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.
The earliest work of applying machine learning on reasoning in large theories is @cite_22 . The most most similar works to ours are TacticToe @cite_20 and GamePad @cite_18 . TacticToe is the first published result on machine learning tackling higher-order theorem proving at a relatively large scale at tactic level @cite_20 . Although TacticToe is a great success that came with significant improvements over previous automated theorem proving systems, they do not propose an easy to use benchmark or environment for machine learning researchers. TacticToe does not employ deep learning nor reinforcement learning. They rely on the HOL4 @cite_34 system that has a significantly less theorems with more complex human proof scripts with a larger number of more elementary tactics.
@cite_18 @cite_34 @cite_22 @cite_20
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Techniques combining machine learning with translation to automated reasoning have recently become an important component of formal proof assistants. Such "hammer" tech- niques complement traditional proof assistant automation as implemented by tactics and decision procedures. In this paper we present a unified proof assistant automation approach which attempts to automate the selection of appropriate tactics and tactic-sequences com- bined with an optimized small-scale hammering approach. We implement the technique as a tactic-level automation for HOL4: TacticToe. It implements a modified A*-algorithm directly in HOL4 that explores different tactic-level proof paths, guiding their selection by learning from a large number of previous tactic-level proofs. Unlike the existing hammer methods, TacticToe avoids translation to FOL, working directly on the HOL level. By combining tactic prediction and premise selection, TacticToe is able to re-prove 39 percent of 7902 HOL4 theorems in 5 seconds whereas the best single HOL(y)Hammer strategy solves 32 percent in the same amount of time. Large computer-understandable proofs consist of millions of intermediate logical steps. The vast majority of such steps originate from manually selected and manually guided heuristics applied to intermediate goals. So far, machine learning has generally not been used to filter or generate these steps. In this paper, we introduce a new dataset based on Higher-Order Logic (HOL) proofs, for the purpose of developing new machine learning-based theorem-proving strategies. We make this dataset publicly available under the BSD license. We propose various machine learning tasks that can be performed on this dataset, and discuss their significance for theorem proving. We also benchmark a set of simple baseline machine learning models suited for the tasks (including logistic regression, convolutional neural networks and recurrent neural networks). The results of our baseline models show the promise of applying machine learning to HOL theorem proving. While several researchers have applied case-based reasoning techniques to games, only Ponsen and Spronck (2004) have addressed the challenging problem of learning to win real-time games. Focusing on Wargus, they report good results for a genetic algorithm that searches in plan space, and for a weighting algorithm (dynamic scripting) that biases subplan retrieval. However, both approaches assume a static opponent, and were not designed to transfer their learned knowledge to opponents with substantially different strategies. We introduce a plan retrieval algorithm that, by using three key sources of domain knowledge, removes the assumption of a static opponent. Our experiments show that its implementation in the Case-based Tactician (CaT) significantly outperforms the best among a set of genetically evolved plans when tested against random Wargus opponents. CaT communicates with Wargus through TIELT, a testbed for integrating and evaluating decision systems with simulators. This is the first application of TIELT. We describe this application, our lessons learned, and our motivations for future work. In this paper, we introduce a system called GamePad that can be used to explore the application of machine learning methods to theorem proving in the Coq proof assistant. Interactive theorem provers such as Coq enable users to construct machine-checkable proofs in a step-by-step manner. Hence, they provide an opportunity to explore theorem proving with human supervision. We use GamePad to synthesize proofs for a simple algebraic rewrite problem and train baseline models for a formalization of the Feit-Thompson theorem. We address position evaluation (i.e., predict the number of proof steps left) and tactic prediction (i.e., predict the next proof step) tasks, which arise naturally in tactic-based theorem proving.