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

ISBN-10: 1447101472

ISBN-13: 9781447101475

ISBN-10: 1447111133

ISBN-13: 9781447111139

In fresh years, man made Intelligence researchers have mostly concentrated their efforts on fixing particular difficulties, with much less emphasis on 'the large photograph' - automating huge scale projects which require human-level intelligence to adopt. the topic of this booklet, automatic idea formation in arithmetic, is one of these huge scale activity. computerized concept formation calls for the discovery of latest strategies, the calculating of examples, the making of conjectures and the proving of theorems. This publication, representing 4 years of PhD paintings through Dr. Simon Colton demonstrates how thought formation should be computerized. development on over twenty years of analysis into developing an automatic mathematician performed in Professor Alan Bundy's mathematical reasoning team in Edinburgh, Dr. Colton has carried out the HR approach as an answer to the matter of forming theories by means of computing device. HR makes use of a number of items of mathematical software program, together with computerized theorem provers, version turbines and databases, to construct a idea from the naked minimal of knowledge - the axioms of a website. the most program of this paintings has been mathematical discovery, and HR has had many successes. particularly, it has invented 20 new forms of variety of enough curiosity to be accredited into the Encyclopaedia of Integer Sequences, a repository of over 60,000 sequences contributed by way of many (human) mathematicians.

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Further work on analysing how BACON worked and the development of a rigorous methodology for domain-independent scientific function finding is given in [Schaffer 90]. 1 The Representation of Mathematical Concepts In [Kerber 91] and more fully in [Kerber 92]' Manfred Kerber argues that a formal logical definition is not enough to fully describe a mathematical concept. In evidence of this, he notes that new concepts are presented in textbooks with a definition, examples and often lemmas about some properties of the concept.

1). %Positive Examples square(l). square(4). square(9). % Negative Examples square(2). square(3). :- square(6). :- square(7). square(S). square(S). :- square (0) . 3 Input to Progol for learning the concept of square numbers The construction of new concepts is achieved by inverting deductive rules of inference to produce inductive rules. One rule of deduction which is inverted is the resolution rule [Robinson 65]. In its simplest form, this states that if we know: A -t Band B -t C then we can infer that: A-tC The first two ways to invert resolution involve inverting a single resolution step.

Literature Survey recently in [Larson 99]. Given a set of well known, interesting graph theory invariants, such as the diameter, independence number, rank and chromatic number, Graffiti uses a database of graphs to empirically check whether one sum of invariants is less than another sum of invariants. If a conjecture passes the empirical test and Fajtlowicz cannot prove it easily, it is recorded in the "writing on the wall" , some of which is publicly available [Fajtlowicz 99] and Fajtlowicz forwards it to interested graph theorists.

### Automated Theory Formation in Pure Mathematics by Simon Colton BSc, MSc, PhD (auth.)

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