Related papers: Mathematics and the formal turn
Matching logic is a formalism for specifying, and reasoning about, mathematical structures, using patterns and pattern matching. Growing in popularity, it has been used to define many logical systems such as separation logic with recursive…
Physics makes powerful use of mathematics, yet the way this use is made is often poorly understood. Professionals closely integrate their mathematical symbology with physical meaning, resulting in a powerful and productive structure. But…
Traditionally, mathematical knowledge is published in printed media such as books or journals. With the advent of the Internet, a new method of publication became available. To date, however, most online mathematical publications do not…
Automatic verification deals with the validation by means of computers of correctness certificates. The related tools, usually called proof assistants or interactive provers, provide an interactive environment for the creation of formal…
In various provers and deductive verification tools, logical transformations are used extensively in order to reduce a proof task into a number of simpler tasks. Logical transformations are often part of the trusted base of such tools. In…
AI for Mathematics (AI4Math) is not only intriguing intellectually but also crucial for AI-driven discovery in science, engineering, and beyond. Extensive efforts on AI4Math have mirrored techniques in NLP, in particular, training large…
In the recent years, we have linked a large corpus of formal mathematics with automated theorem proving (ATP) tools, and started to develop combined AI/ATP systems working in this setting. In this paper we first relate this project to the…
The learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which…
Mathematical knowledge is a central component in science, engineering, and technology (documentation). Most of it is represented informally, and -- in contrast to published research mathematics -- subject to continual change. Unfortunately,…
Recent progress in artificial intelligence (AI) is unlocking transformative capabilities for mathematics. There is great hope that AI will help solve major open problems and autonomously discover new mathematical concepts. In this essay, we…
We commonly think of mathematics as bringing precision to application domains, but its relationship with computer science is more complex. This experience report on the use of Racket and Haskell to teach a required first university CS…
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such…
Despite the vast body of research literature proposing algorithms with formal guarantees, the amount of verifiable code in today's systems remains minimal. This discrepancy stems from the inherent difficulty of verifying code, particularly…
We present an approach for representing abstract argumentation frameworks based on an encoding into classical higher-order logic. This provides a uniform framework for computer-assisted assessment of abstract argumentation frameworks using…
This book explores an alternative to the current dominant paradigm where a discrete computer model is constructed as an attempt to approximate some continuum theory. We focus on a class of discrete computer models that are based on simple…
We state the defining characteristic of mathematics as a type of symmetry where one can change the connotation of a mathematical statement in a certain way when the statement's truth value remains the same. This view of mathematics as…
Formal grammars are extensively used in Computer Science and related fields to study the rules which govern production of a language. The use of these grammars can be extended beyond mere language production. One possibility is to view…
I analyse differences in style between traditional prose mathematics writing and computer-formalised mathematics writing, presenting five case studies. I note two aspects where good style seems to differ between the two: in their…
The last fifty years have seen an impressive development of mathematical methods for the analysis and processing of digital images, mostly in the context of photography, biomedical imaging and various forms of engineering. The arts have…
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…