Abstract
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alur, R., Peled, D.A. (eds.): CAV 2004. LNCS, vol. 3114. Springer, Heidelberg (2004)
Barrett, C., Berezin, S.: CVC Lite: A New Implementation of the Cooperating Validity Checker. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 515–518. Springer, Heidelberg (2004)
Colton, S.: Automated Theory Formation in Pure Mathematics. Springer, Heidelberg (2002)
Colton, S., Meier, A., Sorge, V., McCasland, R.: Automatic Generation of Classification Theorems for Finite Algebras. In: Basin, D., Rusinowitch, M. (eds.) IJCAR 2004. LNCS (LNAI), vol. 3097, pp. 400–414. Springer, Heidelberg (2004)
Colton, S., Muggleton, S.: Mathematical Applications of Inductive Logic Programming. Machine Learning Journal (forthcoming, 2006)
Falconer, E.: Isotopy Invariants in Quasigroups. Transactions of the American Mathematical Society 151(2), 511–526 (1970)
Ganzinger, H., Hagen, G., Nieuwenhuis, R., Oliveras, A., Tinelli, C.: DPLL(T): Fast Decision Procedures. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 175–188. Springer, Heidelberg (2004)
GAP Group: Groups, Algorithms, and Programming, v4.7 (2002), gap-system.org
McCune, W.: Single axioms for groups and Abelian groups with various operations. Journal of Automated Reasoning 10(1), 1–13 (1993)
McCune, W.: Mace4 Reference Manual and Guide. Argonne Nat. Laboratory (2003)
McKay, B., Meynert, A., Myrvold, W.: Counting Small Latin Squares. In: Europ. Women in Mathematics Int. Workshop on Groups and Graphs, pp. 67–72 (2002)
Meier, A., Sorge, V.: Applying SAT Solving in Classification of Finite Algebras. JAR 34(2), 34 (2005)
Moskewicz, M., Madigan, C., Zhao, Y., Zhang, L., Malik, S.: chaff: Engineering an efficient SAT Solver. In: PROC of DAC-2001, pp. 530–535 (2001)
Pflugfelder, H.: Quasigroups and Loops: Introduction, volume 7 of Sigma Series in Pure Mathematics. Heldermann Verlag, Berlin Germany (1990)
Riazanov, A., Voronkov, A.: Vampire 1.1. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 376–380. Springer, Heidelberg (2001)
Schulz, S.: E: A Brainiac theorem prover. J. of AI Comm. 15(2–3), 111–126 (2002)
Schwenk, J.: A classification of Abelian quasigroups. Rendiconti di Matematica, Serie VII 15, 161–172 (1995)
Slaney, J.: FINDER, Notes and Guide. Center for Information Science Research, Australian National University (1995)
Slaney, J., Fujita, M., Stickel, M.: Automated reasoning and exhaustive search: Quasigroup existence problems. Computers and Mathematics with Applications 29, 115–132 (1995)
Sutcliffe, G.: The IJCAR-2004 Automated Theorem Proving Competition. AI Communications 18(1), 33–40 (2005)
Weidenbach, C., Brahm, U., Hillenbrand, T., Keen, E., Theobald, C., Topic, D.: SPASS Version 2. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 275–279. Springer, Heidelberg (2002)
Zhang, J., Zhang, H.: SEM User’s Guide. Dep.of Comp. Science, UIowa (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sorge, V., Meier, A., McCasland, R., Colton, S. (2006). Automatic Construction and Verification of Isotopy Invariants. In: Furbach, U., Shankar, N. (eds) Automated Reasoning. IJCAR 2006. Lecture Notes in Computer Science(), vol 4130. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11814771_5
Download citation
DOI: https://doi.org/10.1007/11814771_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37187-8
Online ISBN: 978-3-540-37188-5
eBook Packages: Computer ScienceComputer Science (R0)