Studying Algebraic Structures Using Prover9 and Mace4

  • Rob Arthan
  • Paulo OlivaEmail author
Part of the Mathematics Education in the Digital Era book series (MEDE, volume 14)


In this chapter we present a case study, drawn from our research work, on the application of a fully automated theorem prover together with an automatic counter-example generator in the investigation of a class of algebraic structures. We will see that these tools, when combined with human insight and traditional algebraic methods, help us to explore the problem space quickly and effectively. The counter-example generator rapidly rules out many false conjectures, while the theorem prover is often much more efficient than a human being at verifying algebraic identities. The specific tools in our case study are Prover9 and Mace4; the algebraic structures are generalisations of Heyting algebras known as hoops. We will see how this approach helped us to discover new theorems and to find new or improved proofs of known results. We also make some suggestions for how one might deploy these tools to supplement a more conventional approach to teaching algebra.

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  1. Blok, W. J., & Ferreirim, I. M. A. (2000). On the structure of hoops. Algebra Universalis, 43(2–3), 233–257.CrossRefGoogle Scholar
  2. Bosbach, B. (1969). Komplementäre Halbgruppen. Axiomatik und Arithmetik. Fundamenta Mathematicae, 64, 257–287.CrossRefGoogle Scholar
  3. Büchi, J. R., & Owens, T. M. (1974). Complemented monoids and hoops. Unpublished manuscript.Google Scholar
  4. Burris, S. (1997). An Anthropomorphized Version of McCune’s machine proof that Robbins Algebras are Boolean algebras. Private communication.Google Scholar
  5. Dahn, B. I. (1998). Robbins algebras are Boolean: A revision of McCune’s computer-generated solution of Robbins problem. Journal of Algebra, 208(2), 526–532.CrossRefGoogle Scholar
  6. De Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17–24.Google Scholar
  7. Łukasiewicz, J., & Tarski, A. (1930). Untersuchungen über den Aussagenkalkül. C. R. Soc. Sc. Varsovie, 23(1930), 30–50.Google Scholar
  8. McCune, W. (2005–2010). Prover9 and Mace4.
  9. McCune, W. (1997). Solution of the Robbins problem. Journal of Automated Reasoning, 19(3), 263–276.CrossRefGoogle Scholar
  10. Moggi, E. (1989). Computational lambda-calculus and monads. In Symposium of Logic in Computer Science, California, June 1989. IEEE.Google Scholar
  11. Winker, S. (1992). Absorption and idempotency criteria for a problem in near-Boolean algebras. Journal of Algebra, 153(2), 414–423.CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Electronic Engineering and Computer ScienceQueen Mary University of LondonLondonUK

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