We develop a self-learning approach for conjecturing of induction predicates on a dataset of 16197 problems derived from the OEIS. These problems are hard for today's SMT and ATP systems because they require a combination of inductive and arithmetical reasoning. Starting from scratch, our approach consists of a feedback loop that iterates between (i) training a neural translator to learn the correspondence between the problems solved so far and the induction predicates useful for them, (ii) using the trained neural system to generate many new induction predicates for the problems, (iii) fast runs of the z3 prover attempting to prove the problems using the generated predicates, (iv) using heuristics such as predicate size and solution speed on the proved problems to choose the best predicates for the next iteration of training. The algorithm discovers on its own many interesting induction predicates, ultimately solving 5565 problems, compared to 2265 problems solved by CVC5, Vampire or Z3 in 60 seconds.

Automated theorem provers and formal proof assistants are general reasoning systems that are in theory capable of proving arbitrarily hard theorems, thus solving arbitrary problems reducible to mathematics and logical reasoning. In practice, such systems however face large combinatorial explosion, and therefore include many heuristics and choice points that considerably influence their performance. This is an opportunity for trained machine learning predictors, which can guide the work of such reasoning systems. Conversely, deductive search supported by the notion of logically valid proof allows one to train machine learning systems on large reasoning corpora. Such bodies of proof are usually correct by construction and when combined with more and more precise trained guidance they can be boostrapped into very large corpora, with increasingly long reasoning chains and possibly novel proof ideas. In this paper we provide an overview of several automated reasoning and theorem proving domains and the learning and AI methods that have been so far developed for them. These include premise selection, proof guidance in several settings, AI systems and feedback loops iterating between reasoning and learning, and symbolic classification problems.

We introduce a self-learning algorithm for synthesizing programs for OEIS sequences. The algorithm starts from scratch initially generating programs at random. Then it runs many iterations of a self-learning loop that interleaves (i) training neural machine translation to learn the correspondence between sequences and the programs discovered so far, and (ii) proposing many new programs for each OEIS sequence by the trained neural machine translator. The algorithm discovers on its own programs for more than 78000 OEIS sequences, sometimes developing unusual programming methods. We analyze its behavior and the invented programs in several experiments.

The search for abstract patterns is one of the principal occupations of mathematicians. The discovery of similar patterns across different mathematical fields often leads to surprising connections. Probably the most famous example of such an unexpected connection in mathematics is the Taniyama-Shimura conjecture proved in 2001 [2]. It relates elliptic curves over the field of rational numbers with a special kind of complex analytical functions known as modular forms. This conjecture became especially famous because a restricted version of it implies Fermat's last theorem [16]. The connections found by the system described in this paper are more modest. For instance, it has created formulas for testing prime numbers based both on Fermat's little theorem

Tactics analyze the current proof state (goal and assumptions) and apply non-trivial proof transformations. Formalized proofs take advantage of different levels of automation which are in increasing order of generality: specialized rules, theory-based strategies and general purpose strategies. Thanks to progress in proof automation, developers can delegate more and more complicated proof obligations to general purpose strategies. Those are implemented by automated theorem provers (ATPs) such as E prover [32]. Communication between an ITP and ATPs is made possible by a "hammer" system [4,14]. It acts as an interface by performing premise selection, translation and proof reconstruction. Yet, ATPs are not flawless and more precise user-guidance, achieved by applying a particular sequence of specialized rules, is almost always necessary to develop a mathematical theory.

The paper describes an implementation of deep reinforcement learning through self-supervised learning within the proof assistant HOL4. 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 tree structures of TNNs. Recursive improvement is possible when a given task is expressed as a search problem. In this case, a Monte Carlo Tree Search (MCTS) algorithm guided by a TNN can be used to explore the search space and produce better examples for training the next TNN. As an illustration, tasks over propositional and arithmetical terms, representative of fundamental theorem proving techniques, are specified and learned: truth estimation, end-to-end computation, term rewriting and term synthesis.