Premise selection is a crucial yet challenging step in mathematical formalization, especially for users with limited experience. Due to the lack of available formalization projects, existing approaches that leverage language models often suffer from data scarcity. In this work, we introduce an innovative method for training a premise retriever to support the formalization of mathematics. Our approach employs a BERT model to embed proof states and premises into a shared latent space. The retrieval model is trained within a contrastive learning framework and incorporates a domain-specific tokenizer along with a fine-grained similarity computation method. Experimental results show that our model is highly competitive compared to existing baselines, achieving strong performance while requiring fewer computational resources. Performance is further enhanced through the integration of a re-ranking module. To streamline the formalization process, we will release a search engine that enables users to query Mathlib theorems directly using proof states, significantly improving accessibility and efficiency. Codes are available at https://github.com/ruc-ai4math/Premise-Retrieval.

But Table 2. shows BDLSTM (average 10 models) improves over [14]. Please also check the same in the line nos.

This paper investigates the role of AI assistants, specifically OpenAI's ChatGPT, in teaching formal methods (FM) to undergraduate students, using the B-method as a formal specification technique. While existing studies demonstrate the effectiveness of AI in coding tasks, no study reports on its impact on formal specifications. We examine whether ChatGPT provides an advantage when writing B-specifications and analyse student trust in its outputs. Our findings indicate that the AI does not help students to enhance the correctness of their specifications, with low trust correlating to better outcomes. Additionally, we identify a behavioural pattern with which to interact with ChatGPT which may influence the correctness of B-specifications.

People are adept at learning a wide variety of structured patterns from small amounts of data, presenting a conundrum from the standpoint of the bias-variance tradeoff: what kinds of representations and algorithms support the joint flexibility and data-paucity of human learning? One possibility is that people "learn by programming": inducing probabilistic models to fit observed data. Here, we experimentally test human learning in the domain of structured 2-dimensional patterns, using a task in which participants repeatedly predicted where a dot would move based on its previous trajectory. We evaluate human performance against standard parametric and non-parametric time-series models, as well as two Bayesian program synthesis models whose hypotheses vary in their degree of structure: a compositional Gaussian Process model and a structured "Language of Thought" (LoT) model. We find that signatures of human pattern learning are best explained by the LoT model, supporting the idea that the flexibility and data-efficiency of human structure learning can be understood as probabilistic inference over an expressive space of programs.

A key challenge for reinforcement learning is solving long-horizon planning problems. Recent work has leveraged programs to guide reinforcement learning in these settings. However, these approaches impose a high manual burden on the user since they must provide a guiding program for every new task. Partially observed environments further complicate the programming task because the program must implement a strategy that correctly, and ideally optimally, handles every possible configuration of the hidden regions of the environment. We propose a new approach, model predictive program synthesis (MPPS), that uses program synthesis to automatically generate the guiding programs.

Autoformalization is the process of automatically translating from natural language mathematics to formal specifications and proofs. A successful autoformalization system could advance the fields of formal verification, program synthesis, and artificial intelligence.While the long-term goal of autoformalization seemed elusive for a long time, we show large language models provide new prospects towards this goal. We make the surprising observation that LLMs can correctly translate a significant portion ( 25.3\%) of mathematical competition problems perfectly to formal specifications in Isabelle/HOL. We demonstrate the usefulness of this process by improving a previously introduced neural theorem prover via training on these autoformalized theorems.

Program synthesis from input-output (IO) examples has been a long-standing challenge. While recent works demonstrated limited success on domain-specific languages (DSL), it remains highly challenging to apply them to real-world programming languages, such as C. Due to complicated syntax and token variation, there are three major challenges: (1) unlike many DSLs, programs in languages like C need to compile first and are not executed via interpreters; (2) the program search space grows exponentially when the syntax and semantics of the programming language become more complex; and (3) collecting a large-scale dataset of real-world programs is non-trivial. As a first step to address these challenges, we propose LaSynth and show its efficacy in a restricted-C domain (i.e., C code with tens of tokens, with sequential, branching, loop and simple arithmetic operations but no library call). More specifically, LaSynth learns the latent representation to approximate the execution of partially generated programs, even if they are incomplete in syntax (addressing (1)). The learned execution significantly improves the performance of next token prediction over existing approaches, facilitating search (addressing (2)).

The recent successful paradigm of solving logical reasoning problems with tool-augmented large language models (LLMs) leverages translation of natural language statements into First-Order Logic~(FOL) and external theorem provers. However, the correctness of FOL statements, comprising operators and text predicates, often goes unverified due to the lack of a reliable evaluation metric for comparing generated and ground-truth FOLs. In this paper, we present a comprehensive study of sensitivity of existing metrics and their alignment with human judgement on FOL evaluation. Using ground-truth FOLs, we carefully designed various perturbations on the ground-truth to assess metric sensitivity. We sample FOL translation candidates for natural language statements and measure the ranking alignment between automatic metrics and human annotators. Our empirical findings highlight oversensitivity in the n-gram metric BLEU for text perturbations, the semantic graph metric Smatch++ for structural perturbations, and FOL metric for operator perturbation. We also observe a closer alignment between BertScore and human judgement. Additionally, we show that combining metrics enhances both alignment and sensitivity compared to using individual metrics.

We introduce an unsupervised learning algorithmthat combines probabilistic modeling with solver-based techniques for program synthesis.We apply our techniques to both a visual learning domain and a language learning problem,showing that our algorithm can learn many visual concepts from only a few examplesand that it can recover some English inflectional morphology.Taken together, these results give both a new approach to unsupervised learning of symbolic compositional structures,and a technique for applying program synthesis tools to noisy data.

Designing neural networks with bounded Lipschitz constant is a promising way to obtain certifiably robust classifiers against adversarial examples. However, the relevant progress for the important \ell_\infty perturbation setting is rather limited, and a principled understanding of how to design expressive \ell_\infty Lipschitz networks is still lacking. In this paper, we bridge the gap by studying certified \ell_\infty robustness from a novel perspective of representing Boolean functions. We derive two fundamental impossibility results that hold for any standard Lipschitz network: one for robust classification on finite datasets, and the other for Lipschitz function approximation. These results identify that networks built upon norm-bounded affine layers and Lipschitz activations intrinsically lose expressive power even in the two-dimensional case, and shed light on how recently proposed Lipschitz networks (e.g., GroupSort and \ell_\infty -distance nets) bypass these impossibilities by leveraging order statistic functions.