By Simon Colton BSc, MSc, PhD (auth.)

In contemporary years, synthetic Intelligence researchers have principally centred their efforts on fixing particular difficulties, with much less emphasis on 'the immense photo' - automating huge scale projects which require human-level intelligence to adopt. the topic of this ebook, computerized idea formation in arithmetic, is any such huge scale activity. automatic conception formation calls for the discovery of recent options, the calculating of examples, the making of conjectures and the proving of theorems. This e-book, representing 4 years of PhD paintings by way of Dr. Simon Colton demonstrates how concept formation may be computerized. construction on over twenty years of study into developing an automatic mathematician conducted in Professor Alan Bundy's mathematical reasoning crew in Edinburgh, Dr. Colton has applied the HR method as an answer to the matter of forming theories by means of laptop. HR makes use of a variety of items of mathematical software program, together with automatic theorem provers, version turbines and databases, to construct a thought from the naked minimal of data - the axioms of a site. the most program of this paintings has been mathematical discovery, and HR has had many successes. particularly, it has invented 20 new kinds of variety of adequate curiosity to be approved into the Encyclopaedia of Integer Sequences, a repository of over 60,000 sequences contributed by way of many (human) mathematicians.

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In [McCune & Padmanabhan 96], equational logic and cubic curves are explored. Furthermore, Otter has been used for discovery tasks, in particular finding single axioms for group theory and other algebraic systems [McCune 92]' [McCune 93], [Padmanabhan & McCune 95]. MACE [McCune 94] is the sister program to Otter. MACE is designed to generate models as counterexamples to false conjectures. MACE takes the same input as Otter, which is an appeal of using these two programs in conjunction. 7 The Encyclopedia of Integer Sequences 25 solutions to satisfiability problems [Davis & Putnam 60], [Yugami 95].

It is unlikely that groups where, say, 17% of pairs of elements commute have been studied, for one of two reasons. Either no-one has yet found a reason to study these concepts, or they have been overlooked because it is likely that nothing interesting will be provable about them. The first reason is more appealing, and it is certainly true in the majority of cases: the proportion of useless concept definitions is very high. However, the second reason becomes more plausible when we note that, for example, concepts which identify objects where every possible occurrence of a phenomena occurs, are common in mathematics.

Another fundamental concept is divisors, where a divisor of an integer, x, is a positive integer which multiplies by another positive integer to give x. For example, the divisors of 12 are {I, 2, 3, 4, 6, 12}. Proper divisors are the divisors other than the number itself. An important concept based on this which we use throughout this book is the number of divisors of an integer. We write T(n) for the number of divisors of n, for example T(12) = 6. Prime numbers are those integers with exactly two divisors, and these appear in countless many theorems in number theory and many other domains of pure and applied mathematics.