[论文解读] Proof Complexity of Linear Logics

Amirhossein Akbar Tabatabai, Raheleh Jalali|arXiv (Cornell University)|Jan 29, 2026

Logic, Reasoning, and Knowledge被引用 0

一句话总结

本论文通过证明各种线性和亚结构演算的指数级和亚指数级下界， isolates structural rules 对证明复杂度的影响，并展示在转向无剪切或弱化形式时的指数级加速。

ABSTRACT

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.

研究动机与目标

• Investigate how removing structural rules (contraction, weakening) affects proof size in linear/substructural logics.
• Demonstrate exponential and sub-exponential lower bounds for FL_e-provable formulas in ALL and RLL variants.
• Show that restoring contraction or weakening yields polynomial-size proofs for certain formulas.
• Exhibit formulas with polynomial FL_e-proofs that require exponential LK-cut-free proofs to illustrate cut's power.

提出的方法

• Use Chu’s translation to transfer lower bounds from intuitionistic substructural logics to classical variants.
• Leverage the connection between provability in FL_ec and counter machine reachability to obtain EXPSPACE-hardness results.
• Apply monotone feasible interpolation (via LK^{-}) to obtain exponential separations, adapting the method to cut-free and substructural settings.
• Construct explicit formula families (A_n) that are FL_e-provable but require exponential ALL proofs, yet have polynomial LK proofs.
• Compare cut-free versus with-cut systems to demonstrate exponential speed-ups in several calculi when cut is present.

实验结果

研究问题

• RQ1 What is the exact role of contraction and weakening in determining proof size for linear and substructural logics?
• RQ2 Can one separate contraction-free and weakening-free systems from LK (classical sequent calculus) via exponential or sub-exponential lower bounds?
• RQ3 Do cut-free calculi inherently require longer proofs than their cut-enabled counterparts in these logics?
• RQ4 Can classical linear logic proofs be efficiently simulated in substructural logics and what are the resulting proof-size consequences?
• RQ5 How do translations like Chu’s construction and connections to vector addition systems inform proof-size lower bounds?

主要发现

• There exist FL_e-provable formulas whose ALL proofs are exponential in size, while LK proofs are polynomial, proving exponential separation for contraction-free calculi.
• There are FL_ec-provable formulas with polynomial LK-proofs, but whose RLL-proofs are sub-exponential in size (for every sub-exponential function f).
• Cut-free LK^{-} provability achieves monotone feasible interpolation, enabling exponential lower bounds via monotone circuit complexity arguments.
• There are formulas with polynomial FL_e-proofs that require exponential-size proofs in cut-free LK, showing exponential speed-ups when cut is available.
• Exponential speed-ups are established between several calculi with and without cut, including FL_e, CFL_e, IMALL, MALL, ILL, CLL, across combinations of contraction and weakening.