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.

There is increasing interest within the Machine Learning community in systems which automatically reformulate their problem representation by defining and constructing new predicates. A previous paper discussed such a system, called CIGOL, and gave a derivation for the mechanism of inverting individual steps in first order resolution proofs. In this paper we describe an enhancement to CIGOL's learning strategy which strongly constrains the formation of new concepts and hypotheses. The new strategy is based on results from algorithmic information theory. Using these results it is possible to compute the probability that the simplifications produced by adopting new concepts or hypotheses are not based on chance regularities within the examples. This can be derived from the amount of information compression produced by replacing the examples with the hypothesised concepts. CIGOL's improved performance, based on an approximation of this strategy, is demonstrated by way of the automatic "di...

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.

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.

All published research on automated mathematical conjecture-making is surveyed, and the ideas underlying the successful programs in this area are outlined. One particularly successful -- and little known -- program is comprehensively described for the first time. The fundamental principle underlying this program can be simply stated: make the strongest conjecture for which no counterexample is known. Conjecture-making may be key to building machines with a wide variety of intelligent behaviors. If so, this principle should prove exceptionally useful.

Abstract. The HR system forms scientific theories, and has found particularly successful application in domains of pure mathematics. Starting with only the axioms of an algebraic system, HR can generate dozens of example algebras, hundreds of concepts and thousands of conjectures, many of which have first order proofs. Given the overwhelming amount of knowledge produced, we have provided HR with sophisticated tools for handling this data. We present here the first full description of these management tools. Moreover, we describe how careful analysis of the theories produced by HR – which is enabled by the management tools – has led us to make interesting discoveries in algebraic domains. We demonstrate this with some illustrative results from HR’s theories about an algebra of one axiom. The results fueled further developments, and led us to discover and prove a fundamental theorem about this domain. 1

Abstract. The first attempt at automating mathematical conjecture-making appeared in the late-1950s. It was not until the mid-1980s though that a program produced statements of interest to research mathematicians and actually contributed to the advancement of mathematics. A central and important idea underlying this program is the Principle of the Strongest Conjecture: make the strongest conjecture for which no counterexample is known. These two programs as well as other attempts to automate mathematical conjecture-making are surveyed—the success of a conjecture-making program, it is found, correlates strongly whether the program is designed to produce statements that are relevant to answering or advancing our mathematical questions. 1.

We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.

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