Proof Complexity of Linear Logics

Proving proof-size lower bounds for $\mathbf{LK}$, the sequent calculus for classical propositional logic, remains a major open problem in proof complexity. We shed new light on this challenge by isolating the power of structural rules, showing that their combination is extremely stronger than any single rule alone. We establish exponential (resp. sub-exponential) proof-size lower bounds for $\mathbf{LK}$ without contraction (resp. weakening) for formulas with short $\mathbf{LK}$-proofs. Concretely, we work with the Full Lambek calculus with exchange, $\mathbf{FL_e}$, and its contraction-extended variant, $\mathbf{FL_{ec}}$, substructural systems underlying linear logic. We construct families of $\mathbf{FL_e}$-provable (resp. $\mathbf{FL_{ec}}$-provable) formulas that require exponential-size (resp. sub-exponential-size) proofs in affine linear logic $\mathbf{ALL}$ (resp. relevant linear logic $\mathbf{RLL}$), but admit polynomial-size proofs once contraction (resp. weakening) is restored. This yields exponential lower bounds on proof-size of $\mathbf{FL_e}$-provable formulas in $\mathbf{ALL}$ and hence for $\mathbf{MALL}$, $\mathbf{AMALL}$, and full classical linear logic $\mathbf{CLL}$. Finally, we exhibit formulas with polynomial-size $\mathbf{FL_e}$-proofs that nevertheless require exponential-size proofs in cut-free $\mathbf{LK}$, establishing exponential speed-ups between various linear calculi and their cut-free counterparts.

💡 Research Summary

The paper tackles one of the most stubborn open problems in proof complexity: establishing non‑trivial lower bounds on proof size for the classical sequent calculus LK. Since direct lower bounds for LK have resisted all known techniques, the authors adopt a different strategy: they gradually weaken the structural discipline of the logic and study how each weakening affects proof size. The central objects of study are the Full Lambek calculus with exchange (FLₑ) and its contraction‑extended variant (FLₑc), which serve as substructural foundations for linear logic. By systematically removing contraction, weakening, and finally the cut rule, the authors obtain three families of separation results that isolate the contribution of each rule.

1. Contraction‑free separation (FLₑ vs. ALL).
   The authors construct a sequence of FLₑ‑provable formulas {Aₙ} that are !‑free (no exponentials) and have polynomial‑size proofs in LK. Using Chu’s categorical translation, each Aₙ is mapped into a formula of affine linear logic (ALL). Because ALL is TOWER‑complete, any proof of Aₙ in ALL must have size at least exponential in |Aₙ|; otherwise one could enumerate all proofs and decide provability within a tower‑bounded time, contradicting the known hardness. Hence, ALL requires exponential‑size proofs for these formulas, while LK does not, establishing LK ≰ FLₑ ALL. The same argument lifts to other contraction‑free systems such as CFLₑ, MALL, AMALL, and CLL, yielding exponential lower bounds for all of them.

2. Weakening‑free separation (FLₑc vs. RLL).
   For the weakening‑free case the authors turn to relevant linear logic (RLL). They define a notion of sub‑exponential functions f (e.g., f(n)=2^{(log n)^k}) and prove that for any monotone sub‑exponential f there exists a family {Bₙ} of FLₑc‑provable formulas with polynomial‑size LK proofs, yet every RLL‑proof of Bₙ exceeds f(|Bₙ|). The proof proceeds by encoding the reachability problem for counter‑addition systems (a restricted class of vector addition systems) into FLₑc formulas. These systems are known to be EXPSPACE‑hard, which translates into the required lower bound for RLL. The same separation holds for the weakening‑free calculi CFLₑc and RMALL.

3. Cut‑free separation (FLₑ vs. LK⁻).
   To assess the impact of the cut rule, the authors construct a sequence of sequents {Sₙ} that admit polynomial‑size proofs in the full FLₑ calculus (contraction and weakening present) but any cut‑free LK proof (LK⁻) must be exponentially large. The argument adapts Krajiček’s exponential separation between LK and LK⁻, which relies on monotone feasible interpolation: LK⁻ enjoys a monotone interpolation property, so any LK⁻ proof of a certain implication yields a monotone circuit of comparable size. By choosing implications whose monotone circuit complexity is known to be exponential, the authors force LK⁻ proofs to be exponential, while FLₑ can simulate the needed reasoning using initial sequents that mimic contraction and weakening without violating the one‑atom restriction needed for the interpolation argument.

The methodological toolbox combines three distinct techniques:

• Chu’s translation – a categorical construction that lifts lower bounds from intuitionistic substructural logics to their classical linear counterparts while preserving the shape of formulas.
• Vector addition / counter‑addition systems – computational models whose reachability problems are known to be hard (EXPSPACE or Ackermann), providing a bridge between proof size and classical complexity.
• Monotone feasible interpolation – a proof‑theoretic tool that translates proof size into circuit size, enabling the authors to import known exponential circuit lower bounds into proof‑size lower bounds for cut‑free systems.

By isolating each structural rule, the paper demonstrates that the combination of contraction, weakening, and cut yields a dramatic, super‑polynomial boost in proof efficiency. Moreover, the results are robust: they apply not only to the specific systems studied (ALL, RLL, LK⁻) but also to a host of related calculi such as MALL, AMALL, CLL, and their intuitionistic counterparts. The authors conclude with a discussion of open cases (e.g., systems where only some of the three rules are present) and suggest that their techniques could be extended to other non‑classical logics, potentially paving the way toward the long‑sought lower bounds for full LK.