Lawrence C. Paulson

ICLR Conference 2021 Conference Paper

IsarStep: a Benchmark for High-level Mathematical Reasoning

• Wenda Li 0001
• Lei Yu
• Yuhuai Wu
• Lawrence C. Paulson

A well-defined benchmark is essential for measuring and accelerating research progress of machine learning models. In this paper, we present a benchmark for high-level mathematical reasoning and study the reasoning capabilities of neural sequence-to-sequence models. We build a non-synthetic dataset from the largest repository of proofs written by human experts in a theorem prover. The dataset has a broad coverage of undergraduate and research-level mathematical and computer science theorems. In our defined task, a model is required to fill in a missing intermediate proposition given surrounding proofs. This task provides a starting point for the long-term goal of having machines generate human-readable proofs automatically. Our experiments and analysis reveal that while the task is challenging, neural models can capture non-trivial mathematical reasoning. We further design a hierarchical transformer that outperforms the transformer baseline.

AAAI Conference 2020 Short Paper

Bayesian Optimisation for Premise Selection in Automated Theorem Proving (Student Abstract)

• Agnieszka Słowik
• Chaitanya Mangla
• Mateja Jamnik
• Sean B. Holden
• Lawrence C. Paulson

Modern theorem provers utilise a wide array of heuristics to control the search space explosion, thereby requiring optimisation of a large set of parameters. An exhaustive search in this multi-dimensional parameter space is intractable in most cases, yet the performance of the provers is highly dependent on the parameter assignment. In this work, we introduce a principled probabilistic framework for heuristic optimisation in theorem provers. We present results using a heuristic for premise selection and the Archive of Formal Proofs (AFP) as a case study.

LPAR Conference 2007 Conference Paper

Extending a Resolution Prover for Inequalities on Elementary Functions

• Behzad Akbarpour
• Lawrence C. Paulson

Abstract Experiments show that many inequalities involving exponentials and logarithms can be proved automatically by combining a resolution theorem prover with a decision procedure for the theory of real closed fields (RCF). The method should be applicable to any functions for which polynomial upper and lower bounds are known. Most bounds only hold for specific argument ranges, but resolution can automatically perform the necessary case analyses. The system consists of a superposition prover (Metis) combined with John Harrison’s RCF solver and a small amount of code to simplify literals with respect to the RCF theory.

I&C Journal 2006 Journal Article

Automation for interactive proof: First prototype

• Jia Meng
• Claire Quigley
• Lawrence C. Paulson

Interactive theorem provers require too much effort from their users. We have been developing a system in which Isabelle users obtain automatic support from automatic theorem provers (ATPs) such as Vampire and SPASS. An ATP is invoked at suitable points in the interactive session, and any proof found is given to the user in a window displaying an Isar proof script. There are numerous differences between Isabelle (polymorphic higher-order logic with type classes, natural deduction rule format) and classical ATPs (first-order, untyped, and clause form). Many of these differences have been bridged, and a working prototype that uses background processes already provides much of the desired functionality.

I&C Journal 2006 Journal Article

Erratum to “Automation for interactive proof: First prototype” [Inform. and Comput. 204 (2006) 1575–1596]

• Jia Meng
• Claire Quigley
• Lawrence C. Paulson