Proofnets and Context Semantics for the Additives

We provide a context semantics for Multiplicative-Additive Linear Logic (MALL), together with proofnets whose reduction preserves semantics, where proofnet reduction is equated with cut-elimination on MALL sequents. The results extend the program of Gonthier, Abadi, and Lévy, who provided a “geometry of optimal λ-reduction” (context semantics) for λ-calculus and Multiplicative-Exponential Linear Logic (MELL). We integrate three features: a semantics that uses buses to implement slicing; a proofnet technology that allows multidimensional boxes and generalized garbage, preserving the linearity of additive reduction; and finally, a read-back procedure that computes a cut-free proof from the semantics, which is closely related to full abstraction theorems. Peut-être que la logique se trompe. —Yves Lafont Linear Logic [4, 7] appeared in 1987 and turned out quickly to be an interesting tool to model programming languages, specifically reasoning that is sensitive to the notion of consumable resources. Indeed the multiplicative fragment of Linear Logic (⊗, ) allows linear products (pairing and unpairing), implementing functions: a context pairs a continuation and an argument, a function unpairs and connects the two. The additive fragment (⊕, &) allows linear sums (injection and case dispatch), implementing features of processes in the style of CSP or CCS [3, 17, 12]. The crucial difference between these two components is the way they take care with consumption of resources. The exponential fragment implements sharing of resources: arguments, control contexts. We can then implement, for example, graph reduction technology for λ-calculus with control operators (call/cc, abort, jumps), and related mechanical proof systems for classical logic—taking care of the sharing and copying, implicit in these calculi [16, 18, 11]. Linear Logic was initially a sequent calculus, but this sequentialized structure was too strong and the design of a nice cut-elimination procedure was complicated. Therefore, proofnets were introduced as a more flexible representation of proofs [8, 13]. Further, Geometry of Interaction (GoI) developed the idea that the reduction of proofs can be seen as a local interaction process [5, 6]. Its intensional features provide a mediating Purgatory between the Heaven of denotational semantics, and the Hell of operational semantics. GoI was simplified in the “geometry of optimal λ-reduction” by Gonthier, Abadi and Lévy [9, 10] in the context of the MELL fragment. They reduced Hilbert spaces to simple data-structures, known as context semantics, and developed a proofnet technology which implemented the context semantics locally. Reduction on proofnets preserves the semantics, and Lamping’s algorithm for optimal reduction of λ-terms [14] is a method of graph reduction. They further indicated how to read back any part of the Böhm tree (normal form) of a λ-term from its context semantics. Can this program be carried out for full Linear Logic? In this paper we extend these result to the MALL fragment (multiplicatives and additives): this may be a step towards a satisfactory proofnet syntax for full Linear Logic with a good characterization of proofs. The MALL fragment is quite problematic since it does not have a nice cutelimination procedure—unlike MLL, which enjoys a straightforward one. Part of the work will be to understand and improve the reduction procedure for MALL. The main contributions of this paper are to provide an integrated development of (1) a context semantics for the MALL fragment; (2) a proofnet technology allowing normalization of MALL proofs, using the ideas of multidimensional boxes and generalized garbage; and (3) a read-back procedure that inputs a valid context semantics and outputs a normalized proofnet. In Section 1, we recall some basic definitions on MALL, proofnets and Linear Logic. We introduce in Section ?? a form of context semantics for MALL proofs, and we derive from it a bus-notation based proofnet syntax in Section 3. Then, we envisage the main problems that come from the reduction on the additives in Section 4. The first difficulty is that the additive cut-elimination is not really linear, since an additive reduction step erases a whole part of the proof (this is also a problem for the locality of the reduction we would like to achieve). The second problem also stems from the additives: the way one should reduce a cut involving auxiliary formulas of two &-links is unclear. The solution to this problem will come from an extension of the MALL syntax. After this adaptation, we will be able to get a much more satisfactory cut-elimination procedure. In Section 5, we show how a normalized proof can be read-back from the context semantics of a proof, and we see how this result can be related to a form of full completeness. In Section 6, we compare our approach with other works. 1 MALL: Proofs, Nets, Reduction Definition 1 (Formula). The formulas of MALL are generated by the grammar F −→ V | V ⊥ | F ⊗F | F F | F&F | F ⊕F where V ranges over variables, ⊗ and are the conjunction and disjunction of the multiplicative component, & and ⊕ are the conjunction and disjunction of the additive component; (−) is the involutive negation on literals. The atomic negation can be extended to the formulas as a defined involutive connector, using the De Morgan identities (A⊗B) = A B and (A&B) = A ⊕ B. We use right-handed sequents, where all sequent formulas play the same role. Definition 2 (Sequent). A sequent is a multiset of formulas ` F0, . . . , Fn−1 (also simply written F0, . . . , Fn−1). Figure 1 gives the MALL rules: Ax and Cut are the identity rules. The rules ⊗ and (resp. &, ⊕0 and ⊕1) are the multiplicative (resp. additive) rules. Ax ` A, A⊥ ` Γ, A ` ∆, A⊥ Cut ` Γ, ∆ ` Γ, A ` ∆, B ⊗ ` Γ, ∆, A⊗B ` Γ, A, B ` Γ, A B ` Γ, A ` Γ, B & ` Γ, A&B ` Γ, A ⊕0 ` Γ, A⊕B ` Γ, B ⊕1 ` Γ, A⊕B Fig. 1. The rules of the Multiplicative and additive fragment Definition 3 (Prooftree). A prooftree (or MALL-prooftree) is a tree whose leaves are sequents, linked by the rules showed in Figure 1. There is exactly one introduction rule for each additive or multiplicative connector (except ⊕); the principal formula of a link is the new formula introduced, and the other formulas are auxiliary. The cut formulas of a cut-link are the two hypotheses that are eliminated in the conclusion (A and A in Figure 1). An immediate cut is a cut link whose cut formulas are the principal formulas of the two links above the cut. The ports of a prooftree are the formulas in the final proof link. Since full linear logic has a cut-elimination procedure, so does MALL (see [4]): Theorem 1 (Cut-elimination). There exists an algorithm which inputs a prooftree π of MALL sequent S, and outputs a prooftree π of the same sequent S without any occurrence of the Cut-rule. A well-known interpretation of the connectives is economic [7]: the formulas of a sequent are the algebraic terms of a trade. Negation represents need, and involution (A = A) means if you need to need, you have. A proof of ` Γ, A means you only need (a proof of) A to produce (a proof of) Γ ; thus the Ax-rule says that you need A to produce A. Similarly, the Cut-rule says if you need a proof of Γ to produce a proof of A, and you need a proof of A to produce a proof of ∆, then transitively “trading” the need and production of A, you need a proof of Γ to produce a proof of ∆— thus ` Γ, ∆ is provable. The multiplicative and additive rules describe two related trading situations (see Figure 1). In the former, we need Γ, ∆ to produce both A and B; in the latter, we need Γ as a sufficient resource to prove either A or B. 1.1 Cut-elimination and proofnets Cut-elimination for MALL is described by a collection of local rewriting rules that push Cut-links upwards and make them disappear; these rules appear in the proof of Theorem 1. The rules are not confluent, partly because the prooftree syntax introduces unnecessary sequentializations. For instance, if we start with the proof: π0 ` Γ, A, B, F ` Γ, A B, F π1 ` ∆, C, F ⊕0 ` ∆, C ⊕D, F Cut ` Γ, ∆, A B, C ⊕D then we can rewrite it in two steps to a proof ending with one of the following sequences of links: ·· ` Γ, ∆, A, B, C ` Γ, ∆, A B, C ⊕0 ` Γ, ∆, A B, C ⊕D ·· ` Γ, ∆, A, B, C ⊕0 ` Γ, ∆, A, B, C ⊕D