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On Automating the Calculus of Relations

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Automated Reasoning (IJCAR 2008)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5195))

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Abstract

Relation algebras provide abstract equational axioms for the calculus of binary relations. They name an established area of mathematics and have found numerous applications in computing. We prove more than hundred theorems of relation algebras with off-the-shelf automated theorem provers. They form a basic calculus from which more advanced applications can be explored. We also present two automation experiments from the formal methods literature. Our results further demonstrate the feasibility of automated deduction with complex algebraic structures. They also open a new perspective for automated deduction in relational formal methods.

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References

  1. http://www.dcs.shef.ac.uk/~georg/ka

  2. Abrial, J.-R.: The B-Book. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  3. Bachmair, L., Ganzinger, H.: Ordered chaining calculi for first-order theories of transitive relations. J. ACM 45(6), 1007–1049 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berghammer, R., Neumann, F.: An OBDD-based computer algebra system for relations. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 40–51. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  5. Berghammer, R., Zierer, H.: Relational algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science 43, 123–147 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bird, R., de Moor, O.: Algebra of Programming. Prentice-Hall, Englewood Cliffs (1996)

    Google Scholar 

  7. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  8. Dang, H.-H., Höfner, P.: First-order theorem prover evaluation w.r.t. relation- and Kleene algebra. In: R. Berghammer, B. Möller, and G. Struth, editors, RelMiCS10/AKA5 - PhD Programme, Technical Report 2008-04, University of Augsburg, pp. 48–52 (2008)

    Google Scholar 

  9. de Roever, W.-P., Engelhardt, K.: Data Refinement: Model-Oriented Proof Methods and their Comparison. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  10. Formisano, A., Omodeo, E.G., Orłowska, E.: A Prolog tool for relational translations of modal logics: A front-end for relational proof systems. In: Beckert, B. (ed.) TABLEAUX 2005. LNCS (LNAI), vol. 3702, pp. 1–11. Springer, Heidelberg (2005)

    Google Scholar 

  11. Henkin, L., Monk, D.J., Tarski, A.: Cylindric Algebras, Part I. North-Holland, Amsterdam (1971)

    Google Scholar 

  12. Huntington, E.V.: Boolean algebra. A correction. Trans. AMS 35, 557–558 (1933)

    Article  MATH  MathSciNet  Google Scholar 

  13. Huntington, E.V.: New sets of independent postulates for the algebra of logic. Trans. AMS 35, 274–304 (1933)

    Article  MATH  MathSciNet  Google Scholar 

  14. Jackson, D.: Software Abstractions: Logic, Language, and Analysis. MIT Press, Cambridge (2006)

    Google Scholar 

  15. Jónsson, B., Tarski, A.: Boolean algebras with operators I. American J. Mathematics 74, 891–939 (1951)

    Article  Google Scholar 

  16. Kahl, W.: Calculational relation-algebraic proofs in Isabelle/Isar. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS 2003. LNCS, vol. 3051, pp. 179–190. Springer, Heidelberg (2004)

    Google Scholar 

  17. MacCaull, W., Orlowska, E.: Correspondence results for relational proof systems with application to the Lambek calculus. Studia Logica 71(3), 389–414 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Maddux, R.: A sequent calculus for relation algebras. Annals of Pure and Applied Logic 25, 73–101 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  19. Maddux, R.: Relation Algebras. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  20. Maddux, R.D.: Relation-algebraic semantics. Theoretical Computer Science 160(1&2), 1–85 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. McCune, W.: Prover9 and Mace4, http://www.cs.unm.edu/~mccune/prover9

  22. McCune, W.: Solution of the Robbins problem. J. Automated Reasoning 19(3), 263–276 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  23. Schmidt, G., Ströhlein, T.: Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer, Heidelberg (1993)

    MATH  Google Scholar 

  24. Sinz, C.: System description: ARA — An automated theorem prover for relation algebras. In: McAllester, D. (ed.) CADE 2000. LNCS, vol. 1831, pp. 177–182. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  25. Spivey, J.M.: Understanding Z. Cambridge University Press, Cambridge (1988)

    MATH  Google Scholar 

  26. Sutcliffe, G., Suttner, C.: The TPTP problem library: CNF release v1.2.1. J. Automated Reasoning 21(2), 177–203 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  27. Sutcliffe, G., Suttner, C.: Evaluating general purpose automated theorem proving systems. Artificial Intelligence 131(1–2), 39–54 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  28. Tarski, A.: On the calculus of relations. Journal of Symbolic Logic 6(3), 73–89 (1941)

    Article  MATH  MathSciNet  Google Scholar 

  29. Tarski, A., Givant, S.R.: A Formalization of Set Theory Without Variables. American Mathematical Society (1987)

    Google Scholar 

  30. von Oheimb, D., Gritzner, T.F.: Rall: Machine-supported proofs for relation algebra. In: McCune, W. (ed.) CADE 1997. LNCS, vol. 1249, pp. 380–394. Springer, Heidelberg (1997)

    Google Scholar 

  31. Weidenbach, C., Schmidt, R.A., Hillenbrand, T., Rusev, R., Topic, D.: System description: SPASS version 3.0. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 514–520. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Institut für Informatik, Universität Augsburg, Germany

    Peter Höfner

  2. Department of Computer Science, University of Sheffield, United Kingdom

    Georg Struth

Authors
  1. Peter Höfner
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  2. Georg Struth
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Editor information

Alessandro Armando Peter Baumgartner Gilles Dowek

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Höfner, P., Struth, G. (2008). On Automating the Calculus of Relations. In: Armando, A., Baumgartner, P., Dowek, G. (eds) Automated Reasoning. IJCAR 2008. Lecture Notes in Computer Science(), vol 5195. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71070-7_5

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  • DOI: https://doi.org/10.1007/978-3-540-71070-7_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-71069-1

  • Online ISBN: 978-3-540-71070-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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