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Learning to Prove with Tactics [article]

Thibault Gauthier, Cezary Kaliszyk, Josef Urban, Ramana Kumar, Michael Norrish
2018 arXiv   pre-print
On a single CPU, with a time limit of 60 seconds, TacticToe proves 66.4 percent of the 7164 theorems in HOL4's standard library, whereas E prover with auto-schedule solves 34.5 percent.  ...  We implement a automated tactical prover TacticToe on top of the HOL4 interactive theorem prover. TacticToe learns from human proofs which mathematical technique is suitable in each proof situation.  ...  Acknowledgments We would like to thank Lasse Blaauwbroek and Yutaka Nagashima for their insightful comments which contributed to improve the quality of this paper.  ... 
arXiv:1804.00596v1 fatcat:twmo2yoiwrfaxmj2onudf6sw7a

Learning to Reason with HOL4 tactics [article]

Thibault Gauthier, Cezary Kaliszyk, Josef Urban
2018 arXiv   pre-print
Techniques combining machine learning with translation to automated reasoning have recently become an important component of formal proof assistants.  ...  By combining tactic prediction and premise selection, TacticToe is able to re-prove 39 percent of 7902 HOL4 theorems in 5 seconds whereas the best single HOL(y)Hammer strategy solves 32 percent in the  ...  The algorithm goes on creating new nodes with new open goals (goal 3 ) until a tactic (tactic 2 ) proves a goal (goal 1 ).  ... 
arXiv:1804.00595v1 fatcat:p6oehxklwfdalokzeez3v6prga

Learned Provability Likelihood for Tactical Search

Thibault Gauthier
2021 Electronic Proceedings in Theoretical Computer Science  
This amelioration in performance together with concurrent updates to the TacticToe framework lead to an improved user experience.  ...  We present a method to estimate the provability of a mathematical formula. We adapt the tactical theorem prover TacticToe to factor in these estimations.  ...  Monte Carlo Tree Search with Tactics We integrate the learned provability estimator into the proof search of TacticToe.  ... 
doi:10.4204/eptcs.342.7 fatcat:xyp7yj4gond3vbw233kaordqoe

Tactic Learning and Proving for the Coq Proof Assistant [article]

Lasse Blaauwbroek, Josef Urban, Herman Geuvers
2020 arXiv   pre-print
When combined with the CoqHammer system, the two systems together prove 56.7% of the library's lemmas.  ...  To do this, it learns from previous tactic scripts and how they are applied to proof states. The performance of the system is evaluated on the Coq Standard Library.  ...  We assume that when the user tries to prove a new lemma, all previous lemmas are already proven and can be used for learning.  ... 
arXiv:2003.09140v1 fatcat:vozpbh5w4fgrboqnrkju7odxza

HOList: An Environment for Machine Learning of Higher-Order Theorem Proving [article]

Kshitij Bansal, Sarah M. Loos, Markus N. Rabe, Christian Szegedy, and Stewart Wilcox
2019 arXiv   pre-print
We also present a deep reinforcement learning driven automated theorem prover, DeepHOL, with strong initial results on this benchmark.  ...  We present an environment, benchmark, and deep learning driven automated theorem prover for higher-order logic.  ...  Acknowledgements We would like to thank Alex Alemi, Geoffrey Irving, Cezary Kaliszyk, Thibault Gauthier, Ramana Kumar, Viktor Toman, and Josef Urban for their insightful comments and contributions to early  ... 
arXiv:1904.03241v3 fatcat:ih4fizuonrbvzk2oyu4pekhftu

The Tactician [chapter]

Lasse Blaauwbroek, Josef Urban, Herman Geuvers
2020 Lecture Notes in Computer Science  
To this end, Tactician learns from previously written tactic scripts and gives users either suggestions about the next tactic to be executed or altogether takes over the burden of proof synthesis.  ...  Tactician's goal is to provide users with a seamless, interactive, and intuitive experience together with robust and adaptive proof automation.  ...  With its current machine learning capabilities, we expect Tactician to help the user with its proving efforts significantly.  ... 
doi:10.1007/978-3-030-53518-6_17 fatcat:u7o7b4ne2zghdctir6cp6zs2lm

Learning to Reason in Large Theories without Imitation [article]

Kshitij Bansal, Christian Szegedy, Markus N. Rabe, Sarah M. Loos, Viktor Toman
2020 arXiv   pre-print
This helps with exploring and learning which premises are relevant for proving a new theorem.  ...  In this paper, we demonstrate how to do automated theorem proving in the presence of a large knowledge base of potential premises without learning from human proofs.  ...  To keep the focus on premise selection, we still learn the tactic selection, and use similar amounts of resources trying to prove statements as in one RL loop.  ... 
arXiv:1905.10501v3 fatcat:ft65xynsgbfdxkzlcy4omg7ov4

Graph Representations for Higher-Order Logic and Theorem Proving [article]

Aditya Paliwal, Sarah Loos, Markus Rabe, Kshitij Bansal, Christian Szegedy
2019 arXiv   pre-print
Interactive, higher-order theorem provers allow for the formalization of most mathematical theories and have been shown to pose a significant challenge for deep learning.  ...  Higher-order logic is highly expressive and, even though it is well-structured with a clearly defined grammar and semantics, there still remains no well-established method to convert formulas into graph-based  ...  We first start with the top level goal which is the top level theorem to be proved.  ... 
arXiv:1905.10006v2 fatcat:kokjdqbvpvgvbfl4blxmbrafpe

Graph Representations for Higher-Order Logic and Theorem Proving

Aditya Paliwal, Sarah Loos, Markus Rabe, Kshitij Bansal, Christian Szegedy
2020 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
Interactive, higher-order theorem provers allow for the formalization of most mathematical theories and have been shown to pose a significant challenge for deep learning.  ...  Higher-order logic is highly expressive and, even though it is well-structured with a clearly defined grammar and semantics, there still remains no well-established method to convert formulas into graph-based  ...  Similar to this work, GamePad (Huang et al. 2018) , TacticToe (Gauthier, Kaliszyk, and Urban 2017) , Coq-Gym (Yang and Deng 2019) , and Proverbot9001 (Sanchez-Stern et al. 2019) use imitation learning  ... 
doi:10.1609/aaai.v34i03.5689 fatcat:7dwvu7ml4zgmdfzcjofyltc2ga

Faster Smarter Induction in Isabelle/HOL [article]

Yutaka Nagashima
2021 arXiv   pre-print
To address this problem, we developed sem_ind. Given inductive problem, sem_ind recommends what arguments to pass to the induct method.  ...  the median value of execution time from 2.79 seconds to 1.06 seconds.  ...  To the best of our knowledge, no project based on deep learning has man-aged to predict arguments to the induct tactic accurately.  ... 
arXiv:2009.09215v4 fatcat:y6eyd4vswrh25pn3fp6nwti3oq

GRUNGE: A Grand Unified ATP Challenge [article]

Chad E. Brown, Thibault Gauthier, Cezary Kaliszyk, Geoff Sutcliffe, Josef Urban
2019 arXiv   pre-print
The formalisms are in higher-order logic (with and without type variables) and first-order logic (possibly with multiple types, and possibly with type variables).  ...  This paper describes a large set of related theorem proving problems obtained by translating theorems from the HOL4 standard library into multiple logical formalisms.  ...  TacticToe is a machine-learning guided prover that searches for a tactical proof by selecting suitable tactics and theorems learned from humanwritten tactical proofs.  ... 
arXiv:1903.02539v1 fatcat:w2py2275rzc4rh5s3v45fff5vi

Deep Reinforcement Learning for Synthesizing Functions in Higher-Order Logic [article]

Thibault Gauthier
2020 arXiv   pre-print
A close interaction between the machine learning modules and the HOL4 library is achieved by the choice of tree neural networks (TNNs) as machine learning models and the internal use of HOL4 terms to represent  ...  We achieve a success rate of 65% on combinator synthesis problems outperforming state-of-the-art ATPs run with their best general set of strategies.  ...  To solve this issue, the tactical prover TacticToe [16, 17] , also implemented in HOL4, learns to apply tactics extracted from existing proof scripts.  ... 
arXiv:1910.11797v3 fatcat:bqtdfbrcxrcc5p6e3tzygogxsy

Generative Language Modeling for Automated Theorem Proving [article]

Stanislas Polu, Ilya Sutskever
2020 arXiv   pre-print
We explore the application of transformer-based language models to automated theorem proving.  ...  GPT-f found new short proofs that were accepted into the main Metamath library, which is to our knowledge, the first time a deep-learning based system has contributed proofs that were adopted by a formal  ...  Finally, the authors would like to thank the whole Metamath community for their support, feedback, and encouragement, in particular, David A.  ... 
arXiv:2009.03393v1 fatcat:fqabfsag6rcv7cdw3xlstdnmdq

LiFtEr: Language to Encode Induction Heuristics for Isabelle/HOL [article]

Yutaka Nagashima
2019 arXiv   pre-print
Proof assistants, such as Isabelle/HOL, offer tools to facilitate inductive theorem proving.  ...  Isabelle experts know how to use these tools effectively; however, there is a little tool support for transferring this expert knowledge to a wider user audience.  ...  P xs (x # ys) ==> P (x # xs) ys) ==> P a0 a1 Essentially, this rule states that to prove a property P of a0 and a1 we have to prove it for two cases where a0 is the empty list and the list with at least  ... 
arXiv:1906.08084v3 fatcat:uehy5lr3rnhgzfwon7bfbqz6o4

PaMpeR: Proof Method Recommendation System for Isabelle/HOL [article]

Yutaka Nagashima, Yilun He
2018 arXiv   pre-print
Deciding which sub-tool to use for a given proof state requires expertise specific to each ITP. To mitigate this problem, we present PaMpeR, a Proof Method Recommendation system for Isabelle/HOL.  ...  Given a proof state, PaMpeR recommends proof methods to discharge the proof goal and provides qualitative explanations as to why it suggests these methods.  ...  Section VIII compares our work with other attempts of applying machine learning and data mining to interactive theorem proving. II. BACKGROUND AND OVERVIEW OF PAMP ER A.  ... 
arXiv:1806.07239v1 fatcat:7uaoocasdzbbfeqsmvyb6lsnhy
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