Modern saturation theorem provers are based on the givenclause algorithm, which iteratively selects new clauses to process. This clause selection has a large impact on the performance of proof search and has been the subject of much folklore. The standard approach is to alternate between selecting the oldest clause and the lightest clause with a fixed, but configurable age/weight ratio (AWR). An optimal fixed value of this ratio is shown to produce proofs significantly more quickly on a given problem, and further that varying AWR during proof search can improve upon a fixed ratio. Several new modes for the Vampire prover which vary AWR according to a "shape" during proof search are developed based on these observations. The modes solve a number of new problems in the TPTP benchmark set.

We present a prototype of a neurally-guided automatic theorem prover for first-order logic with equality. The prototype uses a neural network trained on previous proof search attempts to evaluate subgoals based directly on their structure, and hence bias proof search toward success. An existing first-order theorem prover is employed to dispatch easy subgoals and prune branches which cannot be solved. Exploration of the search space is asynchronous with respect to both the evaluation network and the existing prover, allowing for efficient batched neural network execution and for natural parallelism within the prover. Evaluation on the MPTP dataset shows that the prover can improve with learning.

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