Automatic Generation of Classification Theorems for Finite Algebras
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.
KeywordsDecision Tree Algebraic Structure Isomorphism Class Automatic Generation Pure Mathematic
Unable to display preview. Download preview PDF.
- 1.Colton, S.: Automated Theory Formation in Pure Mathematics. Springer, Heidelberg (2002)Google Scholar
- 2.Colton, S.: The HR program for theorem generation. In: Voronkov Google Scholar
- 3.The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.3 (2002), http://www.gap-system.org
- 6.Kronecker, L.: Auseinandersetzung einiger Eigenschaften der Klassenanzahl idealer komplexer Zahlen. Monatsbericht der Berliner Akademie, pp. 881–889 (1870)Google Scholar
- 9.McCune, W.: Mace4 Reference Manual and Guide. Argonne National Laboratory, ANL/MCS-TM-264 (2003)Google Scholar
- 10.McKay, B., Meinert, A., Myrvold, W.: Counting small latin squares. In: European Women in Mathematics Int. Workshop on Groups and Graphs, pp. 67–72 (2002)Google Scholar
- 12.Quinlan, R.: C4.5: Programs for Machine Learning. Morgan Kaufmann, San Francisco (1993)Google Scholar
- 16.Weidenbach, C., Brahm, U., Hillenbrand, T., Keen, E., Theobald, C., Topic, D.: SPASS version 2.0. In: Voronkov , pp. 275–279Google Scholar