# Symbolic Intermediaries as a Linguistic-Numerical Interface for LLM-Driven Geometric Reasoning

arXiv: https://arxiv.org/abs/2505.17607 · HTML: https://mlsanity.com/paper/2505.17607/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 30 of its 66 references that are in the map.
- First posted on arXiv 23 May 2025.

## Placed in

- LLM reasoning / [Spatial and embodied reasoning](https://mlsanity.com/lr/spatial-and-embodied-reasoning/)

## Builds on (most cited first, 30 of 30)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks](https://mlsanity.com/paper/2005.11401/) (2020)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [Reflexion: Language Agents with Verbal Reinforcement Learning](https://mlsanity.com/paper/2303.11366/) (2023)
- [Toolformer: Language Models Can Teach Themselves to Use Tools](https://mlsanity.com/paper/2302.04761/) (2023)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Self-Refine: Iterative Refinement with Self-Feedback](https://mlsanity.com/paper/2303.17651/) (2023)
- [A Survey on Large Language Model based Autonomous Agents](https://mlsanity.com/paper/2308.11432/) (2023)
- [ToolLLM: Facilitating Large Language Models to Master 16000+ Real-world APIs](https://mlsanity.com/paper/2307.16789/) (2023)
- [Rethinking the Role of Demonstrations: What Makes In-Context Learning Work?](https://mlsanity.com/paper/2202.12837/) (2022)
- [CAMEL: Communicative Agents for "Mind" Exploration of Large Language Model Society](https://mlsanity.com/paper/2303.17760/) (2023)
- [What Can Transformers Learn In-Context? A Case Study of Simple Function Classes](https://mlsanity.com/paper/2208.01066/) (2022)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [SayPlan: Grounding Large Language Models using 3D Scene Graphs for Scalable Robot Task Planning](https://mlsanity.com/paper/2307.06135/) (2023)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [AgentSquare: Automatic LLM Agent Search in Modular Design Space](https://mlsanity.com/paper/2410.06153/) (2024)
- [Transformer-based Planning for Symbolic Regression](https://mlsanity.com/paper/2303.06833/) (2023)
- [SNIP: Bridging Mathematical Symbolic and Numeric Realms with Unified Pre-training](https://mlsanity.com/paper/2310.02227/) (2023)
- [The RBO Dataset of Articulated Objects and Interactions](https://mlsanity.com/paper/1806.06465/) (2018)
- [LimSim++: A Closed-Loop Platform for Deploying Multimodal LLMs in Autonomous Driving](https://mlsanity.com/paper/2402.01246/) (2024)
- [Grounding LLMs For Robot Task Planning Using Closed-loop State Feedback](https://mlsanity.com/paper/2402.08546/) (2024)
- [Towards Cognitive AI Systems: a Survey and Prospective on Neuro-Symbolic AI](https://mlsanity.com/paper/2401.01040/) (2024)
- [AgentKit: Structured LLM Reasoning with Dynamic Graphs](https://mlsanity.com/paper/2404.11483/) (2024)
- [Retrospex: Language Agent Meets Offline Reinforcement Learning Critic](https://mlsanity.com/paper/2505.11807/) (2025)
- [LINKS: A dataset of a hundred million planar linkage mechanisms for data-driven kinematic design](https://mlsanity.com/paper/2208.14567/) (2022)
- [LInK: Learning Joint Representations of Design and Performance Spaces through Contrastive Learning for Mechanism Synthesis](https://mlsanity.com/paper/2405.20592/) (2024)
- [Learning Kinematic Descriptions using SPARE: Simulated and Physical ARticulated Extendable dataset](https://mlsanity.com/paper/1803.11147/) (2018)

## Built on by (most cited first, 2 of 2)

- [Language Models Refine Mechanical Linkage Designs Through Symbolic Reflection and Modular Optimisation](https://mlsanity.com/paper/2604.27962/) (2026)
- [R-APS: Compositional Reasoning and In-Context Meta-Learning for Constrained Design via Reflective Adversarial Pareto Search](https://mlsanity.com/paper/2606.04823/) (2026)

## Abstract (arXiv)

Large Language Models (LLMs) display reasoning capabilities over linguistic and symbolic objects but have limited capabilities to directly interpret the continuous numerical outputs of physics simulators, e.g., distances, curvatures, and trajectories that resist discrete tokenisation. Across spatially grounded engineering reasoning tasks, from mechanism design to motion planning, this defines a fundamental gap, which limits the wider application of LLMs within broader geometrical domains, for exmaple interfacing with physics simulators. We propose symbolic intermediaries, compact analytical expressions discovered via symbolic regression, as a structured interface that translates a simulator's numerical traces into a symbolic form, which language models can interpret, compare, and critique while preserving the original geometric semantics. Around this interface we build an agentic coordination-and-refinement loop: a design agent maps natural-language specifications to executable simulation code, a critique agent reasons over the shared symbolic vocabulary, and a revision step turns this feedback into grounded refinement decisions, enabling inference-time generalization without parameter updates. On the MSynth benchmark for planar mechanism synthesis, all three evaluated LLM agents outperform a budget-matched genetic-algorithm baseline by 19-53% (up to 63% lower median error with feedback), and analysis of the critique entries across three model architectures shows that the interface shifts reasoning from generic structural commentary to grounded geometric verification. The principle of translating continuous simulation outputs into symbolic forms generalises to any domain where simulator behaviour must be interpreted linguistically.