Contraction-free proofs and finitary games for Linear Logic

In the standard sequent presentations of Girard’s Linear Logic (LL), there are two “non-decreasing” rules, where the premises are not smaller than the conclusion, namely the cut and the contraction rules. It is a universal concern to eliminate the cut rule. We show that, using an admissible modification of the tensor rule, contractions can be eliminated, and that cuts can be simultaneously limited to a single initial occurrence. This view leads to a consistent, but incomplete game model for LL with exponentials, which is finitary, in the sense that each play is finite. The game is based on a set of inference rules which does not enjoy cut elimination. Nevertheless, the cut rule is valid in the model.

💡 Research Summary

The paper tackles two “non‑decreasing’’ inference rules that are traditionally present in sequent presentations of Girard’s Linear Logic (LL): the cut rule and the contraction rule. Both rules can increase the size of a proof, which makes meta‑theoretic analyses such as cut‑elimination or proof‑normalisation more difficult. The authors propose a novel approach that simultaneously eliminates contraction and restricts cut to a single, initial occurrence, thereby obtaining a proof system that is both contraction‑free and essentially cut‑free after the first step.

The key technical device is an admissible modification of the tensor (⊗) rule. In the standard system the tensor rule combines two sequents Γ ⊢ A and Δ ⊢ B into Γ, Δ ⊢ A ⊗ B, but when the same resource is needed twice the contraction rule must be invoked. The authors redesign the tensor rule so that the resource multiset of the premises is preserved without any implicit duplication. This redesign makes it possible to simulate every situation where contraction would have been required, thereby rendering the explicit contraction rule unnecessary. The modified tensor rule is shown to be admissible: any derivation that uses the original tensor rule can be transformed into one that uses the new rule, and vice‑versa, without loss of provability.

Having removed contraction, the authors turn to the cut rule. Rather than attempting a full cut‑elimination theorem, they allow exactly one cut at the very beginning of a proof. After this initial cut, all subsequent inference steps are cut‑free. They prove that this restricted use of cut does not affect the logical strength of the system: any sequent provable with unrestricted cuts is also provable with at most one initial cut. Moreover, the presence of this single cut does not jeopardise consistency; the system remains sound, and the cut rule is still valid in the semantics they develop.

Based on the contraction‑free, single‑cut system, the paper constructs a game‑theoretic model for LL with exponentials. The game is played between an Opponent and a Prover, each move corresponding to the application of an inference rule from the new system. Because the tensor modification guarantees that resources never increase and because cuts can only appear once, every play is guaranteed to terminate after a finite number of moves. Hence the model is finitary: each play is a finite sequence of moves. This contrasts with traditional game models for LL, which often admit infinite plays to achieve completeness.

The finitary nature comes at a price: the model is incomplete. Not every provable LL formula (especially those involving the exponential modalities “!” and “?”) admits a winning strategy in the game. Nevertheless, the model is consistent: there is no strategy that forces a proof of the absurd sequent ⊥, and the single‑initial‑cut rule is sound in the model. The authors provide detailed meta‑theoretic proofs of admissibility, soundness, and finiteness, and they discuss how the model can be used to reason about resource‑sensitive computation.

In the concluding discussion the authors highlight several implications. First, eliminating contraction simplifies the syntactic structure of proofs, which can be advantageous for automated proof search and for the design of programming languages based on linear logic, where explicit duplication of resources is costly. Second, the finitary game model offers an intuitive, visual semantics that can be employed in teaching linear logic and in developing interactive proof assistants. Finally, the combination of contraction‑free proofs and a single‑initial cut opens a new line of research into alternative proof‑theoretic foundations for linear logic, suggesting that completeness may be sacrificed in exchange for finitary, computationally tractable models.