Optimizing Proof-Search via Linearization for Gödel-Löb Logic with Tree-Hypersequents

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.

• The paper introduces a depth-first linearization method with disjunctive branching that reduces tree-hypersequent proof search to polynomial space and achieves an O(N⁴) space bound, matching Gödel–Löb logic’s PSPACE-completeness.
• The paper proves correctness by extracting genuine proofs from successful computation trees and composing stable failed branches into finite countermodels, while the algorithm itself returns a Boolean validity verdict.
• The paper establishes that every valid formula has a line-like, end-active proof, creating a route to linear nested sequent calculi while leaving implementation, time optimization, and generalization to other modal logics open.

Background and motivation

Gödel–Löb logic ($GL$) is the provability logic of Peano Arithmetic, axiomatized as $\mathsf{K}$ extended with Löb's axiom $$\Box(\Box\varphi \rightarrow \varphi) \rightarrow \Box\varphi$$, and is sound and complete for transitive, conversely well-founded Kripke frames [Seg71]. Solovay's arithmetical completeness theorem identifies $GL$ with the modal principles PA can prove about its own provability predicate [Sol76]. Structurally, $GL$ admits cut-free Gentzen calculi due to Sambin and Valentini [SamVal80, SamVal82], and a labeled sequent calculus $\mathsf{G3KGL}$ due to Negri [Neg14]. Poggiolesi introduced the tree-hypersequent system $\mathit{CS}_{\mathit{GL}}$ [Pog09b]—a notational variant of nested sequents [Kas94, Bul92, Bru09]—and posed an open question regarding a syntactic decidability proof for $GL$ within that system. Independently, Maggesi and Perini Brogi implemented a decision procedure for $GL$ in HOL Light based on $\mathsf{G3KGL}$, but observed that their procedure lies in $\mathsf{K}$0 and identified a $\mathsf{K}$1 procedure as "the ideal goal" [MagPer23].

The paper by Lyon and Taher (2606.03484) answers both questions simultaneously. Working with a labeled-syntax variant of $\mathsf{K}$2 (tree sequents, per Goré and Ramanayake's observation that tree-restricted labeled sequents coincide with tree-hypersequents [GorRam12]), it presents a proof-search algorithm that decides validity of $\mathsf{K}$3 formulae in $\mathsf{K}$4. Since $\mathsf{K}$5 is $\mathsf{K}$6-complete [ChaZak97], this bound is optimal. Moreover, because proofs between $\mathsf{K}$7 and $\mathsf{K}$8 are inter-translatable in $\mathsf{K}$9 [GorRam12, LyoOst24], the technique transfers to labeled sequent proof-search, resolving the Maggesi–Perini Brogi challenge.

The linearization method

Naive proof-search in nested sequent systems generates derivations that are "trees of trees," typically causing exponential blowup in both time and space. The central technical contribution is linearization: a depth-first strategy that constructs only a single branch of a derivation—and only a single branch of each tree sequent—at any time. To the authors' knowledge this is the first such procedure for a tree (nested) sequent system. Two design choices underpin the space savings:

• Forgoing direct counter-model extraction. Prior procedures output counter-models on failure, which are exponentially large; the new algorithm outputs only a Boolean verdict, treating counter-model construction as a theoretical correctness device rather than an algorithmic feature.
• Disjunctive branching at two levels. Motivated by Mints [Min00], the algorithm employs a disjunctive inference rule $$\Box(\Box\varphi \rightarrow \varphi) \rightarrow \Box\varphi$$0 whose conclusion is provable whenever at least one premise is provable. Disjunctive branching occurs both across derivation branches (standard) and within tree sequents themselves—the latter being, according to the authors, a new idea. This contrasts with tableau-style existential rules [AbaGor09], which branch only at one level.

The calculus and the algorithm

The paper recasts Poggiolesi's $$\Box(\Box\varphi \rightarrow \varphi) \rightarrow \Box\varphi$$1 over tree sequents $$\Box(\Box\varphi \rightarrow \varphi) \rightarrow \Box\varphi$$2, where $$\Box(\Box\varphi \rightarrow \varphi) \rightarrow \Box\varphi$$3 is a set of relational atoms forming a directed tree. The rules comprise initial rules ($$\Box(\Box\varphi \rightarrow \varphi) \rightarrow \Box\varphi$$4, $$\Box(\Box\varphi \rightarrow \varphi) \rightarrow \Box\varphi$$5, $$\Box(\Box\varphi \rightarrow \varphi) \rightarrow \Box\varphi$$6), propositional rules ($$\Box(\Box\varphi \rightarrow \varphi) \rightarrow \Box\varphi$$7, $$\Box(\Box\varphi \rightarrow \varphi) \rightarrow \Box\varphi$$8), propagation rules ($$\Box(\Box\varphi \rightarrow \varphi) \rightarrow \Box\varphi$$9, $GL$0), and the modal rule $GL$1, which introduces a fresh label $GL$2 with relational atom $GL$3 and auxiliary formulae $GL$4 (the diagonal formula) and $GL$5. A simplification replaces Poggiolesi's original two-premise $GL$6 rule with a single-premise variant, justified by provability of the discarded premise via $GL$7, $GL$8, and $GL$9. Weakening and cut are admissible, all non-initial rules are height-preserving invertible, and validity of $GL$0 coincides with provability of $GL$1 [Pog09b, ManKas24].

The recursive algorithm prove operates on line sequents. It halts with true on initial sequents and false on stable sequents (saturated sequents with no box formulae in the consequent at leaves). Implication handling uses conjunctive branching with short-circuit evaluation, so at most one derivation branch exists at a time. Saturation triggers simultaneous applications of $GL$2, encoded as the disjunctive rule $GL$3: given saturated leaf $GL$4 with consequent box formulae $GL$5, up to $GL$6 fresh successors $GL$7 are generated, each premise unpacking exactly one box formula while retaining the others as diagonal formulae. Crucially, the diagonal formulae $GL$8 ($GL$9) ensure termination without loop-checking despite transitivity—a phenomenon previously observed for Gentzen and labeled calculi [SamVal80, Neg14].

Correctness

Correctness is established through the notion of a computation tree: the full tree structure traced by a terminating run of prove, viewed as a derivation in $\mathsf{G3KGL}$0, with nodes labeled by their Boolean outcome.

On success, pruning the computation tree—retaining all true nodes, both premises of $\mathsf{G3KGL}$1 applications, and a single true premise of each $\mathsf{G3KGL}$2 application (thereby converting it into a $\mathsf{G3KGL}$3 application)—yields a genuine $\mathsf{G3KGL}$4 proof. Because each $\mathsf{G3KGL}$5 premise unpacks one box formula at a leaf, every sequent in the extracted proof is a line sequent, and every rule application is end-active (auxiliary formulae at leaves). This yields the structural result that every valid formula admits a line-like, end-active proof. Since line sequents are notational variants of linear nested sequents (LNS) [Lel15, LelPim15], these proofs translate into variants of those in the recently introduced LNS calculus $\mathsf{G3KGL}$6 [Lyo25]; the paper also exhibits the LNS calculus directly extractable from its proof-search. This connects depth-first proof-search in tree sequent systems to linear nested sequent calculi, suggesting a general method for deriving sound and complete LNS systems from tree sequent procedures.

On failure, the usual counter-model extraction methods are obstructed because each branch yields only a fragment of a would-be model. The paper's second conceptual contribution shows how to overcome this: pruning the computation tree to the downward closure of false nodes (retaining one false premise per $\mathsf{G3KGL}$7 application, all premises of $\mathsf{G3KGL}$8 applications), composing the resulting stable line sequent leaves via sequent composition, and reading off a finite model whose worlds are labels, whose accessibility relation is the transitive closure of the composed relational atoms, and whose valuation places $\mathsf{G3KGL}$9 at world $\mathit{CS}_{\mathit{GL}}$0 iff $\mathit{CS}_{\mathit{GL}}$1 occurs in some antecedent. A mutual induction establishes that antecedent formulae are satisfied and consequent formulae falsified, so failed proof-search certifies invalidity. Notably, the composition step relies on freshness of labels introduced by $\mathit{CS}_{\mathit{GL}}$2 and on end-activeness to argue that shared labels cannot carry contradictory atomic information across distinct stable leaves—an argument where the linearization makes correctness genuinely nontrivial rather than immediate.

Termination and complexity

The termination argument exploits the subformula property together with the pigeonhole principle. Letting $\mathit{CS}_{\mathit{GL}}$3, suppose a line sequent of length exceeding $\mathit{CS}_{\mathit{GL}}$4 were generated; then some box subformula $\mathit{CS}_{\mathit{GL}}$5 was principal twice along the branch. By saturation and the $\mathit{CS}_{\mathit{GL}}$6 condition, the later occurrence $\mathit{CS}_{\mathit{GL}}$7 already appears in the antecedent when the earlier principal formula is unpacked, making the sequent an instance of $\mathit{CS}_{\mathit{GL}}$8—contradicting the applicability of $\mathit{CS}_{\mathit{GL}}$9. Hence every line sequent has at most $GL$0 relational atoms, bounding its size by $GL$1, and branches contain at most $GL$2 rule applications. Depth-first execution therefore consumes $GL$3 space, i.e., polynomial in $GL$4, establishing the $GL$5 upper bound and matching the known lower bound. The implication is that the $GL$6 behavior observed in the HOL Light implementation [MagPer23] is an artifact of the search strategy, not inherent to expressive sequent formalisms.

Limitations and open questions

The paper concedes several boundaries of its results. First, the algorithm itself returns only a Boolean verdict; counter-model extraction from computation trees is proved correct but is not integrated as an output feature, so users of the procedure do not obtain witnesses of invalidity directly. Second, the complexity analysis gives a $GL$7 space bound without tighter practical constants or time analysis, and no implementation or empirical evaluation is provided. Third, the claim that linearization "appears adaptable" to other modal logics remains conjectural: the termination argument depends specifically on the diagonal-formula mechanism of $GL$8 and the conversely well-founded semantics of $GL$9, and the paper does not demonstrate the method for logics lacking these features. Finally, the extracted LNS calculus for $GL$0 is presented without a systematic study of its structural properties—admissibility of weakening and contraction, invertibility, and syntactic cut admissibility are explicitly left open, as is a comparison with the independently introduced $GL$1 beyond noting that the produced proofs are variants thereof.

Conclusion

This paper delivers a complexity-optimal, $GL$2 decision procedure for Gödel–Löb logic formulated in Poggiolesi's tree-hypersequent framework, thereby answering Poggiolesi's decidability question and achieving the $GL$3 goal set by Maggesi and Perini Brogi. Its two methodological contributions—linearization via depth-first, disjunctively branched proof-search, and fragment-composition-based counter-model extraction for correctness on failure—are technically independent of interest, and the corollary that valid formulae admit line-like, end-active proofs provides a new route to linear nested sequent completeness. The open questions concerning generalization to other modal logics and the metatheory of the extracted LNS calculus delineate the natural continuation of this work.