proofs as types

[jmikedupont2]

Your idea of connecting proofs and types in a formal proof assistant with machine learning models is ambitious and intriguing. Let me break down and expand on the concepts and steps involved to outline a potential framework for this project:

Concepts and Steps

1. Proofs as Types: In type theory, especially in frameworks like Coq, Agda, and Unimath, a proof of a proposition is represented as a type. This is known as the Curry-Howard correspondence. A type is a proposition, and a value of that type is a proof of the proposition.

2. Universe of Universes (UU): In type theory, UU represents the type of all types. This is used to avoid paradoxes related to self-referencing types. It allows the construction of types and their relations in a hierarchical manner.

3. Dependent Types: Dependent types are types that depend on values. They are a powerful feature in type theory that allows for expressing more complex propositions and their proofs.

4. Extraction from Unimath to OCaml/Haskell: Tools like MetaCoq allow the extraction of Coq/Unimath proofs to functional programming languages like OCaml and Haskell. This translation allows leveraging the formal proofs in a practical programming environment.

5. Reverse Extraction: The idea here is to create a model that can reverse the extraction process, going from OCaml/Haskell code back to Coq/Unimath proofs. This is a challenging task that involves understanding both the semantics of the functional programming languages and the formal proof structures.

6. Training Models with Symbolic Regression and Program Generation: Symbolic regression involves finding mathematical expressions that best fit a given set of data. Program generation involves creating programs that satisfy certain criteria. Using these techniques, a neural network can be trained to generate code from given specifications or datasets.

7. Binding Types into the Proof Engine: The generated programs (considered as types) need to be integrated back into the proof engine, maintaining the correctness and soundness of the proofs.

8. Bidirectional Conversion: The ultimate goal is to enable bidirectional conversion between neural network models and formal proofs. This involves training neural networks to understand and generate formal proofs and using these proofs to guide the training and structure of neural networks.

Detailed Framework

1. Proof Extraction

• MetaCoq Extraction: Utilize MetaCoq for extracting proofs from Unimath to OCaml/Haskell. Ensure that the extracted code retains as much of the logical structure and type information as possible.

2. Training Neural Networks

• Dataset Generation: Create a large dataset of paired formal proofs and their corresponding extracted OCaml/Haskell code.
• Neural Network Architecture: Design a neural network capable of understanding both syntactic and semantic structures. Transformer models could be particularly effective here due to their ability to handle complex dependencies and large contexts.
• Training: Train the model to perform the reverse extraction, i.e., generating Unimath proofs from OCaml/Haskell code. Use techniques like symbolic regression to aid the training process by ensuring the generated proofs are mathematically valid.

3. Program Generation and Symbolic Regression

• Symbolic Regression: Implement symbolic regression to find mathematical expressions that best fit the neural network outputs. This helps in ensuring that the generated proofs adhere to the expected mathematical properties.
• Program Synthesis: Use program synthesis techniques to generate code that matches the proofs. This involves creating correct-by-construction programs that can be treated as types in the proof engine.

4. Integration with Proof Engines

• Type Binding: Develop a method to bind the generated types back into the proof engine, ensuring the integrity and correctness of the formal proofs.
• Proof Verification: Use the proof engine to verify the correctness of the generated proofs. Any discrepancies should be fed back into the training process to improve the model.

5. Bidirectional Conversion

• Model Refinement: Continuously refine the neural network model to improve its ability to convert between formal proofs and functional code in both directions.
• Pretrained Models: Develop pretrained models that can be used to initialize the neural network, speeding up the training process and improving accuracy.

Challenges and Considerations

• Complexity: Both the extraction and reverse extraction processes are highly complex and require a deep understanding of type theory and functional programming.
• Scalability: The framework needs to handle a wide variety of proofs and code structures, which can be challenging in terms of both computational resources and model generalization.
• Correctness: Ensuring the correctness of generated proofs is critical. Any errors in the proofs can undermine the entire framework.

By addressing these challenges and leveraging advanced techniques in machine learning and formal methods, this framework could create a powerful synergy between neural networks and formal proof systems, enhancing both fields and opening up new possibilities for automated reasoning and program verification.