Showing papers by "Lawrence C. Paulson published in 2020"

Modelling High-Level Mathematical Reasoning in Mechanised Declarative Proofs

Abstract: Mathematical proofs can be mechanised using proof assistants to eliminate gaps and errors. However, mechanisation still requires intensive labour. To promote automation, it is essential to capture high-level human mathematical reasoning, which we address as the problem of generating suitable propositions. We build a non-synthetic dataset from the largest repository of mechanised proofs and propose a task on causal reasoning, where a model is required to fill in a missing intermediate proposition given a causal context. Our experiments (using various neural sequence-to-sequence models) reveal that while the task is challenging, neural models can indeed capture non-trivial mathematical reasoning. We further propose a hierarchical transformer model that outperforms the transformer baseline.

Formalising Ordinal Partition Relations Using Isabelle/HOL

Abstract: This is an overview of a formalisation project in the proof assistant Isabelle/HOL of a number of research results in infinitary combinatorics and set theory (more specifically in ordinal partition relations) by Erdős--Milner, Specker, Larson and Nash-Williams, leading to Larson's proof of the unpublished result by E.C. Milner asserting that for all $m \in \mathbb{N}$, $\omega^\omega\arrows(\omega^\omega, m)$. This material has been recently formalised by Paulson and is available on the Archive of Formal Proofs; here we discuss some of the most challenging aspects of the formalisation process. This project is also a demonstration of working with Zermelo-Fraenkel set theory in higher-order logic.

Algebraically Closed Fields in Isabelle/HOL.

Abstract: A fundamental theorem states that every field admits an algebraically closed extension Despite its central importance, this theorem has never before been formalised in a proof assistant We fill this gap by documenting its formalisation in Isabelle/HOL, describing the difficulties that impeded this development and their solutions

Evaluating Winding Numbers and Counting Complex Roots Through Cauchy Indices in Isabelle/HOL

Abstract: In complex analysis, the winding number measures the number of times a path (counter-clockwise) winds around a point, while the Cauchy index can approximate how the path winds. We formalise this approximation in the Isabelle theorem prover, and provide a tactic to evaluate winding numbers through Cauchy indices. By further combining this approximation with the argument principle, we are able to make use of remainder sequences to effectively count the number of complex roots of a polynomial within some domains, such as a rectangular box and a half-plane.

Bayesian optimisation of solver parameters in CBMC

Abstract: Satisfiability solvers can be embedded in applications to perform specific formal reasoning tasks. CBMC, for example, is a bounded model checker for C and C++ that embeds SMT and SAT solvers to check internally generated formulae. Such solvers will be solely used to evaluate the class of formulae generated by the embedding application and therefore may benefit from domain-specific parameter tuning. We propose the use of Bayesian optimisation for this purpose, which offers a principled approach to black-box optimisation within limited resources. We demonstrate its use for optimisation of the solver embedded in CBMC specifically for a collection of test harnesses in active industrial use, for which we have achieved a significant improvement over the default parameters.

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

Abstract: 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

IsarStep: a Benchmark for High-level Mathematical Reasoning

Abstract: 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.