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DOI: 10.4230/LIPIcs.LICS.2026.10

Hypersequent Calculi Have Ackermann Complexity

Authors: A. R. Balasubramanian, Vitor Greati, and Revantha Ramanayake

Published in: LIPIcs, Volume 380, 41st Annual Symposium on Logic in Computer Science (LICS 2026)

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).

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A. R. Balasubramanian, Vitor Greati, and Revantha Ramanayake. Hypersequent Calculi Have Ackermann Complexity. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 10:1-10:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{balasubramanian_et_al:LIPIcs.LICS.2026.10,
  author =	{Balasubramanian, A. R. and Greati, Vitor and Ramanayake, Revantha},
  title =	{{Hypersequent Calculi Have Ackermann Complexity}},
  booktitle =	{41st Annual Symposium on Logic in Computer Science (LICS 2026)},
  pages =	{10:1--10:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-434-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{380},
  editor =	{Faggian, Claudia and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.LICS.2026.10},
  URN =		{urn:nbn:de:0030-drops-267970},
  doi =		{10.4230/LIPIcs.LICS.2026.10},
  annote =	{Keywords: Hypersequent calculi, Substructural logics, Ackermann complexity, Well-quasi-orders}
}

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Hypersequent Calculi Have Ackermann Complexity

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