arXiv subjects
Publications and source records attributed to Lawrence C. Paulson.
A 2021 proof of a special case of the Union-Closed Conjecture, by Aaronson, Ellis and Leader, has been formalised in the proof assistant Isabelle/HOL. Our discussion involves sketching their proof and displaying snippets from the Isabelle version of the proof, illustrating the extent to which mathematical reasoning can be rendered clearly in a formal language.
Formalised libraries of combinatorial mathematics have rapidly expanded over the last five years, but few use one of the most important tools: probability. How can often intuitive probabilistic arguments on the existence of combinatorial structures, such as hypergraphs, be translated into a formal text? We present a modular framework using locales in Isabelle/HOL to formalise such probabilistic proofs, including the basic existence method and first formalisation of the Lovász local lemma, a fundamental result in probability. The formalisation focuses on general, reusable formal probabilistic lemmas for combinatorial structures, and highlights several notable gaps in typical intuitive probabilistic reasoning on paper. The applicability of the techniques is demonstrated through the formalisation of several classic lemmas on the existence of hypergraphs with certain colourings.
A paper on ordinal partitions by Erdős and Milner (1972) has been formalised using the proof assistant Isabelle/HOL, augmented with a library for Zermelo-Fraenkel set theory. The work is part of a project on formalising the partition calculus. The chosen material is particularly appropriate in view of the substantial corrections later published by its authors, illustrating the potential value of formal verification.
We have formalised Szemerédi's Regularity Lemma and Roth's Theorem on Arithmetic Progressions, two major results in extremal graph theory and additive combinatorics, using the proof assistant Isabelle/HOL. For the latter formalisation, we used the former to first show the Triangle Counting Lemma and the Triangle Removal Lemma: themselves important technical results. Here, in addition to showcasing the main formalised statements and definitions, we focus on sensitive points in the proofs, describing how we overcame the difficulties that we encountered.
The formalisation of mathematics is continuing rapidly, however combinatorics continues to present challenges to formalisation efforts, such as its reliance on techniques from a wide range of other fields in mathematics. This paper presents formal linear algebraic techniques for proofs on incidence structures in Isabelle/HOL, and their application to the first formalisation of Fisher's inequality. In addition to formalising incidence matrices and simple techniques for reasoning on linear algebraic representations, the formalisation focuses on the linear algebra bound and rank arguments. These techniques can easily be adapted for future formalisations in combinatorics, as we demonstrate through further application to proofs of variations on Fisher's inequality.
Informal arguments that cryptographic protocols are secure can be made rigorous using inductive definitions. The approach is based on ordinary predicate calculus and copes with infinite-state systems. Proofs are generated using Isabelle/HOL. The human effort required to analyze a protocol can be as little as a week or two, yielding a proof script that takes a few minutes to run. Protocols are inductively defined as sets of traces. A trace is a list of communication events, perhaps comprising many interleaved protocol runs. Protocol descriptions incorporate attacks and accidental losses. The model spy knows some private keys and can forge messages using components decrypted from previous traffic. Three protocols are analyzed below: Otway-Rees (which uses shared-key encryption), Needham-Schroeder (which uses public-key encryption), and a recursive protocol by Bull and Otway (which is of variable length). One can prove that event $ev$ always precedes event $ev'$ or that property $P$ holds provided $X$ remains secret. Properties can be proved from the viewpoint of the various principals: say, if $A$ receives a final message from $B$ then the session key it conveys is good.
A formalisation of Gödel's incompleteness theorems using the Isabelle proof assistant is described. This is apparently the first mechanical verification of the second incompleteness theorem. The work closely follows {Ś}wierczkowski (2003), who gave a detailed proof using hereditarily finite set theory. The adoption of this theory is generally beneficial, but it poses certain technical issues that do not arise for Peano arithmetic. The formalisation itself should be useful to logicians, particularly concerning the second incompleteness theorem, where existing proofs are lacking in detail.
An Isabelle/HOL formalisation of Gödel's two incompleteness theorems is presented. The work follows Świerczkowski's detailed proof of the theorems using hereditarily finite (HF) set theory. Avoiding the usual arithmetical encodings of syntax eliminates the necessity to formalise elementary number theory within an embedded logical calculus. The Isabelle formalisation uses two separate treatments of variable binding: the nominal package is shown to scale to a development of this complexity, while de Bruijn indices turn out to be ideal for coding syntax. Critical details of the Isabelle proof are described, in particular gaps and errors found in the literature.
We give an overview of our formalizations in the proof assistant Isabelle/HOL of certain irrationality and transcendence criteria for infinite series from three different research papers: by Erdős and Straus (1974), Hančl (2002), and Hančl and Rucki (2005). Our formalizations in Isabelle/HOL can be found on the Archive of Formal Proofs. Here we describe selected aspects of the formalization and discuss what this reveals about the use and potential of Isabelle/HOL in formalizing modern mathematical research, particularly in these parts of number theory and analysis.
The proof of the relative consistency of the axiom of choice has been mechanized using Isabelle/ZF. The proof builds upon a previous mechanization of the reflection theorem. The heavy reliance on metatheory in the original proof makes the formalization unusually long, and not entirely satisfactory: two parts of the proof do not fit together. It seems impossible to solve these problems without formalizing the metatheory. However, the present development follows a standard textbook, Kunen's Set Theory, and could support the formalization of further material from that book. It also serves as an example of what to expect when deep mathematics is formalized.
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 $m \in \mathbb{N}$, $ω^ω\arrows(ω^ω, m)$. 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.
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.
Heuristics in theorem provers are often parameterised. Modern theorem provers such as Vampire 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 probablistic framework for heuristics optimisation in theorem provers. We present results using a heuristic for premise selection and The Archive of Formal Proofs (AFP) as a case study.
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.
Interactive theorem provers have developed dramatically over the past four decades, from primitive beginnings to today's powerful systems. Here, we focus on Isabelle/HOL and its distinctive strengths. They include automatic proof search, borrowing techniques from the world of first order theorem proving, but also the automatic search for counterexamples. They include a highly readable structured language of proofs and a unique interactive development environment for editing live proof documents. Everything rests on the foundation conceived by Robin Milner for Edinburgh LCF: a proof kernel, using abstract types to ensure soundness and eliminate the need to store proofs. Compared with the research prototypes of the 1970s, Isabelle is a practical and versatile tool. It is used by system designers, mathematicians and many others.
A quotient construction defines an abstract type from a concrete type, using an equivalence relation to identify elements of the concrete type that are to be regarded as indistinguishable. The elements of a quotient type are \emph{equivalence classes}: sets of equivalent concrete values. Simple techniques are presented for defining and reasoning about quotient constructions, based on a general lemma library concerning functions that operate on equivalence classes. The techniques are applied to a definition of the integers from the natural numbers, and then to the definition of a recursive datatype satisfying equational constraints.
Internet browsers use security protocols to protect sensitive messages. An inductive analysis of TLS (a descendant of SSL 3.0) has been performed using the theorem prover Isabelle. Proofs are based on higher-order logic and make no assumptions concerning beliefs or finiteness. All the obvious security goals can be proved; session resumption appears to be secure even if old session keys have been compromised. The proofs suggest minor changes to simplify the analysis. TLS, even at an abstract level, is much more complicated than most protocols that researchers have verified. Session keys are negotiated rather than distributed, and the protocol has many optional parts. Nevertheless, the resources needed to verify TLS are modest: six man-weeks of effort and three minutes of processor time.
Many problems in computer algebra and numerical analysis can be reduced to counting or approximating the real roots of a polynomial within an interval. Existing verified root-counting procedures in major proof assistants are mainly based on the classical Sturm theorem, which only counts distinct roots. In this paper, we have strengthened the root-counting ability in Isabelle/HOL by first formally proving the Budan-Fourier theorem. Subsequently, based on Descartes' rule of signs and Taylor shift, we have provided a verified procedure to efficiently over-approximate the number of real roots within an interval, counting multiplicity. For counting multiple roots exactly, we have extended our previous formalisation of Sturm's theorem. Finally, we combine verified components in the developments above to improve our previous certified complex-root-counting procedures based on Cauchy indices. We believe those verified routines will be crucial for certifying programs and building tactics.