Graph Representations for Higher-Order Logic and Theorem Proving (2019)

Graph Representations for Higher-Order Logic and Theorem Proving (2019)(arxiv.org)

104 points by brzozowski 5 years ago | 15 comments

westurner 5 years ago |

ONNX (and maybe RIF) are worth mentioning.

> ONNX is an open format built to represent machine learning models. ONNX defines a common set of operators - the building blocks of machine learning and deep learning models - and a common file format to enable AI developers to use models with a variety of frameworks, tools, runtimes, and compilers

RIF (~FOL): https://en.wikipedia.org/wiki/Rule_Interchange_Format

Datalog (not Turing-complete): https://en.wikipedia.org/wiki/Datalog

westurner 5 years ago |

HOList Benchmark: https://sites.google.com/view/holist/home

"HOList: An Environment for Machine Learning of Higher-Order Theorem Proving" (2019) https://arxiv.org/abs/1904.03241

> Abstract: We present an environment, benchmark, and deep learning driven automated theorem prover for higher-order logic. Higher-order interactive theorem provers enable the formalization of arbitrary mathematical theories and thereby present an interesting, open-ended challenge for deep learning. We provide an open-source framework based on the HOL Light theorem prover that can be used as a reinforcement learning environment. HOL Light comes with a broad coverage of basic mathematical theorems on calculus and the formal proof of the Kepler conjecture, from which we derive a challenging benchmark for automated reasoning. We also present a deep reinforcement learning driven automated theorem prover, DeepHOL, with strong initial results on this benchmark.

thomasahle 5 years ago |

I wonder why they don't mention any work based on transformer architectures? The recent work on solving differential equations based on expressions in reverse polish notation seemed like a reasonable idea to apply to theorem proving as well.

Really cool work though!

Tainnor 5 years ago | |

Solving differential equations symbolically[^1] is a classic pattern matching problem though. It takes a lot of practice recognising which equations can be solved with which method/heuristic, but in the end, the available heuristics are somewhat limited in number and so it's not unreasonable that an ML model would perform well here. Notice that you will still need to actually check your result in some way or the other, because ML models can generate false positives and you really don't want that in maths; so it's not that you're replacing rule-based systems, you're just augmenting them with other approaches (and of course, verifying the solution to a differential equation is basically a solved problem in symbolic analysis, since taking derivatives is a simple rule-based procedure[^2]).

I think theorem proving is much different. The space of possible heuristics you may apply to a proof is basically infinite. It can take a tremendous amount of creativity and intuition to come up with a complicated and novel proof. While I can see ML models being of use for simpler proofs or lemmas within bigger proofs (such as simple epsilon-delta proofs etc.), I have a hard time imagining that they will really be able to do real proofs anytime soon.

[^1]: I emphasise "symbolically" because it is my understanding that outside of simple situations and university lectures, most people don't bother with that, instead solving differential equations numerically.

[^2] There are some subtleties around the fact that, afaik, deciding whether two expressions are equal is undecidable in the general case because at some point you will have to compare e.g. coefficients and the equality of real numbers is undecidable. In practice, you will consider two numbers to be equal if their difference is below a certain threshold, which could in theory also yield false positives, but I find it unlikely that this would occur in practice in the situation of solving an equation.

thomasahle 5 years ago | | |

Actually OpenAI just published an article about using Transformer for theorem proving: https://arxiv.org/abs/2009.03393

riku_iki 5 years ago | | |

I think they can use transformer in the same way for proof path generation: do pattern matching for complex logical statements to find those where specific rule can be applied, similarly as it has be done in differential equations paper.

thomasahle 5 years ago | |

This is the differential equations paper I was thinking about https://medium.com/analytics-vidhya/solving-differential-equ...

The_rationalist 5 years ago | |

A transformer is unable to really represent logic, let alone higher order logic and theorem proving.

6gvONxR4sf7o 5 years ago | | |

A transformer is a universal function approximator. The question is whether it can do so reasonably efficiently. Trained on natural language linear sequences, I’m with you. Trained on abstract logical graph representations? I don’t think that question’s answered yet, unless I’m missing something.

neolog 5 years ago | | |

Where can I read more about this?

Findeton 5 years ago |

It looks like Graph NNs are going places.