The aim of this project is to develop an understanding of what makes one diagrammatic proof more “readable” than another, and to apply that understanding in diagrammatic tools.
To begin unpacking that last sentence, whenever we use the word “readable” we ought to explain what we mean by it. Readability is, in general, a subjective quality and a matter of taste of course. I think you’d be right to be deeply sceptical of an algorithm to identify readability in fiction, for instance. Fortunately, we’re talking about formal proofs, where each step of the proof is generated from the ones that went before it according to a limited set of proof rules. For each (true) theorem, there are infinitely many proofs, each of them formally equivalent. From among those proofs, we want to find ones which users find easier to understand and there are some straightforward ways in which that can be measured — for instance, we could say that one proof, P, is more readable than another, Q, if users can identify which rules are applied in each step in P, but they are less able to do that for Q. The length of time it takes a user to read and understand a proof is part of readability too.
There is a rich history of logical diagrams, reaching back to Leibniz. In recent years, there has been a lot of interest in logical diagrams as a way to enable non-logicians to make precise statements, for instance when producing conceptual models such as ontologies. Our project focusses on Euler diagrams because they are well-known and many people find them easy to understand. This is the kind of theorem you can state and prove using Euler diagrams:
Euler diagrams themselves aren’t expressive enough for conceptual modelling tasks but more expressive systems that extend Euler diagrams software exist, such as spider diagrams and concept diagrams. So, if we can understand readability in Euler diagrams, we have a good chance of understanding it in systems that are expressive enough for real-world tasks too.
Another good reason to choose Euler diagrams is that there are already software reasoning tools, or theorem provers, that work with them. Theorem provers are either automated (enter a theorem and the tool generates a proof, if one exists) or interactive (enter the theorem then choose which rule or tactic should be applied next). EDITH is an automated Euler diagram theorem prover developed by our colleagues Gem Stapleton, Jean Flower and others at the Visual Modelling Group. Speedith, an interactive spider diagram prover, is a descendant of EDITH. We are planning to produce a version of Speedith that, like its predecessor, works automatically and do generates proofs with readability and the human reader in mind. We’re calling it Readith 🙂
The fact that diagrams can be an expressive and accessible way to communicate is widely recognised and valued, but what makes one diagram more effective than another is seldom measured. Formal diagrams attempt to combine the effectiveness of graphics with mathematical logic, and have the potential to bring the benefits of this expressiveness and accessibility to the field of logical reasoning. Diagrammatic reasoning (or visual logic) consists of using diagrams to represent logical theorems and to create diagrammatic proofs or counter-arguments for those theorems. In this project we will analyse and measure, for the first time, the factors that affect comprehension in diagrammatic reasoning. Our research question is as follows: is it possible to develop a systematic understanding of readability in diagrammatic proofs? (We use the term “readable” to mean easy to understand and use.) To be effective, such understanding would yield strategies to be used to automatically construct readable proofs. Furthermore, these strategies should be general enough to be applied to any visual logic and possess predictive qualities that can inform the design of future visual logics.
Euler diagrams are a simple visual logic which is used to represent data in a great many contexts. As well as being used informally or semi-formally in fields such as education, Euler diagrams have been used as a formal logic since the 1990s. The widespread use of Euler diagrams (for instance, in teaching set theory to school children) testifies to the fact that they are generally considered easy to understand. Logicians, philosophers and cognitive scientists have attempted to describe the origins of this ease of understanding, but have not done so in the specific context of the use of Euler diagrams in proofs, or in a way that yields general strategies for creating Euler diagram proofs which are easily understood.
The aims of this project are to provide a detailed, formal understanding of readability in diagrammatic proofs and to provide ways of automatically generating readable diagrammatic proofs. We will achieve the second of these aims by adapting an existing automated Euler diagram theorem prover. The aims allow us to address what we believe to be two of the most challenging issues for the diagrams community today: the need to explore when diagrammatic reasoning is understandable by people, and the need to produce effective software tools that make the benefits of diagrams available to a wide range of people. The project addresses the need to categorise and to empirically measure the features that make diagrammatic proofs more, or less, easy to understand. Identifying these features and the ways in which they interact will help us to make better use of existing notations and design more effective logics in the future.