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

In fresh years, synthetic Intelligence researchers have mostly centred their efforts on fixing particular difficulties, with much less emphasis on 'the substantial photograph' - automating huge scale initiatives which require human-level intelligence to adopt. the topic of this publication, automatic concept formation in arithmetic, is this kind of huge scale activity. automatic thought formation calls for the discovery of latest options, the calculating of examples, the making of conjectures and the proving of theorems. This ebook, representing 4 years of PhD paintings through Dr. Simon Colton demonstrates how concept formation could be computerized. construction on over two decades of study into developing an automatic mathematician performed 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 way 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 idea from the naked minimal of knowledge - the axioms of a site. the most software of this paintings has been mathematical discovery, and HR has had many successes. specifically, it has invented 20 new different types of variety of adequate curiosity to be authorized into the Encyclopaedia of Integer Sequences, a repository of over 60,000 sequences contributed by means of many (human) mathematicians.

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**Example text**

In pure mathematics, the arguments are not just demonstrations that the statement is likely to be true based on all available evidence. Rather, an argument is meant to be irrefutable and everyone who understands it should agree it proves that the statement is true. While there have been incorrect proofs accepted as true for many years, it is generally hoped that the truth will prevail, and that it is not possible to fool all of the mathematicians all of the time. It is common to call the arguments presented in mathematics "proofs" with the statements made being termed "open conjectures" until a proof of their truth is found,3 after which time they are called "theorems".

The procedure uses unit propagation to improve performance. This method chooses a variable in a unit clause (a clause containing a single literal) and assigns a value which satisfies the clause. To demonstrate the usage of MACE, the conjecture that all groups are Abelian is supplied in the same format as for Otter, as follows: all abc (a * (b * c) = (a * b) * c). all b (a * id = b & b * a = b). all a (inv(a) * a = id & a * inv(a) = id). -(all abc (a * b = C -> b * a = c». We must also specify the size of the example we want.

Problems and results from previous centuries are constantly providing inspiration and new challenges for mathematicians today. A good example of this is provided by the area of Diophantine equations. Diophantus of Alexandria (c. 200-284), was perhaps the first person to try to solve equations such as: a3 _ b3 = c3 + d 3 by finding integer values of a, b, c and d for which the equation is true. Fermat (1601-1665) looked at a general case and conjectured that there are no values of n greater than 2 for which: has a solution in non-zero positive integers a, band c.