Markus N. Rabe

Yuhuai Wu, Albert Q. Jiang, Wenda Li et al. · cambridge

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. Our methodology results in a new state-of-the-art result on the MiniF2F theorem proving benchmark, improving the proof rate from $29.6\%$ to $35.2\%$.

Emily First, Markus N. Rabe, Talia Ringer et al.

Formally verifying software properties is a highly desirable but labor-intensive task. Recent work has developed methods to automate formal verification using proof assistants, such as Coq and Isabelle/HOL, e.g., by training a model to predict one proof step at a time, and using that model to search through the space of possible proofs. This paper introduces a new method to automate formal verification: We use large language models, trained on natural language text and code and fine-tuned on proofs, to generate whole proofs for theorems at once, rather than one step at a time. We combine this proof generation model with a fine-tuned repair model to repair generated proofs, further increasing proving power. As its main contributions, this paper demonstrates for the first time that: (1) Whole-proof generation using transformers is possible and is as effective as search-based techniques without requiring costly search. (2) Giving the learned model additional context, such as a prior failed proof attempt and the ensuing error message, results in proof repair and further improves automated proof generation. (3) We establish a new state of the art for fully automated proof synthesis. We reify our method in a prototype, Baldur, and evaluate it on a benchmark of 6,336 Isabelle/HOL theorems and their proofs. In addition to empirically showing the effectiveness of whole-proof generation, repair, and added context, we show that Baldur improves on the state-of-the-art tool, Thor, by automatically generating proofs for an additional 8.7% of the theorems. Together, Baldur and Thor can prove 65.7% of the theorems fully automatically. This paper paves the way for new research into using large language models for automating formal verification.

Yuhuai Wu, Markus N. Rabe, DeLesley Hutchins et al.

Language models typically need to be trained or finetuned in order to acquire new knowledge, which involves updating their weights. We instead envision language models that can simply read and memorize new data at inference time, thus acquiring new knowledge immediately. In this work, we extend language models with the ability to memorize the internal representations of past inputs. We demonstrate that an approximate kNN lookup into a non-differentiable memory of recent (key, value) pairs improves language modeling across various benchmarks and tasks, including generic webtext (C4), math papers (arXiv), books (PG-19), code (Github), as well as formal theorems (Isabelle). We show that the performance steadily improves when we increase the size of memory up to 262K tokens. On benchmarks including code and mathematics, we find that the model is capable of making use of newly defined functions and theorems during test time.

Kshitij Bansal, Sarah M. Loos, Markus N. Rabe et al.

We present an environment, benchmark, and deep learning driven automated theorem prover for higher-order logic. Higher-order interactive theorem provers enable the formalization of arbitrary mathematical theories and thereby present an interesting, open-ended challenge for deep learning. We provide an open-source framework based on the HOL Light theorem prover that can be used as a reinforcement learning environment. HOL Light comes with a broad coverage of basic mathematical theorems on calculus and the formal proof of the Kepler conjecture, from which we derive a challenging benchmark for automated reasoning. We also present a deep reinforcement learning driven automated theorem prover, DeepHOL, with strong initial results on this benchmark.

Markus N. Rabe, Judith Clymo, Zheren Dong

We introduce a simple modification to the embedding layer. The key change is to infuse token embeddings with information about their spelling. Models trained with these embeddings improve not only on spelling, but also across standard benchmarks. We conduct scaling studies for models with 40M to 800M parameters, which suggest that the improvements are equivalent to needing about 8% less compute and data to achieve the same test loss.

Markus N. Rabe, Charles Staats

We present a very simple algorithm for attention that requires $O(1)$ memory with respect to sequence length and an extension to self-attention that requires $O(\log n)$ memory. This is in contrast with the frequently stated belief that self-attention requires $O(n^2)$ memory. While the time complexity is still $O(n^2)$, device memory rather than compute capability is often the limiting factor on modern accelerators. Thus, reducing the memory requirements of attention allows processing of longer sequences than might otherwise be feasible. We provide a practical implementation for accelerators that requires $O(\sqrt{n})$ memory, is numerically stable, and is within a few percent of the runtime of the standard implementation of attention. We also demonstrate how to differentiate the function while remaining memory-efficient. For sequence length 16384, the memory overhead of self-attention is reduced by 59X for inference and by 32X for differentiation.

Frederik Schmitt, Christopher Hahn, Markus N. Rabe et al.

We train hierarchical Transformers on the task of synthesizing hardware circuits directly out of high-level logical specifications in linear-time temporal logic (LTL). The LTL synthesis problem is a well-known algorithmic challenge with a long history and an annual competition is organized to track the improvement of algorithms and tooling over time. New approaches using machine learning might open a lot of possibilities in this area, but suffer from the lack of sufficient amounts of training data. In this paper, we consider a method to generate large amounts of additional training data, i.e., pairs of specifications and circuits implementing them. We ensure that this synthetic data is sufficiently close to human-written specifications by mining common patterns from the specifications used in the synthesis competitions. We show that hierarchical Transformers trained on this synthetic data solve a significant portion of problems from the synthesis competitions, and even out-of-distribution examples from a recent case study.

Markus N. Rabe, Dennis Lee, Kshitij Bansal et al.

We examine whether self-supervised language modeling applied to mathematical formulas enables logical reasoning. We suggest several logical reasoning tasks that can be used to evaluate language models trained on formal mathematical statements, such as type inference, suggesting missing assumptions and completing equalities. To train language models for formal mathematics, we propose a novel skip-tree task. We find that models trained on the skip-tree task show surprisingly strong mathematical reasoning abilities, and outperform models trained on standard skip-sequence tasks. We also analyze the models' ability to formulate new conjectures by measuring how often the predictions are provable and useful in other proofs.

Christopher Hahn, Frederik Schmitt, Jens U. Kreber et al.

We study two fundamental questions in neuro-symbolic computing: can deep learning tackle challenging problems in logics end-to-end, and can neural networks learn the semantics of logics. In this work we focus on linear-time temporal logic (LTL), as it is widely used in verification. We train a Transformer on the problem to directly predict a solution, i.e. a trace, to a given LTL formula. The training data is generated with classical solvers, which, however, only provide one of many possible solutions to each formula. We demonstrate that it is sufficient to train on those particular solutions to formulas, and that Transformers can predict solutions even to formulas from benchmarks from the literature on which the classical solver timed out. Transformers also generalize to the semantics of the logics: while they often deviate from the solutions found by the classical solvers, they still predict correct solutions to most formulas.

Dennis Lee, Christian Szegedy, Markus N. Rabe et al.

We design and conduct a simple experiment to study whether neural networks can perform several steps of approximate reasoning in a fixed dimensional latent space. The set of rewrites (i.e. transformations) that can be successfully performed on a statement represents essential semantic features of the statement. We can compress this information by embedding the formula in a vector space, such that the vector associated with a statement can be used to predict whether a statement can be rewritten by other theorems. Predicting the embedding of a formula generated by some rewrite rule is naturally viewed as approximate reasoning in the latent space. In order to measure the effectiveness of this reasoning, we perform approximate deduction sequences in the latent space and use the resulting embedding to inform the semantic features of the corresponding formal statement (which is obtained by performing the corresponding rewrite sequence using real formulas). Our experiments show that graph neural networks can make non-trivial predictions about the rewrite-success of statements, even when they propagate predicted latent representations for several steps. Since our corpus of mathematical formulas includes a wide variety of mathematical disciplines, this experiment is a strong indicator for the feasibility of deduction in latent space in general.

Kshitij Bansal, Christian Szegedy, Markus N. Rabe et al.

In this paper, we demonstrate how to do automated theorem proving in the presence of a large knowledge base of potential premises without learning from human proofs. We suggest an exploration mechanism that mixes in additional premises selected by a tf-idf (term frequency-inverse document frequency) based lookup in a deep reinforcement learning scenario. This helps with exploring and learning which premises are relevant for proving a new theorem. Our experiments show that the theorem prover trained with this exploration mechanism outperforms provers that are trained only on human proofs. It approaches the performance of a prover trained by a combination of imitation and reinforcement learning. We perform multiple experiments to understand the importance of the underlying assumptions that make our exploration approach work, thus explaining our design choices.

Marcell Vazquez-Chanlatte, Markus N. Rabe, Sanjit A. Seshia

In this paper, we systematize the modeling of probabilistic systems for the purpose of analyzing them with model counting techniques. Starting from unbiased coin flips, we show how to model biased coins, correlated coins, and distributions over finite sets. From there, we continue with modeling sequential systems, such as Markov chains, and revisit the relationship between weighted and unweighted model counting. Thereby, this work provides a conceptual framework for deriving #SAT encodings for probabilistic inference.

Gil Lederman, Markus N. Rabe, Edward A. Lee et al.

We demonstrate how to learn efficient heuristics for automated reasoning algorithms for quantified Boolean formulas through deep reinforcement learning. We focus on a backtracking search algorithm, which can already solve formulas of impressive size - up to hundreds of thousands of variables. The main challenge is to find a representation of these formulas that lends itself to making predictions in a scalable way. For a family of challenging problems, we learned a heuristic that solves significantly more formulas compared to the existing handwritten heuristics.

Bernd Finkbeiner, Markus N. Rabe, César Sánchez

Hyperproperties, as introduced by Clarkson and Schneider, characterize the correctness of a computer program as a condition on its set of computation paths. Standard temporal logics can only refer to a single path at a time, and therefore cannot express many hyperproperties of interest, including noninterference and other important properties in security and coding theory. In this paper, we investigate an extension of temporal logic with explicit path variables. We show that the quantification over paths naturally subsumes other extensions of temporal logic with operators for information flow and knowledge. The model checking problem for temporal logic with path quantification is decidable. For alternation depth 1, the complexity is PSPACE in the length of the formula and NLOGSPACE in the size of the system, as for linear-time temporal logic.