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There are several approaches for using computers in deriving mathematical proofs. For their illustration, we provide an in-depth study of using computer support for proving one complex combinatorial conjecture -- correctness of a strategy…
Since its existence, the computer tool has often supported mathematicians, whether it is to implement an approximation method (numerical calculation of a root, of an integral, ...) or to simulate a phenomenon (geometric in nature,…
We compare the values associated with (traditional) community based proof verification to those associated with computer proof verification. We propose ways that computer proofs might incorporate successful strategies from human…
Interactive proof assistants make it possible for ordinary mathematicians to write definitions and theorems in a formal proof language, like a programming language, so that a computer can parse them and check them against the rules of a…
In this short essay, we show how computer experiments, and especially visualization, allowed for the investigation and discovery of phenomena which would have passed unnoticed. We shall also highlight the importance of interactivity between…
A proof is one of the most important concepts of mathematics. However, there is a striking difference between how a proof is defined in theory and how it is used in practice. This puts the unique status of mathematics as exact science into…
We discuss the idea that computers might soon help mathematicians to prove theorems in areas where they have not previously been useful. Furthermore we argue that these same computer tools will also help us in the communication and teaching…
Noting that lemmas are a key feature of mathematics, we engage in an investigation of the role of lemmas in automated theorem proving. The paper describes experiments with a combined system involving learning technology that generates…
Large Language Models (LLMs) are transforming programming practices, offering significant capabilities for code generation activities. While researchers have explored the potential of LLMs in various domains, this paper focuses on their use…
Computers have already changed the way that humans do mathematics: they enable us to compute efficiently. But will they soon be helping us to reason? And will they one day start reasoning themselves? We give an overview of recent…
We report the results of a game-theoretic experiment with human players who solve the problems of increasing complexity by cooperating in groups of increasing size. Our experimental environment is set up to make it complicated for players…
The evolution of mathematics is shaped importantly by interestingness: researchers choose which problems to pursue, and students choose which problems to engage with, based on expectations of interest and challenge. As AI systems,…
One of the main uses of computers is to do statistical analysis of data. But, so far, the theory of statistics, and its noble mother, Probability theory, were all discovered and developed by lowly humans. No more! Computers can also develop…
Many concurrent and distributed systems are safety-critical and therefore have to provide a high degree of assurance. Important properties of such systems are frequently proved on the specification level, but implementations typically…
A computer model of "a sense of humour" suggested previously [arXiv:0711.2058,0711.2061], relating the humorous effect with a specific malfunction in information processing, is given in somewhat different exposition. Psychological aspects…
Human psychology plays an important role in organizational performance. However, understanding our employees is a difficult task due to issues such as psychological complexities, unpredictable dynamics, and the lack of data. Leveraging…
Math is constructed by people for people: just as natural language corpora reflect not just propositions but the communicative goals of language users, the math data that models are trained on reflects not just idealized mathematical…
This paper is an experimental exploration of the relationship between the runtimes of Turing machines and the length of proofs in formal axiomatic systems. We compare the number of halting Turing machines of a given size to the number of…
Large language models (LLMs) can surpass humans in certain forecasting tasks. What role does this leave for humans in the overall decision process? One possibility is that humans, despite performing worse than LLMs, can still add value when…
Computational experiments have emerged as a valuable method for studying complex systems, involving the algorithmization of counterfactuals. However, accurately representing real social systems in Agent-based Modeling (ABM) is challenging…