A Classical Linear $λ$-Calculus based on Contraposition

We present a novel linear $λ$-calculus for Classical Multiplicative Exponential Linear Logic (\MELL) along the lines of the propositions-as-types paradigm. Starting from the standard term assignment for Intuitionistic Multiplicative Linear Logic (IMLL), we observe that if we incorporate linear negation, its involutive nature implies that both $A\multimap B$ and $B^\perp\multimap A^\perp$ should have the same proofs. The introduction of a linear modus tollens rule, stating that from $B^\perp\multimap A^\perp$ and $A$ we may conclude $B$, allows one to recover classical MLL. Furthermore, a term assignment for this elimination rule, {the study of proof normalization in a $λ$-calculus with this elimination rule} prompts us to define the novel notion of contra-substitution $t { a \backslash!\backslash s }$. Introduced alongside linear substitution, contra-substitution denotes the term that results from “grabbing” the unique occurrence of $a$ in $t$ and “pulling” from it, in order to turn the term $t$ inside out (much like a sock) and then replacing $a$ with $s$. We call the one-sided natural deduction presentation of classical MLL, the $λ_{\rm MLL}$-calculus. Guided by the behavior of contra-substitution in the presence of the exponentials, we extend it to a similar presentation for MELL. We prove that this calculus is sound and complete with respect to MELL and that it satisfies the standard properties of a typed programming language: subject reduction, confluence and strong normalization. Moreover, we show that several well-known term assignments for classical logic can be encoded in $λ_{\rm MLL}$.

💡 Research Summary

The paper introduces a new one‑sided natural‑deduction presentation of Classical Multiplicative‑Exponential Linear Logic (MELL) that stays within the propositions‑as‑types paradigm while preserving the classical symmetry of contraposition. Starting from the standard term assignment for Intuitionistic Multiplicative Linear Logic (IMLL), the authors observe that linear negation is involutive, so the two implications (A⊸B) and (B^{⊥}⊸A^{⊥}) should be identified. To recover classical MLL they add a linear modus‑tollens rule: from a term of type (B^{⊥}⊸A^{⊥}) together with a term of type (A) one may infer a term of type (B). This rule is the only genuinely classical rule in the system.

To give a computational meaning to modus‑tollens they introduce contra‑application (t\ ▼\ s) (the term‑level counterpart of the new elimination rule) and a novel substitution operation called contra‑substitution (t{a\!\ s}). While ordinary linear substitution implements the usual β‑reduction ((λa.t)·s → t{a:=s}), contra‑substitution implements the reduction ((λa.t) ▼ s → t{a\!\ s}). Intuitively, the unique free occurrence of the variable (a) in (t) is “grabbed”, the term is turned inside‑out, and the variable is replaced by (s). The definition proceeds by structural induction on terms and relies crucially on linearity (each variable occurs exactly once).

The resulting calculus, named λ_{MLL}, contains the usual linear λ‑calculus constructs (variables, tensor pair, tensor elimination, λ‑abstraction, application) plus the new contra‑application construct. Typing rules are given in the familiar introduction/elimination style; the two elimination rules for implication are m‑e⊸1 (modus ponens) and m‑e⊸2 (modus tollens). The authors prove:

• Soundness: If Γ ⊢ t : A in λ_{MLL} then the sequent ⊢ Γ^{⊥}, A holds in MLL.
• Completeness: For any MLL‑valid sequent ⊢ Γ, picking any formula as the conclusion yields a typing derivation Γ^{⊥} ⊢ t : A in λ_{MLL}. The construction is effective, extracting a term directly from the proof tree.
• Subject reduction: Both ordinary substitution and contra‑substitution preserve types.
• Confluence and strong normalization: Reduction is defined via case‑contexts (a form of reduction at a distance) and enjoys the usual properties; a structural equivalence ≡ is introduced and shown to be a strong bisimulation.

The paper also shows that without the modus‑tollens rule the system loses completeness, providing explicit counter‑examples.

Extending to exponentials, the authors define λ_{MELL} by adding the usual ! (of course) and ? (why‑not) rules, together with appropriate typing for !‑promotion and ?‑weakening/dereliction/contraction. They demonstrate that contra‑substitution interacts well with these modalities, preserving linearity and normalization. Consequently, λ_{MELL} is sound and complete for full MELL, and retains strong normalization and confluence.

A significant contribution is the demonstration that several well‑known classical term assignments can be encoded in λ_{MELL}:

• Parigot’s λµ calculus (via the Danos‑Joinet‑Schellinx translations),
• Curien‑Herbelin’s λµ˜µ calculus,
• Hasegawa’s µ‑DCLL.

These encodings are compositional and respect the structural equivalence, showing that λ_{MELL} subsumes existing classical λ‑calculi while offering a more symmetric, substitution‑based operational semantics.

Finally, the authors discuss a structural equivalence relation defined as a strong bisimulation, which guarantees that any two terms related by ≡ have the same normal form. This provides a robust notion of program equality suitable for reasoning about programs derived from classical linear proofs.

In summary, the paper presents a novel, elegant computational interpretation of classical linear logic based on a new contra‑substitution mechanism. It achieves a fully functional, strongly normalizing, confluent λ‑calculus for MELL, preserves the classical contraposition symmetry, and unifies several existing classical λ‑calculi under a single, principled framework. This work opens avenues for functional programming languages and proof assistants that exploit the resource‑sensitive nature of linear logic while retaining classical reasoning capabilities.