A Tool for Producing Verified, Explainable Proofs

Mathematicians are reluctant to use interactive theorem provers. In this thesis I argue that this is because proof assistants don't emphasise explanations of proofs; and that in order to produce good explanations, the system must create proofs in a manner that mimics how humans would create proofs. My research goals are to determine what constitutes a human-like proof and to represent human-like reasoning within an interactive theorem prover to create formalised, understandable proofs. Another goal is to produce a framework to visualise the goal states of this system. To demonstrate this, I present HumanProof: a piece of software built for the Lean 3 theorem prover. It is used for interactively creating proofs that resemble how human mathematicians reason. The system provides a visual, hierarchical representation of the goal and a system for suggesting available inference rules. The system produces output in the form of both natural language and formal proof terms which are checked by Lean's kernel. This is made possible with the use of a structured goal state system which interfaces with Lean's tactic system which is detailed in Chapter 3. In Chapter 4, I present the subtasks automation planning subsystem, which is used to produce equality proofs in a human-like fashion. The basic strategy of the subtasks system is break a given equality problem in to a hierarchy of tasks and then maintain a stack of these tasks in order to determine the order in which to apply equational rewriting moves. This process produces equality chains for simple problems without having to resort to brute force or specialised procedures such as normalisation. This makes proofs more human-like by breaking the problem into a hierarchical set of tasks in the same way that a human would. To produce the interface for this software, I also created the ProofWidgets system for Lean 3. This system is detailed in Chapter 5. The ProofWidgets system uses Lean's metaprogramming framework to allow users to write their own interactive, web-based user interfaces to display within the VSCode editor and in an online web-editor. The entire tactic state is available to the rendering engine, and hence expression structure and types of subexpressions can be explored interactively. The ProofWidgets system also allows the user interface to interactively edit the proof document, enabling a truly interactive modality for creating proofs; human-like or not. In Chapter 6, the system is evaluated by asking real mathematicians about the output of the system, and what it means for a proof to be understandable to them. The user group study asks participants to rank and comment on proofs created by HumanProof alongside natural language and pure Lean proofs. The study finds that participants generally prefer the HumanProof format over the Lean format. The verbal responses collected during the study indicate that providing intuition and signposting are the most important properties of a proof that aid understanding.EPSR

Landscapes of the American Past: Visualizing Emancipation

The Digital Scholarship Lab at the University of Richmond proposes Landscapes of the American Past,an online atlas of American history, as a tool for organizing and interpreting a part of the outpouring of digital materials over the past twenty years and as a tool for thinking spatially about the past. In the start-up period, we will produce "Landscapes of Emancipation," the first detailed map of emancipation yet published, and answer questions about when, where, and how emancipation emerged from the Civil War. In doing so, we will also address a question of increasing interest in the digital humanities: how can we produce maps that rely on and support open resources while at the same time creating effective and elegant visualizations that convey scholarly arguments? We will publish our findings online as a mapping application, in peer-reviewed essays, as freely accessible data and metadata, and in a white paper addressing the methodology of visualizing historical arguments

Everyman as Master (Book Review)

Ayers, Edward L. Review of Tombee: Portrait of a Cotton Planter, by Theodore Rosengarten. Charlottesville: University of Virginia Press,1987

The United States on the Eve of the Civil War

The four-year war that eventually descended on the nation seemed impossible only months before it began. Powerful conflicts pulled the United States apart in the decades before 1860, but shared interests, cultures, and identities tied the country together, sometimes in new ways. So confident were they in the future that Americans expected that the forces of cohesion would triumph over the forces of division

The Trials of Robert Ryland

Robert Ryland tried to behave in a generous Christian way with the African-American people among whom he lived all his life even as he presided over what he recognized was a compromised form of the church. He faced skepticism and criticism from all sides, and experienced considerable doubt, but he pressed on

An Overview: The Difference Slavery made: A Close Analysis of Two American Communities

Using digital media, we wanted to give readers full access to a scholarly argument, the historiography about it, and the evidence for it. Our early models of the article contained neat squares and lines and carefully arranged explanations of the links from one part to another. Through two sets of readings by peer reviewers and presentations to a range of audiences, we have revised our presentation and our argument while maintaining the original purpose of the article. This essay introduces the electronic article and explains its development, as well as our intentions for it

Living Monuments: Confederate Soldiers\u27 Homes in the New South (Book Review)

Review of the book, Living Monuments: Confederate Soldiers\u27 Homes in the New South by R.B Rosenburg. Chapel Hill: University of North Carolina Press, 1993

Virginia History as Southern History: The Nineteenth Century

This essay briefly surveys some of the best work that has been done over the last ten years or so in the field of nineteenth-century Virginia and southern history in general, hoping to supply inspiration for histories yet to be written

Momentous Events in Small Places : The Coming of the Civil War in Two American Communities

Historians, professional and otherwise, have written thousands of regimental histories, county histories, and town histories of the Civil War years. These studies make the coming of the war concrete and compelling. Inspired by such accounts, it seemed to me that two local portrayals could be even better than one, that exploring communities on both sides of the Mason-Dixon Line as they each confronted the events from the late fifties to the late sixties might make both sides more comprehensible

Black American Intellectuals in the 1990s

As everyone who has followed the leading American periodicals in 1995 can tell you, a group of black academics has been much on the country\u27s mind recently. Rather breathless articles have several times announced the arrival of America\u27s New Public Intellectuals. One commentator argues that the recent burst of publishing and attention signals nothing less than the arrival of the Third Black Intellectual Renaissance, fit to be compared with those of the 1920s and the civil rights era