Paper: Hypersequent Calculi Have Ackermann Complexity (at LICS 2026)

Authors: A. R. Balasubramanian Vitor Greati Revantha Ramanayake

Open access: https://doi.org/10.4230/LIPIcs.LICS.2026.10

Abstract

For substructural logics with contraction or weakening admitting cut-free sequent calculi, proof search was analyzed using well-quasi-orders on ℕ^d (Dickson’s lemma), yielding Ackermann 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 Ackermann to hyper-Ackermann complexity in the fast-growing hierarchy, suggesting that cut-free hypersequent calculi for extensions of the commutative Full Lambek calculus with contraction or weakening (FL_ec/FL_ew) inherently entail hyper-Ackermann upper bounds. We show that this intuition does not hold: every extension of FL_ec and FL_ew admitting a cut-free hypersequent calculus has an Ackermann 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 MTL. Our Ackermann upper bound is optimal for the contraction case (realized by the logic FL_ec).

BibTeX

  @InProceedings{BalasubramanianGrea-HypersequentCalculi,
    author = 	 {A. R. Balasubramanian and Vitor Greati and Revantha Ramanayake},
    title = 	 {Hypersequent Calculi Have Ackermann Complexity},
    booktitle =  {Proceedings of the Forty-First Annual Symposium on Logic in Computer Science (LICS 2026)},
    year =	 {2026},
    month =	 {July}, 
    pages =      {10:1--10:26},
    location =   {Lisbon, Portugal}, 
    publisher =	 {Schloss Dagstuhl -- Leibniz-Zentrum für Informatik},
    doi =        {10.4230/LIPIcs.LICS.2026.10}
  }