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arxiv: 2602.21265 · v2 · pith:PQSDYDBFnew · submitted 2026-02-24 · 💻 cs.CL · cs.LG· cs.SE

ToolMATH: A Diagnostic Benchmark for Long-Horizon Tool Use under Systematic Tool-Catalog Constraints

Hyeonje Choi , Jeongsoo Lee , Hyojun Lee , Jay-Yoon Lee This is my paper

Pith reviewed 2026-05-21 11:54 UTC · model grok-4.3

classification 💻 cs.CL cs.LGcs.SE
keywords benchmarktool uselong-horizonlanguage modelsdistractorsadaptabilityrobustness
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The pith

ToolMATH converts MATH problems into tool chains to test how models adapt to changing tool catalogs over long sequences.

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

The paper presents ToolMATH as a diagnostic benchmark that converts MATH dataset solutions into reusable Python tools with natural-language descriptions. It constructs environments with gold tools and graded distractors to control catalog difficulty. Evaluation focuses on three axes: adaptability to full distractor replacement, robustness to added noise, and connectivity across long tool chains. Behavior-conditioned metrics and failure traces distinguish model profiles like tool avoidance or adaptive substitution. This provides a testbed for long-horizon tool use beyond final accuracy.

Core claim

ToolMATH converts stepwise MATH solutions into reusable Python tools with natural-language descriptions and typed schemas, pairing each problem with a tool environment that requires sequential use and intermediate reuse, while controlling availability with gold tools and graded distractors to measure adaptability, robustness, and tool connectivity.

What carries the argument

The construction of gold tools and graded distractors with varying similarity, combined with behavior-conditioned metrics for diagnostic evaluation beyond final accuracy.

If this is right

  • Distinct model profiles emerge: reliable tool use, tool avoidance, adaptive substitution, and impacts of unreliable catalogs.
  • Trace-level failure analyses characterize failures under each tool-catalog condition.
  • Models are tested on preserving accuracy over long executed tool-call chains.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could help develop better tool-selection strategies in agents facing dynamic tool environments.
  • Extending the benchmark to non-math domains might reveal if the observed profiles are domain-specific.
  • Unreliable tool catalogs may require models to have built-in verification mechanisms for tool outputs.

Load-bearing premise

Stepwise solutions from the MATH dataset can be converted into reusable Python tools with natural-language descriptions and typed schemas while preserving the original logical structure and intermediate reuse requirements.

What would settle it

If models maintain similar performance across all tool-catalog conditions without showing the expected distinct profiles, the diagnostic power of the benchmark would be undermined.

Figures

Figures reproduced from arXiv: 2602.21265 by Hyeonje Choi, Hyojun Lee, Jay-Yoon Lee, Jeongsoo Lee.

Figure 1
Figure 1. Figure 1: TOOLMATH construction and evaluation pipeline. Step 1: Tool extraction and validation. We convert MATH solution steps into schema-specified Python tools and retain only tools whose described semantics are consistent with their executed behavior via validation. Step 2: Tool-grounded evaluation with controlled redundancy. For each problem, we form a tool environment by combining its gold tools with distracto… view at source ↗
Figure 2
Figure 2. Figure 2: GPT-4o-mini average accuracy across logical-hop groups(averaging the number of distractors k). Curves correspond to No tools, Gold-only, and gold + distractors with similarity level 1-5. Accuracy degrades nearly monotonically with hop count, while higher levels amplify variance in higher-hop regimes [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Failure cases under Gold-present (Level 3, k=5). Counts from 100 failed instances per model; multiple labels per instance allowed. Failure-type taxonomy. We label failures using the fol￾lowing non-exclusive categories (multiple labels may apply). Unless stated otherwise, each label is assigned if the condi￾tion occurs at least once in the trace. (see [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: TOOLMATH vs. TOOLMATH-HARD: hop-wise accuracy under tool availability and insufficiency (Llama 3-8B and Qwen 2.5-7B). Left: TOOLMATH. Right: TOOLMATH-HARD. We report accuracy by logical-hop group under No tools, Gold-only, and Distractors-only. For all settings involving distractors, we fix the distractor list to pure random sampling (Level 2) with k = 10 tools [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Hop-wise framework comparison across models on both TOOLMATH and TOOLMATH-HARD under the Gold￾only tool list. Across the two datasets, framework differences separate strongly at higher hops, where Plan+ReAct most reliably maintains high accuracy, highlighting long-horizon plan coherence as the key bottleneck. (Hop 1 is absent in TOOLMATH-HARD.) planning does not uniformly help at low hops and can be compar… view at source ↗
Figure 6
Figure 6. Figure 6: Gold-setting results for GPT-4o-mini (full grid). The grid reports accuracy over logical hops (columns) and distractor set size k (rows), for distractor sampling levels 1–5. Baselines (No Tools, Gold Only) are shown at the top. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Gold-setting results for Qwen 2.5-7B (full grid). 11 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Gold-setting results for Llama 3-8B (full grid). 12 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Gold(Level 1) vs Distractors-only(Level 1) for GPT-4o-mini (full grid) [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Gold(Level 1) vs Distractors-only(Level 1) for Llama 3-8B (full grid) [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Gold(Level 1) vs Distractors-only(Level 1) for Qwen 2.5-7B (full grid). C. Detailed Evaluation Results for TOOLMATH-HARD [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Hard-set frameworks for GPT-4o-mini (heatmap): no tools vs. ReAct vs. DFSDT vs. Plan+ReAct [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Hard-set frameworks for Llama 3-8B (heatmap): no tools vs. ReAct vs. DFSDT vs. Plan+ReAct. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Hard-set frameworks for Qwen 2.5-7B (heatmap): no tools vs. ReAct vs. DFSDT vs. Plan+ReAct. D. Tool Extraction and Logical-Hop Annotation Details This appendix describes (i) how we extract reusable tools from annotated MATH solutions, (ii) the tool schema and storage format, and (iii) how we annotate each problem with a logical-hop count used for evaluation. D.1. Tool Extraction Prompt We extract tools by… view at source ↗
Figure 15
Figure 15. Figure 15: TOOLMATH vs. TOOLMATH-HARD: hop-wise accuracy under tool availability and insufficiency (all models). Left: TOOLMATH. Right: TOOLMATH-HARD. We report accuracy by logical-hop group under No tools, Gold-only, and Distractors-only. For all settings involving distractors, we fix the distractor list to pure random sampling (Level 2) with k = 10 tools. I. Representative Correct Traces in Distractors-only Settin… view at source ↗
read the original abstract

We introduce \ToolMATH, a math-grounded diagnostic benchmark for evaluating long-horizon tool use under controllable tool-catalog conditions. \ToolMATH converts stepwise MATH solutions into reusable Python tools with natural-language descriptions and typed schemas, and pairs each problem with a tool environment requiring sequential tool use, intermediate-output reuse, and logically connected tool-call chains. \ToolMATH controls tool availability and catalog difficulty by constructing gold tools and graded distractors with varying similarity to gold tools. \ToolMATH also incorporates behavior-conditioned metrics, enabling diagnostic evaluation beyond final accuracy. Building on these measurements, \ToolMATH emphasizes three evaluation axes: (1) \emph{Adaptability} measures how much Gold-only success is retained when gold tools are replaced entirely by distractors; (2) \emph{Robustness} measures stability under adding distractors as a noise; and (3) \emph{Tool Connectivity} measures whether models preserve accuracy over long executed tool-call chains. Furthermore, trace-level failure analyses characterize how models fail under each tool-catalog condition. Together, these diagnostics reveal distinct model profiles: reliable tool use, tool avoidance, adaptive substitution, and impacts of unreliable tool catalogs. Overall, \ToolMATH provides a controlled testbed for evaluating how language models adapt to changing tool availability, remain robust to distractors, and maintain correctness across long-horizon tool-use trajectories.

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

2 major / 2 minor

Summary. The paper introduces ToolMATH, a diagnostic benchmark that converts stepwise solutions from the MATH dataset into reusable Python tools equipped with natural-language descriptions and typed schemas. Each problem is paired with a controlled tool environment that requires sequential calls, intermediate-output reuse, and logically connected chains. Tool availability is varied through gold tools plus graded distractors of differing similarity; three axes are defined—Adaptability (retention of Gold-only success under full distractor replacement), Robustness (stability when distractors are added as noise), and Tool Connectivity (accuracy across long executed chains)—together with trace-level failure analyses that aim to distinguish model profiles such as reliable tool use, tool avoidance, and adaptive substitution.

Significance. If the conversion process demonstrably preserves the original dependency structure and forces intermediate reuse, ToolMATH would supply a useful, controllable testbed for isolating specific failure modes in long-horizon tool use that current benchmarks do not systematically vary. The emphasis on behavior-conditioned metrics and catalog-difficulty controls is a constructive addition to the tool-use evaluation literature.

major comments (2)
  1. [§3] §3 (Benchmark Construction): the manuscript states that MATH solution steps are converted into Python tools whose schemas enforce sequential use and intermediate reuse, yet supplies neither concrete conversion examples nor any verification (e.g., dependency-graph statistics or manual inspection) that the resulting gold-tool chains are minimal and non-redundant. Without such evidence the three diagnostic axes risk measuring something other than the intended properties.
  2. [§4] §4 (Evaluation Metrics): the definitions of Adaptability and Tool Connectivity presuppose that distractors cannot duplicate functionality or allow direct subproblem solving; the paper should report quantitative checks confirming that gold chains remain the shortest and that distractors do not create bypass routes.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'behavior-conditioned metrics' is introduced without a one-sentence gloss; a brief parenthetical definition would aid readers.
  2. [Figure 1] Figure 1 or §3.2: the tool-catalog construction diagram would benefit from an explicit legend distinguishing gold tools, similarity-graded distractors, and distractors that duplicate functionality.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (Benchmark Construction): the manuscript states that MATH solution steps are converted into Python tools whose schemas enforce sequential use and intermediate reuse, yet supplies neither concrete conversion examples nor any verification (e.g., dependency-graph statistics or manual inspection) that the resulting gold-tool chains are minimal and non-redundant. Without such evidence the three diagnostic axes risk measuring something other than the intended properties.

    Authors: We agree that explicit examples and verification evidence are necessary to substantiate the benchmark construction. In the revised manuscript we will add concrete conversion examples showing how individual MATH solution steps are turned into Python tools (including their natural-language descriptions and typed schemas). We will also report dependency-graph statistics (e.g., average and maximum chain lengths, number of intermediate outputs) together with the results of a manual inspection of a random sample of problems confirming that the gold-tool chains are minimal and contain no redundant steps. These additions will directly demonstrate that the three diagnostic axes measure the intended properties of sequential use and intermediate reuse. revision: yes

  2. Referee: [§4] §4 (Evaluation Metrics): the definitions of Adaptability and Tool Connectivity presuppose that distractors cannot duplicate functionality or allow direct subproblem solving; the paper should report quantitative checks confirming that gold chains remain the shortest and that distractors do not create bypass routes.

    Authors: We accept that quantitative validation of these assumptions is required. In the revision we will add explicit checks: (1) a comparison of the shortest executable path length using only gold tools versus the full catalog (gold + distractors) across all problems, and (2) an enumeration of whether any distractor combination permits solving a subproblem without following the gold chain. These statistics will be reported in §4 and will confirm that distractors neither duplicate gold functionality nor create bypass routes, thereby supporting the presuppositions underlying the Adaptability and Tool Connectivity metrics. revision: yes

Circularity Check

0 steps flagged

No circularity in ToolMATH benchmark construction or evaluation axes

full rationale

The paper introduces ToolMATH by explicitly constructing a benchmark from the external MATH dataset: stepwise solutions are converted into Python tools with descriptions and schemas, then paired with controlled tool environments that include gold tools and graded distractors. The three evaluation axes (Adaptability, Robustness, Tool Connectivity) and trace-level metrics are defined directly from these construction choices and catalog variations rather than derived from any equations, fitted parameters, or prior results. No load-bearing step reduces a claimed diagnostic property back to its inputs by construction, and the work contains no self-citation chains or uniqueness theorems that would force the central claims. The derivation is therefore self-contained as a benchmark proposal.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central contribution rests on the assumption that MATH stepwise solutions can be turned into faithful tool representations; no free parameters, new physical entities, or ad-hoc constants are introduced.

axioms (1)
  • domain assumption Stepwise solutions from the MATH dataset can be converted into reusable Python tools with natural-language descriptions and typed schemas without loss of logical connectivity.
    Invoked directly in the benchmark construction described in the abstract.

pith-pipeline@v0.9.0 · 5792 in / 1210 out tokens · 53572 ms · 2026-05-21T11:54:07.917162+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 3 internal anchors

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    work page internal anchor Pith review Pith/arXiv arXiv 2022

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    API-Bank: A Comprehensive Benchmark for Tool-Augmented LLMs

    Association for Computational Linguistics. doi: 10.18653/v1/2023.emnlp-main.187. arXiv:2304.08244. Patil, S. G., Zhang, T., Wang, X., and Gonzalez, J. E. Go- rilla: Large language model connected with massive APIs. InAdvances in Neural Information Processing Systems (NeurIPS), 2024. Poster; arXiv:2305.15334. Patil, S. G., Mao, H., Yan, F., Ji, C. C.-J., S...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.18653/v1/2023.emnlp-main.187 2023

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    Toolformer: Language Models Can Teach Themselves to Use Tools

    arXiv:2302.04761. Shen, Y ., Song, K., Tan, X., Li, D., Lu, W., and Zhuang, Y . Hugginggpt: Solving AI tasks with chatgpt and its friends in hugging face. InAdvances in Neural Information Processing Systems 36: Annual Conference on Neural Information Processing Systems 2023, NeurIPS 2023, New Orleans, LA, USA, December 10–16, 2023, 2023. arXiv:2303.17580....

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  4. [4]

    name" - concise snake_case identifier -

    arXiv:2311.18760. Trevi˜no, E., Contant, H., Ngai, J., Neubig, G., and Wang, Z. Z. Benchmarking failures in tool-augmented language models. InAnnual Conference of the Nations of the Americas Chapter of the Association for Computational Linguistics (NAACL), 2025. arXiv:2503.14227. Xu, Q., Hong, F., Li, B., Hu, C., Chen, Z., and Zhang, J. On the tool manipu...

    work page arXiv 2025

  5. [5]

    I will give you a math problem

    work page

  6. [6]

    At each step, analyze the current state and choose the next action by calling ONE function

    work page

  7. [7]

    You will receive the function result (observation), then repeat

    work page

  8. [8]

    give_answer

    When you have enough to answer, call Finish with return_type="give_answer" and include a complete final_answer

    work page

  9. [9]

    give_up_and_restart

    If you cannot proceed effectively, call Finish with return_type=" give_up_and_restart". Rules: - Keep your thoughts concise (max 5 sentences). - Prefer tool calls for computation, algebra, simplification, verification. - Make one attempt per idea; don’t repeat the same call with the same arguments. User prompt (problem wrapper; verbatim). Task description...

    work page

  10. [10]

    Vertical asymptotes occur at real $x$ values where the denominator $xˆ3-3xˆ2+3x-1$ equals $0$

    work page

  11. [11]

    To find the real roots of the denominator, use the tool ‘[TOOL: solve_rational_equation_quadratic]‘, which numerically returns real solutions of a cubic polynomial determined by its parameters

    work page

  12. [12]

    Choose parameters so the tool’s internal cubic matches $xˆ3-3xˆ2+3x-1=0$

    work page

  13. [13]

    name":"solve_rational_equation_quadratic

    Count the number of distinct real roots returned. Thought: Find real zeros of the denominator via a distractor tool and count distinct roots. Action: {"name":"solve_rational_equation_quadratic","arguments":{"a":-8.0,"b":-1.0,"c ":1.0,"d":-1.0,"k":-4.0}} Observation (solve_rational_equation_quadratic, {"a":-8.0,"b":-1.0,"c":1.0,"d":-1.0,"k ":-4.0}): [np.fl...

    work page

  14. [14]

    Rewrite each complex number as a 2D vector of its real and imaginary parts: $a \mapsto [2,-1]$ and $b \mapsto [-1,1]$

    work page

  15. [15]

    - $3b = b+b+b$ using ‘[TOOL:add_vectors_2d]‘ twice

    Compute the scaled vectors by repeated addition: - $2a = a+a$ using ‘[TOOL:add_vectors_2d]‘. - $3b = b+b+b$ using ‘[TOOL:add_vectors_2d]‘ twice

    work page

  16. [16]

    Add the results to obtain $2a+3b$ as a 2D vector $[\,\Re,\,\Im\,]$

    work page

  17. [17]

    name":"add_vectors_2d

    Convert the final 2D vector back to complex form $\Re+\Im i$. Thought: Represent $a,b$ as 2D vectors and use repeated vector-addition to form $2a+3 b$. Action: {"name":"add_vectors_2d","arguments":{"v1":[2,-1],"v2":[2,-1]}} Observation (add_vectors_2d, {"v1":[2,-1],"v2":[2,-1]}): [4,-2] Thought: Now, let’s build $3b$ by adding $b$ twice more starting from...

    work page