KELPS: A New Era in Autoformalization of Mathematics with Multilingual Precision
Table of Contents
Introduction
Background: The Need for Autoformalization in Modern Mathematics
The Challenges of Multilingual Formalization
Overview of KELPS: Bridging Natural and Formal Language
Knowledge Equations (KEs): The Core of KELPS
Semantic-Syntactic Alignment: Key to Verified Translation
From KEs to Formal Theorem Provers: Lean, Coq, and Isabelle
Data Generation and Parallel Corpus Creation
Empirical Results: KELPS vs. DeepSeek-V3, Herald, and Others
Evaluation on MiniF2F
Comparison with State-of-the-Art Models
Symbolic-Neural Synergy: How KELPS Combines the Best of Both Worlds
Code Availability and Reproducibility
Real-World Applications of KELPS
Implications for Education and Knowledge Access
Limitations and Future Directions
Conclusion
1. Introduction
The formalization of mathematics is undergoing a revolution powered by artificial intelligence. Large Language Models (LLMs) have demonstrated remarkable capabilities in transforming informal, human-written mathematical statements into rigorously verified formal theorems. However, their success has largely been monolingual and constrained, often restricted to English and limited domains.
Enter KELPS — the Knowledge-Equation based Logical Processing System — a cutting-edge neuro-symbolic framework designed to take autoformalization to the next level. By introducing a new intermediate logical form, called Knowledge Equations (KEs), and focusing on multilingual, verified formalization, KELPS opens the door to a scalable and accurate system that bridges informal and formal mathematics.
2. Background: The Need for Autoformalization in Modern Mathematics
Modern mathematics, while rigorous in its core, is often written informally in natural language across books, papers, and educational materials. Translating these informal descriptions into formal proofs in theorem provers (like Coq, Lean, or Isabelle) is vital for:
Machine verification of mathematical claims
Building knowledge graphs for scientific discovery
Enhancing educational technologies
Enabling AI systems to reason about complex formal logic
Yet, manual formalization is slow, difficult, and error-prone. Even expert mathematicians struggle to formalize basic theorems. Hence, there's an urgent need for automatic, accurate, and language-agnostic tools.
3. The Challenges of Multilingual Formalization
While recent LLMs (e.g., GPT-4, DeepSeek-V3) can perform basic formalization in English, the problem becomes exponentially harder in a multilingual setting:
Lack of high-quality multilingual datasets
Variations in syntax and logical constructs across languages
Lack of consistency in how mathematical ideas are phrased
Different formal theorem provers (Lean vs Coq vs Isabelle) have different syntaxes
Most current systems fail to retain semantic integrity during translation, especially across languages. KELPS addresses this by introducing an intermediate, logic-grounded representation.
4. Overview of KELPS: Bridging Natural and Formal Language
KELPS is a multi-stage framework that processes informal mathematical statements and outputs their formal equivalents in multiple theorem proving languages. It works as follows:
Translation: Convert natural language statements into Knowledge Equations (KEs) — a logic-based intermediate language.
Synthesis: Use defined rules to transform KEs into formal representations in Lean, Coq, and Isabelle.
Filtering: Discard invalid or ambiguous formalizations using symbolic logic checks.
This pipeline ensures both syntactic accuracy (valid code) and semantic preservation (correct meaning).
5. Knowledge Equations (KEs): The Core of KELPS
KEs are the heart of KELPS.
What are Knowledge Equations?
KEs are logic expressions rooted in assertional logic, a formal system designed to represent mathematical knowledge clearly. Each KE has:
A logical structure that encodes the assertion (e.g., ∀x ∈ ℕ, x + 0 = x)
Semantic tags that identify the type of mathematical object
Alignment hooks for converting to multiple formal languages
KEs act as a universal representation, enabling one-to-many translation from natural language to multiple formal languages.
6. Semantic-Syntactic Alignment: Key to Verified Translation
A major innovation in KELPS is its semantic-syntactic alignment mechanism. This involves:
Mapping natural language phrases to formal constructs (e.g., “there exists” → ∃)
Using contextual reasoning to resolve ambiguities (e.g., distinguishing between “a function” and “a function defined on X”)
Ensuring that each translated component maintains meaning fidelity across languages
This method allows for bidirectional mapping, enabling both formalization and de-formalization for explainability.
7. From KEs to Formal Theorem Provers: Lean, Coq, and Isabelle
Once a statement is represented in KE form, it can be translated into three major formal languages using rule-based transformations:
Lean (community-driven, highly expressive syntax)
Coq (widely used in software verification)
Isabelle/HOL (powerful higher-order logic prover)
Each target language has its own syntax and logic constructs, but KELPS ensures uniformity by:
Preserving operator precedence
Mapping quantifiers, assumptions, and types precisely
Ensuring formal statements are machine-verifiable
8. Data Generation and Parallel Corpus Creation
Using the KELPS framework, the authors generated a multilingual parallel corpus of:
✅ 60,000+ formal-informal problem pairs
🌍 Spanning multiple languages (including Chinese, Spanish, and French)
🧠 Covering algebra, logic, calculus, and discrete mathematics
🧪 Compatible with MiniF2F and other benchmark datasets
This corpus can be used to train and evaluate multi-language mathematical LLMs and is available open-source for community use.
9. Empirical Results: KELPS vs. DeepSeek-V3, Herald, and Others
On MiniF2F, a popular benchmark for math formalization:
KELPS achieved 88.9% pass@1 syntactic accuracy
This outperforms DeepSeek-V3 (81.0%), Herald (81.3%), and other top models
In multilingual settings, KELPS consistently produced valid and executable code
The authors also tested on formal verification tasks and code synthesis, where KELPS maintained robust performance.
10. Evaluation on MiniF2F
MiniF2F tests models on formalizing math competition problems. The key challenges are:
Complex reasoning
Informal-to-formal language mapping
Long input-output sequences
KELPS’s structured pipeline gives it a clear edge over purely neural models that hallucinate or misinterpret logic.
11. Comparison with State-of-the-Art Models
| Model | MiniF2F Accuracy (Pass@1) | Multilingual Support | Formal Language Output |
|---|---|---|---|
| KELPS | 88.9% | ✅ Full | ✅ Lean, Coq, Isabelle |
| DeepSeek-V3 | 81.0% | Partial | ✅ Lean |
| Herald | 81.3% | ❌ | ✅ Coq |
| GPT-4 (API) | ~85.2% | ✅ Partial | ❌ Inconsistent |
KELPS proves that neuro-symbolic methods + multilingual grounding outperform monolithic LLMs.
12. Symbolic-Neural Synergy: How KELPS Combines the Best of Both Worlds
KELPS is not purely a neural model. It integrates:
🤖 Neural models for understanding informal inputs
📐 Symbolic systems for reasoning and logic validation
⚙️ Rule-based systems for syntactic transformation
This hybrid approach leads to better:
Explainability
Verifiability
Multilingual generalization
13. Code Availability and Reproducibility
All code, datasets, and trained models are available via:
GitHub repository (link in the paper)
Pretrained KE-to-Coq/Lean models
Docker and Jupyter deployment options
API demo to test custom formalizations
This makes KELPS highly reproducible and extensible.
14. Real-World Applications of KELPS
Mathematical Research: Autoformalizing papers for theorem verification
Education: Generating quizzes and formal exercises from textbooks
Multilingual STEM tools: Translate math for diverse global users
AI theorem proving: Enhance proof assistants with automatic formalization
Knowledge bases: Structuring math for querying and retrieval
15. Implications for Education and Knowledge Access
KELPS could democratize access to formal logic by:
Supporting non-English learners
Making math more rigorous and explorable
Enhancing platforms like Khan Academy, Wolfram Alpha, and ChatGPT plugins
Creating AI tutors capable of verified feedback
This could revolutionize mathematical education globally.
16. Limitations and Future Directions
While KELPS is groundbreaking, it has challenges:
⚠️ Manual rule engineering is complex
📉 Some ambiguous language inputs still fail
🔣 Non-Latin script support is limited
🧪 Needs deeper symbolic inference for long proofs
Future work includes:
MoE-based scaling of KELPS
Adding formal languages like HOL-Light, Agda, and Mizar
Integrating with LangChain and DeepSeek for RAG applications
17. Conclusion
KELPS represents a paradigm shift in the field of mathematics autoformalization. By introducing Knowledge Equations and combining symbolic reasoning with LLMs, KELPS achieves state-of-the-art performance, supports multiple languages, and lays the groundwork for verifiable AI mathematicians.
With strong results on challenging benchmarks, transparent design, and open-source availability, KELPS is poised to accelerate formal science, education, and global knowledge access.
