On the Notion of Interestingness in Automated Mathematical Discovery

Deciding whether something is interesting or not is of central importance in automated mathematical discovery, as it helps determine both the search space and search strategy for finding and evaluating concepts and conjectures

The "Artificial Mathematician" Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer

Considerations on Approaches and Metrics in Automated Theorem Generation/Finding in Geometry

The pursue of what are properties that can be identified to permit an automated reasoning program to generate and find new and interesting theorems is an interesting research goal (pun intended). The automatic discovery of new theorems is a goal in itself, and it has been addressed in specific areas, with different methods. The separation of the "weeds", uninteresting, trivial facts, from the "wheat", new and interesting facts, is much harder, but is also being addressed by different authors using different approaches. In this paper we will focus on geometry. We present and discuss different approaches for the automatic discovery of geometric theorems (and properties), and different metrics to find the interesting theorems among all those that were generated. After this description we will introduce the first result of this article: An undecidability result proving that having an algorithmic procedure that decides for every possible Turing Machine that produces theorems, whether it is able to produce also interesting theorems, is an undecidable problem. Consequently, we will argue that judging whether a theorem prover is able to produce interesting theorems remains a non deterministic task, at best a task to be addressed by program based in an algorithm guided by heuristics criteria. Therefore, as a human, to satisfy this task two things are necessary: An expert survey that sheds light on what a theorem prover/finder of interesting geometric theorems is, and-to enable this analysis- other surveys that clarify metrics and approaches related to the interestingness of geometric theorems. In the conclusion of this article we will introduce the structure of two of these surveys -the second result of this article- and we will discuss some future work.</p

Showing Proofs, Assessing Difficulty with GeoGebra Discovery

In our contribution we describe some on-going improvements concerning the Automated Reasoning Tools developed in GeoGebra Discovery, providing different examples of the performance of these new features. We describe the new ShowProof command, that outputs both the sequence of the different steps performed by GeoGebra Discovery to confirm a certain statement, as well as a number intending to grade the difficulty or interest of the assertion. The proposal of this assessment measure, involving the comparison of the expression of the thesis (or conclusion) as a combination of the hypotheses, will be developed.Comment: In Proceedings ADG 2023, arXiv:2401.1072

Mathematical applications of inductive logic programming

Accepted versio

GeoGebra discovery at EGMO 2022

Este estudo mostrará a capacidade (ou incapacidade) da GeoGebra Discovery de lidar com problemas de geometria euclidiana propostos na recente Olimpíada Europeia de Matemática das Meninas (Hungria, 6 a 12 de abril de 2022). Após uma breve introdução ao contexto desta Olimpíada e ao programa GeoGebra Discovery, os problemas serão descritos e será feita uma tentativa de resolvê-los com a GeoGebra Discovery, finalmente apontando a relação entre as dificuldades encontradas pelos membros da equipe e pela GeoGebra, que podem contribuir para o estabelecimento de critérios sobre o interesse (e complexidade) dos resultados obtidos automaticamente

A constructive theory of automated ideation

In this thesis we explore the field of automated artefact generation in computational creativity with the aim of proposing methods of generation of ideas with cultural value. We focus on two kinds of ideas: fictional concepts and socially embedded concepts. For fictional concepts, we introduce a novel method based on the non-existence-conjectures made by the HR automated theory formation system. We further introduce the notion of typicality of an example with respect to a concept into HR. This leads to methods for ordering fictional concepts with respect to three measurements: novelty, vagueness and stimulation. We ran an experiment to produce thousands of definitions of fictional animals and then compared the software's evaluations of the non-fictional concepts with those obtained through a survey consulting sixty people. The results showed that two of the three measurements have a correlation with human notions.For socially embedded concepts, we apply a typicality-based classification method, the Rational Model of Classification (RMC), to a set of data obtained from Twitter. The aim being the creation of a set of concepts that naturally associate to an initial topic. We applied the RMC to four sets of tweets, each corresponding to one of four initial topics. The result was a set of clusters per topic, each cluster having a definition consisting of a set of words that appeared recurrently in the tweets. A survey was used to ask people to guess the topic given a set of definitions and to rate the artistic relevance of these definitions. The results showed both high association percentage and high relevance scores. A second survey was used to compare the rankings on the social impact of each of the definitions. The results obtained show a weak positive correlation between the two rankings. Our experiments show that it is possible to automatically generate ideas with the purpose of using them for artefact generation. This is an important step for the automation of computational creativity because most of the available artefact generation systems do not explicitly undertake idea generation. Moreover, our experiments introduce new ways of using the notion of typicality and show how these uses can be integrated in both the generation and evaluation of ideas.Open Acces