Neural Precedence Recommender

Abstract

The state-of-the-art superposition-based theorem provers for first-order logic rely on simplification orderings on terms to constrain the applicability of inference rules, which in turn shapes the ensuing search space. The popular Knuth-Bendix simplification ordering is parameterized by symbol precedence—a permutation of the predicate and function symbols of the input problem’s signature. Thus, the choice of precedence has an indirect yet often substantial impact on the amount of work required to complete a proof search successfully.This paper describes and evaluates a symbol precedence recommender, a machine learning system that estimates the best possible precedence based on observations of prover performance on a set of problems and random precedences. Using the graph convolutional neural network technology, the system does not presuppose the problems to be related or share a common signature. When coupled with the theorem prover Vampire and evaluated on the TPTP problem library, the recommender is found to outperform a state-of-the-art heuristic by more than 4 % on unseen problems.

Graph Sequence Learning for Premise Selection

“…This is essential because the chance of deriving a contradiction reduces considerably as the number of clauses grows. Machine learning is currently being used to discover efficient heuristics [11,12,26], and to intelligently operate internal heuristic components [4,6,8,25].…”

How Much Should This Symbol Weigh? A GNN-Advised Clause Selection

“…In the clause and symbol nodes, the initial embedding is augmented with the metadata extracted from the input problem. 2 Four message-passing layers follow: In each of these layers, each node embedding is updated by aggregating messages incoming from the node's neighbors. Each message is the output of a trainable affine transformation of the embedding of the source node.…”

“…More precisely, the initial embedding of a node is the concatenation of a node-specific metadata vector (empty in all nodes except clause and symbol nodes; see Section 4.1), and a trainable vector common to all nodes of one type 3. The architecture and hyperparameters of the GNN are the same as those used in our earlier work on recommending symbol precedences[2] which we refer the reader to for additional details.…”