This session aims to connect a constellation of topics related to processing corpora of mathematical texts. NLP on informal mathematical documents presents many interesting challenges, and can be aided by the presence of corresponding formal documents (interpreted broadly). Creating and maintaining these corpora presents even more challenges. We hope to bring together researchers thinking about the retrieval, curation, and interpretation of formal and informal mathematical texts, as well as the underlying languages used in these projects and the connections between them. There are many exciting projects underway in all of these areas, and a combined session will spur fruitful discussions and help to align the work of different groups.
The Lean 3 Mathematical Library
Mario Carneiro, Carnegie Mellon University
The Lean 3 “mathlib” library has been rewritten from the ground up to take advantage of the new features and tactics framework. In this article we take stock of the meteoric growth of the library in the past several months, which covers elementary number theory, algebra, ring theory, topology and analysis, set theory, as well as support for verified pure-functional programming with lists, maps, arrays, quotient types, and coinductive types. This is supplemented with additional tactics, written in Lean on top of the Lean 3 tactic framework, for proving arithmetic equalities and inequalities, deciding the equational theory of commutative rings, as well as general automation tactics based on heuristic instantiation and resolution proving.
NLP and Large-scale Information Retrieval on Mathematical Texts
Yihe Dong, Wolfram Research
We present a recommender system covering math and math physics papers from the arXiv to date, to assist researchers quickly retrieve theorems and discover similar results from this vast corpus. The retrieval aims to discover not just syntactic, but also semantic similarity. We will discuss the challenges encountered and the experimental methodologies used.
A New Style of Mathematical Proof
William Farmer, McMaster University
Mathematical proofs will play a crucial role in building a universal digital mathematics library (UDML). Traditional and formal style proofs do not adequately fulfill all the purposes that mathematical proofs have. We propose a new style of proof that achieves seven purposes of mathematical proofs. We believe this style of proof is needed to build a highly interconnected UDML.
Is Univalence Inevitable?
Krzysztof Kapulkin, University of Western Ontario
Voevodsky’s Univalence Axiom (UA) is a major contribution to the area of computer formalization of mathematics. It significantly extends depended type theory, the underlying system of proof assistants such as Coq, Agda, or Lean; for instance, it is known that UA implies the principle of Functional Extensionality. In practical terms, it brings foundations of mathematics closer to human intuition, providing a foundation that justifies common abuses of notation, e.g., writing G=H when the groups G and H were only proven to be isomorphic. The use of the Univalence Axiom was instrumental in formalization of category theory (cf. Ahrens-Kapulkin-Shulman 2015), an area that had previously proven particularly ill-suited for formalization.
This talk will be a survey of Univalence, covering the necessary mathematical background, advantages it gives in formalization, and predictions for the future. I will also report on the progress in developing a new generation of proof assistant, based on cubical type theory and created in an attempt to give a computational interpretation of the Univalence Axiom.
A Bi-directional Extensible Ad Hoc Interface between Lean and Mathematica
Robert Y. Lewis, Vrije Universiteit Amsterdam
We implement a user-extensible ad hoc connection between the Lean proof assistant and the computer algebra system Mathematica. By reflecting the syntax of each system in the other and providing a flexible interface for extending translation, our connection allows for the exchange of arbitrary information between the two systems. We show how to make use of the Lean metaprogramming framework to verify certain Mathematica computations, so that the rigor of the proof assistant is not compromised. We also establish a connection in the opposite direction, using Mathematica to import and process proof terms from Lean. This allows one to process and explore the mathematical library of the proof assistant, using the data processing and display tools of the CAS.
Set-Theoretic Type Theory
David McAllester, Toyota Technological Institute at Chicago
The talk will present a set-theoretic model of dependent type theory handling isomorphism. In this model expressions are given their naive set-theoretic compositional meaning. For example a dependent pair type Sigma x:A B[x] denotes (literally) the set (or class) of pairs (c,d) with c a member of A and d a member of B[c]. This has the consequence that semantic values do not have unique types — Abelian groups and permutation groups are also (literally) groups as implied by the standard naive set-theoretic definition of a group. Contrasts with homotopy type theory will be discussed from both semantic and proof-theoretic perspectives.
First Experiments with Neural Translation of Informal Mathematics to Formal
Josef Urban, Czech Technical University
The talk will give a summary of our first experiments to train deep neural networks that automatically translate informalized LATEX-written Mizar texts into the formal Mizar language. This is joint work with Qingxiang Wang and Cezary Kaliszyk, submitted to CICM’18. Using Luong et al.’s neural machine translation model (NMT), we tested our aligned informal-formal corpora against various hyperparameters and evaluated their results. Our experiments show that NMT is able to generate correct Mizar statements on more than 60 percent of the inference data, indicating that formalization through artificial neural network is a promising approach for automated formalization of mathematics. We present several case studies to illustrate our results.