LLM-SRBench: LLM Reasoning & Discovery

• LLM-SRBench is a suite of benchmarks systematically evaluating LLMs in scientific equation discovery, logical reasoning, and autograding for retrieve/generate tasks.
• It enforces strict anti-memorization with synthetic data, variable transformations, and curriculum-based complexity to ensure genuine reasoning and discovery.
• Empirical results reveal LLM limitations in symbolic accuracy, out-of-distribution generalization, and recursive logic, guiding next-generation hybrid model designs.

LLM-SRBench refers to several ambitious benchmarks converging on the systematic evaluation of LLMs’ abilities to reason, search, retrieve, and discover in tasks spanning scientific equation discovery, logical reasoning, and retrieval/QA grading. Distinct LLM-SRBench initiatives include (1) a scientific equation-discovery benchmark spanning symbolic and data-driven “discovery” challenges (Shojaee et al., 14 Apr 2025), (2) a scalable, curriculum-based logical reasoning suite built from synthesized inductive logic programming tasks (Helff et al., 18 Jun 2025), and (3) a rubric-based workbench enabling automated and partially human-in-the-loop evaluation of retrieve/generate systems using LLM autograding (Dietz, 2024). All variants are explicitly designed to preclude training set memorization, support fine-grained metric reporting, and calibrate model evaluations to reflect genuine reasoning or discovery rather than recitation or heuristic exploitation.

1. Scientific Equation Discovery Benchmark: Composition and Methodology

LLM-SRBench for scientific equation discovery is a purpose-built benchmark containing 239 problems distributed across four scientific domains: chemistry (36), biology (24), physics (43), and materials science (25) (Shojaee et al., 14 Apr 2025). The benchmark consists of two primary categories:

• LSR-Transform: 111 problems derived from the Feynman physics collection, each presented in an algebraically transformed form (e.g., solving for non-standard variables) to prevent model reliance on memorized template equations. Each transformation is analytically verified (using SymPy), and only invertible, valid-domain variants are used.
• LSR-Synth: 128 fully synthetic, discovery-driven problems constructed by combining canonical (“known”) and LLM-generated (“synthetic”) terms within each scientific context. Each synthetic problem is numerically verified for analytic solvability and domain expert-vetted for scientific plausibility.

Problem generation is rigorously multi-step: transformation problems require re-filtering of data points for validity after variable inversion, and synthetic problems are created via prompt-driven collection of known/synthetic terms, symbolic composition, LLM-based novelty checking, and ODE numerical integration or closed-form sampling as appropriate for the equation type.

Evaluation employs several fidelity and reasoning metrics:

• Numeric accuracy within tolerance $\tau$:

$$\mathrm{Acc}_\tau = \frac{1}{N_{\text{test}}} \sum_{i=1}^{N_{\text{test}}} \mathbf{1}\left( |y_i - \hat{y}_i| \le \tau \right)$$

(typically $\tau=0.1$).

• Normalized MSE (NMSE).
• Symbolic correctness: GPT-4o-based symbolic equivalence check, parameter-invariant, with protocol validated to 94.6% agreement with human experts on a random 130-problem sample.

2. Empirical Results and Analysis for Equation Discovery

Benchmarked equation-discovery frameworks—Direct Prompting, SGA (bilevel optimizer), LaSR (LLM-guided genetic search), and LLM-SR (evolutionary Python-code islands with LLM-driven code synthesis)—were evaluated with Llama-3.1-8B-Instruct, GPT-3.5-turbo, and GPT-4o-mini backbones under a 1,000 LLM-call problem cap.

Key findings (Shojaee et al., 14 Apr 2025):

• Direct prompting achieves <5% symbolic accuracy.
• SGA and LaSR reach NMSE < $10^{-3}$ on some domains but rarely >20% symbolic correctness.
• LLM-SR with GPT-4o-mini attains up to 31.5% symbolic accuracy on LSR-Transform and 31.5% (chemistry), 11.1% (biology), 16.7% (physics), and 20.2% (material science) within LSR-Synth.
• Average symbolic accuracy: ~25% on LSR-Transform, ~22% on LSR-Synth. OOD numerical generalization degrades significantly, especially for chemistry/biology ODEs.

Observed model errors indicate fragility under variable transformation (memorization collapse), limited ability to combine synthetic terms, and poor OOD extrapolation. Symbolic accuracy is not strongly predicted by numeric fit, highlighting the need for multi-faceted evaluation.

3. SLR-Bench: Scalable Logical Reasoning Benchmark Construction and Curriculum

The logical reasoning branch (also referred to as SLR-Bench or MetaBench) is constructed automatically using the SLR (Scalable Logical Reasoning) synthesis framework (Helff et al., 18 Jun 2025). Each of its 19,000+ tasks arises from the following pipeline:

• Formal ILP (Inductive Logic Programming) instance generation:

$\mathcal{I} = (B, E^+, E^-)$

with $B$ as background atoms, $E^+$ positive queries, and $E^-$ negative queries; solution rules $H$ must entail all $q\in E^+$ and not $q\in E^-$.

• Rule synthesis: Uniform random or LLM-guided generation of single definite-clause rules $R^\star$ of specified length; LLM-guided synthesis introduces arithmetic and recursion at higher levels.
• Background generation: Sampling policies (“mirror” for trivial, “uniform” for unconstrained) ensure precise control of difficulty.
• Prompt generation: Both natural language and Prolog-like instructions, with level-dependent vocabulary, background size, and verbosity.
• Executable judge: Symbolic (Prolog) kernel that deterministically verifies candidate rules (binary and partial credit computed).

The 20-level curriculum systematically scales relational complexity (number of predicates/constants), arithmetic complexity (inclusion of numeric tests), and recursive complexity (multi-clause/recursive rules). Each tier (Basic, Easy, Medium, Hard) introduces novel reasoning patterns:

• Level 1: Single-car train, one predicate, single literal rules.
• Level 8: Two trains, three cars, rule length 2, numeric comparison.
• Level 13: Multiple predicates, rule length 4, categorical tests.
• Level 19: Recursive/multi-clause rules (e.g., transitive closure).

No ground-truth rule repeats between training, development, and test sets; combinatorial coverage grows from $10^3$ at Level 1 to $10^{919}$ at Level 20, enforcing novelty and preventing overfitting.

4. Experimental Outcomes for Logical Reasoning

Seventeen LLMs were evaluated zero-shot on 600 SLR-Bench tasks (20 levels, 30 tasks/level) (Helff et al., 18 Jun 2025). Metrics reported include:

• Logical-Reasoning Level (LRL): weighted per-level accuracy.
• Syntactic correctness.
• Tier (Basic/Easy/Medium/Hard) accuracy, total completion tokens, and API cost.

Summary results:

• o3 (presumably OpenAI’s flagship) achieves LRL 15.5, 99%/93%/74%/45% (Basic/Easy/Medium/Hard), 80% syntax accuracy (>$200 API cost for batch).
• Gemini-FlashThinking: LRL 8.6.
• Llama-3-8B: LRL 5.0, major dropoff after Basic (82%→17% Easy; 1% on Medium/Hard).
• Fine-tuned Llama-3-8B (“FFT”): LRL 9.4, 92% (Basic), 77% (Easy), 17% (Medium), 2% (Hard), outperforming Gemini-FlashThinking on Basic/Easy at >100× lower compute than o3.

Logic-tuning (parameter fine-tuning on SLR tasks) nearly quadruples Easy/Medium-level accuracy. Hard levels remain an open challenge (max 2%). The fully automated synthesis and symbolic judging allow for repeated novel task generation without memorization risk.

5. LLM-SRBench for Autograding Retrieve/Generate Systems

A separate “LLM-SRBench” paradigm arises from the rubric-graded workbench for autograding retrieve/generate QA systems (Dietz, 2024). The proposed pipeline is as follows:

1. Test bank generation: Semi-automatic (ChatGPT-initialized, human-refined) generation of per-query “nuggets” (key facts) and “exam questions.”
2. LLM grading: Self-rating (0–5 scale with $\tau=4$$default), answer-extraction plus gold verification, and direct binary grading. Grading uses <a href="https://www.emergentmind.com/topics/instruction-tuned-models" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">instruction-tuned models</a> (e.g., FLAN-T5).</li> <li><strong>Manual oversight</strong>: Curators inspect highly-graded artifacts and test-bank “gaps” or “spurious” items, then revise nuggets/questions as needed.</li> <li><strong>Evaluation</strong>: <ul> <li><strong>Rubric-Cover@k</strong>: Fraction of test items covered in top-$$k$$passages.</li> <li><strong>Rubric-Qrels</strong>: qrels file exported from self-ratings or correct-answers, enabling trec_eval computation (e.g., MRR, nDCG, R@k).</li> </ul></li> </ol> <p>In controlled TREC DL 2020 experiments, question-based grading at$$\tau=4$$achieves Spearman/Kendall correlation of .941/.810 with the official leaderboard, matching direct-grading baselines. Nugget-based scores saturate, revealing that fact granularity is crucial. <a href="https://www.emergentmind.com/topics/inter-annotator-agreement-iaa-bf6a0eff-29b6-496f-804f-9d738b6ed1c2" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Inter-annotator agreement</a> with manual judgments is modest (Cohen’s$$\kappa$$≈ .25 for questions, .16 for nuggets) but improves with iterative test-bank refinement.</p> <p>The primary software artifact is a modular, CLI-driven Python library supporting JSON-lines interchange, with explicit auditability at every stage in the rubric pipeline. Best practices emphasize the necessity of human-verified test banks, careful setting of self-rating thresholds, and hybridized direct and rubric approaches for optimal reliability.</p> <h2 class='paper-heading' id='design-principles-and-novelty-guarantees-across-llm-srbench-variants'>6. Design Principles and Novelty Guarantees Across LLM-SRBench Variants</h2> <p>All LLM-SRBench (in all senses) are designed with stringent anti-memorization provisions:</p> <ul> <li>Datasets are either procedurally synthesized (SLR-Bench, equation discovery synthetic) or explicitly transformed to prevent template recall (LSR-Transform).</li> <li>Test/train/dev splits are mutually exclusive at the rule/equation level (no$$R^\star$$or symbolic form repeats).</li> <li>Grammar filtering and novelty-checking (including LLM adversarial prompts) eliminate superficial or redundant tasks.</li> <li>Task combinatorics for logic exceed$$10^{900}$ at high curricula, precluding any realistic coverage by model pretraining.
3. For retrieve/generate evaluation, per-query test banks are curated to avoid spurious generalities and ensure coverage/reportability.

This architectural and procedural transparency allows for benchmarking of authentic model inductive/deductive reasoning and scientific discovery—critical capabilities poorly measured by rote recall or autofill metrics.

7. Outlook and Recommendations

LLM-SRBench (across all iterations) exposes enduring weaknesses in current LLMs, including poor symbolic transformation, limited synergy with synthetic novel terms, sharp OOD generalization dropoff, and shallow performance scaling for increasingly recursive or arithmetic logic tasks (Shojaee et al., 14 Apr 2025, Helff et al., 18 Jun 2025). Recommendations for future work emerging from these frameworks include:

• Expansion to differential-algebraic and PDE problem classes.
• More granular partial-credit symbolic match metrics.
• Stronger OOD/robustness regimes.
• Hybrid neural-symbolic architectures combining LLM priors with differentiable program synthesis.
• Curriculum-driven prompt engineering to teach explicit algebraic/logic rules.
• Improved human-in-the-loop semantic verification for automated grading.

LLM-SRBench thus constitutes a collection of methodologies and resources that enable transparent, high-fidelity evaluation of reasoning, search, and discovery capabilities in LLMs, supporting the systematic advancement of the field.