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Register a new userENIGMA Anonymous: Symbol-Independent Inference Guiding Machine (system description)
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Added by
Josef Urban
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We describe an implementation of gradient boosting and neural guidance of saturation-style automated theorem provers that does not depend on consistent symbol names across problems. For the gradient-boosting guidance, we manually create abstracted features by considering arity-based encodings of formulas. For the neural guidance, we use symbol-independent graph neural networks (GNNs) and their embedding of the terms and clauses. The two methods are efficiently implemented in the E prover and its ENIGMA learning-guided framework. To provide competitive real-time performance of the GNNs, we have developed a new context-based approach to evaluation of generated clauses in E. Clauses are evaluated jointly in larger batches and with respect to a large number of already selected clauses (context) by the GNN that estimates their collectively most useful subset in several rounds of message passing. This means that approximative inference rounds done by the GNN are efficiently interleaved with precise symbolic inference rounds done inside E. The methods are evaluated on the MPTP large-theory benchmark and shown to achieve comparable real-time performance to state-of-the-art symbol-based methods. The methods also show high complementarity, solving a large number of hard Mizar problems.
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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.
Several algorithms for solving constraint satisfaction problems are based on survey propagation, a variational inference scheme used to obtain approximate marginal probability estimates for variable assignments. These marginals correspond to how frequently each variable is set to true among satisfying assignments, and are used to inform branching decisions during search; however, marginal estimates obtained via survey propagation are approximate and can be self-contradictory. We introduce a more general branching strategy based on streamlining constraints, which sidestep hard assignments to variables. We show that streamlined solvers consistently outperform decimation-based solvers on random k-SAT instances for several problem sizes, shrinking the gap between empirical performance and theoretical limits of satisfiability by 16.3% on average for k=3,4,5,6.
In this paper, we consider the problem of learning a first-order theorem prover that uses a representation of beliefs in mathematical claims to construct proofs. The inspiration for doing so comes from the practices of human mathematicians where plausible reasoning is applied in addition to deductive reasoning to find proofs. Towards this end, we introduce a representation of beliefs that assigns probabilities to the exhaustive and mutually exclusive first-order possibilities found in Hintikkas theory of distributive normal forms. The representation supports Bayesian update, induces a distribution on statements that does not enforce that logically equivalent statements are assigned the same probability, and suggests an embedding of statements into an associated Hilbert space. We then examine conjecturing as model selection and an alternating-turn game of determining consistency. The game is amenable (in principle) to self-play training to learn beliefs and derive a prover that is complete when logical omniscience is attained and sound when beliefs are reasonable. The representation has super-exponential space requirements as a function of quantifier depth so the ideas in this paper should be taken as theoretical. We will comment on how abstractions can be used to control the space requirements at the cost of completeness.
Automated theorem provers have traditionally relied on manually tuned heuristics to guide how they perform proof search. Deep reinforcement learning has been proposed as a way to obviate the need for such heuristics, however, its deployment in automated theorem proving remains a challenge. In this paper we introduce TRAIL, a system that applies deep reinforcement learning to saturation-based theorem proving. TRAIL leverages (a) a novel neural representation of the state of a theorem prover and (b) a novel characterization of the inference selection process in terms of an attention-based action policy. We show through systematic analysis that these mechanisms allow TRAIL to significantly outperform previous reinforcement-learning-based theorem provers on two benchmark datasets for first-order logic automated theorem proving (proving around 15% more theorems).
Automated theorem proving in first-order logic is an active research area which is successfully supported by machine learning. While there have been various proposals for encoding logical formulas into numerical vectors -- from simple strings to more involved graph-based embeddings -- little is known about how these different encodings compare. In this paper, we study and experimentally compare pattern-based embeddings that are applied in current systems with popular graph-based encodings, most of which have not been considered in the theorem proving context before. Our experiments show that the advantages of simpler encoding schemes in terms of runtime are outdone by more complex graph-based embeddings, which yield more efficient search strategies and simpler proofs. To support this, we present a detailed analysis across several dimensions of theorem prover performance beyond just proof completion rate, thus providing empirical evidence to help guide future research on neural-guided theorem proving towards the most promising directions.
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