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Automated Theory Formation (ATF) is a novel machine learning and discovery approach which has been developed over 15 years, and which is implemented in the HR systems (HR1 in Prolog, HR2 and HR3 in Java). Given some background knowledge, HR forms new concepts from old ones using a set of production rules, and then makes conjectures which relate the concepts, by appealing to empirical patterns in the examples of the concepts. HR then uses third party systems to prove/disprove the conjectures (usually the Otter theorem prover and the MACE model generator). HR also interacts with computer algebra systems such as Maple and Gap, in order to calculate values for concepts. To drive a heuristic search, HR uses a weighted sum - with the weights set by the user - of measures of interestingness for concepts, i.e., having decided which concept is most interesting, HR builds new concepts from this. The book arising from Simon Colton's PhD work is the main reference text for the early work on Automated Theory Formation. |
Here are three talks about the HR project, which give an overview of the project and different contexts for it:
The papers below describe how the HR systems have been developed, the
applications to which they have been applied and a context in AI for
the project.
The following papers describe some of the fundamental aspects of
automated theory formation. The first paper is the main reference for
ATF in the HR2 system, described as an Inductive Logic Programming system.
The other papers are from quite early on in the development of ATF
in the HR1 and HR2 systems.
We have taken Automated Theory Formation as the basis for more
in-depth studies into how mathematical theories can be formed
automatically. In addition to providing extensions to the basic
automated theory formation model and providing more background to the
subject, these projects have led to more sophisticated systems for
mathematical invention and machine learning in general, which take
into account philosophical and psychological perspectives on theory
formation. The following papers describe some of our projects in this
area:
As described above, the HR system for theory formation routinely
appeals to third party software as part of its core routine. This led
us to address the more general question of when it is possible to
combine AI techniques so that the whole is more than a sum of the
parts. In total, we have experimented with various combinations of
around 20 different AI systems, including descriptive and predictive
machine learning systems, model generators, constraint solvers,
satisfiability solvers, theorem provers and computer algebra
systems. Many of the applications described below make use of a
combination of reasoning systems. We have also looked at some more
generic ways to combine AI systems. The following papers describe some
of our projects in this area:
Pure mathematics is a unique domain for AI research, as mathematical
enquiry involves many diverse forms of reasoning, hence we can look at
the question of theory formation in pure mathematics and study
computational systems which combine different AI techniques. In
addition, the data in pure mathematics is usually error free, hence we
can concentrate on pure forms of reasoning without (usually) requiring
statistical interpretations. On numerous occasions, we have shown that
HR and other systems can make mathematical discoveries of genuine
value in graph theory, number theory and various algebraic domains of
pure mathematics. In addition, by combining HR with multiple other AI
systems, we have achieved new partial classifications of algebraic
domains, which were previously beyond any computer (or human). The
following papers describe some of our projects in this area:
1. The Development of Automated Theory Formation
2. More Sophisticated Mathematical Theory Formation Models
3. The Combination of Reasoning Systems
4. Applications to Discovery Tasks in Pure Mathematics