. We shall introduce a “true concurrency” version of games in which the global polarization is abandoned. Local 1 All the games considered in this paper, and in the relevant literature to date, are two-person games.
Research supported by the EPSRC grant “Foundational Structures”.
1
plex notion of proof net it requires [Gir95]. Our proof of Full Completeness is correspondingly lengthy and complex. (We can only give an outline in this extended abstract; a detailed account is given in a draft full paper [AM98].) However, we believe that our methods and results will extend to the exponentials as well, thus yielding a complete analysis of Linear Logic. Independently, Girard has obtained a form of Full Completeness result using a game semantics [Gir98a,b]. His methods, and the details of his results, appear very different to our’s. We are not yet familiar enough with his work to make a detailed comparison. The structure of the rest of the paper is as follows. In Section 2, we present the concurrent games model. In Section 3 we show how MALL proof structures are constructed from strategies, and in Section 4 we outline the proofs of the correctness criteria for these proof structures. Finally, Section 5 gives the main result.
decisions are still polarized, in the sense that they are for one player or the other to make, but globally the two players act in a distributed, asynchronous fashion. At any given time, both may be active in different parts of the “game board”. Moreover, these concurrent games are a strict generalization of the usual sequential games. Remarkably, this generalization and apparent complication can be formalized in a simple and robust way, arguably more elegant and mathematically tractable than the current formalizations of sequential games. In fact, the key ideas of the formalization were present in Abramsky and Jagadeesan’s “New Foundations for the Geometry of Interaction” [AJ92a]. What was missing in that work was the gametheoretic interpretation of the mathematics, which in turn suggests a more intuitive presentation using closure operators. This basic formalization, combined with a suitable form of “Classical Linear Realizability”, which also contains a number of important novel ingredients, leads to a model of the whole system of Classical Linear Logic—and hence to models of Intuitionistic and Classical Logic and various calculi, via the by now well-established interpretations of these systems into Linear Logic [Gir87, DJS97]. But how good is this model?
2 The concurrent games model As a convenient point of departure, we begin with Blass games [Bla92]. These have the form
Y i2I
Full Completeness The usual notion of completeness for a logic is with respect to provability; Full Completeness is with respect to proofs. Let M be a model of the formulas and proofs of a logic L. Typically this means that M is a category with structure of an appropriate kind, such that the formulas of L denote objects of M, proofs in L of entailments A ` B denote morphisms
Ai or
X i2I
Ai
where Q each Ai is (co)inductively a Blass game. The idea is that in i2I Ai , Opponent starts P by playing some i 2 I , and play then proceeds as in Ai . i2I Ai is defined dually, with Player making the initial move i 2 I . Thus these games are trees; at each stage, it is (exactly) one player’s turn to move—the “global schedule”. Play proceeds as a sequence of moves tracing a path through the tree. This commitment to a sequential format forces an interleaving interpretation of the multiplicatives (analogous to the Expansion Theorem of CCS), which leads inexorably to the failure of composition of strategies to be associative, as shown in [AJ92b]. We begin our development of concurrent games with the idea that game trees can be viewed as partial orders, in which x y means that the position y can be reached from the position x by playing some additional moves. This is a natural “information ordering” as in Domain theory [AJ94]. If we add “limit points” corresponding to the infinite branches in the game tree, we obtain a complete partial order D. Viewed in these terms, the construction of sums and products of games as in Blass games can be described as lifted sums as far as the underlying domains of positions are concerned:
[ ]] : [ A] ?! [ B ] ;
and the convertibility of proofs in L with respect to cutelimination is soundly modelled by the equations between morphisms holding in M. We say that M is fully complete for L if for all formulas A, B of L, every morphism f : [ A] ! [ B ] in M is the denotation of some proof of A ` B in L: f = [ ]]. Thus the full completeness of M means that it characterizes “what it is to be a proof in L” in a very strong sense. If M is defined in a syntax-independent way, this is a true semantic characterization of the “space” of proofs spanned by L. The notion of Full Completeness was introduced in [AJ92b], and a Full Completeness theorem was proved for a game semantics of Multiplicative Linear Logic (with the MIX rule). This was followed by a series of papers which established full completeness results for a variety of models with respect to various versions of Multiplicative Linear Logic (MLL) [HO92, BS96, Loa94a, Loa94b]. However, there have been no results for logics beyond the (very weak) multiplicative fragment of Linear Logic. In this paper, we make a first significant extension beyond the multiplicative fragment, by proving that the concurrent games model is fully complete for Multiplicative-Additive Linear Logic (MALL). MALL is already a much richer system than MLL, as shown by the much more sophisticated and com-
DPi2I Ai = DQi2I Ai = (
X i2I
DAi )? :
We shall represent strategies as functions on these domains of positions: f : D ! D, where f (x) is the position obtained from x by extending it with whatever moves the strategy makes in that position. It is then immediate that f (x) w x. Moreover, those positions where f has no 2
moves to make (e.g. because “it is not its turn”) are exactly the fixpoints of f . In the usual way, we require computationally reasonable strategies to be monotonic and continuous. Finally, as a useful normalizing condition, we require strategies to be idempotent: f 2 = f . To understand this, consider f applied to f (x). The only moves made in f (x) which were not already made in x are those made by f itself: f (x) contains no more information supplied by the Opponent (i.e. the environment) than x did. Hence anything f decides to do at f (x) it should have already been able to decide to do at x, and we require that f (f (x)) = f (x). Of course, this allows several moves to be made in a block by a player. This possibility already exists in Blass games, e.g. Opponent must move twice initially in
This corresponds exactly to the idea that Player must first wait for Opponent to choose an i 2 I , and then plays according to some strategy for Ai .
SQ i2I Ai = fini () j 2 SA i ; i 2 I g
where
)(?) = ini ((?)) )(ini(x)) = ini ((x)) )(inj (x)) = >; (i 6= j ):
ini( ini( ini(
Again, this corresponds to the idea that a strategy for Opponent will firstly choose some i, then play according to a strategy for Opponent in Ai . Note that the last case above covers “unreachable states” for Pini(). WeQcan then)?define sums by De Morgan duality: i2I Ai = ( i2I A? i . For the strategies for tensor, we define
(A B ) (C D): An important point is that strategies may not be welldefined at all positions. In general there are some positions that can never be reached by following that strategy. To mesh with the requirement that strategies are increasing functions, we adjoin a top element to the domain of positions D, writing this as D> . We represent f being undefined at x by f (x) = >. In summary, strategies (for either player) are represented as continuous closure operators on D> , which under modest assumptions on D (bounded completeness) is a complete lattice. We can completely specify a game as a structure (D; S; S ), where D is the domain of positions, S is the set of legal strategies for Player, and S is the set of legal counter-strategies, i.e. strategies for Opponent. This strictly generalizes Blass games: for such games, D is the domain of finite and infinite sequences under the prefix ordering corresponding to the paths through the game tree, and the conventions about who is to play are formalized by saying that for all x 2 D, either (x) = x for all 2 S (it is Opponent’s turn to move), or (x) = x for all 2 S (it is Player’s turn). However, this is a very special case of our general setting; and we will overcome the problems with Blass games precisely by allowing situations in which both players can move. To do this, we shall interpret the game boards for the multiplicatives differently to Blass: by a true concurrency rather than an interleaving representation. In our setting, this is simply a matter of defining
SA B = f j 2 SA ; 2 SB g where (x; y) = ((x); (y)). (This is really smash product with respect to >; if either (x) = > or (y) = >, then the result is >.) This exactly captures the idea of informational independence between Player’s actions in A and in B (cf. [AJ92b]). How Player moves in A depends only on the information available in A, and similarly for B . In order to define the counter-strategies for the Tensor (and hence the strategies for Par and Linear implication, and eventually the morphisms in the category of concurrent games), we introduce the most important feature of our formalization: the elegant treatment it affords of composition of strategies. Suppose firstly that 2 S and 2 S in a game (D; S; S ). How do we play off against ? We define
W k hj i = Y W ( ) = k k2! ( ) (?) = ( ) (?) = Y( ): k2!
The fact that these two least fixpoints coincide follows easily from the fact that and are continuous closures; in fact, this is a special case of the construction of the join of two closure operators. Thus hj i 2 D is the position we reach as a result of playing against . The equality of the two formulas above also shows that this is independent of all questions about “who starts”. Now given closure operators on (D E )> and on (E F )> , we want to “compose” them to get a closure ; on (D F )> . We define this as follows:
DA B = DA................. B = DA DB ; the cartesian product of domains. How should the Linear Logic connectives be interpreted , as acting on sets of strategies? We define SA? = SA SA? = SA , which corresponds to interchanging the rˆoles of Player and Opponent. Clearly then A?? = A. For products, we define
; (x; z ) = (1 (x; y); 2 (y; z )) where
y = h2 (x; ?)j1 (?; z )i: That is, given input in D and F , we play and off against each other in E relative to this input, and obtain their ex-
SQi2I Ai = fhiji 2 I i j 8i 2 I; i 2 SAi g
ternal response taking into account their interaction with each other. In particular, if is a closure on (D E )> , it induces an “action” taking closures on D> to closures on E > , 7! ; , and a “coaction” taking closures on E > back to
where
hi ji 2 I i(?) = ? hi ji 2 I i(ini (x)) = ini(i (x)): 3
closures on D> , 7! ; . E.g. where y = h2 (x; ?)j i. Now we can define
than taking a game to be (D; S; S ), where S , S are simply predicates picking out the sets of strategies and counterstrategies, we will take games of the form (D; E; E ), where E and E are partial equivalence relations on the (stable) closure operators on D> . We simply adapt the definitions given above for the Linear connectives from unary predicates to binary relations, and check that they preserve symmetry and transitivity. If A is a game of this form, we write : A to mean that EA . We are now in a position to define the category R of concurrent games. Objects are structures (D; E; E ) of a domain and two partial equivalence relations on stable closure operators, as already explained, subject to the following condition: if : E and : E have the same maximal fixpoints, then E , and similarly for E . A morphism from A to B is a closure operator such that : A B . Composition is defined as above. Identities are given by:
; (x) = 1 (x; y),
SA B = f : (DA DB )> j 8 2 SA : ; 2 SB ^ 8 2 SB : ; 2 SA g: Again, by De Morgan duality,
A................. B = (A? B ? )? ; A ( B = A? ................. B: In particular,
SA(B = f : (DA DB )> j 8 2 SA : ; 2 SB ^ 8 2 SB : ; 2 SA g:
(
This is a symmetric, “classical” version of the familiar logical relations condition: ’s action carries Player strategies on A to Player strategies on B , and its coaction carries counter-strategies on B to counter-strategies on A. We round out our account by defining the units, which all have the one-point poset as their underlying domain. For the tensor unit 1, S1 = f(? 7! ?)g, while S1 = f(? 7! ?); (? 7! >)g. For the unit for Plus (i.e. the initial object) we have S0 = ?, S0 = f(? 7! >)g. The exponentials can also be defined in our model (the basic ideas are along the lines of [AJ92a]), but for lack of space we will refrain from doing so. The reader will really understand our model by checking that composition is associative, and seeing how the problems with Blass games simply do not arise in our setting. We are now almost ready to define our category of concurrent games. Two further refinements are needed.
x; y) = (x _ y; x _ y):
idA (
They can be understood as “symmetric, bidirectional copycat strategies” as in [AJ92b]. We can then define E as the “extensional quotient of R”, in which a morphism from A to B is a partial equivalence class of EA B .
(
Proposition 2.1 its and colimits.
E is a ?-autonomous category with all lim-
The model of MALL The fragment of Linear Logic we will consider in this paper consists of formulas built from propositional atoms X and their negations X ? with the ....... binary connectives (Tensor), ........... (Par), & (With) and (Plus). We refer to this fragment as MALL. A sequent ` A1 ; : : :; Ak will be interpreted as the formula A1 .................. .................. Ak . Since proofs of propositional formulas should be uniform over all substitution instances, we will treat propositional atoms as variables, so that in effect we are viewing propositional logic as the 1 fragment of second-order propositional logic. In a by-now standard fashion [BFSS88], a MALL formula in the propositional atoms X1 ; : : :; Xn can be interpreted as a mixed-variance functor
Stability Rather than taking all continuous closure operators as possible strategies, we will impose the domaintheoretic condition of stability [AC98]. (This will turn out to be important for our proof of Full Completeness, although we don’t know if it is a necessary condition; it is deeply related to the “monomial condition” in [Gir95].) But what does it mean for a closure operator to be stable? Define the domain dom of a closure operator on D to be the set of all x 2 D such that (x) 6= >. An output function for is a continuous function f : D ! D such that, for all x in the domain of , (x) = x _ f (x). We say that is stable if it has a stable output function. (The link between concurrent games and the NFGOI [AJ92a] is here; the output functions for the strategies interpreting proofs will be exactly the denotations of proofs in [AJ92a]. Moreover, composition of closure operators is “tracked” by the composition of the corresponding output functions, defined as in [AJ92a].) We shall henceforth restrict ourselves to stable closure operators. This needs some conditions on the underlying domains: it suffices to assume that they are bounded complete, and distributive in a suitable sense. For details, see [AC98].
F : (Rop R)n ?! R: The set of such functors will form the objects of a category M[n], and collectively we will obtain an indexed ?autonomous category M with all limits and colimits. This will provide the right algebraic structure to model propositional MALL. We define the morphisms in M[n] in two stages. Firstly, a dinatural family for a functor F in M[n] is a family of strategies (A~ )A~ 2Rn indexed by n-tuples of games such ~ , A~ EF (A~ ) A~ , and moreover for all tuples of that, for all A strategies ~
: A~ ! B~ ,
A~ ; F (idA~ ;~ ) = B~ ; F (~ ; idB~ ): We define a per EF for each such functor by ~ A~ EF (A~) A~ ; (A~ )EF ( A~ ) () 8A:
Extensionality To ensure that we get a genuine model of Linear Logic in which all the required equations hold, we adapt classical ideas from realizability to our setting. Rather 4
and similarly for EF . We then define morphisms in M[n] as partial equivalence classes of families of strategies, generalizing what we did for E . Defining the various connectives and constructions for the Linear types pointwise on the functors and families in M[n], we obtain a ?-autonomous category with all limits and colimits. There is a fine point here; as is well known, dinaturality is not in general preserved by composition. However, after we have proved Full Completeness—which will not use the assumption that dinaturality is preserved by composition (!!)— then by pulling back to syntax it will be easy to see that (for definable functors) closure under composition does hold, and hence each M[n] (restricted to definable functors) is indeed a category. We could avoid this logical detour by using a stronger property than dinaturality, namely Reynolds-style relational parametricity [BFSS88], for which closure under composition can be proved directly; however, this would complicate the description of the model, and dinaturality is sufficient to prove Full Completeness. Another hypothesis we will need to prove Full Completeness is that the closure operators are symmetric, in the sense of having output functions f satisfying f 3 = f . Again, the only problem with this condition is proving closure under composition; and again, once Full Completeness is proved, we obtain closure under composition as a corollary.
By restricting to unfolded Petri nets, the accessibility relation becomes an order. We call root any place p of the unfolded Petri net N such that p does not appear in any p ost(t), t 2 T . We single out the state ?N P consisting of all root places of N , and call a state x of N accessible when it is accessible from that special state ?N . Now, the accessibility relation N defines a domain DN on the set of accessible states of N . We call Petri game any game A whose domain of position DA is of the form DN for an unfolded Petri net N , and call N the board of A. Intuitively, a Petri game A with board N has ?N as least position, and every state x of N obtained after a sequence tn x as position. t1 ::: ?! of transition ?N ?! Now, we show how to construct the board of A B and A&B given the boards NA and NB of A and B . For the multiplicatives, NA B = NA................. B is the Petri net obtained by juxtaposing the Petri nets NA and NB . In particular, ?NA B is the union of ?NA and ?NB . NA
NA
B
For the additives, NA&B = NAB is the juxtaposition of NA and NB with a new place p and two new transitions tleft and tright with pre-conditions:
p re(tleft ) = p re(tright ) = fpg
and post-conditions:
Petri games We would like to describe very concretely the (dinatural) strategy [ ] interpreting a cut-free MALL proof of the formula ? in the game-model. The first step in that direction is to see each position of a game as the state of a Petri net. Of course, this is only possible on very special games, what we call Petri games and define below. Formally, a Petri net is a quadruple N = (P; T; p re; p ost) where P is the set of places and T the set of transitions. To every transition t is associated two nonempty sets p re(t) P of pre-conditions p ost(t) P of post-conditions. A state in a Petri net N is a subset of P . t States are related by transition relations: We write x ?! y when x = z + p re(t) and y = z + p ost(t) for some state z and transition t, where + means disjoint union. A state y is accessible from x, what we write x N y, when there exists a sequence of transitions t1 ; :::; tn such that tn y. t1 ?! x ?! A Petri net is called unfolded when
p ost(tleft ) = ?NA p ost(tright ) = ?NB Graphically NA
NB
NA & B
p *
For instance, given two Petri games A1 and A2 with boards NA1 and NA2 : A1
A2
the Petri game B has board:
1. p re(t) is a singleton for every t 2 T , 2. p ost(t1 ) \ p ost(t2 ) t1; t2 2 T .
NB
. ...... ? = ((A1 A2)&(A1 A2 )).............. A? 1 ...... A2 .....
.....
6= ; implies t1 = t2 , for every
Graphically, this means that the patterns (1) and (2) are forbidden, in other words that the Petri net N looks like a forest.
(( A
1
A2 ) & ( A 1
A2 ) )
&
(2)
A1
&
(1)
A2
where the least position ?B is indicated by the three tokens. 5
A2 ))................ A?1 ................ A?2 , which plays as below:
The concrete interpretation Every formula ? on the atoms X1 ; :::; Xn, is interpreted as a mixed-variance functor
[ ?]] : (Rop R)n ?! R: ~ A~ ) and every proof of ? defines a strategy [ ] A~ of ?(A; ~ = A1 ; :::; An of games in R. In the particfor every tuple A ular case of Petri games A1 ; :::; An, the strategy [ ] A~ plays ~ A~ ) and defined on the board N?(A; ~ A~ ) associated to ?(A; from the boards NAi ’s of the Ai ’s as in the previous paragraph. Let us describe briefly how this strategy plays on N?(A;~ A~) . The first step is to consider that the proof is translated as a proof-net with additive boxes, see [Gir87]. Then, to observe that every place p of N?(A; ~ A~ ) is associated either to an additive link L of ?, or to the residual of a place among the NAi ’s. In the first case, each of the valuation “left” and L “right” of L corresponds to a transition tL left and tright with pre-condition fpg. By definition, every position x 2 D?(A; ~ A~ ) induces a transition sequence
tn x t1 ?! ?N?(A;~ A~) ?!
3 Proof-structures from strategies ? ) is total when EA and E ? A game A = (DA ; EA; EA A are nonempty and hj i is maximal for every : A and : A? . The class of total games is closed under all the ....... MALL constructions: , ........... , , & and (?)? . Observe that ....... every strategy of A B is also a strategy of A .......... B when A and B are total games. A strategy : A is civil when for every : A? , hj i is ? ) is civil when it contains not >. A game A = (DA ; EA; EA a civil strategy and a civil counter-strategy. Again, the class of civil games is closed under all the MALL constructions. Observe that every civil game is inhabited by a strategy and a counterstrategy. Total games are examples of civil games. C denotes the full subcategory of civil games in R. Let ? be a MALL formula interpreted as a functor in M[n]. Every valuation v of the &-links of ? defines a MLL formula ?v , which may be interpreted in turn as a functor in M[n]. We leave to the reader the inductive definition of the associated natural transformation
(1)
which may not be unique, yet induces a partial valuation vx of the &-links of ? which does not depend on the sequence (1): The valuation valuates the &-links L left or right when L one of the transition tL left or tright appears among the ti ’s in the sequence (1). Once this partial valuation vx of the &-link of ? is computed from x, a new proof-net x with additive boxes is constructed by removing every additive box of whose principal door is a &-formula valuated by vx . At this point, the strategy = [ ] A~ shall determine its answer (x) from the information it reads in x, considering every remaining additive box in x as a “black box” with no possibility to peep inside. Observe that the proof-net x verifies the two following fundamental properties:
proj? v
using the projection maps
A&B ?! B in R.
1 : A&B ?! A and 2 :
One important step of the construction, see lemma 3.3, requires us to restrict dinaturality to the category C of civil games. So, given ? in M[n], define a C -dinatural family for ? as a family of strategies (A~ )A~ 2Cn indexed by tuples A~ = A1 ; :::; An of civil games such that, for all A~ 2 C n , ~ A~ ), and moreover such that A~ : ?(A;
1. every -link visible in x (=not in a black box) is either left or right,
A~ ; ?(idA~ ;~ ) = B~ ; ?(~ ; idB~ ): for all tuple ~ = 1 ; :::; n of morphisms i : Aj ?! Bj in
2. every visible atom of x is related to another visible atom with an axiom link. Now, the position (x) is defined as the least position above x in D?(A; ~ A~) such that:
? : ? ?! v
C.
y
A C -dinatural family for ? is uniform when the closure operator A~ only depends on the domain DAj for 1 j n. By the following lemma, every dinatural family defines a uniform C -dinatural family.
1. the valuation of every -link in x appears in y,
2. given an axiom link, the position on one side is the same as the position on the other side (concurrent copy-cat).
~a Lemma 3.1 (uniformity) Let a dinatural family and A tuple A1 ; :::; An of games. The strategy A~ does not depend ? , only on the domains DA . on the pers EAi and EA i i
For instance, the only proof of formula
P ROOF As in [Loa94b].
.. .. .. .. ((X1 X2 )&(X1 X2 ))............ X1? .............. X2?
(resp. Yj? ) in a MALL formula ?(X1 ; Y1; :::; Xn; Yn), and tuples A~ , B~ of games, we associate to any x 2 Aj (resp. x 2 Bj ) the corresponding ele? ? ?i ment (Xji 7! x)A; ~ B~ ) . ~ B~ (resp. ((Yj ) 7! x)A; ~ B~ ) in D?(A; Given an instance of
is interpreted as a dinatural strategy [ ] whose instance = [ ] A~ on the Petri games A~ = A1; A2 (see previous paragraph) is a strategy of B = ((A1 A2)&(A1
6
Xj
i : DB> (i) , we associate to every partial valuation v of the
To do so, we introduce first the notion of annotated formula. It is a pair (?; index) consisting of a MALL formula ? and a one-to-one function index associating an integer i to every occurrence of Xj and Yj? in ?. An annotated formula (?; index) is best seen as a formula ? where occurrences of Xj (resp. Yj? ) associated to i are replaced by Xji (resp. (Yj? )i ). Now, given a literal Xji in an annotated formula ?, and A~ , B~ two tuples of games, we associate to x 2 DAj an ? element (Xji 7! x)A; ~ B~ ) by induction on ? (the ~ B~ of D?(A; ? i definition is similar for (Yj ) )):
X i0
j0 (Xji 7! x)A; ~ B~ =
x
?
&-links of ? a closure operator
> d ; e?v : D?( ~ B~ ) A; as follows:
i
and ?2 :
d ; e?v11 d ; e?v22 d ; ev?1 ?2 : ? 7! ?; inlx 7! inld ; e?v11 x; inrx 7! inrd ; e? v22 x:
if (i; j ) = (i0 ; j 0) otherwise
......
=
=
(
When v does not assign a value to the root &-link:
?1
i ((Xji 7! x)A; ~ B~ ; ?) if Xj is in ?1 ? 2 i (?; (Xji 7! x)A; ~ B~ ) if Xj is in ?2
1 ?2 i (Xji 7! x)?A; ~ B~ = (Xj
..........
d ; e?v 1 ......... ?2 = d ; ev?1 ?2
is equal to the smash product
. ..... 1 ?2 i 7! x)?1 ........ ?2 = ( X (Xji 7! x)?A; ~ B~ ~ B~ j A;
(
?i
d ; eX; j = i and d ; e(;Yj ) = i , Letting v1 and v2 be the respective restrictions of v to ?1
d ; ev?1 &?2 : (x; y) 7! (x; y) When v assigns the value left to the root &-link:
?1&?2
7! x)A;~ B~
d ; ev?1 &?2 : ? 7! inld ; e?v11 ?; inlx 7! inld ; e? v11 x; inrx 7! >:
1 i (inl(Xji 7! x)?A; ~ B~ )? if Xj is in ?1 ? 2 i (inr(Xji 7! x)A; ~ B~ )? if Xj is in ?2
When v assigns the value right to the root &-link:
(Xji 7! x)?A; ~ B~ is easily adapted to asso? of D ciate a prime element (L 7! left)A; ~ B~ ~ B~ ) to the ?(A; valuation (L 7! left) of an additive link L of ?. The base
d ; ev?1 &?2 : ? 7! inrd ; e?v22 ?; inlx 7! >; inrx 7! inrd ; e? v22 x:
The definition of
The main reason for the construction is that given n-tuples A~ and B~ of total games, a total valuation v, and strategies i : A?(i) , i : B ?(i) for all 1 i m, the closure ~ B~ )? . operator d ; e? is a strategy of ?(A;
case of the induction is:
?1 L?2
(L 7! left)A; ~ B~ = (inl?)? ? The prime element (L 7! right)A; ~ B~ corresponding to (L 7! right) is defined similarly. Observe that contrary ? ? and to (Xji 7! x)A; ~ B~ ~ B~ , the prime elements (L 7! left)A; ? (L 7! right)A; ~ B~ do not depend on the annotation of ?.
v
3.1
The multiplicatives
Starting with the case of a multiplicative formula ?, we construct the multiplicative proof-structure associated to the C -dinatural family . This construction does not require any of the game-theoretic properties yet, and we simply follow the steps of R. Loader who carried out the construction in the relational model, see [Loa94b].
We fix in sections 3.1, 3.2, 3.4 and 3.5: a formula ? built from......X ; :::; Xn and Y1? ; :::; Yn? .... 1 . . with the connectives , . .... , , &, a uniform C -dinatural family for the functor ? in M[n].
Lemma 3.2 There exists a fixpoint-free involution : ~ = that for any tuple A A1 ; :::; An of civil games, the set of maximal fixpoints of A~ is
Let m be the number of literal occurrences in ?. All through this section, we consider the formula ? annotated in f1; :::; mg in such a way that each index i appears to the left of the index i + 1. Given this “canonical” annotation, two functions : f1; :::; mg ! f1; :::; ng and : f1; :::; mg ! f+; ?g indicate which literal the index i annotates: X i (i) when (i) = +, and (Y ?(i) )i when (i) = ?. We write P (resp. N ) for the set of indices i 2 f1; :::; mg occurring positively: (i) = + (resp. negatively: (i) = ?) in ?. For each j = 1; :::; n, we write Nj = fi 2 N j i = j g, and Pj = fi 2 P j i = j g. Given the annotated formula ?, m-tuples ~ = 1 ; :::; m > and and ~ = 1 ; :::; m of closure operators i : DA (i)
f1; :::; mg ! f1; :::; mg such
f(x1; :::; xm) j xi = x (i)
3.2
for all
1 i mg
The MLL fragment
We suppose now that the formula ? is constructed from and . We associate a &-free proof-structure to the uniform C -dinatural family by reducing that problem to the multiplicative case.
X1 ; :::; Xn and Y1? ; :::; Yn? with the connectives , .................
7
Let us observe first that a total valuation v of the -links of ? is also a valuation w of the &-links of ?? . Thus, such a valuation v induces a functor ?v : (C op C )n ?! C and a natural transformation inj? v : ?v ?! ? defined as:
?v = ((?? )w )?
inj? v
if L is 1 , then w(L) = w(A), if L is 2 , then w(L) = w(B ), if L is a &-link, then w(A) = w(L):pL w(B ) = w(L)::pL
? = (proj?w )?
Our condition 4 is equivalent to the requirement appearing in [Gir95] that for any element of the boolean algebra generated by the weights occurring in , and any &-link L, w::w(L) does not depend on pL.
The main result of the section (factorization) implies with lemma 3.2 that the uniform C -dinatural family describes a MLL proof-structure. Lemma 3.3 (factorization) There exists a valuation the -link of ? such that the morphism A~ factors as
A~
1
/
~ A~ ) ?v (A;
(inj?v )A; ~ A~ /
3.4
v of
We associate an event structure EVENT ? to the formula
~ A~ ) (2) ?(A;
quence of formulas and links. The second step is to replace every subtree (A; B ) by two subtrees 1 (A) and 2(B ) connected to the same ancestor. The resulting tree TREE ? is labelled with
; ................... ; 1 ; 2; &-links and MALL formulas (the literals annotated). A node in TREE ? means either a link or a formula in the tree. Nodes are ordered by the tree-nesting ordering. An axiom link in TREE ? is a 2-element set fXji1 ; (Yj? )i2 g of annotated literals in TREE? . An additive boundary is a &-link or a formula A B in TREE ? . The event structure EVENT ? is defined as follows: 1. the events are the nodes and axiom links of TREE ? ,
P ROOF Let 4 = 4? be the total game with domain f?g. ~ ;4 ~ ) off against the counter-strategy We play 4 ~ : ?(4 (x 7! x). The maximal element we obtain as a result indicates which valuation v of the -links in ? to choose. We prove factorization (2) at every instance by establishing that (inj?v )A; ~ B~ is a split mono in the case of tuples A~ and B~ of civil games (using that every civil game is inhabited). This is where the notion of C -dinaturality appears in the proof.
F
MALL proof-structures
2. We recall the definition of a MALL proof-structure in [Gir95]. A proof-structure consists of: 1. a set of formula occurrences,
3.
2. a set of links; each of these links takes its premise(s) and conclusion(s) among the formula occurrences of ;
3. for each formula occurrence A of , a weight w(A), i.e. a non-zero element of the boolean algebra generated by the eigenweights p1 ; :::; pn of the &-links of
Observe that any maximal state in EVENT ? describes a multiplicative proof-structure. The equivalence relation relates two nodes of TREE ? when there is a path between them that does not cross an additive boundary. More formally, it is the least equivalence relation on the events of EVENT ? such that:
satisfying the following conditions: 1. each formula occurrence is the premise of at most one link and the conclusion of at least one link
L A when A is a formula premise of L not a &-link,
2. if A is a conclusion of , then w(A) = 1,
3. if w is any weight occurring in , then w is a monomial of eigenweights and negations of eigenweights,
5.
6.
w(A) =
e e0 when e and e0 are nodes and e nests e0 , or when e is a node and e0 is an axiom link containing a literal nested by e, or e and e0 are the same axiom link. e#e0 when e and e0 are nodes and e ^ e0 is an additive boundary, when e is a node and e0 is an axiom link containing a literal Xji such that e#Xji , when e and e0 are axiom links whose intersection e \ e0 is singleton, or when e and e0 are axiom links containing incompatible literals.
4. if pL:w is any weight occurring in , then w
Main construction
?(X1 ; Y1; :::; Xn; Yn). The first step is to consider ? canon........ ically annotated. So, ? is a tree of ; ; .......... ; &-links and MALL formulas, with the formula ? at its root, the literals Xji and (Yj? )i at its leaves, and every path an alternating se-
~ of civil games. Moreover, the family at every instance A for ?v in (2) is uniform and C -dinatural.
3.3
and
A L when A is the conclusion of a , ................. L.
w(L),
or &-link
In particular, every axiom link is the only element of its class. Let Zbe the total game associated based on the flat domain of integers with strategies all (? 7! n), n 2 Z. Let Z2 be Z................ Z?. This instance Z2 will play a particular role in the construction of the proof-structure corresponding to . Every prime element of the form
P
F
w(L), the sum being taken over the set of links with conclusion A. It is required here that the sum is disjoint. w(L) 6= 0, moreover, if L is any non-axiom link, with premises A (or) B then
if L is any or ................ , then w(L) = w(A) = w(B ),
(Xji1 8
ZZ
7! (i2 ; ?))~?2;~ 2
or
((Yj? )i1
ZZ
7! (?; i2))~?2;~ 2
ZZ ZZ D ZZ v D ZZ
is called an input of D?(~ 2;~ 2) and every prime element of the form ? ? (Xji1 7! (?; i2))~ 2;~ 2 or ((Yj? )i1 7! (i2 ; ?))~ 2;~ 2
2. the weight w(e; v) is computed from the valuation v as follows:
an output of ?(~ 2;~ 2) . Given a valuation of the &-links in ?, the closure op> erator v on ?( ~ ;~ ) is defined as
where the first product is taken over the &-links (L; v1) such that v extends v1 + (L 7! left), and the second product over the &-links (L; v2 ) such that v extends v2 + (L 7! right).
ZZ
w(e; v) =
2 2
v = d ; e?v
Z
Lemma 3.5
3.5
ZZ
= &, the prime (L 7! left)~? 2;~ 2 is sent
Z
ZZ
ZZ
ZZ ZZ
Some graph-theoretic notation will be useful here. A graph is a pair G = (V; E ) of sets such that every element of E is a 2-element subset of V . The elements of V are the vertices of G, the elements of E are its edges. One often writes AB for the edge fA; B g. A path p is a sequence A1 :::An of distinct vertices of G such that Ai Ai+1 is an edge for 1 i n ? 1. A cycle is a sequence A1 :::An of vertices with n 3 such that A1 = An, A1:::An?1 is a path and An?1A1 is an edge.
4.1
Z
MALL proof-nets
Let ' be a valuation of the &-links of a proof-structure . ' induces a function (still denoted ') from the weights of to f0; 1g. The slice '() is obtained by restricting to the formulas A verifying '(w(A)) = 1, with the obvious modification for the remaining &-links: only one premise is
: v 7! EV+? h~ 2j v i
F The proof-structure is constructed as follows: 1. its formula occurrences (resp. links) are the elements 2 + of tr(PRF + ), the trace of the stable function PRF , of the form (e; v) where e is a formula occurrence (resp. a link) of EVENT ? . Every node (e; v) is labelled as e, 2 It
Z
? : v 7! EV? h~ j v i ? 2
PRF
4 Correctness criteria
partial valuation v of the &-links in ? a state in EVENT ? : PRF +
Symmetry
leads to the same proof-structure. This is the case when the instance ~ 2 of is symmetric. The proof is not straightforward and we omit it for lack of space.
premise, for L = , the prime (L 7! left)~? 2;~ 2 is sent to the -class of the corresponding 1 -link, and the prime (L 7! right)~?2;~ 2 is sent to the -class of the corresponding 2 -link, the output prime (Xji1 7! (?; i2))~? 2;~ 2 for an index i2 2 Nj of some literal (Yj? )i2 is sent by EV+? to the axiom link fXji1 ; (Yj? )i2 g, the output prime ((Yj? )i1 7! (i2 ; ?))~? 2;~ 2 for an index i2 2 Pj of some literal Xji2 is sent by EV?? to the axiom link f(Yj? )i1 ; Xji2 g, any other prime is sent to the empty state. The following (stable) function PRF + associates to any
defines a MALL proof-structure.
As mentioned earlier, a closure operator : D> is called symmetric when one of its output function f : D ! D is stable and verifies f 3 = f . By extension, a dinatural family ( A~ )A~ 2Rn for the functor ? in M[n] is called symmetric ~ 2 Rn . when A~ is symmetric at every instance A Before embarking to the proof that is a proof-net, it is vital to establish that the alternative definition of as the trace of
-class? of L’s first premise, andZZ the prime (L 7! right)~Z to the
-class of L ’s second 2;~Z 2
for L to the
:p(L;v2)
For lack of space, we omit the lengthy verification that
P ROOF The proof relies heavily on the connection between our model and NFGOI, the definition of stable closure operators, the fact that the category of domains and stable maps is cartesian-closed, and finally the choice of the i’s and i ’s. ? We introduce two linear functions EV+ ? and EV? that associate a state of EVENT ? to any element of D?(~ 2;~ 2) . These functions are defined as follows: ? is sent to the union of the -class of the conclusion,
Y
4. a formula occurrence (A; v1 ) is the conclusion of a link (L; v2 ) when A is a conclusion of L in ? and v2 extends v1 .
i : (x; y) 7! (x _ i; y) i : (x; y) 7! (x; y _ i) Note that v is a counter-strategy of ?(~ Z2; ~Z2) when the valuation v is total. Moreover, Lemma 3.4 (stability) The map v 7! h~ 2j v i from valuations to D?(~ 2;~ 2) is stable.
F
p(L;v1)
3. a formula occurrence (A; v1 ) is the premise of a link (L; v2 ) when A is a premise of L in ? and v1 extends v2 ,
(here ? is considered canonically annotated) with the strategies i : Z2 and i : Z? 2 for all 1 i m:
ZZ
Y
present. A switching of a proof-structure consists in 1. the choice of a valuation 'S for ,
is here that stability is used in our proof.
9
2. the selection of a choice S (L) 2 fl; rg for all ......... -links of 'S ()
P ROOF (sketch) Every switching S defines a normal switching S0 with the same valuation and the same choice ........ of ........ -links. By hypothesis, S0 is a tree. As S and S0 have the same number of vertices and edges, a simple graph-theoretic argument tells that:
........
3. the selection for each &-link L of 'S () of an occurrence S (L), the jump of L, depending on pL in 'S (). There is always a normal choice of jump for L, namely the premise A of L such that 'S (w(A)) = 1. Any other choice is called proper.
S connected () S acyclic Suppose that is not a proof-net, and let m be the minimum number of proper jumps among the S containing a
A normal switching is a switching with no proper jump.
cycle. By hypothesis, this number m is at least 1. We show that every cycle of a switching S with m proper jumps is oriented. Let C be a cycle in S . Suppose that C is not oriented. There are two &-links L; M and two occurrences A; B defining proper jumps LA and MB , and a path ApB contained in C = LApBMqL such that ApB contains no proper jump. Call LC and MD the normal jumps of L and M respectively. We write SL for the switching S where L is switched to C instead of A, SM for S where M is switched to D instead of B , and SLM for S where L and M are switched to C and D instead of A and B . By definition of m, SL , SM and SLM define trees SL , SM and T = SLM . In T , the occurrence A is in TLC and B in TMD by lemma 4.3. By definition of a path BMqLA, the path MqL and the proper jumps LA and MB are disjoint. Thus, MqL is the only path from M to L in T . Because MqL, LC and MD are disjoint, the two subtrees TLC and TMD are disjoint themselves. So, the graph S obtained from T by replacing the edges LC and MD by LA and MB is still a tree. This yields the required contradiction with the definition of S . The same kind of argument is used to prove that C may be chosen canonical in some switching S .
Definition 4.1 Let S be a switching of a proof-structure . We define the graph S as follows: 1. the vertices of 'S (),
S are the occurrences and links of
2. for all links of 'S (), we draw an edge between the link and each of its conclusions, 3. for all i -links of 'S (), we draw an edge between the link and its premise, 4. for all -links of 'S (), we draw an edge between the link and its left premise, and between the link and its right premise, ........
5. for all ......... -links L of 'S (), we draw an edge between the link and the premise (left or right) selected by S (L), 6. for all &-links L of 'S (), we draw an edge between the link and the jump S (L) selected by S .
A proof-net is a proof-structure such that for all switchings S , the induced graph is acyclic and connected.
4.2
Oriented cycles
Lemma 4.4 Assume that S contains a cycle for a normal switching S . Then there exists a normal switching S 0 such that S 0 contains a flat (hence canonical and oriented) cycle.
Every pair of distinct vertices A; B in a tree T defines a maximal subtree containing A, not containing B . This maximal subtree is denoted TBA . Lemma 4.2 Let T be a tree with an edge LA and a vertex B . Let T 0 be the tree T where the edge LA is erased and replaced by an edge LB , not in T . Then T 0 is a tree if and only if B is in the subtree TLA of T .
4.3
Acyclicity
This excursion into graph-theory shows that a proofstructure is a proof-net when: 1. acyclicity: whatever the switching S of , there is no oriented and canonical cycle in S , 2. connectedness: the graph S is connected when the switching S is normal.
We call normal (resp. proper) jump an edge LA in S where L is a &-link and A is the normal (resp. proper) jump of L. An oriented cycle in S is a cycle in which there is no path LApBM where L, M are &-links and LA, MB are proper jumps. A flat cycle is a cycle in which there is no path ApB where A and B are nested and p is not the path of that nesting in TREE ?. A cycle C is canonical when it is flat and every proper jump in C is a jump to an axiom link.
We prove in this section and the next that the proof-structure defined in section 3 satisfies these two conditions: acyclicity by exhibiting a deadlock on the assumption of a cycle, connectedness by propagating the “error” value > around the net on the assumption that there is more than one connected component.
Lemma 4.3 Let be a proof-structure such that every normal switching of defines a tree. Then, one of the following cases occurs:
We fix in subsections 4.3 and 4.4: a formula ? built from......X ; :::; Xn and Y1? ; :::; Yn? .... 1 . . . . . . . with the connectives , . , , &, a symmetric dinatural family for the functor ? in M[n].
is a proof-net there is a switching S such that S contains an oriented and canonical cycle.
10
the totality of ?(~ Z2; ~Z2). Calling T the set f 0 ; :::; N ?1g the strategy is defined as d ; e? v;T where the i’s and i ’s are defined as in section 3.4. Suppose that x is an element of D?(~ 2;~ 2) verifying
Given two total games A and B , and two strategies : A? >, and : B ? such that is stable as a function DA ! DA ? we construct the strategy ( ) : (A B ) as follows:
ZZ
8 if x = > or x = > (x; y) 7! (x; y) if x 6= > is not maximal : (x; y) otherwise Let T be any set of disjoint -links in the formula ?, v a total valuation of the &-nodes of ?, strategies i : Ai and i : Bi? stable as functions DAi ! DA>i and DBi ! DB>i for all 1 i m. The inductive definition of the strategy d ; e?v;T : ?(A; B )?
Observe that every p2k+1 is an output and every p2k an input or the prime associated to the valuation of a &-link of ?. This implies that for all 0 k N ? 1, p2k+1 6 x and p2k 6 ~ 2x. We then deduce from the construction of and the definition of that:
Z
8k 2 f0; :::; 2N ? 1g;
? is similar to the definition of d ; e? v : ?(A; B ) , except for the case of a -link in T : letting v1 and v2 (resp. T1 and T2) be the respective restrictions of v (resp. of S ) to ?1 and ?2: ? = d ; ev;T
(
d ; e?v11;T1 d ; e?v22;T2 d ; e?v11;T1 d ; e?v22;T2
For instance, d ; e? v
Z
4.4
if is in T otherwise
ax
ax &
Let us label k each -link appearing as a “rebouncing” tensor in the figure: 0, ..., N ?1 . We have to consider two possible cases for any k 2 f0; :::; N ? 1g: ax
ax &L
k
k+ 1
k
k+ 1
(3) In the first case, each of the literals Xji1 and (Yj? )i2 forming the axiom link is nested by one of k or k+1 . We define p2k (resp. p2k+1) as the input prime (resp. output prime) of the literal nested by k (resp. k+1 ). In the second case, k nests the &-link L, k+1 nests one of the literals forming the axiom link, and the jump from L to the axiom link connects the cycle. We define p2k+1 as the output prime of the literal nested by k+1 , and p2k as the ? prime (L 7! 2)~ 2;~ 2, where 2 is the valuation of L (left or right) compatible with the valuation v associated to S . We prove that the cycle C does not exist by constructing a strategy : ?(~ Z2; ~Z2)? whose interaction against ~ 2 does not reach a maximal element (i.e. deadlocks), contradicting
ZZ
defines a con-
P ROOF (sketch) By projection, the lemma reduces to proving that every MLL proof-structure defined from the semantics has connected switchings. We restrict ourself to the multiplicative case here, for lack of space. Although the adaptation of the proof to MLL is not entirely straightforward, the essence of the proof appears below. Let (A~ )A~ 2Rn be a dinatural family for a multiplicative formula ? interpreted in M[n], = the corresponding proof-structure, and S a switching such that S is nonconnected. Composing with the natural transformations A
..... C ................ (A
B ................. (A C ) and A (B ................. C ) ?! (B ............. C ) ?! . . . B ) transform into a dinatural family for a functor....... corresponding to the switching S . Here, is a L-fold ............ product of -products of literals, and the proof-structure associated to by lemma 3.2 is homeomorphic to the “switching” graph S . Define the game A = A? with DA the flat domain of integers, and strategies the closure operators (x 7! x _ i), for i any integer, with EA the equality on them. Observe that A A k ..............is not total, not even civil. We use the notation ............... ... k or A to mean respectively the k-fold or .... -product of a game A. For every pair of integers k 1 and N , there is a strategy ............... . k . k . N of A whose set of maximal fixpoints is exactly:
P ROOF (sketch) Given a switching S of , every oriented canonical cycle C in the “switching” graph ( )S has the following shape:
&
Connectedness
Lemma 4.6 Every normal switching of nected graph.
Lemma 4.5 There is no oriented and canonical cycle in any switching of .
ax
Z ZZ
p 6 x h j i; k v ~2 ~2 pk 6 x h v j~ 2 i:
We conclude that h j~ 2i is not maximal.
= d ; e?v;T when T is empty.
ax
Z
8k 2 f0; :::; 2N ? 1g; pk 6 x h v j~ 2i:
f(x1 ; :::; xk) j
i=k X i=1
xi = N g
~ A~ ) Now, suppose that S is disconnected, and that (A; ...... ....... ....... is the .......... -product of A k1 .......... ........... A kL . We may assume without loss of generality that the first l -powers A ki and axiom links form a connected component of S . ....... ....... Set the strategy : (A k1 ........... .......... A kL?1 )? as
Z
1 = kN11 kNL? L?1
11
(4)
Here, we are careful to choose N1 ; :::; Nl such that
i=l X i=1
Ni
is odd
Gabbay and T. S. E. Maibaum, editors. Oxford University Press 1994. (5)
[AM98] S. Abramsky and P.-A. Melli`es. Concurrent games and full completeness. Draft paper, 37 pages, 1998.
Let be any counter-strategy of A kL . A simple argu~ m ; A~ m )? ment shows that the result of playing : (A against A~ is either a maximal element (x1; :::; xm) or >. Assume that it is a maximal element (x1 ; :::; xm). By lemma 3.2, there exists an involution of f1; :::; mg such that xi = x (i) for every i 2 f1; :::; mg. The involution describes the axiom links of S , therefore restricts by definition of l to an involution of f1; :::; klg where P P kl = ii==1l ki. We obtain that the sum ii==1kl xi is even, which forbids by (5) (x1; :::; xm) to be a fixpoint of . We conclude that the result of playing against A~ is necessarily >. Let be the strategy of A kL obtained by composing and A~ . We have just shown that > is the result of playing
[AC98] R. Amadio, P.-L. Curien, Domains and Lambdacalculi. Cambridge University Press, 1998. [BDER97] P. Baillot and V. Danos and T. Ehrhard and L. Regnier. Believe it or not, AJM’s games is a Model of Classical Linear Logic. Proceedings of LiCS ‘97, pp. 68–75, 1997. [BFSS88] E. S. Bainbridge and P. J. Freyd and A. Scedrov and P. J. Scott. Functorial Polymorphism. Theoretical Computer Science 70,1:35–64, 1988. [Bla92] A. Blass, A game semantics for linear logic, Annals of Pure and Applied Logic, 56:183-220, NorthHolland, 1992.
..........
off against any strategy : A ........ kL . This implies that is (x 7! >), hence not a strategy of A kL , yielding the
[BS96] R. Blute and P. J. Scott. Linear L¨auchli Semantics. Annals of Pure and Applied Logic, 1996.
required contradiction.
Theorem 4.7
[DJS97] V. Danos and J.-B. Joinet and H. Schellinx. A new deconstructive logic: Linear Logic. Journal of Symbolic Logic, 1997.
is a MALL proof-net.
5 Main result
[Gir87] J.-Y. Girard. Linear Logic. Theoretical Computer Science, 50:1–102, 1987.
Theorem 5.1 (Full Completeness) Let ? and be MALL formulas. Every morphism [ ?]] ?! [ ]] in M[n] is the interpretation [ ]] of a MALL proof of the formula ? .
(
[Gir95] J.-Y. Girard. Proof-nets: the parallel syntax for proof theory. 1995. [Gir98a,b] J.-Y. Girard. On the Meaning of Logical Rules I, II. Unpublished papers, 1998.
P ROOF By theorem 4.7 and the Sequentialization Theorem in [Gir95], every symmetric dinatural family in the perclass [] : [ ?]] ?! [ ]] defines a proof of ? . In turn, the proof is interpreted as a per-class [ ] = [ ]] of symmetric dinatural families from [ ?]] to [ ]]. The proof is reduced to showing that the two classes [] and [ ] are equal. A simple argument on the slices of shows that and have the same maximal fixpoints at every instance A~ . We then use the fact that two strategies of a game are equivalent when they have the same maximal fixpoints, and conclude that [] = [ ].
(
[HO92] M. Hyland and C.-H. L. Ong. Fair Games and Full Completeness for Multiplicative Linear Logic without the MIX rule. Unpublished Manuscript, 1992. [Loa94a] R. Loader. Linear Logic, Totality and Full Completeness. In Proceedings of LiCS ‘94. [Loa94b] R. Loader, Models of lambda calculi and linear logic: structural, equational and proof-theoretic characterisations, PhD Thesis, Oxford, 1994.
References [AJ92a] S. Abramsky, R. Jagadeesan, New foundations for the geometry of interaction, Information and Computation, 111(1):53-119, 1994. Conference version appeared in LiCS ‘92. [AJ92b] S. Abramsky and R. Jagadeesan. Games and Full Completeness for Multiplicative Linear Logic, Journal of Symbolic Logic, (1994), vol. 59, no. 2, 543–574. Conference version appeared in FSTTCS ‘92. [AJ94] S. Abramsky and A. Jung. Domain Theory. Handbook of Logic in Computer Science. Volume III: Semantic Structures, pages 1–168. S. Abramsky, D. M. 12
