Premise Selection for Mathematics by Corpus Analysis and Kernel Methods

Alama, Jesse; Heskes, Tom; Kühlwein, Daniel; Tsivtsivadze, Evgeni; Urban, Josef

doi:10.1007/s10817-013-9286-5

Cited by 102 publications

(114 citation statements)

References 33 publications

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“…Premise selection in large theories seems to be an important instance of the general proof guidance problem. It has been shown that proper design and choice of knowledge selection heuristics can change the overall success of large-theory ATP techniques by tens of percents [1]. This paper continues our work on machine learning algorithms for premise selection [1,5].…”

Section: Introductionmentioning

confidence: 63%

“…It has been shown that proper design and choice of knowledge selection heuristics can change the overall success of large-theory ATP techniques by tens of percents [1]. This paper continues our work on machine learning algorithms for premise selection [1,5]. We investigate how the knowledge of different proofs can be integrated in the machine learning algorithms for premise selection, and how it influences the performance of the ATPs.…”

Section: Introductionmentioning

confidence: 88%

“…Because of its good performance in earlier evaluations, we used the Multi-Output-Ranking (MOR) learning algorithm [1] for the experiments. For each conjecture, MOR is allowed to train on all proofs that were (in the chronological order of MML) done up to that conjecture.…”

Section: Methodsmentioning

confidence: 99%

“…We start with the setting introduced in [1,5]. Γ denotes the set of all first order formulas (usually axioms, definitions and theorems) that appear in a given (fixed) large mathematical corpus (MPTP2078 in this paper).…”

Section: The Machine Learning Framework and The Datamentioning

confidence: 99%

“…This proof matrix, together with the formula features, is used for training machine learning algorithms in [1,5], using MML proofs as ∆. [1] and [5] also contain more details on using machine learning for premise selection.…”

Section: The Machine Learning Framework and The Datamentioning

confidence: 99%

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Learning from Multiple Proofs: First Experiments

Kühlwein

Urban

EPiC Series in Computing

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Mathematical textbooks typically present only one proof for most of the theorems. However, there are infinitely many proofs for each theorem in first-order logic, and mathematicians are often aware of (and even invent new) important alternative proofs and use such knowledge for (lateral) thinking about new problems.In this paper we start exploring how the explicit knowledge of multiple (human and ATP) proofs of the same theorem can be used in learning-based premise selection algorithms in large-theory mathematics. Several methods and their combinations are defined, and their effect on the ATP performance is evaluated on the MPTP2078 large-theory benchmark. Our first findings are that the proofs used for learning significantly influence the number of problems solved, and that the quality of the proofs is more important than the quantity.

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Section: Introductionmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 88%

Section: Methodsmentioning

confidence: 99%

Section: The Machine Learning Framework and The Datamentioning

confidence: 99%

Section: The Machine Learning Framework and The Datamentioning

confidence: 99%

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Learning from Multiple Proofs: First Experiments

Kühlwein

Urban

EPiC Series in Computing

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ENIGMAWatch: ProofWatch Meets ENIGMA

Goertzel

Jakubův

Urban

2019

Lecture Notes in Computer Science

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In this work we describe a new learning-based proof guidance -ENIGMAWatch -for saturation-style first-order theorem provers. ENIGMAWatch combines two guiding approaches for the given-clause selection implemented for the E ATP system: ProofWatch and ENIGMA. ProofWatch is motivated by the watchlist (hints) method and based on symbolic matching of multiple related proofs, while ENIGMA is based on statistical machine learning. The two methods are combined by using the evolving information about symbolic proof matching as an additional information that characterizes the saturation-style proof search for the statistical learning methods. The new system is experimentally evaluated on a large set of problems from the Mizar library. We show that the added proof-matching information is considered important by the statistical machine learners, and that it leads to improvements in E's performance over ProofWatch and ENIGMA.

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ENIGMA-NG: Efficient Neural and Gradient-Boosted Inference Guidance for E

Chvalovský

Jakubův

Suda

et al. 2019

Lecture Notes in Computer Science

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We describe an efficient implementation of clause guidance in saturation-based automated theorem provers extending the ENIGMA approach. Unlike in the first ENIGMA implementation where fast linear classifier is trained and used together with manually engineered features, we have started to experiment with more sophisticated state-of-the-art machine learning methods such as gradient boosted trees and recursive neural networks. In particular the latter approach poses challenges in terms of efficiency of clause evaluation, however, we show that deep integration of the neural evaluation with the ATP data-structures can largely amortize this cost and lead to competitive real-time results. Both methods are evaluated on a large dataset of theorem proving problems and compared with the previous approaches. The resulting methods improve on the manually designed clause guidance, providing the first practically convincing application of gradient-boosted and neural clause guidance in saturation-style automated theorem provers. IntroductionAutomated theorem provers (ATPs) [32] have been developed for decades by manually designing proof calculi and search heuristics. Their power has been growing and they are already very useful, e.g., as parts of large interactive theorem proving (ITP) verification toolchains (hammers) [5]. On the other hand, with small exceptions, ATPs are still significantly weaker than trained mathematicians in finding proofs in most research domains.Recently, machine learning over large formal corpora created from ITP libraries [37,28,19] has started to be used to develop guidance of ATP systems [39,25,2]. This has already produced strong systems for selecting relevant facts for proving new conjectures over large formal libraries [1,4,9]. More recently, machine learning has also started to be used to guide the internal search of the ATP systems. In sophisticated saturation-style provers this has been done by feedback loops for strategy invention [38,16,33] and by using supervised learning [14,26] to select the next given clause [27]. In the simpler connection tableau systems such as leancop [29], supervised learning has been used to choose ⋆ Supported by the ERC Consolidator grant no. 649043 AI4REASON, and by the Czech project AI&Reasoning CZ.02.1.01/0.0/0.0/15 003/0000466 and the European Regional Development Fund.

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Premise Selection for Mathematics by Corpus Analysis and Kernel Methods

Cited by 102 publications

References 33 publications

Learning from Multiple Proofs: First Experiments

Learning from Multiple Proofs: First Experiments

ENIGMAWatch: ProofWatch Meets ENIGMA

ENIGMA-NG: Efficient Neural and Gradient-Boosted Inference Guidance for E

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