Learning to Prove Theorems via Interacting with Proof Assistants

Humans prove theorems by relying on substantial high-level reasoning and problem-specific insights. Proof assistants offer a formalism that resembles human mathematical reasoning, representing theorems in higher-order logic and proofs as high-level tactics. However, human experts have to construct proofs manually by entering tactics into the proof assistant. In this paper, we study the problem of using machine learning to automate the interaction with proof assistants. We construct CoqGym, a large-scale dataset and learning environment containing 71K human-written proofs from 123 projects developed with the Coq proof assistant. We develop ASTactic, a deep learning-based model that generates tactics as programs in the form of abstract syntax trees (ASTs). Experiments show that ASTactic trained on CoqGym can generate effective tactics and can be used to prove new theorems not previously provable by automated methods. Code is available at https://github.com/princeton-vl/CoqGym.Comment: Accepted to ICML 201

Phobos: A front-end approach to extensible compilers (long version)

This paper describes a practical approach for implementing certain types of domain-specific languages with extensible compilers. Given a compiler with one or more front-end languages, we introduce the idea of a "generic" front-end that allows the syntactic and semantic specification of domain-specific languages. Phobos, our generic front-end, offers modular language specification, allowing the programmer to define new syntax and semantics incrementally

Code Generation = A* + BURS

A system called BURS that is based on term rewrite systems and a search algorithm A* are combined to produce a code generator that generates optimal code. The theory underlying BURS is re-developed, formalised and explained in this work. The search algorithm uses a cost heuristic that is derived from the termrewrite system to direct the search. The advantage of using a search algorithm is that we need to compute only those costs that may be part of an optimal rewrite sequence

Formal Compiler Implementation in a Logical Framework

The task of designing and implementing a compiler can be a difficult and error-prone process. In this paper, we present a new approach based on the use of higher-order abstract syntax and term rewriting in a logical framework. All program transformations, from parsing to code generation, are cleanly isolated and specified as term rewrites. This has several advantages. The correctness of the compiler depends solely on a small set of rewrite rules that are written in the language of formal mathematics. In addition, the logical framework guarantees the preservation of scoping, and it automates many frequently-occurring tasks including substitution and rewriting strategies. As we show, compiler development in a logical framework can be easier than in a general-purpose language like ML, in part because of automation, and also because the framework provides extensive support for examination, validation, and debugging of the compiler transformations. The paper is organized around a case study, using the MetaPRL logical framework to compile an ML-like language to Intel x86 assembly. We also present a scoped formalization of x86 assembly in which all registers are immutable

Application-tailored Linear Algebra Algorithms: A search-based Approach

In this paper, we tackle the problem of automatically generating algorithms for linear algebra operations by taking advantage of problem-specific knowledge. In most situations, users possess much more information about the problem at hand than what current libraries and computing environments accept; evidence shows that if properly exploited, such information leads to uncommon/unexpected speedups. We introduce a knowledge-aware linear algebra compiler that allows users to input matrix equations together with properties about the operands and the problem itself; for instance, they can specify that the equation is part of a sequence, and how successive instances are related to one another. The compiler exploits all this information to guide the generation of algorithms, to limit the size of the search space, and to avoid redundant computations. We applied the compiler to equations arising as part of sensitivity and genome studies; the algorithms produced exhibit, respectively, 100- and 1000-fold speedups