A Promising Path Towards Autoformalization and General Artificial Intelligence

An autoformalization system is an AI that learns to read natural language content and to turn it into an abstract, machine verifiable formalization, ideally by bootstrapping from unlabeled training data with minimum human interaction. This is a difficult task in general, one that would require strong automated reasoning and automated natural language processing capabilities. In this paper, it is argued that autoformalization is a promising path for systems to learn sophisticated, general purpose reasoning in all domains of mathematics and computer science. This could have far reaching implications not just for mathematical research, but also for software synthesis. Here I provide the outline for a realistic path towards those goals and give a survey of recent results that support the feasibility of this direction.

[1]  John McCarthy,et al.  Computer programs for checking mathematical proofs , 1962 .

[2]  W. Bledsoe,et al.  Checking number theory proofs in natural language , 1990 .

[3]  Melvin Fitting,et al.  First-Order Logic and Automated Theorem Proving , 1990, Graduate Texts in Computer Science.

[4]  John Harrison,et al.  HOL Light: A Tutorial Introduction , 1996, FMCAD.

[5]  Armin Biere,et al.  Symbolic Model Checking without BDDs , 1999, TACAS.

[6]  Niklas Sörensson,et al.  An Extensible SAT-solver , 2003, SAT.

[7]  Josef Urban.  Translating Mizar for First Order Theorem Provers , 2003, MKM.

[8]  Josef Urban,et al.  MPTP 0.2: Design, Implementation, and Initial Experiments , 2006, Journal of Automated Reasoning.

[9]  Csaba Szepesvári,et al.  Bandit Based Monte-Carlo Planning , 2006, ECML.

[10]  Josef Urban,et al.  MaLARea: a Metasystem for Automated Reasoning in Large Theories , 2007, ESARLT.

[11]  N. Nisan.  Introduction to Mechanism Design (for Computer Scientists) , 2007 .

[12]  Georges Gonthier,et al.  Formal Proof—The Four- Color Theorem , 2008 .

[13]  Michael Norrish,et al.  A Brief Overview of HOL4 , 2008, TPHOLs.

[14]  Tobias Nipkow,et al.  The Isabelle Framework , 2008, TPHOLs.

[15]  Joseph N. Wilson,et al.  Twenty Years of Mixture of Experts , 2012, IEEE Transactions on Neural Networks and Learning Systems.

[16]  Jeremy Avigad,et al.  A Machine-Checked Proof of the Odd Order Theorem , 2013, ITP.

[17]  Cezary Kaliszyk,et al.  HOL(y)Hammer: Online ATP Service for HOL Light , 2013, Math. Comput. Sci..

[18]  Jeremy Avigad,et al.  The Lean Theorem Prover (System Description) , 2015, CADE.

[19]  Dumitru Erhan,et al.  Going deeper with convolutions , 2014, 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[20]  Cezary Kaliszyk,et al.  Learning to Parse on Aligned Corpora (Rough Diamond) , 2015, ITP.

[21]  Victor W. Marek,et al.  Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer , 2016, SAT.

[22]  Nate Soares,et al.  Logical Induction , 2016, Electron. Colloquium Comput. Complex..

[23]  Cezary Kaliszyk,et al.  Hammering towards QED , 2016, J. Formaliz. Reason..

[24]  Jian Sun,et al.  Deep Residual Learning for Image Recognition , 2015, 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[25]  Gisbert Schneider,et al.  Deep Learning in Drug Discovery , 2016, Molecular informatics.

[26]  Leon A. Gatys,et al.  Image Style Transfer Using Convolutional Neural Networks , 2016, 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[27]  Josef Urban,et al.  DeepMath - Deep Sequence Models for Premise Selection , 2016, NIPS.

[28]  Jian Wang,et al.  Premise Selection for Theorem Proving by Deep Graph Embedding , 2017, NIPS.

[29]  Lukasz Kaiser,et al.  Attention is All you Need , 2017, NIPS.

[30]  Marcin Andrychowicz,et al.  Hindsight Experience Replay , 2017, NIPS.

[31]  Cezary Kaliszyk,et al.  System Description: Statistical Parsing of Informalized Mizar Formulas , 2017, 2017 19th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC).

[32]  Quoc V. Le,et al.  Neural Architecture Search with Reinforcement Learning , 2016, ICLR.

[33]  Frank Nielsen,et al.  DeepBach: a Steerable Model for Bach Chorales Generation , 2016, ICML.

[34]  Demis Hassabis,et al.  Mastering the game of Go without human knowledge , 2017, Nature.

[35]  Alexei A. Efros,et al.  Unpaired Image-to-Image Translation Using Cycle-Consistent Adversarial Networks , 2017, 2017 IEEE International Conference on Computer Vision (ICCV).

[36]  Thibault Gauthier,et al.  TacticToe: Learning to Reason with HOL4 Tactics , 2017, LPAR.

[37]  Tobias Nipkow,et al.  A FORMAL PROOF OF THE KEPLER CONJECTURE , 2015, Forum of Mathematics, Pi.

[38]  Cezary Kaliszyk,et al.  Deep Network Guided Proof Search , 2017, LPAR.

[39]  Luke S. Zettlemoyer,et al.  Deep Contextualized Word Representations , 2018, NAACL.

[40]  John Salvatier,et al.  When Will AI Exceed Human Performance? Evidence from AI Experts , 2017, ArXiv.

[41]  Quoc V. Le,et al.  QANet: Combining Local Convolution with Global Self-Attention for Reading Comprehension , 2018, ICLR.

[42]  Guillaume Lample,et al.  Unsupervised Machine Translation Using Monolingual Corpora Only , 2017, ICLR.

[43]  Elad Eban,et al.  MorphNet: Fast & Simple Resource-Constrained Structure Learning of Deep Networks , 2017, 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition.

[44]  Demis Hassabis,et al.  A general reinforcement learning algorithm that masters chess, shogi, and Go through self-play , 2018, Science.

[45]  Quoc V. Le,et al.  EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks , 2019, ICML.

[46]  Dawn Xiaodong Song,et al.  GamePad: A Learning Environment for Theorem Proving , 2018, ICLR.

[47]  Philip S. Yu,et al.  A Comprehensive Survey on Graph Neural Networks , 2019, IEEE Transactions on Neural Networks and Learning Systems.

[48]  Sarah M. Loos,et al.  HOList: An Environment for Machine Learning of Higher-Order Theorem Proving (extended version) , 2019, ArXiv.

[49]  Jia Deng,et al.  Learning to Prove Theorems via Interacting with Proof Assistants , 2019, ICML.

[50]  Ming-Wei Chang,et al.  BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding , 2019, NAACL.

[51]  Yiming Yang,et al.  XLNet: Generalized Autoregressive Pretraining for Language Understanding , 2019, NeurIPS.

[52]  Sarah M. Loos,et al.  Graph Representations for Higher-Order Logic and Theorem Proving , 2019, AAAI.

[53]  Sarah M. Loos,et al.  Mathematical Reasoning in Latent Space , 2019, ICLR.

[54]  Guillaume Lample,et al.  Deep Learning for Symbolic Mathematics , 2019, ICLR.

[55]  Cezary Kaliszyk,et al.  Towards Finding Longer Proofs , 2019, TABLEAUX.