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.

Introduction Adding constraints to a basic CSP model can signi cantly reduce search, e.g. for Golomb rulers [6]. The generation process is usually performed by hand, although some recent work has focused on automatically generating symmetry breaking constraints [4] and (less so) on generating implied constraints [5]. We describe an approach to generating implied, symmetry breaking and specialisation constraints and apply this technique to quasigroup construction [10]. Given a problem class parameterised by size, we use a basic model to solve small instances with the Choco constraint programming language [7]. We then give these solutions to the HR automated theory formation program [1] which detects implied constraints (proved to follow from the speci cations) and induced constraints (true of a subset of solutions). Interpreting HR's results to reformulate the model can lead to a reduction in search on larger instances. It is often more ecient to run HR, interpret the results and so

We report on the application of the HR program (Colton, Bundy, & Walsh 1999) to the problem of automatically inventing integer sequences. Seventeen sequences invented by HR are interesting enough to have been accepted into the Encyclopedia of Integer Sequences (Sloane 2000) and all were supplied with interesting conjectures about their nature, also discovered by HR. By extending HR, we have enabled it to perform a two stage process of invention and investigation. This involves generating both the definition and terms of a new sequence, relating it to sequences already in the Encyclopedia and pruning the output to help identify the most surprising and interesting results.

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...

We consider aspects pertinent to evaluating creativity to be input, output and the process by which the output is achieved. These issues

The HR program, Colton et al. (1999), performs theory formation in domains of pure mathematics. Given only minimal information about a domain, it invents concepts, make conjectures, proves theorems and finds counterexamples to false conjectures. We present here a multi-agent version of HR which may provide a model for how individual mathematicians perform separate investigations but communicate their results to the mathematical community, learning from others as they do. We detail the exhaustive categorisation problem to which we have applied a multi-agent approach. 1

Recently, many programs have been written to perform tasks which are usually regarded as requiring creativity in humans. We can

Automated Theorem Proving (ATP) researchers who always use the same problems for testing their systems, run the risk of producing systems that can solve only those problems, and are weak on new problems or applications. Furthermore, as the state-of-the-art in ATP progresses, existing test problems become too easy, and testing on them provides little useful information. It is thus important to regularly nd new and harder problems for testing ATP systems. HR is a program that performs automated theory formation in mathematical domains, such as group theory, quasigroup theory, and ring theory. Given the axioms of the domain...

Automated theory formation involves the production of examples, concepts and hypotheses about the concepts. The HR program performs automated theory formation and has used to form theories in mathematical domains. In addition to providing a plausible model for automated theory formation, HR has been applied to some applications in machine learning. We discuss HR's application to inducing de nitions from examples, scienti c discovery, problem solving and puzzle generation. For each problem, we look at how theory formation was applied, and mention some initial results from using HR.

Broadly speaking, machine learning programs are asked to identify a single concept given a set of examples and some background knowledge. Mathematical theory formation programs, such as the AM program, (Davis & Lenat 1982) and the HR program, (Colton, Bundy, & Walsh 1999), are also given a set of examples and some background knowledge. However, they are not asked to find a single concept, but rather to explore the domain and attempt to gain some understanding of it. Because the domain is mathematics, there are a range of ways by which the program can gain an understanding of the domain, including inventing concepts, performing calculations, making conjectures, proving theorems and finding counterexamples. The HR system is able to perform all of these activities in domains such as group theory, where it employs the Otter theorem prover, (McCune 1990), and MACE counterexample finder, (McCune 1994), to prove and disprove theorems. Whereas machine learning programs are in general goal directed, theory formation programs perform forward chaining. To control this, measures of interestingness for concepts are derived and employed so that an effective best first search through the space of concepts can be achieved. For example, measures based on the clarity of the definition of a concept are common in theory formation programs and help increase the yield of understandable concepts. Theory formation programs are not limited to just mathematics. We have recently enabled HR to work in “train theory”, in which it invents concepts which involve properties of trains as described in the original Michalski problem, (Michalski & Larson 1977). For example, it invents the concept of trains where each carriage has a different shape. In general, the kinds of concepts which machine learning programs and theory formation programs can invent in pure mathematics are very similar, but theory formation programs have to avoid many uninteresting concepts. A good example of this is that the Progol ILP program, (Muggleton 1995), Copyright c