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

This paper presents a PSPACE-optimal proof-search algorithm for Gödel-Löb Logic using a "linearization method" on tree-hypersequents that resolves open questions regarding syntactic decidability and complexity, while establishing a connection to linear nested sequents and providing a mechanism for extracting finite counter-models.

Original authors: Tim S. Lyon, Omar Taher

Published 2026-06-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tim S. Lyon, Omar Taher

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a very tricky logic puzzle. The puzzle is based on a system called Gödel-Löb logic (GL), which is essentially the mathematics of "provable truth." Think of it as a rulebook for figuring out what can be proven within a specific system, like a game with strict rules about what moves are allowed.

For a long time, mathematicians have had a few different rulebooks (called "calculi") to solve these puzzles. One popular rulebook is called CSGL. It's powerful, but it has a big problem: when you try to use it to solve a puzzle, the process can get incredibly messy and huge, like a tree that keeps branching out into millions of tiny twigs. If you try to follow every single twig, you run out of memory (space) very quickly, making it impossible to solve complex puzzles on a standard computer.

Two researchers, Poggiolesi and Maggesi & Perini Brogi, asked a specific question: "Can we use this powerful rulebook (CSGL) to solve these puzzles efficiently, without running out of memory?"

This paper says yes, and here is how they did it, using some clever tricks:

1. The "One-Path-at-a-Time" Trick (Linearization)

Imagine you are exploring a giant, dark cave system (the logic puzzle). The old way of doing this was to send out a thousand explorers at once, each taking a different path. Eventually, the cave fills up with explorers, and you can't remember who is where. This is what happens in the old proof-search methods: they try to build the whole "tree of trees" at once, which explodes in size.

The authors' new method is like sending out one single explorer who walks down one path, checks if it works, and if it hits a dead end, they backtrack and try the next path. They call this "linearization."

Instead of building a massive, branching tree, they build a single, long line (like a snake) of steps.
They only keep one path in their memory at a time.
This is like reading a book one page at a time instead of trying to hold the entire book open in your hands. It saves a massive amount of space.

2. The "Magic Stop Sign" (The Diagonal Formula)

In logic puzzles, there's a risk of getting stuck in an infinite loop, like walking in circles forever. Usually, you need a complex system to check if you've been somewhere before to stop this.

The authors found a clever shortcut. In their specific rulebook, there is a special "magic stop sign" built into the rules (called the diagonal formula).

Every time the explorer tries to go deeper into the cave, this sign checks the history.
If the explorer tries to use a rule they've already used in a specific way, the sign stops them.
This guarantees that the explorer will never walk in an infinite circle. The path must end eventually. This means the puzzle is guaranteed to be solved (or proven unsolvable) in a reasonable amount of time.

3. The "Scrapbook" Method (Counter-Models)

What happens if the explorer tries every possible path and none of them work? In logic, this means the puzzle is actually a trick question (it's invalid). Usually, to prove this, you need to build a giant "counter-example" (a fake world where the rules break).

Because the authors are only walking one path at a time, they don't have the whole picture to build a giant fake world immediately.

The Solution: They treat each failed path as a small "scrap" of a puzzle.
When the search is over, they take all these little scraps and stitch them together like a patchwork quilt.
This stitched-together quilt becomes the proof that the original puzzle was indeed a trick question. It's a theoretical tool to say, "We tried everything, and here is the proof that it doesn't work."

4. The "Straight Line" Discovery

Here is a surprising bonus: The authors found that if a puzzle is solvable, you don't actually need the complex, branching tree structure at all.

Every valid puzzle can be solved using a straight line of steps.
This connects their method to a newer, simpler style of logic called Linear Nested Sequents. It's like discovering that even though the map looked like a forest, the solution was actually just a straight highway all along.

The Bottom Line

The authors have created a super-efficient detective for logic puzzles.

Before: The detective tried to map the whole forest at once, which took too much memory (EXPSPACE).
Now: The detective walks one path at a time, uses a magic stop sign to avoid loops, and stitches together scraps if the path fails.
Result: They can solve these puzzles using the minimum amount of memory possible (PSPACE), which matches the theoretical limit of how hard these puzzles are.

They answered the questions posed by other mathematicians by showing that you don't need to sacrifice power for efficiency; you just need to change how you look for the answer.

Problem Statement
The paper addresses two specific open questions regarding the proof-search complexity of Gödel-Löb logic (GL). First, it responds to a query by Poggiolesi regarding how to establish a syntactic decidability proof for GL within the tree-hypersequent system CSGL. Second, it resolves a challenge identified by Maggesi and Perini Brogi, who implemented a decision procedure for GL in HOL Light based on labeled sequent calculus (G3KGL) but observed that their procedure operates in EXPSPACE. They identified achieving a PSPACE complexity bound as the "ideal goal." While GL is known to be PSPACE-complete, existing proof-search algorithms for expressive sequent formalisms (such as tree-hypersequents or labeled sequents) often suffer from exponential space blowup due to the "tree of trees" structure of derivations and the requirement to output counter-models upon failure.

Methodology
The authors work with a notational variant of Poggiolesi's CSGL system, formulated using labeled tree sequents. To achieve PSPACE complexity, they introduce a linearization method for proof-search. Unlike naive approaches that explore the entire derivation tree simultaneously, this method constructs only a single branch of a derivation and a single branch of the underlying tree sequent at any given time.

Key technical mechanisms include:

Depth-First Search with Disjunctive Branching: The algorithm (prove) operates in a depth-first manner. For logical rules like $\to L$ , it uses conjunctive branching (both premises must succeed). However, for the $\square R$ rule, which typically generates multiple branches, the authors employ a disjunctive inference rule ( $\square_{DB}$ ). This rule allows the algorithm to explore premises sequentially; if one premise yields a proof, the search halts for that branch. This ensures that at most one branch of the tree sequent is active in memory at any time.
Diagonal Formula Termination: The algorithm relies on the "diagonal formula" ( $y : \square \phi$ ) introduced by the $\square R$ rule. This formula provides a natural bound on the depth of tree sequents generated during proof-search. The presence of this formula prevents infinite loops and eliminates the need for explicit loop-checking mechanisms, even in the presence of transitivity (the $4L$ rule).
Computation Trees: The authors define a "computation tree" structure that records the execution of the algorithm. This structure is used theoretically to extract proofs when the search succeeds and counter-models when it fails.

Key Contributions and Results

PSPACE Complexity: The authors present a proof-search algorithm that decides the (in)validity of GL formulae in PSPACE. This matches the known PSPACE-completeness of GL, achieving complexity-optimality. Since CSGL and G3KGL are inter-translatable in polynomial time, this approach also resolves the PSPACE goal for labeled sequent-based proof-search.
Counter-Model Construction: Because the linearization method generates only single branches, standard methods for extracting counter-models from failed searches are obstructed. The authors demonstrate how to systematically combine fragments of tree sequents generated during a failed search to construct a finite counter-model. This serves as a theoretical device to prove the algorithm's correctness when it returns "false."
Line-Like Proofs and Linear Nested Sequents: The authors show that every valid formula admits a proof in CSGL where every tree sequent is a "line sequent" (a tree sequent where the relational atoms form a single path). These line sequents are notational variants of Linear Nested Sequents (LNS). Consequently, the depth-first proof-search procedure effectively extracts a Linear Nested Sequent Calculus (specifically, a variant of the recently introduced LNGL) for GL.
Trade-off on Counter-Model Extraction: The paper highlights a specific design trade-off: the algorithm forgoes the immediate output of counter-models during execution to reduce space consumption. Counter-model extraction is performed only as a post-hoc theoretical step to verify correctness, rather than as a runtime requirement that would cause exponential space usage.

Significance
The paper claims to answer the specific questions posed by Poggiolesi and Maggesi and Perini Brogi by demonstrating that the EXPSPACE blowup observed in previous implementations is not inherent to expressive sequent formalisms but can be overcome with suitable proof-search techniques.

Furthermore, the work provides new insights into the relationship between depth-first proof-search and linear nested sequent calculi. It establishes that sound and complete LNS systems can be extracted directly from depth-first proof-search procedures formulated in tree (nested) sequent calculi. The results offer a complexity-optimal decision procedure for GL and a theoretical framework for understanding how linearization techniques can be applied to modal logics to manage space complexity while maintaining the structural benefits of tree-hypersequents.

1. The "One-Path-at-a-Time" Trick (Linearization)

2. The "Magic Stop Sign" (The Diagonal Formula)

3. The "Scrapbook" Method (Counter-Models)

4. The "Straight Line" Discovery

The Bottom Line

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