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arxiv: 2606.03484 · v1 · pith:ZSUHAIVWnew · submitted 2026-06-02 · 💻 cs.LO · math.LO

Optimizing Proof-Search via Linearization for G\"odel-L\"ob Logic with Tree-Hypersequents

Tim S. Lyon , Omar Taher This is my paper

Pith reviewed 2026-06-28 07:53 UTC · model grok-4.3

classification 💻 cs.LO math.LO
keywords Gödel-Löb logictree-hypersequentsproof-searchPSPACE complexitylinearization methodlinear nested sequentscounter-modelsmodal logic
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The pith

A linearization method decides validity for Gödel-Löb logic in tree-hypersequent systems in PSPACE.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a proof-search algorithm for Gödel-Löb logic that runs in PSPACE by using a linearization technique in a tree-hypersequent calculus. This technique builds derivations one branch at a time instead of exploring the full tree structure at once. A reader would care because it matches the known complexity lower bound and provides a way to construct counter-models when a formula is invalid. It also shows that valid formulas have proofs using only linear sequents, linking to other sequent systems. This resolves open questions about efficient syntactic decision procedures in expressive modal proof systems.

Core claim

We introduce a linearization method for the tree-hypersequent system CSGL that constructs only a single branch of a derivation and of a tree sequent at a time. This yields a PSPACE proof-search algorithm for deciding validity in Gödel-Löb logic. When proof-search fails, fragments of tree sequents can be combined to extract finite counter-models. Every valid formula has a proof consisting solely of line sequents, which correspond to linear nested sequents.

What carries the argument

The linearization method, which constructs only a single branch of a derivation and of a tree sequent at a time.

Load-bearing premise

The linearization method preserves completeness and allows systematic extraction of finite counter-models from failed searches.

What would settle it

A valid Gödel-Löb formula for which the linearization search fails to find a line-sequent proof, or an invalid formula where combining search fragments does not produce a finite counter-model that refutes it.

Figures

Figures reproduced from arXiv: 2606.03484 by Omar Taher, Tim S. Lyon.

Figure 1
Figure 1. Figure 1: Tree Sequent Calculus CSGL for GL. The □R rule is subject to a side condition †, namely, the rule is applicable only if the label y is fresh. occurring in all formulae of Γ and ∆. We adopt standard tree terminology when discussing tree sequents (e.g. root, branch, ancestor, leaf; see [23, Chapter 11]). We define a flat sequent to be a tree sequent of the form Γ ⊢ ∆, that is, a flat sequent is a sequent Γ ⊢… view at source ↗
Figure 2
Figure 2. Figure 2: Admissible rules. auxiliary. We also refer to the auxiliary formula y : □ϕ as the diagonal formula in □R. In the subsequent section, we will explain how the diagonal formula helps ensure the termination of proof-search. Remark 3.2. Poggiolesi’s original system included the following □L ′ rule rather than the □L rule. (NB. We have expressed this rule in labeled notation.) However, the left premise of the □L… view at source ↗
Figure 3
Figure 3. Figure 3: Computation tree and successful proof-search example. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Failed proof-search and counter-model extraction example. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A linear nested sequent calculus extracted from proof-search. [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
read the original abstract

We answer a question posed by Poggiolesi concerning a syntactic decidability proof for GL in the tree-hypersequent system CSGL, and resolve a challenge identified by Maggesi and Perini Brogi, who sought a PSPACE proof-search algorithm for GL in expressive sequent-based formalisms. We work with a notational variant of CSGL formulated in terms of (labeled) tree sequents. Our answer is complexity-optimal: we present a proof-search algorithm that decides the (in)validity of formulae and runs in PSPACE, matching the known PSPACE-completeness of GL. To achieve this, we introduce a "linearization method," which constructs only a single branch of a derivation and of a tree sequent at a time, avoiding the exponential blowup typical of naive proof-search in sequent formalisms. We show how to systematically combine fragments of tree sequents generated during proof-search to extract finite counter-models, which serves as a theoretical device for establishing the correctness of the algorithm when proof-search fails. Finally, we show that every valid formula admits a proof consisting solely of line sequents, which correspond to linear nested sequents. This establishes a connection between depth-first proof-search and linear nested sequent calculi. Our results not only answer the aforementioned questions, but also provide new insights into proof-search and correctness arguments in tree sequent systems for modal logics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper claims to resolve open questions on syntactic decidability and PSPACE proof-search for Gödel-Löb logic (GL) by presenting a proof-search algorithm on a labeled tree-sequent variant of the CSGL system. The algorithm uses a linearization method that constructs only a single branch of a derivation and of a tree sequent at a time to avoid exponential blowup, decides (in)validity in PSPACE (matching the known lower bound), systematically combines fragments of tree sequents to extract finite counter-models when search fails (serving as the correctness device), and shows that every valid formula has a proof consisting solely of line sequents (corresponding to linear nested sequents).

Significance. If the linearization method and counter-model extraction are shown to preserve completeness, the result would be significant: it supplies a complexity-optimal syntactic decision procedure for GL in an expressive tree-hypersequent formalism, directly answering questions posed by Poggiolesi and by Maggesi-Perini Brogi. The connection drawn between depth-first search and linear nested sequents, together with the explicit use of counter-model construction as a theoretical device, would constitute a concrete contribution to proof-search techniques for modal logics.

major comments (1)
  1. [Abstract] Abstract (linearization method): the claim that fragments generated by single-branch search can be 'systematically combined' to extract every finite GL countermodel is load-bearing for the PSPACE-completeness result, yet the description provides no explicit invariant (e.g., a simulation relation between the linearized search tree and the full branching hypersequent tree) that would guarantee coverage of countermodels whose shortest realization requires simultaneous exploration of two sibling subtrees.
minor comments (1)
  1. [Abstract] The abstract refers to 'line sequents' without an immediate definition or pointer to the section where the correspondence to linear nested sequents is formalized; a brief inline gloss would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading and for identifying this point about the abstract's presentation of the linearization method. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (linearization method): the claim that fragments generated by single-branch search can be 'systematically combined' to extract every finite GL countermodel is load-bearing for the PSPACE-completeness result, yet the description provides no explicit invariant (e.g., a simulation relation between the linearized search tree and the full branching hypersequent tree) that would guarantee coverage of countermodels whose shortest realization requires simultaneous exploration of two sibling subtrees.

    Authors: We agree that the abstract is concise and does not itself exhibit the invariant. The manuscript body (Sections 3–5) supplies the required details: the linearization procedure is defined so that each single-branch search fragment records the necessary labels and modal accessibility information; the combination step is shown to reconstruct any finite countermodel by a simulation relation that maps the collected fragments onto the full tree-hypersequent tree. This relation is proved to be total for failed searches, thereby covering countermodels that would require simultaneous exploration of sibling subtrees. The PSPACE bound follows directly from the space-efficient representation of the single branch plus the polynomial-time combination routine. Nevertheless, to make the abstract self-contained on this load-bearing claim, we will revise it to include a one-sentence reference to the simulation invariant used in the counter-model extraction. revision: yes

Circularity Check

0 steps flagged

No circularity: linearization method and PSPACE bound rest on independent completeness argument and external complexity result

full rationale

The paper introduces a linearization technique for single-branch search in labeled tree sequents for GL, proves correctness via systematic counter-model extraction from failed searches, and shows that valid formulas have line-sequent proofs. The PSPACE upper bound is matched directly against the known PSPACE-completeness of GL (an external result). No equation or definition reduces the central claim to a fitted parameter, self-referential renaming, or load-bearing self-citation chain; the derivation is self-contained against the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the soundness and completeness of the underlying CSGL tree-hypersequent system for GL together with the correctness of the new linearization transformation and counter-model extraction procedure.

axioms (1)
  • domain assumption The tree-hypersequent system CSGL is sound and complete for Gödel-Löb logic
    The algorithm is developed inside this system; its properties are inherited from the base calculus.

pith-pipeline@v0.9.1-grok · 5788 in / 1230 out tokens · 26901 ms · 2026-06-28T07:53:28.548346+00:00 · methodology

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Works this paper leans on

31 extracted references · 15 canonical work pages

  1. [1]

    55–67, doi:https://doi.org/10.1016/j.entcs.2009.02.029

    Pietro Abate & Rajeev Goré (2009):The Tableau Workbench.Electronic Notes in Theoretical Com- puter Science231, pp. 55–67, doi:https://doi.org/10.1016/j.entcs.2009.02.029. Available athttps://www. sciencedirect.com/science/article/pii/S1571066109000346. Proceedings of the 5th Workshop on Methods for Modalities (M4M5 2007)

    work page doi:10.1016/j.entcs.2009.02.029 2009

  2. [2]

    935–942, doi:10.2307/2274147

    Arnon Avron (1984):On modal systems having arithmetical interpretations.Journal of Symbolic Logic 49(3), p. 935–942, doi:10.2307/2274147

    work page doi:10.2307/2274147 1984

  3. [3]

    551–577, doi:10.1007/s00153-009-0137-3

    Kai Brünnler (2009):Deep Sequent Systems for Modal Logic.Archive for Mathematical Logic48(6), pp. 551–577, doi:10.1007/s00153-009-0137-3

    work page doi:10.1007/s00153-009-0137-3 2009

  4. [4]

    Bull (1992):Cut Elimination for Propositional Dynamic Logic without *.Zeitschrift für Mathe- matische Logik und Grundlagen der Mathematik38(2), pp

    Robert A. Bull (1992):Cut Elimination for Propositional Dynamic Logic without *.Zeitschrift für Mathe- matische Logik und Grundlagen der Mathematik38(2), pp. 85–100

    1992

  5. [5]

    Oxford University Press

    Alexander Chagrov & Michael Zakharyaschev (1997):Modal logic. Oxford University Press

    1997

  6. [6]

    In:abstract, British Logic Colloquium

    Rajeev Goré & Jack Kelly (2007):Automated proof search in Gödel-Löb provability logic. In:abstract, British Logic Colloquium

    2007

  7. [7]

    In Thomas Bolander, Torben Braüner, Silvio Ghilardi & Lawrence S

    Rajeev Goré & Revantha Ramanayake (2012):Labelled Tree Sequents, Tree Hypersequents and Nested (Deep) Sequents. In Thomas Bolander, Torben Braüner, Silvio Ghilardi & Lawrence S. Moss, editors: Advances in Modal Logic 9, College Publications, pp. 279–299. Available athttp://www.aiml.net/ volumes/volume9/Gore-Ramanayake.pdf

    2012

  8. [8]

    In Nicola Olivetti, editor:Automated Reasoning with Analytic Tableaux and Related Methods,Lecture Notes in Computer Science4548, Springer Berlin Heidelberg, Berlin, Heidelberg, pp

    Ryo Ishigaki & Kentaro Kikuchi (2007):Tree-Sequent Methods for Subintuitionistic Predicate Logics. In Nicola Olivetti, editor:Automated Reasoning with Analytic Tableaux and Related Methods,Lecture Notes in Computer Science4548, Springer Berlin Heidelberg, Berlin, Heidelberg, pp. 149–164

    2007

  9. [9]

    Ryo Kashima (1994):Cut-free sequent calculi for some tense logics.Studia Logica53(1), pp. 119–135

    1994

  10. [10]

    In Hans De Nivelle, editor: Automated Reasoning with Analytic Tableaux and Related Methods,Lecture Notes in Computer Science 9323, Springer International Publishing, Cham, pp

    Björn Lellmann (2015):Linear Nested Sequents, 2-Sequents and Hypersequents. In Hans De Nivelle, editor: Automated Reasoning with Analytic Tableaux and Related Methods,Lecture Notes in Computer Science 9323, Springer International Publishing, Cham, pp. 135–150

    2015

  11. [11]

    Björn Lellmann & Elaine Pimentel (2015):Proof Search in Nested Sequent Calculi. In Martin Davis, Ansgar Fehnker, Annabelle McIver & Andrei V oronkov, editors:Logic for Programming, Artificial Intelligence, and Reasoning, Springer Berlin Heidelberg, Berlin, Heidelberg, pp. 558–574

    2015

  12. [12]

    In Yale Weiss & Romina Birman, editors:Saul Kripke on Modal Logic, Springer International Publishing, Cham, pp

    Björn Lellmann & Francesca Poggiolesi (2024):Nested Sequents or Tree-Hypersequents—A Survey. In Yale Weiss & Romina Birman, editors:Saul Kripke on Modal Logic, Springer International Publishing, Cham, pp. 243–301, doi:10.1007/978-3-031-57635-5_11

    work page doi:10.1007/978-3-031-57635-5_11 2024

  13. [13]

    Tim Lyon, Alwen Tiu, Rajeev Goré & Ranald Clouston (2020):Syntactic interpolation for tense logics and bi-intuitionistic logic via nested sequents. In Maribel Fernández & Anca Muscholl, editors:28th EACSL Annual Conference on Computer Science Logic (CSL 2020),Leibniz International Proceedings in Informat- ics (LIPIcs)152, Schloss Dagstuhl – Leibniz-Zentru...

    work page doi:10.4230/lipics.csl.2020.28 2020

  14. [14]

    Lyon (2025):Unifying Sequent Systems for Gödel-Löb Provability Logic via Syntactic Transforma- tions

    Tim S. Lyon (2025):Unifying Sequent Systems for Gödel-Löb Provability Logic via Syntactic Transforma- tions. In Jörg Endrullis & Sylvain Schmitz, editors:33rd EACSL Annual Conference on Computer Sci- ence Logic (CSL 2025),Leibniz International Proceedings in Informatics (LIPIcs)326, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, pp....

    work page doi:10.4230/lipics.csl.2025.42 2025

  15. [15]

    Lyon & Lucía Gómez Álvarez (2022):Automating Reasoning with Standpoint Logic via Nested Sequents

    Tim S. Lyon & Lucía Gómez Álvarez (2022):Automating Reasoning with Standpoint Logic via Nested Sequents. In:Proceedings of the 19th International Conference on Principles of Knowledge Representation and Reasoning, pp. 257–266, doi:10.24963/kr.2022/26

    work page doi:10.24963/kr.2022/26 2022

  16. [16]

    Lyon & Piotr Ostropolski-Nalewaja (2024):Foundations for an Abstract Proof Theory in the Context of Horn Rules.arXiv preprint

    Tim S. Lyon & Piotr Ostropolski-Nalewaja (2024):Foundations for an Abstract Proof Theory in the Context of Horn Rules.arXiv preprint. Available athttps://arxiv.org/abs/2304.05697. 16Optimizing Proof-Search via Linearization for Gödel-Löb Logic with Tree-Hypersequents

    arXiv 2024

  17. [17]

    Journal of Automated Reasoning67(3), p

    Marco Maggesi & Cosimo Perini Brogi (2023):Mechanising Gödel–Löb Provability Logic in HOL Light. Journal of Automated Reasoning67(3), p. 29, doi:10.1007/s10817-023-09677-z. Available athttps:// doi.org/10.1007/s10817-023-09677-z

    work page doi:10.1007/s10817-023-09677-z 2023

  18. [18]

    Akinori Maniwa & Ryo Kashima (2024):Syntactic Cut-Elimination for Provability Logic GL via Nested Se- quents. In Andrzej Indrzejczak & Michał Zawidzki, editors:Proceedings of the Workshop on Non-Classical Logics: Theory and Applications (NCL 2024),Electronic Proceedings in Theoretical Computer Science415, pp. 93–108, doi:10.4204/EPTCS.415.11

    work page doi:10.4204/eptcs.415.11 2024

  19. [19]

    Springer New York, NY , doi:https://doi.org/10.1007/b115304

    Grigori Mints (2000):A Short Introduction to Intuitionistic Logic. Springer New York, NY , doi:https://doi.org/10.1007/b115304

    work page doi:10.1007/b115304 2000

  20. [20]

    25–60, doi:10.1007/s11787-014-0097-1

    Sara Negri (2014):Proofs and Countermodels in Non-Classical Logics.Logica Universalis8(1), pp. 25–60, doi:10.1007/s11787-014-0097-1

    work page doi:10.1007/s11787-014-0097-1 2014

  21. [21]

    In David Makinson, Jacek Malinowski & Heinrich Wansing, editors:Towards Mathematical Philosophy,Trends in logic28, Springer, pp

    Francesca Poggiolesi (2009):The Method of Tree-Hypersequents for Modal Propositional Logic. In David Makinson, Jacek Malinowski & Heinrich Wansing, editors:Towards Mathematical Philosophy,Trends in logic28, Springer, pp. 31–51, doi:10.1007/978-1-4020-9084-4_3

    work page doi:10.1007/978-1-4020-9084-4_3 2009

  22. [22]

    593–611, doi:10.1017/S1755020309990244

    Francesca Poggiolesi (2009):A Purely Syntactic and Cut-free Sequent Calculus for the Modal Logic of Provability.The Review of Symbolic Logic2(4), p. 593–611, doi:10.1017/S1755020309990244

    work page doi:10.1017/s1755020309990244 2009

  23. [23]

    Rosen (2012):Discrete Mathematics and Its Applications, 7 edition

    Kenneth H. Rosen (2012):Discrete Mathematics and Its Applications, 7 edition. McGraw–Hill, New York

    2012

  24. [24]

    Sambin & S

    G. Sambin & S. Valentini (1980):A Modal Sequent Calculus for a Fragment of Arithmetic.Studia Logica: An International Journal for Symbolic Logic39(2/3), pp. 245–256. Available athttp://www.jstor.org/ stable/20014984

    arXiv 1980

  25. [25]

    The Sequential Approach.Jour- nal of Philosophical Logic11(3), pp

    Giovanni Sambin & Silvio Valentini (1982):The Modal Logic of Provability. The Sequential Approach.Jour- nal of Philosophical Logic11(3), pp. 311–342. Available athttp://www.jstor.org/stable/30226252

    arXiv 1982

  26. [26]

    Uppsala: Filosofiska Föreningen och Filosofiska Institutionen vid Uppsala Universitet

    Krister Segerberg (1971):An Essay in Classical Modal Logic. Uppsala: Filosofiska Föreningen och Filosofiska Institutionen vid Uppsala Universitet

    1971

  27. [27]

    D. S. Shamkanov (2014):Circular proofs for the Gödel-Löb provability logic.Mathematical Notes96(3), pp. 575–585, doi:10.1134/S0001434614090326

    work page doi:10.1134/s0001434614090326 2014

  28. [28]

    Alex K Simpson (1994):The Proof Theory and Semantics of Intuitionistic Modal Logic. Ph.D. thesis, Uni- versity of Edinburgh. College of Science and Engineering. School of Informatics

    1994

  29. [29]

    Solovay (1976):Provability interpretations of modal logic.Israel Journal of Mathematics25(3), pp

    Robert M. Solovay (1976):Provability interpretations of modal logic.Israel Journal of Mathematics25(3), pp. 287–304, doi:10.1007/BF02757006. Available athttps://doi.org/10.1007/BF02757006

    work page doi:10.1007/bf02757006 1976

  30. [30]

    In Thomas Bolander, Torben Braüner, Silvio Ghilardi & Lawrence S

    Alwen Tiu, Egor Ianovski & Rajeev Goré (2012):Grammar Logics in Nested Sequent Calculus: Proof Theory and Decision Procedures. In Thomas Bolander, Torben Braüner, Silvio Ghilardi & Lawrence S. Moss, editors:Advances in Modal Logic 9, College Publications, pp. 516–537

    2012

  31. [31]

    Springer Science & Business Media

    Luca Viganò (2000):Labelled Non-Classical Logics. Springer Science & Business Media. A Additional Material for Section 4 Computation Tree.For the sake of completeness, we add the formal definition of the computation treect(x:ϕ) = (V,E,L)corresponding toprove(⊢x:ϕ). We define(V,E)root-first based on the number of recursive calls inprove(⊢x:ϕ). Initially, o...

    2000