This paper considers the development of an AI-based provably-correct mathematical proof tutor. While Large Language Models (LLMs) allow seamless communication in natural language, they are error prone. Theorem provers such as Lean allow for provable-correctness, but these are hard for students to learn. We present a proof-of-concept system (LeanTutor) by combining the complementary strengths of LLMs and theorem provers. LeanTutor is composed of three modules: (i) an autoformalizer/proof-checker, (ii) a next-step generator, and (iii) a natural language feedback generator. To evaluate the system, we introduce PeanoBench, a dataset of 371 Peano Arithmetic proofs in human-written natural language and formal language, derived from the Natural Numbers Game.

NetKAT is a domain-specific programming language and logic that has been successfully used to specify and verify the behavior of packet-switched networks. This paper develops techniques for automatically learning NetKAT models of unknown networks using active learning. Prior work has explored active learning for a wide range of automata (e.g., deterministic, register, Büchi, timed etc.) and also developed applications, such as validating implementations of network protocols. We present algorithms for learning different types of NetKAT automata, including symbolic automata proposed in recent work. We prove the soundness of these algorithms, build a prototype implementation, and evaluate it on a standard benchmark. Our results highlight the applicability of symbolic NetKAT learning for realistic network configurations and topologies.

We present an incremental syntactic representation that consists of assigning a single discrete label to each word in a sentence, where the label is predicted using strictly incremental processing of a prefix of the sentence, and the sequence of labels for a sentence fully determines a parse tree.Our goal is to induce a syntactic representation that commits to syntactic choices only as they are incrementally revealed by the input, in contrast with standard representations that must make output choices such as attachments speculatively and later throw out conflicting analyses.Our learned representations achieve 93.72 F1 on the Penn Treebank with as few as 5 bits per word, and at 8 bits per word they achieve 94.97 F1, which is comparable with other state of the art parsing models when using the same pre-trained embeddings.We also provide an analysis of the representations learned by our system, investigating properties such as the interpretable syntactic features captured by the system and mechanisms for deferred resolution of syntactic ambiguities.