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A Hierarchy of Algebras for Boolean Subsets

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Relational and Algebraic Methods in Computer Science (RAMiCS 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12062))

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Abstract

We present a collection of axiom systems for the construction of Boolean subalgebras of larger overall algebras. The subalgebras are defined as the range of a complement-like operation on a semilattice. This technique has been used, for example, with the antidomain operation, dynamic negation and Stone algebras. We present a common ground for these constructions based on a new equational axiomatisation of Boolean algebras. All results are formally proved in Isabelle/HOL.

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References

  1. Balbes, R., Horn, A.: Stone lattices. Duke Math. J. 37(3), 537–545 (1970)

    Article  MathSciNet  Google Scholar 

  2. Birkhoff, G.: Lattice Theory, Colloquium Publications, vol. XXV, 3rd edn. American Mathematical Society, Providence (1967)

    Google Scholar 

  3. Blanchette, J.C., Böhme, S., Paulson, L.C.: Extending Sledgehammer with SMT solvers. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 116–130. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22438-6_11

    Chapter  Google Scholar 

  4. Blanchette, J.C., Nipkow, T.: Nitpick: a counterexample generator for higher-order logic based on a relational model finder. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 131–146. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14052-5_11

    Chapter  Google Scholar 

  5. Byrne, L.: Two brief formulations of Boolean algebra. Bull. Am. Math. Soc. 52(4), 269–272 (1946)

    Article  MathSciNet  Google Scholar 

  6. Dang, H.H., Höfner, P.: First-order theorem prover evaluation w.r.t. relation- and Kleene algebra. In: Berghammer, R., Möller, B., Struth, G. (eds.) PhD Programme at RelMiCS10/AKA5, pp. 48–52. Report 2008-04, Institut für Informatik, Universität Augsburg (2008)

    Google Scholar 

  7. Desharnais, J., Jipsen, P., Struth, G.: Domain and antidomain semigroups. In: Berghammer, R., Jaoua, A.M., Möller, B. (eds.) RelMiCS/AKA 2009. LNCS, vol. 5827, pp. 73–87. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04639-1_6

    Chapter  MATH  Google Scholar 

  8. Desharnais, J., Möller, B.: Fuzzifying modal algebra. In: Höfner, P., Jipsen, P., Kahl, W., Müller, M.E. (eds.) RAMICS 2014. LNCS, vol. 8428, pp. 395–411. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-06251-8_24

    Chapter  Google Scholar 

  9. Desharnais, J., Möller, B., Struth, G.: Kleene algebra with domain. ACM Trans. Comput. Logic 7(4), 798–833 (2006)

    Article  MathSciNet  Google Scholar 

  10. Desharnais, J., Struth, G.: Domain axioms for a family of near-semirings. In: Meseguer, J., Roşu, G. (eds.) AMAST 2008. LNCS, vol. 5140, pp. 330–345. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-79980-1_25

    Chapter  Google Scholar 

  11. Desharnais, J., Struth, G.: Modal semirings revisited. In: Audebaud, P., Paulin-Mohring, C. (eds.) MPC 2008. LNCS, vol. 5133, pp. 360–387. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70594-9_19

    Chapter  MATH  Google Scholar 

  12. Desharnais, J., Struth, G.: Internal axioms for domain semirings. Sci. Comput. Program. 76(3), 181–203 (2011)

    Article  MathSciNet  Google Scholar 

  13. Frink, O.: Pseudo-complements in semi-lattices. Duke Math. J. 29(4), 505–514 (1962)

    Article  MathSciNet  Google Scholar 

  14. Frink Jr., O.: Representations of Boolean algebras. Bull. Am. Math. Soc. 47(10), 755–756 (1941)

    Article  MathSciNet  Google Scholar 

  15. Givant, S., Halmos, P.: Introduction to Boolean Algebras. Springer, New York (2009). https://doi.org/10.1007/978-0-387-68436-9

    Book  MATH  Google Scholar 

  16. Grätzer, G.: Lattice Theory: First Concepts and Distributive Lattices. W. H. Freeman and Co., San Francisco (1971)

    MATH  Google Scholar 

  17. Guttmann, W.: Algebras for iteration and infinite computations. Acta Inf. 49(5), 343–359 (2012)

    Article  MathSciNet  Google Scholar 

  18. Guttmann, W.: Verifying minimum spanning tree algorithms with Stone relation algebras. J. Log. Algebraic Methods Program. 101, 132–150 (2018)

    Article  MathSciNet  Google Scholar 

  19. Guttmann, W., Struth, G., Weber, T.: Automating algebraic methods in Isabelle. In: Qin, S., Qiu, Z. (eds.) ICFEM 2011. LNCS, vol. 6991, pp. 617–632. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-24559-6_41

    Chapter  Google Scholar 

  20. Haftmann, F., Wenzel, M.: Constructive type classes in Isabelle. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, pp. 160–174. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74464-1_11

    Chapter  Google Scholar 

  21. Hollenberg, M.: An equational axiomatization of dynamic negation and relational composition. J. Logic Lang. Inform. 6(4), 381–401 (1997)

    Article  MathSciNet  Google Scholar 

  22. Huntington, E.V.: Boolean algebra. A correction. Trans. Am. Math. Soc. 35(2), 557–558 (1933)

    MathSciNet  MATH  Google Scholar 

  23. Jackson, M., Stokes, T.: Semilattice pseudo-complements on semigroups. Commun. Algebra 32(8), 2895–2918 (2004)

    Article  MathSciNet  Google Scholar 

  24. Kozen, D.: Kleene algebra with tests. ACM Trans. Program. Lang. Syst. 19(3), 427–443 (1997)

    Article  MathSciNet  Google Scholar 

  25. Maddux, R.D.: Relation-algebraic semantics. Theor. Comput. Sci. 160(1–2), 1–85 (1996)

    Article  MathSciNet  Google Scholar 

  26. McCune, W.: Prover9 and Mace4 (2005–2010). https://www.cs.unm.edu/~mccune/prover9/. Accessed 16 Jan 2020

  27. McCune, W., Veroff, R., Fitelson, B., Harris, K., Feist, A., Wos, L.: Short single axioms for Boolean algebra. J. Autom. Reason. 29(1), 1–16 (2002)

    Article  MathSciNet  Google Scholar 

  28. Meredith, C.A., Prior, A.N.: Equational logic. Notre Dame J. Formal Logic 9(3), 212–226 (1968)

    Article  MathSciNet  Google Scholar 

  29. Möller, B., Desharnais, J.: Basics of modal semirings and of Kleene/omega algebras. Report 2019-03, Institut für Informatik, Universität Augsburg (2019)

    Google Scholar 

  30. Möller, B., Struth, G.: Algebras of modal operators and partial correctness. Theor. Comput. Sci. 351(2), 221–239 (2006)

    Article  MathSciNet  Google Scholar 

  31. Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45949-9

  32. Paulson, L.C., Blanchette, J.C.: Three years of experience with Sledgehammer, a practical link between automatic and interactive theorem provers. In: Sutcliffe, G., Ternovska, E., Schulz, S. (eds.) Proceedings of the 8th International Workshop on the Implementation of Logics, pp. 3–13 (2010)

    Google Scholar 

  33. Wampler-Doty, M.: A complete proof of the Robbins conjecture. Archive of Formal Proofs (2016, first version 2010)

    Google Scholar 

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Acknowledgement

We thank Andreas Zelend and the anonymous referees for their helpful comments.

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Authors and Affiliations

  1. Department of Computer Science and Software Engineering, University of Canterbury, Christchurch, New Zealand

    Walter Guttmann

  2. Institut für Informatik, Universität Augsburg, Augsburg, Germany

    Bernhard Möller

Authors
  1. Walter Guttmann
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  2. Bernhard Möller
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Corresponding author

Correspondence to Walter Guttmann .

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Editors and Affiliations

  1. Laboratoire d’informatique (LIX), École Polytechnique, Palaiseau, France

    Uli Fahrenberg

  2. Faculty of Mathematics, Chapman University, Orange, CA, USA

    Peter Jipsen

  3. Department of Computer Science, Brock University, St. Catharines, ON, Canada

    Michael Winter

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Guttmann, W., Möller, B. (2020). A Hierarchy of Algebras for Boolean Subsets. In: Fahrenberg, U., Jipsen, P., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2020. Lecture Notes in Computer Science(), vol 12062. Springer, Cham. https://doi.org/10.1007/978-3-030-43520-2_10

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  • DOI: https://doi.org/10.1007/978-3-030-43520-2_10

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  • Online ISBN: 978-3-030-43520-2

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