Simon Colton

... described in § 2. To begin with, TM checks whether the conjecture is true, ie, A⇒ C. It does this by ... a, b, c ((a∗ b= a∗ c)⇒ b= c)). We took out the identity and inverse axioms to generate a non ... x P (x)⇒ Q (x), and a con-cept C (x) which covers all the supporting examples and no ...

We believe that AI programs written for discovery tasks will need to simultaneously employ a variety of reasoning techniques such as induction, abduction, deduction, calculation and invention. We describe the HR system which performs a novel ILP routine called automated theory formation. This combines inductive and deductive reasoning to form clausal theories consisting of classification rules and association rules. HR generates definitions using a set of production rules, interprets the definitions as classification rules, then uses the success sets of the definitions to induce hypotheses from which it extracts association rules. It uses third party theorem provers and model generators to check whether the association rules are entailed by a set of user supplied axioms. HR has been applied to a range of predictive, descriptive and subgroup discovery tasks in domains of pure mathematics. We describe these applications and how they have led to some interesting mathematical discoveries. Our main aim here is to provide a thorough overview of automated theory formation. A secondary aim is to promote mathematics as a worthy domain for ILP applications, and we provide pointers to mathematical datasets.

The HR program by Colton et al. (1999) performs theory formation in mathematics by exploring a space of mathematical concepts. By enabling HR to determine when it has found a particular concept, and by adding a forward looking mechanism, we have applied HR to the ...

Classifying finite algebraic structures has been a major motivation behind much research in pure mathematics. Automated techniques have aided in this process, but this has largely been at a quantitative level. In contrast, we present a qualitative approach which produces verified theorems, which classify algebras of a particular type and size into isomorphism classes. We describe both a semi-automated and a fully automated bootstrapping approach to building and verifying classification theorems. In the latter case, we have implemented a procedure which takes the axioms of the algebra and produces a decision tree embedding a fully verified classification theorem. This has been achieved by the integration (and improvement) of a number of automated reasoning techniques: we use the Mace model generator, the HR and C4.5 machine learning systems, the Spass theorem prover, and the Gap computer algebra system to reduce the complexity of the problems given to Spass. We demonstrate the power of this approach by classifying loops, groups, monoids and quasigroups of various sizes.

Adding constraints to a basic CSP model can significantly reduce search,e.g. for Golomb rulers [6]. The generation process is usually performed by hand, although some recent work has focused on automatically generating symmetry breaking constraints [4]and (less so)on generating implied constraints [5]. We describe an approach to generating implied,symmetry breaking and specialisation constraints and apply this technique to quasigroup construction [10].

An ability to reason at a meta-level is widely regarded as an important aspect of human creativity which is often missing from creative computer programs. We discuss recent experiments with the HR theory formation program where it formed meta-theories about previously formed theories. We report how HR re-invented aspects of how it forms theories and reflected on the nature of

The invention of suitable concepts to characterise mathematical structures is one of the most challenging tasks for both human mathematicians and automated theorem provers alike. We present an approach where automatic concept formation is used to guide non-isomorphism proofs in the residue class domain. The main idea behind the proof is to automatically identify discriminants for two given structures to show that they are not isomorphic. Suitable discriminants are generated by a theory formation system; the overall proof is constructed by a proof planner with the additional support of traditional automated theorem provers and a computer algebra system.

We extend our previous study of the automatic construction of isomorphic classification theorems for algebraic domains by considering the isotopy equivalence relation, which is of more importance than isomorphism in certain domains. This extension was not straightforward, and we had to solve two major technical problems, namely generating and verifying isotopy invariants. Concentrating on the domain of loop theory, we have developed three novel techniques for generating isotopic invariants, by using the notion of universal identities and by using constructions based on substructures. In addition, given the complexity of the theorems which verify that a conjunction of the invariants form an isotopy class, we have developed ways of simplifying the problem of proving these theorems. Our techniques employ an intricate interplay of computer algebra, model generation, theorem proving and satisfiability solving methods. To demonstrate the power of the approach, we generate an isotopic classification theorem for loops of size 6, which extends the previously known result that there are 22. This result was previously beyond the capabilities of automated reasoning techniques.