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.

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

Building experimental games offers an alternative methodology for researching and understanding games, beyond what can be understood by playing and studying existing games alone. Through a simultaneous process of research and artmaking in the construction of the interactive drama Façade, new theoretical and design insights into several game studies questions were realized, including the hotly debated question of ludology vs. narratology. This paper describes some of the ways that building games can inform researchers about what game scholarship should be focused on and why, and ways that building games can offer new perspectives on existing forms and genres.

We describe the HR concept formation program which invents mathematical definitions and conjectures in finite algebras such as group theory and ring theory. We give the methods behind and the reasons for the concept formation in HR, an evaluation of its performance in its training domain, group theory, and a look at HR in domains other than group theory.

Learning general truths from the observation of simple domains and, further, learning how to use this knowledge are essential capabilities for any intelligent agent to understand and execute informed actions in the real world. The aim of this work is the investigation of the automatic learning of mathematical structures from visual observation. This research was conducted upon a system that combines computer vision with inductive logic programming that was first designed to learn protocol behaviour from observation. In this paper we show how transitivity, reflexivity and symmetry axioms could be induced from the noisy data provided by the vision system. Noise in the data accounts for the generation of a large number of possible generalisations by the ILP system, most of which do not represent interesting concepts about the observed domain. In order to automatically choose the best answers among those generated by induction, we propose a method for combining the results of multiple ILP processes by ranking the most interesting answers. 1

An ability to reason at a meta-level is widely regarded as an important aspect of human creativity which is often missing from creative computer programs. We discuss recent experiments with the HR theory formation program where it formed meta-theories about previously formed theories. We report how HR re-invented aspects of how it forms theories and reflected on the nature of the theories it produces. Additionally, the meta-theories contains higher level concepts than those produced using HR normally. We discuss how HR's meta-level abilities were enabled by changing domains, rather than writing new programs, which was the model previously employed in the Meta-DENDRAL and Eurisko programs. These experiments suggest an improved model of theory formation where meta-theories are produced alongside theories, with information from the meta-theory being used to improve the search in the original theory. 1

Many interesting concepts in mathematics are essentially ‘cross-domain ’ in nature, relating objects from more than one area of mathematics, e.g. prime order groups. These concepts are often vital to the formation of a mathematical theory. Often, the introduction of cross-domain concepts to an investigation seems to exercise a mathematician’s creative ability. The HR program, (Colton et al., 1999), proposes new concepts in mathematics. Its original implementation was limited to working in one mathematical domain at a time, so it was unable to create cross-domain concepts. Here, we describe an extension of HR to multiple domains. Cross-domain concept formation is facilitated by generalisation of the data structures and heuristic measures employed by the program, and the implementation of a new production rule. Results achieved include generation of the concepts of prime order groups, graph nodes of maximal degree and an interesting class of graph. 1

One of the key areas for the application of Artificial Intelligence to the game domain is in the design of challenging artificial opponents for human players. Complex simulations such as historical wargames can be seen as natural extensions of classical games where AI techniques such as planning or learning have already proved powerful. Yet the parallel nature of more recent games introduce new levels of complexity which can be tackled at various levels. This paper focuses on the question of finding good representations for the AI design, which implies finding relevant granularities for the various tasks involved, for a popular historical wargame. This work is based on the partially automated use of the rules of the game, as well as some common sense and historical military knowledge, to design relevant heuristics. The resulting gain in representation complexity will help the application of techniques such as Reinforcement Learning.