Compressing Polarized Boxes Beniamino Accattoli Carnegie Mellon University Pittsburgh, PA, USA
Abstract—The sequential nature of sequent calculus provides a simple definition of cut-elimination rules that duplicate or erase sub-proofs. The parallel nature of proof nets, instead, requires the introduction of explicit boxes, which are global and synchronous constraints on the structure of graphs. We show that logical polarity can be exploited to obtain an implicit, compact, and natural representation of boxes: in an expressive polarized dialect of linear logic, boxes may be represented by simply recording some of the polarity changes occurring in the box at level 0. The content of the box can then be recovered locally and unambiguously. Moreover, implicit boxes are more parallel than explicit boxes, as they realize a larger quotient. We provide a correctness criterion and study the novel and subtle cut-elimination dynamics induced by implicit boxes, proving confluence and strong normalization.
I NTRODUCTION Gentzen’s sequent calculus is a standard formalism for a wide variety of logics. However, sequent calculus forces an order among deduction steps even when they are evidently independent, a drawback called bureaucracy. Proof nets, a graphical syntax introduced by Girard in the context of linear logic [1], [2], are a parallel and bureaucracy-free representation of proofs. They also provide new and elegant cut-elimination methods, drastically reducing the need of commutative steps, the non-interesting burden in every sequent calculus proof of cut-admissibility. Sequents, proof nets, and boxes. Some cut-elimination rules require us to duplicate or erase whole sub-proofs, typically the rules for the ! modality in linear logic. Proofs in sequent calculus are tree-shaped and bear a clear notion of last rule, the root of the tree. This property has an obvious but important consequence: given a !-rule r in a proof, there is an evident sub-proof ending on r, the sub-tree rooted in r. Therefore, non-linear cut-elimination rules can easily be defined by duplicating or erasing sub-trees, as in Figure 1.a. Switching to proof nets, the situation radically changes. Proofs are now represented as graphs, retaining only the causality between rules and without any bureaucracy. A proof net in general has many last rules, one for each formula in the last sequent. Unfortunately, given a rule r it is not clear how to find a sub-proof net ending on r, because the causality retained by proof nets does not provide enough information. Thus, in order to eliminate cuts such as the one in Figure 1.b—which require sub-proof nets—some information has to be added. The typical solution is to re-introduce part of the bureaucracy, pairing each !-rule with an explicit box containing the subproof to duplicate or erase. The rule is then implemented as in Figure 1.c.
There has been extensive research on boxes, regarding how to represent them [3]–[6], how to decompose their evaluation [4], [5], [7]–[9], how they control cut-elimination [10], [11], how to see them categorically [12], [13], how to use them for the additive connectives [14], and concerning their relationship with explicit substitutions [6], [15]. In particular, the study of boxes and their dynamics has contributed greatly to the understanding of optimal reductions [8], [9], [16], and has also resulted in a new approach to implicit computational complexity [10], [11]. More recently, the discovery of simple implicit boxes for λ-calculus [6] led to a renewal of the theory of explicit substitutions [17]–[19], which in turn led to new results on λ-calculus [17], [20]–[22]. Compressing polarized boxes. In this paper we show that in an expressive polarized dialect of linear logic it is possible to take advantage of the polarized structure of proofs to compress explicit !-boxes. The idea is to associate with every explicit box a subset of its content from which it may be reconstructed, exploiting polarity to keep this subset compact. The logic is Laurent’s Polarized Multiplicative and Exponential Linear Logic (MELLP) [23], which may be understood as the fragment of linear logic interpreting classical logic or Parigot’s λµ-calculus [24]. This logic derives from Girard’s classical system LC [25], [26] and it is built around the concept of polarity. It has strong connections to important topics in proof theory, as game semantics [27]–[29], focalization [30], [31], ludics [32], [33], realizability [34], and the compact representations of proof certificates [35]. But MELLP is also important for programming languages: beyond the λµ-calculus, it is related to CPS translations [36], delimited continuations [37], and the π-calculus [38]. In MELLP, formulas can be either positive or negative, and the exponential modalities ! and ? have a special role in that they change the polarity of a formula. Sequents have an intuitionistic shape: they can have many negative formulas but never more than one positive formula. This structural property forbids permutations involving two positive rules, i.e. it forbids positive bureaucracy. Something interesting happens when one looks at MELLP via proof nets: the rigid positive structure is preserved by the removal of bureaucracy. The important consequence is that now there is a notion of last positive rule. Parallelism does not disappear, though: it now concerns only the negatives. The rigid structure of positives is already at work in Laurent’s proof nets presentation of MELLP. However, even if polarity provides more structure to his proof nets there is still something missing: the refined coding of causality has yet to allow
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for sub-proof retrieval. Indeed, Laurent uses explicit boxes to implement cut-elimination. In this paper we show that explicit !-boxes can be replaced by a more compact representation, what we call an implicit box. An implicit box is a set of causal information, a set of rules which have to be included in the sub-proof associated with a ! and which are sufficient to reconstruct it. The key point is that polarity provides the structure to keep this set remarkably compact: it is enough to retain the polarity changes (from positive to negative only, given by derelictions and cuts), all other rules can be easily retrieved by the polarity constraints. Implicit boxes add enough information to locally reconstruct a sub-proof—what we call the induced box—for every positive rule l, which is simply given by the causal past of l. For negative rules there is no induced box, but this is not a problem: !-rules are positive, thus having sub-proofs for positives is enough to implement cut-elimination. We show that our approach is not an ad hoc technique, as we provide a correctness criterion, i.e. we characterize proof nets with implicit boxes without any reference to explicit boxes nor to sequent calculus. Moreover, our criterion is a natural extension of Laurent’s criterion for explicit proof nets. The key technical point here is a representation of implicit boxes as additional edges, usually called jumps (introduced by Girard in [39]). It can be claimed that our solution is canonical: in the case of cut-free proof nets, the implicit box of a ! contains only one dereliction and this dereliction is uniquely determined by the polarized structure, independently of the chosen syntactic formalism. Thus, we are simply taking advantage of an already existing structure, that has already been used in semantical studies [28], [40], but never exploited syntactically before. Our results reveal that polarity has also a fundamental geometric role: it internalizes boxes. Let us point out that via the Curry-Howard correspondence between MELLP and λµ-calculus this work provides implicit boxes for classical logic. In addition, this apparently abstract work has the potential of concrete application: it provides a compact way for representing terms in graphical implementations. Cut-elimination. We define cut-elimination on the implicit representation, obtaining a new interesting operational semantics. Implicit proof nets are a quotient of Laurent’s explicit
proof nets, given by the fact that box borders are not stored into implicit boxes. This fact induces new cut-elimination rules, with some good properties and some unexpected behavior: • No commutative rule: since the border of explicit boxes is not explicitly represented, the commutative box-box rule disappears and cut-elimination for implicit proof nets contains only so-called key or principal cases. • Automatic push of negative rules: weakenings and contractions created by cut-elimination are dynamically pushed out of explicit boxes, without any additional rule (necessary instead in [15], [41], see also [42], [43]), simply as a consequence of the implicit representation. • Complex critical pairs: the absence of explicit borders and their new dynamic shrinking also generates some difficulties with respect to confluence. In particular, there is a new problematic critical pair given by the cutelimination rules for axioms. We prove strong normalization and confluence modulo associativity and commutativity of contractions in the propositional case. We first study cut-elimination in the case with atomic axioms only, by relating our formalism to Laurent’s original syntax [23], [24], and obtain strong normalization of implicit proof nets by simulating them with explicit ones. Strong normalization is in turn used to prove confluence. Then we reduce the case with arbitrary axioms to the case with atomic axioms by introducing η-expansion rules. Related work. As previously discussed, alternative representations of boxes have been investigated at length in a variety of studies. At first sight our technique may look similar to the interaction nets encoding of linear logic by Mackie [5], which connects a ! with its border, but in fact it is completely different: the border is not stored in the implicit box. Most research on boxes is simply concerned with evaluation, while ours belongs rather to the those studies concerning the problem of a correctness criterion for the alternative representation (which is a difficult problem). Lamarche’s essential nets [3], [44]–[46] were an initial source of inspiration. However, essential nets put jumps on weakenings and box borders, while we place them on polarity changes (i.e. derelictions and cuts), allowing weakening to freely float in our implicit nets. This paper extends our previous work with Guerrini on λ- calculus [6] to a classical logical setting (MELLP encodes λµ-calculus [24]), but rather than a slight generalization it is a complete re-
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I. MELLP N ETS WITH E XPLICIT B OXES
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elaboration with radical differences (see the end of Section II for more details). Our jumps are placed in precise places while those of Di Giamberardino and Faggian [47], [48] (related to L-nets and game semantics [49]) are used in a more liberal way. They can tame floating rules only by attaching them with jumps or by introducing the MIX rules, while our approach circumvent their use. Plan of the paper. The next section summarizes Laurent’s MELLP proof nets with explicit boxes. In Section II we introduce nets with implicit boxes, give a correctness criterion, prove properties of the induced boxes, and relate them statically to explicit boxes. In Section III we study cutelimination, proving strong normalization and Church-Rosser modulo associativity and commutativity of contractions.
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X and X ⊥ are atomic formulas. The ! connective turns a negative formula into a positive one, and ? does the opposite action. The sequents of MELLP can have two shapes: ` Γ ; P or ` Γ ; , where Γ is a multiset of negative formulas and P , if present, is a positive formula; the place of the eventual positive formula, separated by a semicolon, is called the stoup. The rules of MELLP are in Figure 3 (left side of every ), where [P ] means that the stoup may or may not contain a positive formula P . Note that the {!, c, w}-rules do not require
negative formulas to be ?-formulas: this is a point on which MELLP departs from linear logic. We deal with MELLP sequent calculus only indirectly: we will relate our syntax to Laurent’s proof nets, which are themselves related to sequent calculus. Nets. Nets for proof systems are usually presented as directed graphs, where the orientation goes from the premise(s) to the conclusion(s) of the logical rules, as in Fig. 1.c. For polarized proof nets Laurent uses a correctness criterion based on directed graphs, but with respect to a different orientation. We prefer to avoid having two orientations, therefore we simplify and use only the polarized orientation, i.e. the one used by Laurent for correctness. Moreover, we hardcode polarities into the representation: negative edges are in dotted (and red) lines, positive ones are in solid (and blue) lines (see the example in Fig. 4.a, or Fig. 3). Nets are labelled directed graphs with pending edges, i.e. some edges may not have a source or a target, but not both. Nodes, called links, represent deductive rules and are labeled with an element of {ax, cut, ⊗, `, 1, d, !, w, c}. Edges are labeled with a MELLP formula, and an edge is positive/negative if its label is a positive/negative formula. The label of a link forces the number and the labels of its incoming/outcoming edges as shown in Figure 2.a (there is no ⊥-link because a weakening can introduce no matter which negative formula, so in particular ⊥). The conclusions (resp. premises) of a link, or its principal (resp. auxiliary) edges, are those represented below (resp. above) the link symbol. For instance the ⊗-link is the source of two edges (labeled P and Q) which are also its premises, and it is the target of one edge (labeled P ⊗ Q) which is also its conclusion. A link is positive (resp. negative) if it is the source or the target of a positive (resp. negative)
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edge. Then the links in {d, !, ax, cut} have both polarities, those in {w, `, c} are only negative, and those in {1, ⊗} are only positive. Definition I.1 (MELLP net with boxes). A (MELLP) net G is a finite set of links from those in Fig. 2.a s.t. every edge is the conclusion of some link. The conclusion edges of G are its pending edges and the conclusion links of G are its links with pending edges. A net with explicit boxes E, or simply an explicit net, is a net plus for any !-link l a subset ebox(l) of the links of G, called the explicit box of l, s.t. l ∈ ebox(l) and: • Subnet: its interior int(l) := ebox(l) \ {l} is an explicit net, whose boxes are inherited from E. • Border: the conclusion edges of int(l) are negative and one of them is the negative premise of l; 0 • Nesting: For any two !-links l and l if ebox(l) ∩ 0 0 ebox(l ) 6= ∅ then ebox(l) ⊆ ebox(l ) or ebox(l0 ) ⊆ ebox(l). The translation from MELLP proofs to explicit nets is in Fig. 3, where the bar on some negative edges denotes a (multi)set of conclusions. Leading example (Fig. 4.a, some labels have been omitted): the standard way of representing boxes is to wrap them into a graphical box as in the example. The conclusion (links) of this explicit net are the three d-links plus the `-link. The level of a link is the number of boxes in which it is contained, and the level of a net E is the maximum level of a link of E (the net in Fig. 4.a has level 2). Paths. A path τ of length n ∈ N, noted τ : l →n l0 (or τ : l → l0 or l → l0 ), is an alternated sequence l = l1 , e1 , l2 , e2 , . . . en , ln+1 = l0 of links and edges s.t. ei has source li and target li+1 for i ∈ {1, . . . , n}. A path τ : l →n l0 is positive (resp. negative), noted l →n◦ l0 (resp. l →n• l0 ), if
all its nodes and edges are positive (resp. negative). A cycle is a path l →n l with n > 0. A positive link l is initial if it is not a cut and there is no positive path τ : l0 →n◦ l with n > 0 (the initial links in Fig. 4.a are the three derelictions). Correctness. Laurent found a simple criterion for MELLP proof nets. Definition I.2 (correct net). Let E be an explicit net. The 0-depth net E 0 of E is the net containing all the non-! links at level 0 plus a generalized axiom genaxl for every !-link l at level 0 in E s.t. genaxl and ebox(l) have the same conclusions, i.e., E 0 is obtained from E by applying the transformation in Figure 2.b, called collapsing, for every !-box at level 0. Then E is correct if: 1) Rooted DAG: E 0 is acyclic and has exactly one initial link. 2) Recursive correctness: int(l) is correct, for every !-link l of E at level 0. Consider the net in Fig. 4.a. Its 0-depth net is in Fig. 4.c. In [23] Laurent proves that an explicit net is the translation of a proof, i.e. it is an explicit proof-net, if and only if it is correct. Polarity induces the following matching property on explicit proof nets (see [23], [24], [50]): In every !-box there is exactly one dereliction at level 0 (and at level 0 there is at most one dereliction) This fact—easily proved by induction on the translation—is a structural invariant that will be crucial in the next section (we invite the reader to verify it on the net in Fig. 4.a). Let us provide an intuition. Derelictions may be seen as variable occurrences and !-links as applications in λ-terms1 . The matching property is the fact that the argument of every applicative 1 In the usual (call-by-name) encoding of λ-calculus in linear logic (mapping A ⇒ B to !A ( B) the argument of every application is in a !-box.
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subterm has a different and unique variable occurrence as head variable. Positive tree. In order to define cut-elimination, Laurent introduces the notion of positive tree. The positive tree is a form of induced box for the positive links {1, ax, ⊗, !}, based on explicit !-boxes. Given a {1, ax, ⊗, !}-link l the positive tree of l, noted T (l), is defined as: • Base cases: if l is a {ax, 1}-link then T (l) is l. If l is a !-link then T (l) is ebox(l). • Inductive case: if l is a ⊗-link then T (l) is {l} ∪ T (l1 ) ∪ T (l2 ), where l1 and l2 are the target links of the auxiliary edges of l. By induction on this definition it can be shown that T (l) is a correct explicit net (in a correct net E). Cut-elimination. The cut-elimination rules for explicit proof nets are in Figure 5, where the box of the positive nodes (generically labeled +) is given by their positive tree; we use →cut for the union of these rules. In [23] Laurent proves that cut-elimination is confluent and strongly normalizing by means of a translation to linear logic proof nets inducing a simulation. In order to later simulate our nets with Laurent’s we are going to extend explicit nets with the rules and congruences in Fig. 6, i.e. associativity and commutativity of contractions, neutrality of weakenings for contractions, and permutations of contractions and weakenings with box borders. We define ≡ac := (∼com ∪ ∼a )∗ , and →− :=→pc ∪ →pw ∪ →n . The following theorem about MELLP explicit proof nets has never been proved:
some connections to the underlying graph. The idea is that it is enough to exploit the matching property of polarized proof nets and simply connect every bang to the dereliction at level 0 in its box. The example then becomes as in Fig. 7.b. Note that we introduced a positive edge (labeled i for implicit box), and that in this way the positive subgraph is connected. With this implicit representation the explicit box of a !-link l is given by the set of links P (l) to which l has a positive path (in the example 1, d and ! itself) plus the negative links which have a negative path to P (l) (the w-link). Applying this procedure to Fig. 7.b (and removing the i-edge) we recover Fig. 7.a. For cut-free proof nets—surprisingly—this is enough to obtain an implicit representation of boxes. Note that we are just taking advantage of an already existing structure, given by the matching property, and there is no arbitrary choice to make. By materializing this implicit structure, the skeleton of explicit boxes is simply given by the set of positive paths plus the negative trees above !-links. The edges given by the matching property do nothing but connect the positive subgraph, because in a cut-free net the only points where polarity changes are derelictions and bangs. Some remarks: • •
Theorem I.3. →cut ∪ →− is strongly normalizing modulo ≡ac . However, in [51] Pagani and Tranquilli show that MELL proof nets extended with the same rules and congruences are strongly normalizing (in [51] the authors consider →pc as a congruence, but the change is harmless since →pc is trivially strongly normalizing), and it is easily seen that Laurent’s translation can still be applied, obtaining a simulation proving the theorem. II. I MPLICIT B OXES : S TATICS Introduction to the technique via simple examples. Consider the proof-net in Fig. 7.a. The content of the explicit box is disconnected. Then, an alternative representation of boxes aiming to provide a local reconstruction procedure has to add
•
Weakenings float. We do not attach weakenings, as Fig. 7.a-b shows (see also Fig. 4.a-b). The implicit box is not the border of the explicit box. The implicit box of a !-link does not contain the auxiliary conclusions of its explicit box. Consider Fig. 7.d-e, where the dereliction at level 0 is not a conclusion of the explicit box, or Fig. 7.c, where the box has two conclusions but only one of them is attached in Fig. 7.b (see also Fig. 4.ab). Implicit boxes provide a quotient. Our technique does not always recover the box one started with. Consider Fig. 7.c: the substitution of the explicit box with the edge given by the matching gives again Fig. 7.b, and the reconstruction gives back Fig. 7.a. The point is that the boxes induced by implicit boxes are obtained from the original ones by pushing weakening, contraction, and par conclusions out of boxes as much as possible (see also Fig. 4.a-b). In particular our representation identifies more proofs than explicit boxes, for instance the nets in Fig. 7.a and c. In other words nets with implicit boxes are a quotient of explicit nets.
Cuts connect links of opposite polarities and break the
continuity of the positive sub-graph, as Fig. 7.f shows. This problem is solved iterating the same idea: we add a positive edge from every !-link l to all the cuts at level 0 in ebox(l), as in Fig. 7.g, so that the positive sub-graph of every box becomes a tree (see also Fig. 4.b). Definition II.1 (implicit net). An implicit net I is a net plus for every !-link l an implicit box ibox(l), i.e. a pair (dl , cutsl ) where dl is a d-link and cutsl is a a set of cut-links, and such that dl 6= dl0 and cutsl ∩ cutsl0 = ∅ for any two !-links l 6= l0 . As already discussed informally, we represent implicit boxes by introducing new edges, so-called jumps: if l is a !-link then there is a positive edge (labeled with i) from l to the dereliction dl and also to every cut-link l0 ∈ cutsl . Correctness. We do more than just represent boxes in this compact way, as we provide a correctness criterion for implicit proof nets. Polarized paths are defined as before, but now they can also use the edges induced by implicit boxes, i.e. the jumps, which are considered as positive edges. Jumps may introduce cycles, as in Fig. 7.g (or Fig. 4.ab). Rather than introducing a notion of correction graph we modify the acyclicity requirement of Laurent’s criterion, and ask that every cycle uses the negative premise of a !. We also need a condition guaranteeing the well-formedness of boxes. Given a positive link l its positive trunk P (l) is the set of links s.t. l has a positive path to them, and its negative arborescence P (l)N is the set of links having a non-empty negative path to a link in P (l) (in Fig. 4.b the negative arborescence of the external ! is given by the axioms, the contraction and two weakenings). Remark that P (l)N may contain positive links (d and ax links, see also Fig. 7.e), because some links are both positive and negative. Definition II.2 (correct net). Let I be an implicit net. I is correct when: • •
Rooted !-graph: there is only one initial link and every cycle passes through the negative edge of a !-link. Box: for any positive link l the positive links in P (l)N are also in P (l)2 .
As an example, the net in Fig. 4.b (considered with the jumps but without the box) is correct, in particular it has only one initial link, which is the dereliction that is not the target of a jump. The rooted !-graph condition allows a correct net to have cycles, but since !-links have only positive outgoing edges we get that the positive (resp. negative) subgraph of a correct implicit net is acyclic. A simple inspection of the shape of the links shows that moreover the positive (resp. negative) subgraph of a correct net is a forest, which is a fundamental property, to be referred as to the forest property. Induced boxes. Implicit boxes induce ordinary explicit boxes, as we are going to show. 2 It is easy to see that it is enough to consider only the case where l is a !-link and to ask that all the positive links with a negative path to l or cutsl (and not to P (l)) are in P (l).
Definition II.3 (induced box). Let I be a correct implicit net, l a positive link which is not a cut. The induced box (l) of l is the set of links P (l) ∪ P (l)N . In Fig. 4.b the boxes induced by implicit boxes are represented with round corners. The premises/conclusions orientation can be recovered from the polarized one by simply reversing positive edges (including jumps). With respect to that orientation the induced box of l is nothing else but the set of links with a path (possibly using jumps) to l, i.e. its causal past (test this fact on Fig. 4.b). Remark also that given a link l the reconstruction of (l) is straightforward, it is enough to follow polarized paths. In particular, the reconstruction is a local process. It is easy to see that (l) is a net. We consider it also an implicit net, with respect to the implicit boxes inherited by I, because by definition any !-link l has a positive path to all the positive links in ibox(l), and so ibox(l) ⊆ P (l) ⊆ (l). Definition II.4 (subnet). Let I be an implicit net. A subnet of I is a subset J of the links of I which is a correct implicit net and s.t. the implicit box of l in J and in I coincide, for any !-link l ∈ J. The following theorem collects the properties of induced boxes. We need the following terminology: a cut which does not belong to any implicit box is a ground cut. Theorem II.5 (properties of induced boxes). Let I be a correct net, l, l0 positive links of I which are not cuts. 1) Positive connected skeleton: l0 ∈ (l) iff l0 ∈ P (l). 2) Containment: (l) ⊇ (l0 ) iff l →◦ l0 . 3) Correctness: (l) is a subnet of I of initial link l; 4) Kingdom: Moreover, it is the smallest such one, 5) Border: the conclusions of (l) are l and a possibly empty set of {d, ax}-links, to which l has a positive path. Moreover, (l) has no ground cut. 6) Nesting: if (l) ∩ (l0 ) 6= ∅ then (l) ⊆ (l0 ) or (l0 ) ⊆ (l). 7) Collapsibility: the collapse of (l) preserves correctness. Relating implicit and explicit boxes. It is easy to read-back an explicit net I e from an implicit correct net I: just define ebox(l) as (l) for every !-link l (and discard implicit boxes). Conversely, given a correct explicit net E its translation is the implicit net E i obtained by defining ibox(l) as the implicit box given by the dereliction and the cuts at level 0 in ebox(l). For instance, the translation of the net in Fig. 4.a is the net in Fig. 4.b. The read-back of Fig. 4.b is the same net just without the jumps but with the grey induced boxes considered as explicit ones. By Theorem II.5.5 the explicit boxes of I e —coming from induced boxes—close on axioms and derelictions only; we say that they have minimal borders. By pushing as much as possible weakening, contractions, and `-links out of explicit boxes, every explicit net E can be unambiguously turned into an explicit net having minimal borders, noted E m .
ax ax
ax
w
1
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1
!N P ⊗Q
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Figure 9.
+ P cut
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η-expansion rules
Figure 8.
ax
ax ax
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⊗
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⊥ ` Q⊥ P ⊗Q P
⊥
1
ax Q
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c
c
... N1 Nk
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N1 . . . Nk
!
00
Rewriting rules for implicit proof nets.
Collapsibility of induced boxes and the nesting properties of explicit/implicit box are used to prove the relation between implicit and explicit correct nets: Theorem II.6 (implicit/explicit correct nets). Let I and E be a correct implicit and explicit net, respectively, l a {1, ax, !, ⊗}link in I and le the corresponding link in I e . Then: 1) Correctness: I e and E i are correct; 2) Explicitation: (I e )i = I, (E i )e = E m and (·)e is injective. 3) Positive trees and induced boxes: T (le ) = ((l))e . Remark that point 2 is in fact a sequentialization theorem: explicit proof nets can be sequentialized, i.e. be read-back as sequent calculus proofs, and so—composing read-backs— implicit proof nets can be sequentialized. Then, from now on a correct implicit net is an implicit proof net. Implicit nets are less bureaucratic and more parallel than explicit ones, i.e. (·)i is not injective. Indeed, if E m = F m then E i = F i . In particular, the translation identifies nets differing for permutations of `-links with box borders (Fig. 4.ab), which is a novelty. We conclude this section by sketching a comparison with our previous work [6], which developed implicit boxes for the λ-calculus with explicit substitutions3 : 1) Orientation: the two works use two incompatible orientations of the proof net. In [6] the initial link correspond to the output (i.e. the outermost constructor) of the λterm, while here it corresponds to the head variable. 2) Jumps: in [6] jumps are associated with all and only the exponential cuts (corresponding to explicit substitutions). While here jumps are associated with derelictions (corresponding to variable occurrences) and not necessarily to exponential cuts, nor necessarily to all cuts. 3) Quotient: the use of jumps in [6] induces a less parallel syntax. Explicit boxes already quotient more than the 3 In [6] the graphical syntax in use is not proof nets, but a proof nets-inspired formalism having the constructors of the λ-calculus as link. Here we compare them as if [6] used proof nets, via the call-by-name encoding of λ-calculus into linear logic.
formalism in [6] (see [52]), and here we quotient more than explicit boxes. 4) Language: MELLP encodes the λµ-calculus (with explicit substitutions), while the approach in [6] is limited to the λ-calculus (with explicit substitutions). Summing up, our study of implicit boxes for MELLP is not a slight extension of [6], but a complete re-elaboration, which catches a more expressive language, in a more parallel way, providing new insights on the structural role of polarity. III. I MPLICIT B OXES : DYNAMICS Simplifying hypothesis. To simplify the study of the dynamics of implicit proof nets we use two assumptions: 1) The net is contained in a box. More precisely, we assume that every cut is in some implicit box. In this way we avoid to double the cut-elimination rules for the cases when the cut is out of every implicit box. This assumption is not essential, it just simplifies the presentation. 2) We allow axioms only with atomic formulas, i.e. we restrict to η-expanded nets (the η-expansion rules are in Figure 8). The general case is discussed after the results on η-expanded nets. Cut-elimination rules. The set of rewriting rules and equivalences for implicit proof nets is in Figure 9. The non-linear steps (→c and →w ) are implemented by duplicating or erasing the induced box of the positive premise of the cut (note the round corners). Cut-elimination directly acts on implicit boxes by erasing, duplicating, or moving jumps. Erasing: the rules →ax− , →ax+ , and →w erase the cut link and also remove it from the implicit box ibox(!), i.e. they erase the jump. Duplicating: the rules →c and →` extend ibox(!), duplicating the jump. Composing: the rule →d is where implicit boxes are composed (an example is in Fig. 10.a). First, the propagated cut enters into (!0 ), which is why it has a jump from !0 and not from !00 . Second, also the content of (!)—where ! is the eliminated !-link—enters (!0 ), which is why the jumps on ! (represented with a double arrow) pass to !0 in the reduct.
w 0 i
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a) Critical pair between →w and →ax+ ; b) Neutrality closes the pair.
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Figure 13.
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Figure 12.
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a) Dynamic pushing of created weakenings; b) →ax+ and box borders.
Tc w←
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Figure 11.
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The rules →d and →ax− merge induced boxes (a is only an example). ax ax w
w
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Figure 10.
a)
+
≡ac
c
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w w ...
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a-c) Associativity, commutativity, and neutrality for contractions; d) Closure of the critical pair between →c and →ax+ .
Cut-elimination also acts on induced boxes. In particular, there is an implicit action on the border of induced boxes. Here is where the novelties about cut-elimination appear. It is not difficult to explicit this indirect action, because we know that borders are given by axioms and derelictions (Theorem II.5.5). There are three actions:
Let → be the union of all the rewriting rules in Fig. 9. As expected we have:
1) Merging of boxes without any commutative rule: both the rules →d and →ax− act on box borders, incorporating a big-step box-box rule, as it becomes evident if one represents induced boxes as in Fig. 10. Remark that there is no rule corresponding to the commutative box-box rule for explicit nets, whose role is subsumed by rules →d and →ax− . Therefore, we get only so-called key or principal cases of cut-elimination. 2) Dynamic pushing of created contractions and weakenings: the non-linear rules →c and →w introduce contractions and weakenings where previously there were dereliction and axioms. But induced boxes cannot close on contractions and weakenings, which are in fact automatically pushed out of boxes as much as possible. Figure 11.a shows the case for weakenings, the case for contractions is analogous. 3) Automatic permutations on the border: the rule →ax+ , acting on the (axiom) border of some induced boxes, induce permutations of negative links with box borders, see Fig. 11.b, where Tc is the (possibly empty) tree of contractions rooted in N . Remark that our assumption on η-expanded nets forbids `s to be in Tc and d-links to be leaves of Tc .
Strong normalization by simulation. We now show that implicit proof nets can be simulated by explicit proof nets. As already explained, the simulation of →d and →ax− steps requires a preliminary sequence of commutative steps, while →w , →c , and →ax+ steps require to push the created weakenings and contractions out of explicit boxes.
Proposition III.1 (Cut-elimination preserves correctness). Let I be an implicit proof net. If I → J then J is an implicit proof net.
Proposition III.2 (Simulation via read-back). Let I be an η-expanded implicit proof net. The read-back (·)e induces a simulation: 1) Rules ax− and d use explicit commutation: I →x J implies I e →∗ →x J e , for x ∈ {ax− , d}. 2) The non-linear rules push created weakenings and contraction: I →w I implies I e →w →∗pw J e , and I →c J implies I e →c →∗pc J e . 3) Rule →ax+ and border permutations: I →ax+ J implies I e →ax+ →∗pc →∗pw J e . 4) Multiplicative case: I →` J implies I e →` J e . From strong normalization of explicit nets (Theorem I.3) and Prop. III.2 we get: Corollary III.3. MELLP η-expanded implicit proof nets are strongly normalizing.
N
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M
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N `M
N
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MN
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M
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N `M
Figure 14.
cut
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a-b) Additional rules for `; c) New problematic critical pair.
Confluence. The new features of cut-elimination have a consequence: they require to work modulo associativity of contractions and to add neutrality of weakenings with respect to contractions. Indeed, the critical pairs between the nonlinear rules and →ax+ can be closed only using associativity and neutrality. Figure 12.a shows the critical pair justifying the neutrality rule (omitting irrelevant details), and Fig. 12.b shows how neutrality closes the pair. The similar pair where the weakening cut is replaced by a contraction cut requires associativity and neutrality in order to be closed (Fig.13.d shows how to close this pair). Implicit proof nets are then considered modulo ≡ac which is the equivalence defined as (∼com ∪ ∼a )∗ , see Fig. 13.a-b (commutativity is not essential), and now → denotes the cutelimination rules plus →n (Fig. 13.c). The next proposition shows that the read-back commutes with ≡ac and →n . Proposition III.4. Let I be an η-expanded implicit proof net. If I ≡ac J (resp. I →n J) then J is correct, η-expanded and I e ≡ac J e (resp. I e →n J e ). Now, from Proposition III.2 and Theorem I.3 it follows: Corollary III.5. η-expanded implicit proof nets are strongly normalizing modulo ≡ac . In the case of reduction modulo an equivalence relation there are various notions of confluence. The strongest is Church-Rosser modulo, which by a lemma due to Huet [53] follows from strong normalization whenever the system is locally confluent modulo and locally coherent modulo [54] (Lemma 14.3.11, pp 773-774). We need a definition: given I and J they are joinable modulo ≡ac , noted I♥J, iff exists I 0 , J 0 s.t. I →∗ I 0 , J →∗ J 0 and I 0 ≡ac J 0 . Proposition III.6. For η-expanded implicit proof nets the relation → is: 1) Locally confluent modulo ≡ac : if I → J1 and I → J2 then J1 ♥J2 . 2) Locally coherent with ∼a and ∼com : if I → J and I ∼a I 0 or I ∼com I 0 then I 0 ♥J. The use of ∼a and ∼com in the statement of local coherence is not an error: Huet’s lemma requires the property only for the generators ∼a and ∼com of ≡ac . We then conclude with: Corollary III.7 (Church-Rosser modulo ≡ac ). For η-expanded implicit proof nets → is Church-Rosser modulo ≡ac : if I(→ ∪ ← ∪ ≡ac )∗ J then I♥J. The general case. We now reduce the non-η-expanded case to the use of atomic axioms via η-expansion. Consider the ηrules in Figure 8, and let →η :=→η1 ∪ →η⊗ ∪ →η! . They
are clearly strongly normalizing (the formulas on the axioms strictly decrease in size) and confluent (all redexes are disjoint) modulo ≡ac (they do not interact with contractions). Then, given an implicit proof net I, it makes sense to consider its ηnormal form η(I). Note that the axiom rule applies to atomic axioms only, non-atomic axioms have to be η-expanded first. Proposition III.8. Let I be an implicit proof net, x ∈ {ax+ , ax− , `, d, w, c, n}, and y ∈ {a, com}. 1) Local commutation: J1 η ← I →x J2 implies exists H s.t. J1 →x H ∗η ← J2 . 2) Local coherence: J1 ≡y I1 →η I2 implies exists J2 s.t. J1 →η J2 ≡y I2 . 3) Projection of →x : If I →x J then η(I) →x η(J). 4) Projection of ≡ac : If I ≡ac J then η(I) ≡ac η(J) It is not difficult to derive strong normalization by projection on the η-expanded case, and then infer Church-Rosser as before. Corollary III.9. On implicit proof nets the reduction → ∪ →η is strongly normalizing and Church-Rosser modulo ≡ac . Despite these are remarkably strong results, there still is space for improvement. A detail that the purist would not judge completely satisfactory is that η-reduction is necessary in order to compute normal forms. Unfortunately, the removal of ηreduction has to face serious difficulties. Axiom rules for nonatomic axioms induce two problems that break the relationship with explicit boxes and forbid to apply standard techniques. The first problem is that the automatic permutation of negative links with box borders, which pushes a tree Tc of contractions out of boxes (Fig. 12.a), may now also involve `-links (i.e. `-links may be part of Tc ). These permutations, which forced us to work modulo associativity and to add the neutrality rule (Fig. 12.b and 13.d), now force to work with a similar equation and a similar rule for `s, shown in Fig. 14.a-b. It is easy to extend Laurent’s nets with these rules, the problem is rather that Laurent’s translation of MELLP to MELL does not simulate them, and so strong normalization cannot be obtained via a simulation. The second problem is that a new problematic critical pair appears, the one in Fig. 14.c (note the difference between ! and !0 in the two reducts). The difficulty is that the critical pair is non-local, in the sense that the pair does not contain enough information in order to be closed, one has to look to the context of the rule, as in [55]–[57]. But in contrast to those cases here the context cannot be rewritten independently of the pair, because reductions depend on implicit boxes, i.e. on where jumps are attached. So even local confluence is nontrivial and requires a new technique. Unfortunately, the state
of the art in terms of proof of strong normalization for proof nets relies on local confluence [51], [58]. On η-expanded nets this second problem also vanishes, because the positive premise of the right cut is always an axiom, and so the critical pair can be closed by simply removing both cuts. C ONCLUSIONS We showed how polarity can be exploited in order to get a purely local and implicit implementation of explicit boxes. Our syntax is supported by a correctness criterion, a detailed analysis of its relationship with explicit boxes, and results about cut-elimination. Acknowledgments. This work was partially supported by the Qatar National Research Fund under grant NPRP 09-11071-168. R EFERENCES [1] J.-Y. Girard, “Linear logic,” Theoretical Computer Science, vol. 50, pp. 1–102, 1987. [2] ——, “Proof-nets: The parallel syntax for proof-theory,” in Logic and Algebra. Marcel Dekker, 1996, pp. 97–124. [3] F. Lamarche, “Proof Nets for Intuitionistic Linear Logic: Essential Nets,” Research Report, 2008. [4] G. Gonthier, M. Abadi, and J.-J. L´evy, “Linear logic without boxes,” in LICS, 1992, pp. 223–234. [5] I. Mackie and J. S. Pinto, “Encoding linear logic with interaction combinators,” Inf. Comput., vol. 176, no. 2, pp. 153–186, 2002. [6] B. Accattoli and S. Guerrini, “Jumping boxes,” in CSL, 2009, pp. 55–70. [7] V. Danos and L. Regnier, “Proof-nets and the Hilbert space,” in Advances in Linear Logic. Cambridge University Press, 1995, pp. 307–328. [8] S. Guerrini, S. Martini, and A. Masini, “Coherence for sharing proof nets,” in RTA, 1996, pp. 215–229. [9] V. Danos and L. Regnier, “Reversible, irreversible and optimal lambdamachines,” Theor. Comput. Sci., vol. 227, no. 1-2, pp. 79–97, 1999. [10] J.-Y. Girard, “Light linear logic,” Inf. Comput., vol. 143, no. 2, pp. 175– 204, 1998. [11] V. Danos and J.-B. Joinet, “Linear logic and elementary time,” Inf. Comput., vol. 183, no. 1, pp. 123–137, 2003. [12] A. Asperti, “Linear logic, comonads and optimal reduction,” Fundam. Inform., vol. 22, no. 1/2, pp. 3–22, 1995. [13] P.-A. Melli`es, “Functorial boxes in string diagrams,” in CSL, 2006, pp. 1–30. [14] L. Tortora de Falco, “The additive mutilboxes,” Ann. Pure Appl. Logic, vol. 120, no. 1-3, pp. 65–102, 2003. [15] R. Di Cosmo, D. Kesner, and E. Polonovski, “Proof nets and explicit substitutions,” Math. Str. in Comput. Sci., vol. 13, no. 3, pp. 409–450, 2003. [16] A. Asperti, V. Danos, C. Laneve, and L. Regnier, “Paths in the lambdacalculus,” in LICS, 1994, pp. 426–436. [17] B. Accattoli and D. Kesner, “The structural λ-calculus,” in CSL, 2010, pp. 381–395. [18] ——, “Preservation of strong normalisation modulo permutations for the structural λ-calculus,” Logical Methods in Computer Science, vol. 8, no. 1, 2012. [19] B. Accattoli, “An abstract factorization theorem for explicit substitutions,” in RTA, 2012, pp. 6–21. [20] B. Accattoli and D. Kesner, “The permutative λ-calculus,” in LPAR, 2012, pp. 23–36. [21] B. Accattoli and U. Dal Lago, “On the invariance of the unitary cost model for head reduction,” in RTA, 2012, pp. 22–37. [22] B. Accattoli and L. Paolini, “Call-by-value solvability, revisited,” in FLOPS, 2012, pp. 4–16. ´ [23] O. Laurent, “Etude de la polarisation en logique,” Th`ese de Doctorat, Universit´e Aix-Marseille II, Mar. 2002. [24] ——, “Polarized proof-nets and λµ-calculus,” Theor. Comput. Sci., vol. 290, no. 1, pp. 161–188, 2003. [25] J.-Y. Girard, “A new constructive logic: Classical logic,” Math. Str. in Comput. Sci., vol. 1, no. 3, pp. 255–296, 1991.
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Theorem II.5 (page 6): 1) Positive connected skeleton: ⇒) if l0 ∈ (l) then either l0 ∈ P (l) or l0 ∈ P (l)N . The former case is exactly what we want to prove. In the latter case by the box condition for I we get that l0 —being positive—is in P (l). ⇐) By definition of induced box. 2) Containment: ⇒) If (l) ⊇ (l0 ) then l0 ∈ (l) and from point 1 we get l0 ∈ (l), i.e. l →◦ l0 . ⇐) From the definition of induced box it immediately follows that if l →◦ l0 then P (l0 ) ⊆ P (l), which implies P (l0 )N ⊆ P (l)N and so (l0 ) ⊆ (l). 3) Correctness: (l) satisfies the condition for being a net, because it is defined via the positive trunk and the negative arborescence. Initial link: by point 1 any positive link of (l) has a path from l. Then the only eventual initial link is l. And by correctness l is indeed initial: if (by contraction) there is l0 ∈ (l) with a nonempty positive path to l then there is a positive cycle, contradicting the forest property. !-graph: the eventual cycles of (l) are cycles of I, and thus they satisfy the rooted !-graph condition. Box condition: it follows from the definition of (l) and from the box condition for I. 4) Kingdom: The fact that a subnet—being a net—contains for every edge the link of which it is a conclusion forces a subnet J of root link l to contain the positive trunk P (l). For the same reason it contains the negative arborescence P (l)N , and so (l) ⊆ J. 5) Border: for a negative and non-positive conclusion, i.e. a {`, w, c}-conclusion, there is no positive link to which it has a negative path. Then, it cannot be a conclusion of (l), because by definition in (l) there is no negative link without a path to P (l). Thus, the conclusions 6= l are all both negative and positive. By point 1 l has a positive path to each one of them. By definition of induced box is also evident that any cut link in (l) belongs to the implicit box of some !-link in (l). 6) Nesting: Suppose that X := (l) ∩ (l0 ) contains a positive link i. Then by point 1 i ∈ P (l) ∩ P (l0 ) and both l and l0 have a positive path to i. By the forest property we get either l →◦ l0 or l0 →◦ l, and so by point 2 either (l) ⊆ (l0 ) or (l0 ) ⊆ (l). Now suppose that X contains a negative link n which is not positive, i.e. a {w, `}-link. By definition n has a negative path to a positive link i ∈ (l) and a negative path to a positive link i0 ∈ (l0 ). These two positive links i and i0 necessarily coincide, because by the forest property a {`, w, c}-link can have a negative path to at most one positive link (necessarily a {!, cut}- link). This implies i = i0 ∈ X. But i is positive, so we reduce to the previous case and conclude. 7) Collapsibility: Let J be I after the collapse of (l). Correctness conditions: •
Root: the collapse replaces a complete positive subtree with a positive node, therefore it cannot create new initial links, nor remove the initial link
•
•
of I without replacing it with an initial link. !-graph: suppose that there is a cycle c of J which is not a cycle of I. The c passed through the genax node of the generalized axiom introduced by collapsing. Let N the negative edge outgoing from genax used by c, and let n be the negative conclusion of (l) corresponding to N . By point 5 n is positive and by point 1 there is a positive path τ : l →◦ n. Now, consider the cycle d of I obtained from c by replacing the positive node genax introduced by the collapsing with τ , i.e. d := c{genax/τ }. By correctness of I, d passes through the negative premise edge of a !-link. But d is obtained from c by only introducing positive edges, therefore c uses the same negative premise of a !-link. Box: Let i be a positive link of J and let us prove the box condition for it. To disambiguate we extend the notations P (l) and P (l)N to PI (l) and PI (l)N , where I is the net where the two sets are computed. If i is the generalized axiom it is obvious, because it implies PJ (i)N = ∅. Then suppose that i is not the generalized axiom (and thus it is also in I), and let i0 be a positive link in PJ (i)N , i.e. having a negative path to a link p ∈ PJ (i). Note that p ∈ PI (i), because PJ (genax)N = ∅. We have to prove i0 ∈ PJ (i). Two cases: a) i0 is the generalized axiom. Hence, there is a negative conclusion n of (l) with a negative path to p, which is also a path of I. Then, by the box condition for I, there is a positive path τ : i0 →◦ i in I. This path cannot use any positive edge of (l), otherwise by point 1 there would be a positive cycle, contradicting the forest property. So, τ is also a path of J, and we conclude. b) i0 is not the generalized axiom, i.e. i0 ∈ / (l) and i0 ∈ I. By point 1 there is no positive path l →◦ i0 in I. By the box condition for I there is a positive path τ : i0 →◦ i, which then cannot pass through l nor any other positive link in (l). Then τ is also a path of J and we conclude.
Theorem II.6 (page 7): Let us define the level of an implicit net I as the maximum number of induced !-boxes in which a link of I is contained. We recall that similarly the level of an explicit net E is the maximum number of explicit !-boxes in which a link of E is contained. 1) Correctness: a) I e is an explicit net) By induction on the level k of I . If k = 0 then implicit and explicit nets are defined in the same way. If k > 0 then the readback of the interior of every induced !-box is an explicit net, because it has level smaller than k. The nesting property for implicit correct nets (Theorem
II.5, point 6) and the shape of induced box borders (same theorem, point 5) guarantees that the nesting and border conditions for explicit nets hold for I e . b) I e is correct) By induction on the level k of I. If k = 0 then there is no !-link, therefore I and I e coincide. Moreover, the correctness conditions coincide. Then let k > 0. Rooted DAG: correctness of (I e )0 follows by the fact that the collapsing of induced boxes preserves correctness (Theorem II.5, point 7). Let J be I where all induced boxes for !-links at level 0 have been collapsed, which is correct. The two nets (I e )0 and J coincide. Moreover, for nets without !-links the two correctness criteria also coincide, and so (I e )0 is a rooted DAG. Recursive correctness: let l be a !-link at level 0 in I e . The interior int(l) comes from the interior J of (l) in I. Correctness of J follows from Theorem II.5.3, and J has level k − 1, so that by i.h. its read-back int(l) is correct. c) E i is an implicit net) the condition about implicit boxes follows from the nesting property of E. And E i is uniquely defined (i.e. there is no ambiguity about the derelictions in the implicit boxes) because of the matching property for correct explicit nets. d) E i is correct) By induction on the level k of E. For k = 0 (which implies no !-link) it follows by correctness of E and the fact that E and E i coincide and the two notions of correctness coincide. Let k > 0 and {Bj }nj=1 be the interiors of the n !-links at level 0 in E. By induction hypothesis every implicit net IBj := (Bj )i is correct and has as initial link the dereliction at level 0 in Bj . Let us explain how E i relates to {IBj }nj=1 and E 0 , where E 0 is the 0-depth graph of E. Remark that: the positive subgraph of E 0 is a forest having exactly one initial link and whose leaves are {ax, 1}-links plus a set {genax}nj=1 of generalized axioms obtained by collapsing the boxes at level 0. The relation: E i is obtained from E 0 by replacing genaxj with !IBj , which is IBj extended with a !link lj plus the jumps of the implicit box ibox(lj ), for j = 1, . . . , n. Clearly, E i has exactly one initial link, the one of E 0 . Correctness conditions: •
!-graph: suppose that c is a cycle of E i which is not a cycle of E 0 nor of IBj for any j = 1, . . . , n (note that in fact by correctness E 0 is acyclic). The only possibility is that c involves some of the introduced !-links l1 , . . . , lj and enters in some IBj . We prove that c cannot contain a path which
•
enters in IBj and then exits from one of its conclusions which is not the premise of a !link. By contradiction. Let τ1 , . . . , τs the eventual sub-paths of c of this form. In E 0 the generalized axioms provides paths τ10 , . . . , τs0 s.t. τh0 is coinitial and cofinal to τh for h = 1, . . . , s. From c and these paths follows that there is a cycle c0 in E 0 , which is absurd. Box: here we rely on the footnote at the end of Definition II.2 (page 6). The only links for which we need to prove the box property are l1 , . . . , lk . Let lj be one of them and let i be a positive link with a negative path to P (lj ). By the nesting property i is in IBJ . Now, observe that by definition of the translation and correctness of IBj the positive sub-graph of !IBj is a tree and has initial link lj : indeed IBj is correct and so the roots of its positive forest are the dereliction and the cuts at level 0, to which lj has a path thanks to its implicit box. Then lj has a positive path to i and the box condition holds.
2) Explicitation: a) (I e )i = I) Let l be a !-link in I. Clearly calculating in I e the implicit box of l consists in taking the dereliction and the cuts at level 0 in (l) which is nothing else but ibox(l) in I. b) (E i )e = E m ) From Theorem II.5.5 follows that (·)e always gives a net with minimal borders. Then assume that if E = E m then (E i )e = E (proved in the next paragraph). Note that since implicit boxes do not record information about box borders one has that E m = F m implies E i = F i . We can now conclude with (E i )e = ((E m )i )e = E m . The proof that E = E m implies (E i )e = E is by induction on the level k of E. The case k = 0 is straightforward because E, E i , and (E i )e coincide. The case k > 0 uses the decomposition used in the proof that E i is correct (point ??) and the i.h.. c) (·)e is injective) Note that (·)e preserves the level, i.e. I and I e have the same level. One proves that I e = J e implies I = J by induction on the level k of I e = J e (and I, J). The case k = 0 is trivial because the absence of !-links implies I e = J e = I = J. Let k > 0. The i.h. applied to the interior of ebox(l) for every !-link l at level 0 in I e gives that I (l1 )\{l1 } and J (l2 )\{l2 } coincide, where l1 and l2 are the links of I and J mapping to l. Then I and J may only differ at level 0, which cannot happen because the translation at level 0 is the identity. Thus I = J. 3) Positive trees and induced boxes: by induction on the definition of positive tree T (le ). If le is a {1, ax}-link it is immediate. If it is a !-link then T (le ) is given by ebox(le ) which is defined as the image of (l). If le is a ⊗-link it follows by the i.h. and the fact that (l)
is exactly given by l and by the induced boxes of its premises. Proposition III.1 (page 8, ’cut-elimination preserves correctness’): In each case, when we deal with the box condition we rely on the footnote at the end of Definition II.2 (page 6). We give a detailed proof of the →d case, which is the complex and new one. The others are treated with less details. • Dereliction-bang. We recall the rule: N⊥
!
0 i
N
d
! cut
!00
i
N⊥ i
→d
N cut
!0
i i
!00
Remark: by correctness there is a positive path τ :!00 →◦ !0 in I, because d has a negative path to P (!00 ) (i.e. it is in P (!00 )N ) and so by correctness d (and consequently !0 ) is in P (!00 ). . – Initial link: The rule does not create new positive links, nor it adds incoming edges to previously initial links. The only positive edge erased by the rule is the jump from !00 , but that link has another outgoing positive edge (its premise), and so it does not get isolated. – !-graph) Convention: to simplify the language we identify the edges in the rule with their types and call the involved links simply !, !0 , !00 and d. The proof is by contradiction: suppose that there is a cycle c in J which is not a cycle of I and which does not pass through the negative edge of a !-link. If c uses one of the jumps j which were on ! then one easily gets a cycle d in I, it is enough to replace j with the path using the jump from !0 the cut and j itself. By correctness d uses a negative premise of a !-link which is necessary also an edge of c. There is only one possibility left: c uses the edges N and N ⊥ . Therefore, c passes through N and N ⊥ and induces a path σ in I from d to !, whose first and last edge are N ⊥ and N , respectively (i.e. the positive premise of d and the negative premise of !). Remark that d is outside (!), otherwise by correctness ! has a positive path to !0 which, using d and the cut, closes a cycle in I with no negative premise of a !-link. Then, the only way d can have a path using the negative premise of ! is by entering in (!) using !. This implies that σ passes through !0 or !00 . Cases: ∗ σ uses the negative edge of !0 or !00 . This is absurd because such edge would be in c, against hypothesis. ∗ σ uses the positive edge entering !0 . Since d is inside (!0 ) this means that σ gets out of (!0 ) from an auxiliary conclusion l which has a path to !0 . Call γ the sub-path l →!0 of σ. By Theorem II.5.5 there is a positive path ρ :!0 →◦ l. Both γ and ρ use no negative premise of a !-link (for γ
•
it follows by being a sub-path of σ for which this is true by hypothesis), and together they form a cycle in I, against correctness: absurd. ∗ σ does not use edges of !0 but it uses the positive edge entering !00 . We reduce to the previous case using the positive path τ :!00 →◦ !0 given by the initial remark: replace the positive sub-path !00 →◦ ! of σ with the positive path !00 →◦ !0 followed by the path !0 →! we get a path σ 0 of I, which closes a cycle d in J and which passes through the positive edge entering !0 . – Box: The proof is in three steps: 1) The box condition for !0 . Let l be a positive link with a negative path σ to a positive link l0 ∈ PJ (!0 ). If l0 is in PI (!0 ) then we conclude by correctness of I. Otherwise l0 is the propagated cut or an element of PI (!). In both cases σ can be seen as a path in I to an element of PI (!). By correctness of I there is a positive path ! →◦ l. The rewriting step is such that every positive path from ! in I induces a cofinal positive path from !0 in J. Therefore, there is a positive path !0 →◦ l in J, and we conclude. 2) The box condition for !00 . Let l be a positive link with a negative path σ to a positive link l0 ∈ PJ (!00 ). Note that PJ (!00 ) ⊆ PI (!00 ), and so σ can be seen as a path in I. By correctness there is a positive path γ :!00 →◦ l in I. If γ does not use the jump from !00 erased by the rule then γ is a path of J and we conclude. Otherwise, γ necessarily passes through !. Then in J there is a positive path ρ which is coinitial and cofinal to γ, which is obtained by replacing the sub-path !00 →◦ ! of γ with the positive path τ :!00 →◦ !0 given by the initial remark. 3) The box condition for every other !-link l: just remark that for l the situation in I and J is identical, and thus the box condition holds in J because it holds in I. Contraction-bang. The rule: P ⊥P ⊥
P⊥ P⊥
c
+ P cut
P⊥
!
i
P cut
(+) ... N1
Nk
→c
+
(+) ...
i i
(+) ...
P cut
!
+ c
c
... N1 Nk
Let us use redex pattern and reduct pattern to refer to the set of links on the LHS and RHS of the rule, respectively – Initial link: clearly the rule cannot alter the unicity of the initial link; – !-graph: consider a cycle c in J which is not a cycle of I and which is not entirely contained in one of the copies of the duplicated induced box. It must be that c contains some subpaths τ1 , . . . τh each one entering in the reduct pattern using P ⊥ or ! and leaving using one among N1 , . . . , Nk . For every such path τi there
•
is a coinitial and cofinal path σi in I, differing only for the initial/final contraction. Substituting τi with σi in c for every i we get a new cycle d in I. By correctness of I, d uses the negative premise of a !-link. By construction, such premise is also used by c and we conclude. – Box: by looking to the figure it is easily seen that the rule cannot break the box condition. All the positive links having a negative path to the reduced cut now have negative paths to the propagated cuts, which are attached to the same !, which can reach the same positive links. Weakening-bang. The rule: w P⊥
cut
!
•
+
w
(+) ...
P
N1
→w
Nk
w
Correctness conditions: – Initial link: the bang in the figure does not get isolated because by definition it has an outgoing jump to a dereliction. So the rule only removes noninitial links, and cannot affect the initiality of any other link. – !-graph: the rule eliminates edges and nodes so it cannot create new cycles. – Box: let l be a ! in J and l0 be a positive link having a negative path τ to P (l) (in J). Then, τ is a path of I, and by correctness there is a positive path σ : l →◦ l0 . This path is also a path of J, because if σ uses edges which have been removed then by the properties of induced boxes one gets l0 ∈ (+), which is absurd because then l0 would not be a link of J. Axiom rule ax− . ax X
X⊥
cut
!
!
!
ax X⊥
N⊥ 0 i
!
i
X cut
Proposition III.2 (page 8, ’simulation via read-back’): We need a definition. The box address of a link l is the set of !-boxes in which it is contained. This definition makes sense on both implicit and explicit proof nets, by taking explicit boxes in the former case and induced boxes in the latter (a !-link is not contained in its own box). The read-back (·)e preserves the box address, i.e. given l ∈ I and the link le ∈ I e have the same box address (more precisely if i1 , . . . , ik is the box address of l then ie1 , . . . , iek is the box address of le ). This can be proved by a straightforward induction on the level of I. It is also clear that if two explicit nets E and F have the same links and every link has the same box address in both nets then E = F . This is the principle used in the proof. 1) I →d,ax− J) We show the →d case, the →ax− case is similar and easier. We recall the rule:
X
→ax−
– Initial link: the bang in the figure does not get isolated because by definition it has an outgoing jump to a dereliction. Clearly, the initial link is preserved. – !-graph: Any cycle c in J can be traced back to a cycle d in I: in case c uses the positive edge in the reduct then d uses the edges of the reduced cut and axiom. Then the condition follows by correctness of I. – Box: Note that by the box condition in I the ! in the figure has a positive path τ to the reduced axiom. Then, whenever the box condition holds in I via a positive path using the removed jump and the reduced cut, it holds in J using τ . For the other !-links nothing changes. Axiom rule Iax+ J. X⊥
•
!
i
X
•
...
N1 . . . Nk
– Box: the case which is not immediate is when the reduced axiom has a negative path to a {!, cut}-link l. Let us assume that l is a !-link (the case of a cut link is similar, having a !-link just reduce the number of links that we have to manage). The subtlety is that any positive link l0 which has a negative path to the reduced cut in I gets a negative path to l in J. But by correctness of I there is a positive path from l to the axiom, and thus there is a positive path τ from l to the ! in the figure. Correctness also implies that the ! in the figure has a positive path σ to l0 . Composing τ and σ we get a positive path l →◦ l0 which is a path of J, and conclude. Par-tensor: the multiplicative case is analogous to the contraction case.
→ax+ !
i
– Initial link: as in the previous rule. – !-graph: as in the previous rule.
X⊥
N
d
! cut
00
!
i
N⊥ i
→d
N cut
!0
i i
!00
Let us abuse notations and simply call d and ! the cut d-link and !-link in I, respectively, and de and !e their corresponding links in I i . Let k be the difference of levels between the two cut links (d has level greater or equal to !, i.e., k ≥ 0). The action of the rule in I changes the box address of all and only the links in the (!), replacing the suffix !00 ; !; s in their box address with the sequence !00 ; . . . ; !0 ; s, where !00 ; . . . ; !0 is the sequence of !-links (of length k+1, where the 1 is due to !00 ) on the unique positive path from !00 to !0 (which exists by the box condition because d has a path to P (!00 )). Thus the links in I (!) increase their level of k (in J). By our hypothesis and definition of the translation, there are k explicit boxes closing on de . Hence applying k → -steps we get an explicit net where de and !e have the same box address, and then we can apply a →d step, getting E, i.e. I e →k →d E. It is clear that such sequence of rules increases the level of all and only the links in I (!) of exactly k, giving them the same box
address they have in J. Then E and J e have the same links with the same box addresses, and thus E = J e . 2) I →w J) We recall the rule: w
+
P⊥
cut
!
w
(+) ...
P
N1
→w
Nk
...
w
N1 . . . Nk
!
i
and its general form with induced boxes: w !
+ cut i
(+)
...
... N1
→w
...
!
w
w ...
N1
Nk
Nk
Let cut be the reduced cut-link in I and + its positive premise link. By Theorem II.6.3 the positive tree T (+e ) in I e is the translation of the corresponding induced box I (+) of + in I. Let E be the explicit net obtained by the reduction I e →w E, which eliminates the cut cute in I e corresponding to cut in I. By how the implicit and explicit weakening rules are defined we get that J and E have the same links, and that all links except the weakenings created by the rule have the same box address in both J and E. Let us focus on these weakenings then. In E they have the same box address of cute . In J they may have a different box address, since they are automatically pushed out of boxes, as the second image shows. It is enough to take the →pw normal form pweak(E) of E, which has minimal borders. It is easily seen that the address of the new weakenings in pw(E) and in J e coincide. I →c J) Essentially as the weakening case, where contraction are pushed out of boxes using pc until one gets a net of minimal borders. Note that there is no need of using pw because the links on the leaves of the created contractions, coming from the reduction of the image of an induced box, are derelictions. 3) I →ax+ J) We recall the rule: X⊥
ax X cut
!
X⊥
→ax+ !
i
X⊥
and its general form with induced boxes: ax ax w
w
!
N⊥
Tc
ax
...
N
!
cut i
→ax+
ax ax ... w
w
Tc
Corollary III.3, (page 8, SN for η-expanded PN): Suppose that there is an infinite reduction τ from an implicit proofnet I. By Prop. III.2 τ induces an infinite reduction τ 0 from I e . But this contradicts theorem I.3: absurd. Prop. III.4, (page 9): All properties about ≡ac and →n stated in the proposition are straightforward. Corollary III.5, (page 9, SN modulo ≡ac for for ηexpanded PN): Strong normalization modulo ≡ac follows from Proposition III.4, Proposition III.2, and Theorem I.3. Prop. III.6, (page 9): Given a cut link C we denote with →C the rewriting step which reduces C. 1) Local confluence modulo. Let C and C 0 be two cuts. The cases where C involves an induced box B (contraction, weakening and dereliction cut) and C 0 is strictly contained in B (in the sense that it is not even on its border) are not considered as critical pairs. However, they close according to the following diagrams: I ↓w L2
→C 0
L1 ↓w L2
=
→C 0
I ↓d L2
→C 0
L1 ↓d J
I →C 0 L1 ↓c ↓c L2 →C 0 →C 0 J In the →d and →c cases there is a slight abuse of notations, because →C 0 does not really make sense on L2 , but clearly we mean a step of the same kind of →C 0 reducing the residual(s) of →C 0 in L2 . There are three types of critical pairs. a) The firs kind of critical pairs is between two cuts C and C 0 , where C is a cut on a positive link l and C 0 is a cut on the border of (l). There are many cases: • C is a weakening cut. Subcases: – C 0 is a cut between a dereliction/axiom on the border of (l) and another positive +, for instance as in the following case:
N N
ax
e
It is clear that the only difference between J and the net E obtained by reducing the corresponding cut in I e (i.e. I e →ax+ E) is the box address of the contraction tree Tc (if any) and of its weakenings leaves (if any). As in the previous point it is enough to push contractions and weakenings to eliminate the difference, getting a net J e . Here the hypothesis about η-expanded nets is crucial, because it implies that Tc does not contain any `-link. 4) I →` J) Note that the multiplicative rule does not alter the box address of any link, both on implicit and explicit nets. Moreover, the two rules act exactly in the same way (modulo the presence of jumps, which are removed by the read-back).
w
P cut
!0
!
+
...
P
(+) ... N1
Nk
i
cut
!00
i
The critical pair closes as follows: I ↓w L2
→C 0 →w
L1 ↓w L3
i.e. after C we reduce the unique residual of C 0 which is now a weakening cut, while after C 0 we reduce the unique residual of C. – The reduction of C 0 is an →ax+ -step, as in Fig. 12.a. The critical pair closes as in
Where C is the cut with a jump from !0 . Note that because we restrict to the η-expanded case we get that the premise of the right cut is necessarily an axiom. The pair is easily seen to close as follows: I →ax+ L1 ↓ax− ↓ax− L2 →ax+ L3
Fig. 12.b, i.e. the diagram has the following form: →w
I ↓ax+ L2 •
L1 = L1
→w →∗n
C is a contraction cut. There are two subcases, as for the weakening cut: – C 0 is a cut between a dereliction/axiom on the border of (l) and another positive +, for instance as in the following case: ax
c
+
...
P
cut
!0
!
(+) ...
P
N1
Nk
i
cut
!00
i
b) The second kind of critical pair is given by the interaction between two non-linear cuts C and C 0 s.t. C is a cut between P and P ⊥ , C 0 is a cut between Q and Q⊥ contained inside (P ), and the borders of (Q) and (P ) have non empty intersection I. Cases: 0 • C is a weakening cut and C is a contraction cut. The diagram closes as follows:
The critical pair closes as follows: →C 0
I ↓c L2
→c →C 0 →C 0
i.e. after C we reduce the unique residual of C 0 which is now a contraction cut. This gives two new cuts of the same kind of C 0 in I, which have to be reduced. After C 0 , instead, we reduce the unique residual of C, which is a contraction cut. – The reduction of C 0 is an →ax+ -step, i.e. we have a situation like: ax ax w w Tc
ax cut
i
c
•
X ...
X⊥
•
X⊥
! cut
The diagram closes as follows: →C 0
I ↓c L2
•
→C 0 →C 0 →∗n ≡ac
L1 ↓c L3
After the →C 0 step one reduces the contraction cut, and get the tree Tc with contraction on the leaves, ad in the rightmost tree in Fig. 13.d. If instead one first reduces the contraction cut then one has to reduce the two residuals of C 0 (which are ax+ -cuts). But what one gets is the leftmost tree in Fig. 13.d. In order to close the pair one needs to transform one tree into the other, which can easily be done using associativity and commutativity of contraction plus neutrality of weakening with respect to contraction, as shown in Fig. 13.d. C is an ax+ -cut. Then the other cut is necessarily a ax− -cut, i.e. we get the following critical pair: ax
X
cut
!
0
i
ax
→w →∗n
→w
I ↓c L2 •
i
L1 ↓c L3
→w →w →∗n
C is a contraction cut and C 0 is a contraction cut. The pair closes as follows: →c
I ↓c L2
L1 ↓c L3
→c →c ≡ac
Depending on the order in which the two contraction cuts are reduced one gets different contraction trees on the auxiliary conclusions of the involved induced boxes. But these trees are easily seen to be equivalent modulo associativity. c) The third kind of critical pair does not involve box borders, and there is only one such critical pair. It is given by the interaction between contraction cuts and the neutrality rule. The pair is as follows: P⊥
P⊥
c
+
P cut
cut
!
L1 = L1
Reducing first the contraction cut we get that the border of (P ) grows: the links in I are duplicated and contracted. If we then reduce the unique residual of C in L2 we get that all the contractions created by the first step have a pair of weakenings as premises. It is enough to apply neutrality to each of these contraction to close the diagram. C is a weakening cut and C 0 is a weakening cut. Trivial. C is a contraction cut and C 0 is a weakening cut. The case is similar to the first one and closes as follows:
w
X⊥
→w
I ↓c L2
L1 ↓c L3
!
i
(+) ... N1
Nk
It closes as follows: →n
I ↓c L2
L1 = L1
→w →∗n
After the contraction steps one gets a weakening cut and another cut, plus contractions on the auxiliary conclusions of the two copies of the duplicated induced box. By reducing the weakening cut one gets many new neutrality redexes, and by reducing all of them one closes the pair. 2) Local coherence. The case of ∼com is straightforward, so we skip it. There are only two cases in which ∼a has a non-trivial interaction with the reduction rules: a) Interaction with duplication: c
c c
+ cut
!
(+) ... N1
∼a
c
+ cut
Nk
!
i
(+) ... N1
Nk
η-step on one side of the diagram and by just doing the non-linear step on the other side. 2) The statement follows by a straightforward induction on the length of the reduction I →∗η η(I) (which is welldefined by the remark), using point 1. 3) Immediate, because there is no interaction between →η and contractions. 4) by a straightforward induction on the length of the reduction I →∗η η(I) (which is well-defined by the remark), using point 2. Corollary III.9 (page 9, SN and CR Modulo ≡ac ): Let ⇒:=→ ∪ →η and for a rewriting relation and an equivalence ≡ let us denote /≡ the reduction modulo ≡, i.e. ≡ ≡. • Strong normalization modulo: suppose (by contradiction) that there is an infinite ⇒/≡ac -reduction sequence τ . Since →η is strongly normalizing modulo ≡ac , τ has the following form:
i
+ ∗ 0 I0 →∗η/≡ac I00 →+ /≡ac I1 →η/≡ac I1 →/≡ac I2 . . .
The diagram closes as follows: →c
I ∼a L2
→c →c ∼∗a
In other words by reducing in both nets the duplication cut and then the created duplication cut, one gets in both cases two contraction for each of the auxiliary conclusions of (+), but disposed differently: just note that for each conclusions its enough to apply associativity in order to close the diagram. b) Interaction with neutrality: w c c
∼a
↓n c
w
c c
↓n =
with an infinite number of →+ /≡ac reductions. By Proposition III.8, τ projects on η-normal forms as the following diverging reduction:
L1 ↓c L3
c
Corollary III.7 (page 9, CR Modulo ≡ac for η-expanded PN): It follows from Huet’s lemma ( [54], Lemma 14.3.11, pp 773-774) whose hypothesis are given by Prop. III.4 and Prop. III.6. Proposition III.8 (page 9, properties of →η ): Remark: Note that any two η-redexes are disjoint, so →η enjoys the diamond property and all reductions to normal form have the same length. 1) η-redexes can be duplicated/erased but they cannot duplicate/erase any other redex. The only cases where the steps do not simply permute are →c and →w , when the induced box to duplicate or erase contains the nonatomic axiom to η-expand. But these spans are closed as usual with non-linear steps: by duplicating/erasing the
+ 0 η(I0 ) ≡ac η(I00 ) →+ /≡ac η(I1 ) ≡ac η(I1 ) →/≡ac η(I2 ) . . .
•
contradicting Theorem III.3, absurd. Church-Rosser modulo: it follows by Huet’s lemma, using strong normalization modulo, local confluence modulo, and local coherence as in the proof of Corollary III.7. Local confluence modulo for η is straightforward, because →η is simply locally confluent, the modulo is not needed to close the diagram. Local coherence of η is given by Proposition III.8.2. For the other rules we rely on Proposition III.6.
