Training neural networks on NP-complete problems typically demands very large amounts of training data and often needs to be coupled with computationally expensive symbolic verifiers to ensure output correctness. In this paper, we present NeuRes, a neuro-symbolic approach to address both challenges for propositional satisfiability, being the quintessential NP-complete problem. By combining certificate-driven training and expert iteration, our model learns better representations than models trained for classification only, with a much higher data efficiency -- requiring orders of magnitude less training data. NeuRes employs propositional resolution as a proof system to generate proofs of unsatisfiability and to accelerate the process of finding satisfying truth assignments, exploring both possibilities in parallel. To realize this, we propose an attention-based architecture that autoregressively selects pairs of clauses from a dynamic formula embedding to derive new clauses.

Statistical language models conventionally implement representation learning based on the contextual distribution of words or other formal units, whereas any information related to the logographic features of written text are often ignored, assuming they should be retrieved relying on the cooccurence statistics. On the other hand, as language models become larger and require more data to learn reliable representations, such assumptions may start to fall back, especially under conditions of data sparsity. Many languages, including Chinese and Vietnamese, use logographic writing systems where surface forms are represented as a visual organization of smaller graphemic units, which often contain many semantic cues. In this paper, we present a novel study which explores the benefits of providing language models with logographic information in learning better semantic representations. We test our hypothesis in the natural language inference (NLI) task by evaluating the benefit of computing multi-modal representations that combine contextual information with glyph information. Our evaluation results in six languages with different typology and writing systems suggest significant benefits of using multi-modal embeddings in languages with logograhic systems, especially for words with less occurence statistics.