Linear lambda Calculus with Explicit Substitutions as Proof-Search in Deep Inference

SBV is a deep inference system that extends the set of logical operators of multiplicative linear logic with the non commutative operator Seq. We introduce the logical system SBVr which extends SBV by adding a self-dual atom-renaming operator to it. We prove that the cut elimination holds on SBVr. SBVr and its cut free subsystem BVr are complete and sound with respect to linear lambda calculus with explicit substitutions. Under any strategy, a sequence of evaluation steps of any linear lambda-term M becomes a process of proof-search in SBVr (BVr) once M is mapped into a formula of SBVr. Completeness and soundness follow from simulating linear beta-reduction with explicit substitutions as processes. The role of the new renaming operator of SBVr is to rename channel-names on-demand. This simulates the substitution that occurs in a beta-reduction. Despite SBVr is a minimal extension of SBV its proof-search can compute all boolean functions, as linear lambda-calculus with explicit substitutions can compute all boolean functions as well. So, proof search of SBVr and BVr is at least ptime-complete.

💡 Research Summary

The paper introduces SBVr, an extension of the deep‑inference system SBV, by adding a self‑dual atom‑renaming operator. SBV already enriches multiplicative linear logic (MLL) with a non‑commutative sequencing connective Seq, allowing proof‑search to be performed deep inside formulas rather than only at the top level. The new renaming operator, denoted ρ, can replace any atom A by another atom B (and simultaneously ¬A by ¬B), thereby providing an on‑demand channel‑name substitution mechanism inside proofs.

The authors first formalise the syntax of SBVr: formulas are built from atoms, the multiplicative connectives ⊗ (tensor) and ⅋ (par), the non‑commutative Seq, and the renaming operator ρ(A↦B). They then present a set of inference rules that includes the usual deep‑inference rules (switch, medial, associativity, commutativity) together with propagation rules for ρ (e.g., ρ distributes over ⊗ and Seq) and a cut rule. A central technical contribution is the cut‑elimination theorem for SBVr. By a structural induction on the size of proofs, they show that any cut can be eliminated using only the propagation rules of ρ, preserving the shape of the surrounding context. This guarantees consistency and normalisation of the system.

Next, the paper turns to linear λ‑calculus with explicit substitutions (LλES). In LλES, substitution is not a meta‑operation but an explicit term constructor, which makes the operational semantics more amenable to a logical encoding. The authors map each λ‑term M to an SBVr formula φ(M) as follows: a variable x becomes an atom x; abstraction λx.M is encoded as Seq(ρ(x↦·), φ(M)), where the dot stands for a fresh placeholder; application (M N) is encoded as the tensor φ(M) ⊗ φ(N). The explicit substitution