Core Concepts

GFLean is an autoformalisation framework that translates simple mathematical statements expressed in a controlled natural language called Simplified ForTheL to corresponding expressions in the Lean theorem prover.

Abstract

GFLean is an ongoing project that aims to automate the process of formalising mathematical statements. It uses a high-level grammar writing tool called Grammatical Framework (GF) for parsing the input and linearizing the output.
The key steps in the GFLean pipeline are:
Parsing the input statement written in Simplified ForTheL, a simplified version of the controlled natural language ForTheL, using a GF grammar to produce an abstract syntax tree (AST).
Simplifying the AST through a series of tree transformations implemented in Haskell.
Translating the simplified AST to an AST for the corresponding Lean expression.
Linearizing the Lean AST to the final Lean expression using another GF grammar.
GFLean can currently formalise 42 out of 62 statements from Chapter 3 of the textbook "Mathematical Proofs" by G. Chartrand, A. D. Polimeni, and P. Zhang. The limitations include a small lexicon, lack of support for certain linguistic constructs like conjunction of predicates, and the inability to dynamically extend the lexicon during runtime.

Stats

example (x : R) (h39 : x < 0) : ((x ^ 2) + 1) > 0 := sorry
example (x : R) (h57 : (((x ^ 2) - (2 * x)) + 2) ≤0) : (x ^ 3) ≥8 := sorry
example (x : R) (h64 : x > 0) (h51 : x < 1) : (((x ^ 2) - (2 * x)) + 2) ≠ 0 := sorry
example (r : Q) (h76 : pos r) (h63 : (((r ^ 2) + 1) / r) ≤1) : (((r ^ 2) + 2) / r) ≤2 := sorry
example (x : R) (h70 : (((x ^ 3) - (5 * x)) - 1) ≥0) : ((x - 1) * (x - 3)) ≥-2 := sorry

Quotes

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Key Insights Distilled From

by Shashank Pat... at **arxiv.org** 04-02-2024

Deeper Inquiries

To dynamically extend the lexicon of Simplified ForTheL in GFLean, we can implement a mechanism that allows users to add new definitions and notations during runtime. This would involve creating a system where users can input new grammar rules and words to expand the lexicon as needed. By incorporating a feature that enables users to define new terms and their corresponding translations into Lean expressions, GFLean can adapt to handle a wider range of mathematical statements. This dynamic extension capability would make the grammar more flexible and versatile, allowing for the formalization of a broader set of mathematical concepts and expressions.

Translating mathematical proofs, as opposed to statements, using a rule-based approach like GFLean presents several challenges. One major challenge is the complexity and intricacy of mathematical proofs, which often involve multiple logical steps, dependencies, and specialized mathematical symbols. Rule-based systems like GFLean may struggle to capture the nuanced reasoning and structure of proofs, especially when dealing with advanced mathematical concepts or intricate logical arguments.
Additionally, the formal language used in mathematical proofs can be highly technical and precise, requiring a deep understanding of mathematical logic and notation. Ensuring that the rule-based system accurately captures the logical flow and formal structure of a proof can be challenging, as it may involve intricate transformations and manipulations of the underlying logic.
Moreover, mathematical proofs often involve the application of specific rules of inference, axioms, and theorems, which may need to be encoded in the rule-based system. Managing the complexity of these rules and ensuring their correct application throughout the translation process can be a daunting task for a rule-based approach like GFLean.

Combining neural network-based translation models with rule-based systems like GFLean can enhance the robustness and effectiveness of autoformalisation frameworks. Neural networks excel at capturing complex patterns and relationships in data, making them well-suited for handling the nuances and variations in natural language expressions. By leveraging neural networks for the initial translation of natural language statements into a formal representation, GFLean can benefit from the neural network's ability to handle diverse linguistic patterns and nuances.
The rule-based system like GFLean can then be used to refine and validate the translations produced by the neural network. GFLean can apply domain-specific rules, logic, and constraints to ensure the accuracy and consistency of the formalized statements and proofs. This combination allows for a more comprehensive and reliable autoformalisation process, where the strengths of both approaches complement each other.
Furthermore, the neural network can assist in generating initial translations quickly, while the rule-based system can provide the necessary domain-specific knowledge and logical reasoning to ensure the correctness and coherence of the formalized output. By integrating neural network-based translation models with rule-based systems, autoformalisation frameworks can achieve a balance between flexibility, accuracy, and efficiency in converting natural language mathematical expressions into formal proofs.

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