Related papers: Fail better: What formalized math can teach us abo…
Teaching logic effectively requires an understanding of the factors which cause logic students to struggle. Formalization exercises, which require the student to produce a formula corresponding to the natural language sentence, are a good…
Computational reductions are an important and powerful concept in computer science. However, they are difficult for many students to grasp. In this paper, we outline a concept for how the learning of reductions can be supported by…
Informal logic is a method of argument analysis which is complementary to that of formal logic, providing for the pragmatic treatment of features of argumentation which cannot be reduced to logical form. The central claim of this paper is…
Recent advances in Automated Theorem Proving have shown the effectiveness of leveraging a (large) language model that generates tactics (i.e. proof steps) to search through proof states. The current model, while trained solely on successful…
"Systems that Explain Themselves" appears a provocative wording, in particular in the context of mathematics education -- it is as provocative as the idea of building educational software upon technology from computer theorem proving. In…
The use of logical systems for problem-solving may be as diverse as in proving theorems in mathematics or in figuring out how to meet up with a friend. In either case, the problem solving activity is captured by the search for an…
When mathematicians present proofs they usually adapt their explanations to their didactic goals and to the (assumed) knowledge of their addressees. Modern automated theorem provers, in contrast, present proofs usually at a fixed level of…
Verifying mathematical proofs is difficult, but can be automated with the assistance of a computer. Autoformalization is the task of automatically translating natural language mathematics into a formal language that can be verified by a…
The idea of assisting teachers with technological tools is not new. Mathematics in general, and geometry in particular, provide interesting challenges when developing educative softwares, both in the education and computer science aspects.…
The quality and correct functioning of software components embedded in electronic systems are of utmost concern especially for safety and mission-critical systems. Model-based testing and formal verification techniques can be employed to…
Mathematical proofs are both paradigms of certainty and some of the most explicitly-justified arguments that we have in the cultural record. Their very explicitness, however, leads to a paradox, because the probability of error grows…
This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes…
Humans prove theorems by relying on substantial high-level reasoning and problem-specific insights. Proof assistants offer a formalism that resembles human mathematical reasoning, representing theorems in higher-order logic and proofs as…
With the rapid rise of generative AI in higher education and the unreliability of current AI detection tools, developing policies that encourage student learning and critical thinking has become increasingly important. This study examines…
This is a reflection on the author's experience in teaching logic at the graduate level in a computer science department. The main lesson is that model building and the process of modelling must be placed at the centre stage of logic…
Many undergraduate students of engineering and the exact sciences have difficulty with their mathematics courses due to insufficient proficiency in what we in this paper have termed clear thinking. We believe that this lack of proficiency…
Recent developments show that AI can prove research-level theorems in mathematics, both formally and informally. This essay urges mathematicians to stay up-to-date with the technology, to consider the ways it will disrupt mathematical…
This panel draws on research of the teaching of mathematical proof, conducted in five countries at different levels of schooling. With a shared view of proof as essential to the teaching and learning of mathematics, the authors present…
Modern machine learning models are opaque, and as a result there is a burgeoning academic subfield on methods that explain these models' behavior. However, what is the precise goal of providing such explanations, and how can we demonstrate…
Extensive research on formal verification of machine learning systems indicates that learning from data alone often fails to capture underlying background knowledge, such as specifications implicitly available in the data. Various neural…