ENIGMA is a learning-based method for guiding given clause selection in saturationbased theorem provers. Clauses from many proof searches are classified as positive and negative based on their participation in the proofs. An efficient classification model is trained on this data, using fast feature-based characterization of the clauses . The learned model is then tightly linked with the core prover and used as a basis of a new parameterized evaluation heuristic that provides fast ranking of all generated clauses. The approach is evaluated on the E prover and the CASC 2016 AIM benchmark, showing a large increase of E's performance. Introduction: Theorem Proving and LearningState-of-the-art resolution/superposition automated theorem provers (ATPs) such as Vampire [15] and E [20] are today's most advanced tools for general reasoning across a variety of mathematical and scientific domains. The stronger the performance of such tools, the more realistic become tasks such as full computer understanding and automated development of complicated mathematical theories, and verification of software, hardware and engineering designs. While performance of ATPs has steadily grown over the past years due to a number of human-designed improvements, it is still on average far behind the performance of trained mathematicians. Their advanced knowledge-based proof finding is an enigma, which is unlikely to be deciphered and programmed completely manually in near future.On large corpora such as Flyspeck, Mizar and Isabelle, the ATP progress has been mainly due to learning how to select the most relevant knowledge, based on many previous proofs [10,12,1,2]. Learning from many proofs has also recently become a very useful method for automated finding of parameters of ATP strategies [22,9,19,16], and for learning of sequences of tactics in interactive theorem provers (ITPs) [7]. Several experiments with the compact leanCoP [18] system have recently shown that directly using trained machine learner for internal clause selection can significantly prune the search space and solve additional problems [24,11,5]. An obvious next step is to implement efficient learning-based clause selection also inside the strongest superposition-based ATPs.In this work, we introduce ENIGMA -Efficient learNing-based Internal Guidance MAchine for state-of-the-art saturation-based ATPs. The method applies fast machine learning algorithms to a large number of proofs, and uses the trained classifier together with simpler heuristics to evaluate the millions of clauses generated during the resolution/superposition proof search. This way, the theorem prover automatically takes into account thousands of previous successes and failures that it has seen in previous problems, similarly to trained humans. Thanks to a carefully chosen efficient learning/evaluation method and its tight integration with the core ATP (in our case the E prover), the penalty for this ubiquitous knowledge-based internal proof guidance is very low. This in turn very significantly improves the per...

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.

We describe an implementation of gradient boosting and neural guidance of saturation-style automated theorem provers that does not depend on consistent symbol names across problems. For the gradient-boosting guidance, we manually create abstracted features by considering arity-based encodings of formulas. For the neural guidance, we use symbol-independent graph neural networks (GNNs) and their embedding of the terms and clauses. The two methods are efficiently implemented in the E prover and its ENIGMA learning-guided framework.To provide competitive real-time performance of the GNNs, we have developed a new context-based approach to evaluation of generated clauses in E. Clauses are evaluated jointly in larger batches and with respect to a large number of already selected clauses (context) by the GNN that estimates their collectively most useful subset in several rounds of message passing. This means that approximative inference rounds done by the GNN are efficiently interleaved with precise symbolic inference rounds done inside E. The methods are evaluated on the MPTP large-theory benchmark and shown to achieve comparable realtime performance to state-of-the-art symbol-based methods. The methods also show high complementarity, solving a large number of hard Mizar problems.

ENIGMA is an efficient implementation of learning-based guidance for given clause selection in saturation-based automated theorem provers. In this work, we describe several additions to this method. This includes better clause features, adding conjecture features as the proof state characterization, better data pre-processing, and repeated model learning. The enhanced ENIGMA is evaluated on the MPTP2078 dataset, showing significant improvements.

Inventing targeted proof search strategies for specific problem sets is a difficult task. State-of-the-art automated theorem provers (ATPs) such as E allow a large number of userspecified proof search strategies described in a rich domain specific language. Several machine learning methods that invent strategies automatically for ATPs were proposed previously. One of them is the Blind Strategymaker (BliStr), a system for automated invention of ATP strategies.In this paper we introduce BliStrTune -a hierarchical extension of BliStr. BliStrTune allows exploring much larger space of E strategies by interleaving search for high-level parameters with their fine-tuning. We use BliStrTune to invent new strategies based also on new clause weight functions targeted at problems from large ITP libraries. We show that the new strategies significantly improve E's performance in solving problems from the Mizar Mathematical Library.