Modal Logic, Linear Logic, Optimal Lambda-Reduction Stefano Guerrini1 , Simone Martini2 , Andrea Masini3 IRCS, University of Pennsylvania, 3401 Walnut Street, Suite 400A, Philadelphia, PA 19104-6228 { USA;
[email protected]. 2 Dipartimento di Matematica e Informatica, Universit a di Udine, Via delle Scienze, 206, I-33100 Udine { Italy;
[email protected]. 3 Dipartimento di Informatica, Universit a di Pisa, Corso Italia, 40, I-56125 Pisa { Italy;
[email protected]. 1
1 Introduction The paper will maintain and justify the following thesis : A purist approach to modal proof-theory may have some (applied) bene ts for the ecient reduction of lambda-terms, an important problem in the implementation of functional programming languages.
1.1 Modal proof-theory The quest for symmetry, concision, and generality is a distinctive trait of mathematics. Proof-theory is a remarkable example of this search. A large class of logical systems may be described with few key ingredients, organized around crucial symmetries (left/right; introduction/elimination). At rst sight, modalities seem to spoil the scene. On one hand, the rules are seldom symmetric, or, when they are, they involve global constraints on their applicability. Moreover, moving from one system to a more powerful one often means to radically change the rules (and the proofs of the main metatheoretical properties, like normalization). To restore symmetry and locality one resorts to indexed systems, where formulas are annotated with additional information. The idea goes back to Kripke [Kri63] and has been developed by many authors (a good review is [Wan94]). What kind of information is added, the way it \codes" the structure of the (Kripke) models of the logic, and the bookkeeping of this information, make crucial di erences among the several approaches. We focus here on 2-sequents, put forward in [Mas92]. It is a class of formal systems where annotations are non-negative indexes, introduced with the simple motivation to mimic the interplay of variables and quanti ers in rst order logic. Di erent systems (in the range K{S4) are obtained with simple side conditions on indexes of a single (elimination) rule [MM96].
In Section 2 we will brie y recall the 2-sequent natural deduction calculus for linear logic (which, as far as modalities are concerned, is similar to S4). Section 2.1 will show how this proposal looks like in the context of linear logic proof-nets.
1.2 Optimal reduction of lambda-terms The -calculus is a paradigmatic functional programming language, whose operational semantics is given by -reduction: (x:M)N ! [N=x]M. Given a normalizable -term M it would be interesting to determine the shortest reduction sequence starting from M and reaching its normal form. However, it is well known that there is no recursive, one-step strategy for this. One has to resort to parallel strategies, reducing several redexes at once. It is a fundamental result of Levy [Lev78] that an optimal recursive parallel strategy exists. The only known \implementations" of Levy's strategy are based on graph-rewriting. A lambda-term M is represented as a graph in which special nodes express the (partial) sharing of common subterms of M. Then, the graph-rewriting rules for -reduction are extended with rules for the explicit bookkeeping of the sharing information. After Lamping's original solution [Lam90], also [GAL92a] and [Asp95b] have presented proposals. Interestingly enough, these approaches may be interpreted as a particular way of performing reduction of linear logic proof nets [GAL92b]. We discuss in Section 4 the improvements that may be obtained from our approach to linear logic. More extended discussion may be found in [GMM96], or [Gue95] for the speci c problem of -calculus. The practical application of these reduction techniques is in the implementation of functional programming languages|see [AG97] for a detailed treatment.
2 Linear logic, from an indexed perspective From a proof-theoretical point of view, linear logic [Gir87] arises by dropping the structural rules of contraction and weakening from the standard formulation of a sequent calculus for classical logic. This very weak system is then strengthen by adding modalities (?, and its dual !) marking those formulas on which weakening and contraction is allowed. As exempli cation of the 2-sequent approach to modalities, we present here a natural deduction system for a fragment of the full logic, see [MM95]. Formulas are built out of atoms (ranged over by p), by means of the binary connectives and , and the (unary) modality !. Each formula is decorated with a nonnegative integer (its level ); the formal system establishes judgments of the shape ? Ak , where ? is a multi-set of indexed formulas, A is a formula, k is the level of A (the rules will ensure that the levels of the formulas in ? are less or equal to k). The propositional rules of this fragment are summarized in Figure 1. To ensure linearity, in I and E each discharging is compulsory
(
`
(
2
Ak
?
Ak Bk A Bk [
[
A Bj
Bk
Ck
E jk
?
A ( Bk Bk
Ak
(I
A ( Bk
]
Ck
I
]
] [
? Ak
Aj B j
?
(E
Figure 1. Propositional rules. and involves exactly one formula occurrence (note that this fragment is linear also in modal formulas: no contraction or weakening is allowed at all). We write ? Ak if there is a deduction whose conclusion is Ak and whose open assumptions are exactly the formulas of ? . Modal (exponential) rules|Figure 2|uses the level machinery to formulate the \context constraint" on the introduction rule. For a set of assumptions ? , de ne #? = maxfk j Ak ? g; #? = -1 when ? is empty. `
2
?
?
Ak+1 Ak
Aj Ak
!
I k#?
!
!
Ekj
!
Figure 2. Exponential rules. It is worth to compare the two exponential rules with the rules for universal quanti cation:
?
?
A x:A
8
x:A A t=x 8
I x62FV (?)
8
[
]
E
8
Indeed, as the introduction of \!" decrements the level of the conclusion of exactly one, so the introduction of binds exactly one variable. The side condition k #? , is the analogous of the usual constraint that x be not free in the active premises of the derivation. Again, as the elimination of \!" raises the level of 8
3
the conclusion of an arbitrary increment, so the elimination of allows the introduction of a new term t with an arbitrary number (possibly zero) of new free variables. This analogy has been a leading idea of the 2-sequents approach and keeps holding when we consider reduction of proofs. In linear proofs, exponential redexes and their reductions may be de ned as follows: 8
D
j 1 k-1 D Ak I reduces to Ak-1+j Ak-1 E Ak-1+j where the (meta) notation n k D means the result of incrementing of n all the levels greater than k in the deduction D. Formally: absorption: If v i n i Dv Dv reindexing: If v > i n i Dv nv+i Dn [ -
]
!
!
!
[
:
[
]
=
:
[
]
=
]
[
]
The side condition on !I ensures correctness of the reduction. Under the analogy \modalities are quanti ers", this process of reindexing corresponds to substitution in rst-order logic (the absorption case corresponding to a test on the freeness of the involved variable).
2.1 Levels in a proof-net formulation The system we are interested in adds contraction to the basic rules of the previous section. It is best described using proof-nets, a formal system for linear logic exploiting (and displaying at its best) the symmetry of the logic (see [Laf95] for a lucid introduction). We only recall that multiplicative proof nets are formulated for the logic based on atoms (propositional letters p and their duals p? ), and the two dual connectives tensor ( ) and par (}). Negation is primitive only on atoms and de ned on formulas by full double negation and DeMorgan rules. Linear implication is de ned as A B A? }B. When exponentials are introduced, the elegance and simplicity of the nets (and of their reduction rules) have to be spoiled by the notion of box|a sub-net representing the context where an !-introduction is applied. Boxes are necessary both for static (i.e., for soundness) and dynamic correctness (i.e., for keeping track of the context during the reduction of an exponential redex). A sketch of the general case (including contraction) is depicted in Figure 3 (forget for the moment the levels of the formulas). Observe, rst, that the box (on the left in the rst line) is duplicated. Second, after the reduction the di erent copies of may have been put inside other boxes (this happens when the ?-node is a secondary door of another box). (The notation [ki - k] means that all the levels of have been incremented by ki - k.)
(
4
Πr
Π’s Π’2
Π2
Π1 k+r
A
Π’1 k+s
Γ1
∆1
A Γ2
∆2 ...
...
∆s
Γr
Π B
k+1 ⊥
i
A
?
!
k
?
k
?B
!A
k
?A
cut
Π’s
Πr Π2
B
i+r-1
Π’2
Π1
Π[r-1] k+r ⊥
k+r
Γ1
A
A
Π’1
Π [s-1] B
i+s-1
k+s ⊥
k+s
A
A
Γ2
∆2 cut
Γr
...
...
cut
∆1
∆s
? k
?B
Figure 3. Box (global) reduction. How does the system of the previous section relates to proof-nets? It is not dicult to establish the following result.
Theorem 2.1. Let ? A0 , where all the assumptions in ? have level 0. Then `
there is a proof-net with conclusions ? ?; A.
Proof. (sketch) Let us rst build the labeled graph underlying . Then we will
take care of boxes. Proceed by induction on ? A0 , as in the usual equivalence between natural deduction and sequents: assumptions go to axioms; introduction rules are mapped into the corresponding links; elimination rules are transformed into cuts between the premise and a suitable proof-net. Now assign boxes: the level of a formula is interpreted as the box-nesting-depth of the corresponding link. The side condition on !I ensures that for each !-link (principal door of a `
5
box) one can nd suitable ? links acting as secondary doors of the corresponding box. This translation extends to normalization: any reduction of a natural deduction derivation is transformed into a valid cut-elimination step of the corresponding proof-net. The reindexing case in the de nition of [n]k D corresponds exactly to the fact that a box, after reduction, may be put inside other boxes; absorption corresponds to the selection of the correct subnet to be reindexed (the reindexing stops at the secondary doors), cf. Figure 3. This relation (which was unexpected when we begun our work on levels is the starting point of the technical contribution of the next section.
3 Leveled nets and their reduction We have just seen that rewriting and reindexing of proof-nets may be done using levels|a local information|instead of explicit boxes|a global information. The last step we perform is the internalization of the meta-operation of reindexing. That is, instead of having an external de nition of [n]k D, we extend our nets and their reductions as to incrementally and locally perform reindexing during the process of cut-elimination. We start with the formal de nition [GMM96] of sharing s`-structures (sharing leveled structures of links) and proof `-structures, our level-based formalization of proof-nets with explicit reindexing operators (the mux links).
De nition 3.1. An s`-structure is a nite connected hypergraph whose nodes are labeled with indexed formulas and hyperedges (also called links) are labeled from the set fcut, ax; }; ; !; ?g fmux[i]j i 0g fdemux[i]j i 0g; the integer i in (de)muxes is the threshold of the link. Allowed links and nodes are drawn in Figure 4. The source nodes of a link are its premises; the target nodes are the conclusions. Premises and conclusions are assumed to be distinguishable (i.e., we will have left/right premises, i-th conclusion and so on), with the exception of ?-links. In an s`-structure, each node must be conclusion of exactly one link and premise of at most one link; those nodes that are not premises of any link are the net conclusions; unary (de)muxes are also called lifts.
[
[
De nition 3.2 (proof `-structure). A proof `-structure is an s`-structure without (de)muxes.
We assume that any axiom of an s`-structure has only atomic conclusions. Such a restriction does not decrease the expressive power of s`-structures. To each (standard) proof-net P we may associate a proof `-structure D [P], the decoration of P: D [P] is obtained by assigning to each node of P the level corresponding to the number of exponential boxes containing that node.
De nition 3.3. A proof `-structure S is a restricted proof `-net i S = D [P] for some weakening-free proof net P. 6
A
k
A
k
A?
cut
A
k
k
+1
A
k
A1 k
k
k1 ;:::;kr
1
r
!
A
!
A?
ax
A B
Ar k
A1
k
?1
A
?
k
}
A
k
1
r
k
A}B A
k
k1 ;:::;kr
B
k
Ar k
A
k
k
k
k
i
i
?
k
B
k
A 1 ?1 1 k
k
k ;:::;kr
r
1
Ar k
Figure 4. Links of s`-structures. By using indexes it is possible to \recognize" exponential boxes: De nition 3.4. Let S be a proof `-structure and let Ak be a premise of an !-link; we call box of Ak a sub-hypergraph bxS [Ak ] of S verifying the following properties: 1. Ak bxS [Ak ] (Ak is the principal door of bxS [Ak ]); 2. bxS [Ak ] is a proof `-net; 3. each net conclusion of bxS[Ak ] di erent from the principal door is a premise, in S, of a ?-link with conclusion at level j < k (such ?-premises are the secondary doors of the box); 4. for each Bj S, if Bj bxS [Ak ], then j k. We denote by BX[S] the set of boxes of S. Because of the de nition of `structure, boxes are connected. Restricted proof `-nets are a complete system for weakening-free classical linear logic; s`-structures captures in a local way the process of cut-elimination, while muxes perform reindexing of boxes. (When weakening is allowed, proofnets become disconnected and there is no local way to perform cut-elimination; in [GMM97] we propose a solution, calling for a simple modi cation of the proofnet syntax). We distinguish the rules in two kinds: the logical (or ) rules (Figure 5), where interaction happens through a cut-link (corresponding to a logical cutelimination step); and the rules (Figures 6, 7, and 8), when one of the interacting nodes is a mux/demux (corresponding to a step of incremental duplication and/or reindexing). Observe, in particular, rule absorption ( abs ), corresponding to the case when the mux reaches the border of a box (through one of its secondary doors) and has therefore exhausted its job. It matches the absorption case in the de nition of [n]k D, in Section 2. 2
2
2
B
3.1 Optimality
Levy's parallel strategy (cf. Section 1.2) for lambda-calculus consists in reducing at any step a certain number of redexes|those belonging to the same family, 7
Ak }
Ak
Ak
cut
Bk
A
?
A}B k
A
?
k1
A
?
?
Ak
Ak
!
cut
A
?
?
cut
A
?
?
?
k
k
k
Bk
cut
B
?
k
Ak+1
kr
k
Bexp
?
B
A k B
!
k
cut
Bmul
Ak+1
Ak
Bide
Ak
ax
?
Ak 1
Ak r
k cut
A
?
cut
A
?
kr
k1
Figure 5. Logical (or ) rules. The symmetric cases are not shown. [Lev78]. Main result of [Lam90] is a graph rewriting implementation in which, representing lambda-terms as graphs with additional (partial) sharing operators (fans, akin to our muxes), each -reduction (that is reductions essentially similar to those of Figure 5) corresponds to a parallel step of Levy's strategy. An analysis essentially similar may be conducted for proof-nets [GAL92b, AL93]: also in this case the number of -reductions of the known sharing graph implementations is equal to the number of parallel reductions in Levy's strategy. Of course, not all the reductions of Figures 6, 7, and 8 may be freely applied if we want an optimal reduction. Indeed, a free application of rule dup would duplicate any redex in its scope. Intuitively, during an optimal reduction a mux must be propagated only if otherwise it would prevent the formation of a new redex. A moment's thought shows that a mux \coming from above" on a logical node must never be propagated. Among all the reduction rules, we thus isolate the set opt = - dup containing the only rules allowed during an optimal reduction. It is well known that proof-nets may encode typed lambda-terms (they may encode also type-free terms, if one forces the type isomorphism D = !D D, or D = !(D D), [Reg92]). If we now take the proof-net encoding of a lambda-term, its optimal reduction (as de ned here) exactly corresponds to its Levy optimal reduction.
B
B
(
(
4 Coherence The approaches to optimal reduction of proof-nets/lambda-terms discussed in 1.2 and the previous section, share a problem of coherence. Suppose that the proof8
A
Ak ?
A
kr
k1
i
Ak
i
Bidup cut
Ak1
Ak ?
cut
Ak1 ? Akr
cut
Akr ?
Ajn Ak1
Akr
i
i
Aj1
Ak1
Ajn
Ajn +k1 ?j1
Bdup
?
Akr
Ajn +kr ?j1
?
?Ak+k1 ?j1
?
?Ak+kr ?j1
i
?Ak
where
i j .
Ah1
Figure 7. Duplication rules (II). The following cases are not shown: in the cases where the logical link is , }, !, instead of why-not (?).
Ak1
Ajr
Aj1
B
odup
Akr
i
i
Ajr
Aj1
Ak
Babs
?Av
?
Banh Ak1
Ak
?
A kr
i
?Av
where
Ak1
i v
Akr
Figure 8. Simpli cation rules. Theorem 4.1 (strong normalization). Let G be a correct s`-structure. 1. The rules are strongly normalizing and con uent on G. The normal form of G is a restricted proof `-net. 2. The + rewriting rules are strongly normalizing and con uent on G. The + normal form of G is a restricted proof `-net.
10
3. The normal form of G reduces by standard cut-elimination to its + normal form.
The third item of Theorem 4.1 ensures the soundness of the system. The result can be stated in a stronger way, as in the following Lemma 4.3, which is indeed a preliminary step in the proof of Theorem 4.1 ( std denotes a standard cut-elimination reduction).
B
De nition 4.2 (read-back). The read-back R G of a correct `-net G is the (
normal form of G.
)
Lemma 4.3 (soundness). Let G be a correct s`-structure and N be the restricted proof `-net s.t. N
B
G. Then N Bstd R G . (
)
According to Section 3.1, there is a strategy minimizing the number of rules.
B
Theorem 4.4 (optimality). The opt rewriting rules are Levy optimal. Theorem 4.5. Let G be a correct s`-structure and N be its normal form. +
Let G 0 be a + opt normal form of G, then R(G 0 ) = N.
+
By Theorem 4.5, normalization of correct `-nets may be performed in two distinct steps: rst optimal reduction ( + opt ), then read-back reduction ().
5 Conclusions Our coherence results are made possible by the presence of the absorption rule. In its turn, the rule could not be formulated without a way to recognize the border of a box and, hence, its secondary doors. This is achieved via a clear separation of the logical and control information in the representation of a net. The logical information takes the form of levels on the formulas of the proofnet; control is expressed by muxes (unifying the fans and brackets of the other approaches, which are also used to express levels), which may be viewed simply as the internalization of the meta-operation of reindexing. It is this separation to allow coherence.
References AG97.
Andrea Asperti and Stefano Guerrini. The Optimal Implementation of Functional Programming Languages. Cambridge Univ. Press, 1997. To appear. AL93. Andrea Asperti and Cosimo Laneve. Interaction systems. In Int. Workshop on Higher Order Algebra, Logic and Term Rewriting, 1993. Asp95a. Andrea Asperti. !" = 1: Optimizing optimal -calculus implementations. In Jieh Hsiang, editor. Rewriting Techniques and Applications, 6th International Conference, RTA-95, LNCS 914, pages 102{116, Kaiserslautern, Germany, April 5{7, 1995. Springer-Verlag.
11
Asp95b. Andrea Asperti. Linear logic, comonads and optimal reductions. Fundamenta infomaticae, 22:3{22, 1995. GAL92a. Georges Gonthier, Martn Abadi, and Jean-Jacques Levy. The geometry of optimal lambda reduction. In Proc. of Nineteenth Principles of Programming Languages (POPL), pages 15{26. ACM. January 1992. GAL92b. Georges Gonthier, Martn Abadi, and Jean-Jacques Levy. Linear logic without boxes. Proc. of 7th Symposium on Logic in Computer Science, (LICS), pages 223{234. IEEE. Santa Cruz, CA, June 1992. Gir87. Jean-Yves Girard. Linear logic. Theoretical Computer Sci., 50:1{102, 1987. GMM96. Stefano Guerrini, Simone Martini, and Andrea Masini. Coherence for sharing proof-nets. In H. Ganzinger, editor, Rewriting Techniques and Applications { RTA'96, volume 1103 of Lecture Notes in Computer Science, pages 215{229. Springer-Verlag, 1996. GMM97. Stefano Guerrini, Simone Martini, and Andrea Masini. Proof nets, garbage, and computation. In R. Hindley, editor, Typed Lambda-Calculus and Applications 1997, LNCS. Springer-Verlag, 1997. To appear. Gue95. Stefano Guerrini. Sharing-graphs, sharing-morphisms, and (optimal) graph reductions. In 1st Tbilisi Symposium on Logic, Language, and Computation, Tblisi, Georgia, October 1995. Gue96. Stefano Guerrini. Theoretical and Practical Aspects of Optimal Implementations of Functional Languages. PhD thesis, Dottorato di Ricerca in Informatica, Pisa{Udine, TD 3/96, January 1996. Kri63. Saul Kripke. Semantical analysis of modal logic I. Zeitschr. f. math. Logik und Grund. d. Mathematik, 9:67{96, 1963. Laf95. Yves Lafont. From proof-nets to interaction nets. In J.-Y. Girard, Y. Lafont, and L. Regnier, editors, Advances in Linear Logic, volume 222 of London Mathematical Society Lecture Note Series, pages 225{247. Cambridge University Press, 1995. Proceedings of the 1993 Cornell Linear Logic Workshop. Lam90. John Lamping. An algorithm for optimal lambda calculus reduction. In Principles of Programming Languages (POPL), pages 16{30. ACM, 1990. Lev78. Jean-Jacques Levy. Reductions Correctes et Optimales dans le lambda-calcul. PhD Thesis, Universite Paris VII, 1978. Mas92. Andrea Masini. 2-sequent calculus: A proof theory of modalities. Annals of Pure and Applied Logic, 58:229{246, 1992. MM95. Simone Martini and Andrea Masini. On the ne structure of the exponential rule. In J.-Y. Girard, Y. Lafont, and L. Regnier, editors, Advances in Linear Logic, volume 222 of London Mathematical Society Lecture Note Series, pages 197{210. Cambridge University Press, 1995. Proceedings of the 1993 Cornell Linear Logic Workshop. MM96. Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof Theory of Modal Logics, pages 213{241. Kluwer, 1996. Reg92. Laurent Regnier. Lambda-Calcul et Reseaux. These de doctorat, Universite Paris 7, 1992. Wan94. H. Wansing. Sequent calculi for normal modal propositional logic. Journal of Logic and Computation, 4(2):125{142, 1994.
12
