Hypersequent Calculi Have Ackermannian Complexity

2026-02-22

AI Summary

This work investigates whether proof search in hypersequent calculi for substructural logics such as FL_ec and FL_ew necessarily incurs hyper-Ackermannian complexity. By introducing a novel mechanism to capture dependencies among sequents—thereby circumventing the traditional reliance on powerset well-quasi-orders—and by integrating controlled bad-sequence arguments, Dickson’s lemma, and Karp–Miller acceleration techniques to handle weakening, the authors effectively tame the growth of computational complexity. The study establishes that all extensions of FL_ec/FL_ew admitting cut-free hypersequent calculi enjoy an Ackermannian upper bound on provability complexity. Moreover, this bound is shown to be tight in the presence of contraction, thereby refining and surpassing prior understandings of the inherent complexity of these logics.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

For substructural logics with contraction or weakening admitting cut-free sequent calculi, proof search was analyzed using well-quasi-orders on $\mathbb{N}^d$ (Dickson's lemma), yielding Ackermannian upper bounds via controlled bad-sequence arguments. For hypersequent calculi, that argument lifted the ordering to the powerset, since a hypersequent is a (multi)set of sequents. This induces a jump from Ackermannian to hyper-Ackermannian complexity in the fast-growing hierarchy, suggesting that cut-free hypersequent calculi for extensions of the commutative Full Lambek calculus with contraction or weakening ($\mathbf{FL_{ec}}$/$\mathbf{FL_{ew}}$) inherently entail hyper-Ackermannian upper bounds. We show that this intuition does not hold: every extension of $\mathbf{FL_{ec}}$ and $\mathbf{FL_{ew}}$ admitting a cut-free hypersequent calculus has an Ackermannian upper bound on provability. To avoid the powerset, we exploit novel dependencies between individual sequents within any hypersequent in backward proof search. The weakening case, in particular, introduces a Karp-Miller style acceleration, and it improves the upper bound for the fundamental fuzzy logic $\mathbf{MTL}$. Our Ackermannian upper bound is optimal for the contraction case (realized by the logic $\mathbf{FL_{ec}}$).

Problem

• hypersequent calculi
• Ackermannian complexity
• substructural logics
• proof search
• FL_ec

Innovation

• hypersequent calculi
• Ackermannian complexity
• substructural logics
• proof search
• Karp-Miller acceleration