On the role of connectivity in Linear Logic proofs

We investigate a property that extends the Danos-Regnier correctness criterion for linear logic proof-structures. The property applies to the correctness graphs of a proof-structure: it states that any such graph is acyclic and the number of its connected components is exactly one more than the number of nodes bottom or weakening. This is known to be necessary but not sufficient in multiplicative exponential linear logic to recover a sequent calculus proof from a proof-structure. We present a geometric condition on untyped proof-structures allowing us to turn this necessary property into a sufficient one: we can thus isolate fragments of linear logic for which this property is indeed a correctness criterion. In a suitable fragment of multiplicative linear logic with units, the criterion yields a characterization of the equivalence induced by permutations of rules in sequent calculus. In intuitionistic linear logic, the property is equivalent to the familiar requirement of having exactly one output conclusion, and it is sufficient for sequentialization in the axiom-free setting.

💡 Research Summary

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The paper investigates a structural property of linear‑logic proof‑structures that refines the classic Danos‑Regnier correctness criterion (ACC). ACC requires that every switching graph of a proof‑structure be acyclic and connected. While this condition is both necessary and sufficient for the purely multiplicative fragment (MLL⁻), it fails to be sufficient in richer fragments such as multiplicative‑exponential linear logic (MELL) because the presence of weakening and the unit ⊥ (bottom) breaks connectivity.

The authors introduce a quantitative extension called ACC♯w: for a given proof‑structure, let cc be the number of connected components of any switching graph and w the number of weakening or bottom nodes. ACC♯w demands the equality cc = w + 1. This relation is shown to be a necessary condition for any proof‑structure that corresponds to a genuine sequent‑calculus proof, but it is not sufficient on its own (a counter‑example is provided in Figure 3a).

To turn ACC♯w into a sufficient condition, the paper defines a geometric restriction denoted (¬ w ⊗). Intuitively, (¬ w ⊗) forbids any tensor (⊗) node from having a premise that is the conclusion of a weakening (or bottom) node. Formally, it can be expressed as a local graph property: for every ⊗‑node, neither of its two premises lies on a directed path that ends at a weakening/bottom node. Lemma 45 characterises (¬ w ⊗) purely in graph‑theoretic terms.

The central result, Theorem 47, states that for untyped proof‑structures satisfying (¬ w ⊗), ACC♯w is both necessary and sufficient for sequentializability: such a structure can be reconstructed as a cut‑free sequent‑calculus proof. This theorem is proved in an untyped setting, emphasizing that the sequentialization process itself does not depend on formula types, although types are later used to identify interesting fragments where (¬ w ⊗) holds automatically.

The paper then applies this framework to several fragments:

1. MLL (multiplicative linear logic with units) – Section 4 shows that in the (¬ w ⊗) fragment of MLL, ACC♯w exactly characterises the equivalence class of proofs modulo rule permutations. Consequently, the decision problem “are two cut‑free MLL proofs equivalent up to permutations?” which is PSPACE‑complete in general, becomes polynomial‑time solvable for (¬ w ⊗)‑proof‑structures (Corollary 66).

2. MELL (multiplicative‑exponential linear logic) – Section 5 extends the results to the exponential fragment. By restricting to cut‑free proof‑structures that satisfy (¬ w ⊗), the authors recover a sequentialization theorem (Theorem 47) and obtain a characterization of the polarized fragment (Corollary 72). This demonstrates that even with exponentials, the simple connectivity condition can be made sufficient when the local (¬ w ⊗) constraint holds.

3. IMELL (intuitionistic MELL) – Section 6 studies intuitionistic variants where (¬ w ⊗) does not hold in general. Proposition 77 shows that for intuitionistic proof‑structures satisfying ACC, the familiar “exactly one output conclusion” condition is equivalent to ACC♯w. Moreover, for the intuitionistic version of COMLL (I‑COMLL), Theorem 82 provides a sequentialization theorem by inserting “canonical jumps”, mirroring the approach used for the classical fragments.

Throughout, the authors discuss the stability of the various properties under cut‑elimination. ACC♯w is shown (Theorem 24) to be stable, while (¬ w ⊗) is not; this motivates the use of types in Sections 4 and 5 to isolate fragments where (¬ w ⊗) is preserved.

The contribution of the paper is twofold. First, it reveals a clean quantitative relationship between the number of weakening/bottom nodes and the connectivity of switching graphs, giving a simple necessary condition for correctness. Second, by pairing this condition with the local (¬ w ⊗) restriction, the authors obtain a sufficient correctness criterion for several important fragments of linear logic, including those with units, exponentials, and intuitionistic constraints. The results have practical implications for proof search, automated proof‑checking, and the meta‑theory of type systems based on linear logic, as they provide a graph‑based, polynomial‑time checkable criterion for a large class of proof‑structures that were previously intractable.