We introduce a theorem proving algorithm that uses practically no domain heuristics for guiding its connection-style proof search. Instead, it runs many Monte-Carlo simulations guided by reinforcement learning from previous proof attempts. We produce several versions of the prover, parameterized by different learning and guiding algorithms. The strongest version of the system is trained on a large corpus of mathematical problems and evaluated on previously unseen problems. The trained system solves within the same number of inferences over 40 more problems than a baseline prover, which is an unusually high improvement in this hard AI domain. To our knowledge this is the first time reinforcement learning has been convincingly applied to solving general mathematical problems on a large scale.
ATPboost is a system for solving sets of large-theory problems by interleaving ATP runs with state-of-the-art machine learning of premise selection from the proofs. Unlike many approaches that use multi-label setting, the learning is implemented as binary classification that estimates the pairwise-relevance of (theorem, premise) pairs. ATPboost uses for this the fast state-of-the-art XGBoost gradient boosting algorithm. Learning in the binary setting however requires negative examples, which is nontrivial due to many alternative proofs. We discuss and implement several solutions in the context of the ATP ML feedback loop, and show significant improvement over the multi-label approach. 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 user-specified 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. Machine Learner for Automated Reasoning (MaLARea) is a learning and reasoning system for proving in large formal libraries where thousands of theorems are available when attacking a new conjecture, and a large number of related problems and proofs can be used to learn specific theorem-proving knowledge. The last version of the system has by a large margin won the 2013 CASC LTB competition. This paper describes the motivation behind the methods used in MaLARea, discusses the general approach and the issues arising in evaluation of such system, and describes the Mizar@Turing100 and CASC'24 versions of MaLARea. BliStr is a system that automatically develops strategies for E prover on a large set of problems. The main idea is to interleave (i) iterated low-timelimit local search for new strategies on small sets of similar easy problems with (ii) higher-timelimit evaluation of the new strategies on all problems. The accumulated results of the global higher-timelimit runs are used to define and evolve the notion of "similar easy problems", and to control the selection of the next strategy to be improved. The technique was used to significantly strengthen the set of E strategies used by the MaLARea, PS-E, E-MaLeS, and E systems in the CASC@Turing 2012 competition, particularly in the Mizar division. Similar improvement was obtained on the problems created from the Flyspeck corpus. A photographic copying machine of the type in which exposure radiation is transmitted through a transparent original onto a photosensitive medium. Arranged in the exposure radiation path is an array of multitudinous controllable-transmissivity elements, the individual transmissivity values of which are individually controllable. A control electronics, preferably incorporating a computer and multiplexer, controls the transmissivity values of the multitude of controllable-transmissivity elements of the array. The transmissivity values to which the multitude of elements are set can be equal, but in accordance with the present invention are not; instead, the array is employed to establish a two-dimensional controllably non-uniform distribution of exposure radiation onto the photosensitive medium. This is done to implement non-uniform density and or color correction during printing, or to superimpose onto the image projected onto the photosensitive medium images of symbols, text, or the like, which are to be appear on the final copy, or to implement border effects, by way of example.