We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathematical discovery. We detail how empirical evidence is used to give plausibility to conjectures, and the dierent ways in which a result can be thought of as novel. We also look at the ways in which the programs assess how surprising and complex a conjecture statement is, and the dierent ways in which the applicability of a concept or conjecture is used. Finally, we note how a user can set tasks for the program to achieve and how this aects the calculation of interestingness. We conclude with some hints on the use of interestingness measures for future developers of discovery programs in mathematics.

The HR program forms concepts and makes conjectures in domains of pure mathematics and uses theorem prover OTTER and model generator MACE to prove or disprove the conjectures. HR measures properties of concepts and assesses the theorems and proofs involving them to estimate the interestingness of each concept and employ a best first search. This approach has led HR to the discovery of interesting new mathematics and enables it to build theories from just the axioms of finite algebras.

This paper concerns the open problem of Lovász and Saks regarding the relationship between the communication complexity of a boolean function and the rank of the associated matrix. We first give an example exhibiting the largest gap known. We then prove two related theorems. 1

An important problem in computational scientific discovery is to identify, among the diversity of discovery programs written in various sciences, a commonality that will take a next step beyond the acknowledged general -- but weak -- framework of heuristic search. We characterize discovery in science as the generation of novel, interesting, plausible, and intelligible knowledge about the objects of study. We then analyze four current machine discovery programs in chemistry, medicine, mathematics, and linguistics according to how their design, or the circumstances of their application, heighten the chances of finding knowledge that has all four properties. Some general patterns emerge, although some strategies seem idiosyncratic. Our candidate for a commonality, which focuses on human factors, can be used pragmatically to evaluate and compare the designs of discovery programs that are intended to be used as collaborators by scientists. 1 1 Introduction Early work on machine scienti...

The HR program by Colton et al. (1999) performs theory formation in mathematics by exploring a space of mathematical concepts. By enabling HR to determine when it has found a particular concept, and by adding a forward looking mechanism, we have applied HR to the problem of identifying mathematical concepts. We illustrate this by using HR to identify and extrapolate integer sequences and by performing a qualitative comparison with the machine learning program Progol. 1. Introduction Extrapolating integer sequences such as 1; 4; 9; 16 : : : is an intelligent activity requiring both understanding and creativity. While there have been attempts in Artificial Intelligence to automatically extrapolate sequences, presently the state of the art is to use a large online 1 database, the Encyclopedia of Integer Sequences. Extrapolating integer sequences generalises to the problem of automatically identifying a property of a set of mathematical objects (the example set) which disting...

In this paper, we review AI research on computational discovery and its recent application to the discovery of new scientific knowledge. We characterize five historical stages of the scientific discovery process, which we use as an organizational framework in describing applications. We also identify five distinct steps during which developers or users can influence the behavior of a computational discovery system. Rather than criticizing such intervention, as done in the past, we recommend it as the preferred approach to using discovery software. As evidence for the advantages of such human-computer cooperation, we report seven examples of novel, computer-aided discoveries that have appeared in the scientific literature. We consider briefly the role that humans played in each case, then examine one such interaction in more detail. We close by recommending that future systems provide more explicit support for human intervention in the discovery process. Running head: Computational Sci...

. In this paper, we review AI research on computational discovery and its recent application to the discovery of new scientific knowledge. We characterize five historical stages of the scientific discovery process, which we use as an organizational framework in describing applications. We also identify five distinct steps during which developers or users can influence the behavior of a computational discovery system. Rather than criticizing such intervention, as done in the past, we recommend it as the preferred approach to using discovery software. As evidence for the advantages of such human-computer cooperation, we report seven examples of novel, computer-aided discoveries that have appeared in the scientific literature, along with the role that humans played in each case. We close by recommending that future systems provide more explicit support for human intervention in the discovery process. 1 Introduction The process of scientific discovery has long been viewed as the pinnacle ...

We add to the discussion of how to assess the creativity of programs which generate artefacts such as poems, theorems, paintings, melodies, etc. To do so, we first review some existing frameworks for assessing artefact generation programs. Then, drawing on our experience of building both a mathematical discovery system and an automated painter, we argue that it is not appropriate to base the assessment of a system on its output alone, and that the way it produces artefacts also needs to be taken into account. We suggest a simple framework within which the behaviour of a program can be categorised and described which may add to the perception of creativity in the system.

We show that the number of points with pairwise different sets of neighbors in a graph is O(2 r/2) where r is the rank of the adjacency matrix. We also give an example that achieves this bound. 1

We believe that AI programs written for discovery tasks will need to simultaneously employ a variety of reasoning techniques such as induction, abduction, deduction, calculation and invention. We describe the HR system which performs a novel ILP routine called automated theory formation. This combines inductive and deductive reasoning to form clausal theories consisting of classi cation rules and association rules.