PIE -- Proving, Interpolating and Eliminating on the Basis of First-Order Logic Artificial Intelligence

PIE is a Prolog-embedded environment for automated reasoning on the basis of first-order logic. It includes a versatile formula macro system and supports the creation of documents that intersperse macro definitions, reasoner invocations and LaTeX-formatted natural language text. Invocation of various reasoners is supported: External provers as well as sub-systems of PIE, which include preprocessors, a Prolog-based first-order prover, methods for Craig interpolation and methods for second-order quantifier elimination.

Extensional Higher-Order Paramodulation in Leo-III Artificial Intelligence

Leo-III is an automated theorem prover for extensional type theory with Henkin semantics and choice. Reasoning with primitive equality is enabled by adapting paramodulation-based proof search to higher-order logic. The prover may cooperate with multiple external specialist reasoning systems such as first-order provers and SMT solvers. Leo-III is compatible with the TPTP/TSTP framework for input formats, reporting results and proofs, and standardized communication between reasoning systems, enabling e.g. proof reconstruction from within proof assistants such as Isabelle/HOL. Leo-III supports reasoning in polymorphic first-order and higher-order logic, in all normal quantified modal logics, as well as in different deontic logics. Its development had initiated the ongoing extension of the TPTP infrastructure to reasoning within non-classical logics.

Designing Game of Theorems Artificial Intelligence

"Theorem proving is similar to the game of Go. So, we can probably improve our provers using deep learning, like DeepMind built the super-human computer Go program, AlphaGo." Such optimism has been observed among participants of AITP2017. But is theorem proving really similar to Go? In this paper, we first identify the similarities and differences between them and then propose a system in which various provers keep competing against each other and changing themselves until they prove conjectures provided by users.

Learning dynamic polynomial proofs Machine Learning

Polynomial inequalities lie at the heart of many mathematical disciplines. In this paper, we consider the fundamental computational task of automatically searching for proofs of polynomial inequalities. We adopt the framework of semi-algebraic proof systems that manipulate polynomial inequalities via elementary inference rules that infer new inequalities from the premises. These proof systems are known to be very powerful, but searching for proofs remains a major difficulty. In this work, we introduce a machine learning based method to search for a dynamic proof within these proof systems. We propose a deep reinforcement learning framework that learns an embedding of the polynomials and guides the choice of inference rules, taking the inherent symmetries of the problem as an inductive bias. We compare our approach with powerful and widely-studied linear programming hierarchies based on static proof systems, and show that our method reduces the size of the linear program by several orders of magnitude while also improving performance. These results hence pave the way towards augmenting powerful and well-studied semi-algebraic proof systems with machine learning guiding strategies for enhancing the expressivity of such proof systems.

Towards Finding Longer Proofs Artificial Intelligence

We present a reinforcement learning (RL) based guidance system for automated theorem proving geared towards Finding Longer Proofs (FLoP). FLoP focuses on generalizing from short proofs to longer ones of similar structure. To achieve that, FLoP uses state-of-the-art RL approaches that were previously not applied in theorem proving. In particular, we show that curriculum learning significantly outperforms previous learning-based proof guidance on a synthetic dataset of increasingly difficult arithmetic problems.

HOList: An Environment for Machine Learning of Higher-Order Theorem Proving (extended version) Artificial Intelligence

We present an environment, benchmark, and deep learning driven automated theorem prover for higher-order logic. Higher-order interactive theorem provers enable the formalization of arbitrary mathematical theories and thereby present an interesting, open-ended challenge for deep learning. We provide an open-source framework based on the HOL Light theorem prover that can be used as a reinforcement learning environment. HOL Light comes with a broad coverage of basic mathematical theorems on calculus and the formal proof of the Kepler conjecture, from which we derive a challenging benchmark for automated reasoning. We also present a deep reinforcement learning driven automated theorem prover, DeepHOL, with strong initial results on this benchmark.

Towards Ranking Geometric Automated Theorem Provers Artificial Intelligence

The field of geometric automated theorem provers has a long and rich history, from the early AI approaches of the 1960s, synthetic provers, to today algebraic and synthetic provers. The geometry automated deduction area differs from other areas by the strong connection between the axiomatic theories and its standard models. In many cases the geometric constructions are used to establish the theorems' statements, geometric constructions are, in some provers, used to conduct the proof, used as counter-examples to close some branches of the automatic proof. Synthetic geometry proofs are done using geometric properties, proofs that can have a visual counterpart in the supporting geometric construction. With the growing use of geometry automatic deduction tools as applications in other areas, e.g. in education, the need to evaluate them, using different criteria, is felt. Establishing a ranking among geometric automated theorem provers will be useful for the improvement of the current methods/implementations. Improvements could concern wider scope, better efficiency, proof readability and proof reliability. To achieve the goal of being able to compare geometric automated theorem provers a common test bench is needed: a common language to describe the geometric problems; a comprehensive repository of geometric problems and a set of quality measures.

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

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.

How to Create AI That Can Safely Navigate Our World - An Interview With Andre Platzer - Future of Life Institute


Over the last few decades, the unprecedented pace of technological progress has allowed us to upgrade and modernize much of our infrastructure and solve many long-standing logistical problems. For example, Babylon Health's AI-driven smartphone app is helping assess and prioritize 1.2 million patients in North London, electronic transfers allow us to instantly send money nearly anywhere in the world, and, over the last 20 years, GPS has revolutionized how we navigate, how we track and ship goods, and how we regulate traffic. However, exponential growth comes with its own set of hurdles that must be navigated. The foremost issue is that it's exceedingly difficult to predict how various technologies will evolve. As a result, it becomes challenging to plan for the future and ensure that the necessary safety features are in place.

Reinforcement Learning of Theorem Proving

Neural Information Processing Systems

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.