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Abstract

Knowledge base theory stimulates numerous applications of computer algebra and symbolic computations. The paper is aimed to explain how the general ideas of Galois theory work for knowledge bases and help to determine the criterion of knowledge bases informational equivalence. This criterion reduces the problem of informational equivalence of knowledge bases to the conjugacy problem for groups. We give a survey of recent results and outline prospective problems.

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References

  1. Aho, A.V., Sagiv, Y., Ullman, J.D.: Equivalences among relational expressions. SIAM J. Comput. 8(2), 218–246 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atzeni, P., Aussiello, G., Batini, C., Moscarini, M.: Inclusion and equivalence between relational database schemes. Theor. Comput. Sci. 19, 267–285 (1982)

    Article  MATH  Google Scholar 

  3. Bancillon, F.: On the completeness of query language for relational database. Lect. Notes Comput. Sci. 64, 112–123 (1978)

    Article  Google Scholar 

  4. Baik, K.H., Miller, L.L.: Topological approach for testing equivalence in heterogenous relational databases. Comput. J. 33(1), 2–10 (1990)

    Article  MathSciNet  Google Scholar 

  5. Beeri, C., Mendelzon A., Sagiv Y., Ullman J.: Equivalence of relational database schemes. In: STOC ’79 Proc. of the 11-th Annual ACM Symp. on Theory of Computing, pp. 319–329. ACM (1979)

  6. Beniaminov, E. M.: Algebraic Invariants of Database Schemes. In: Proceedings of the 2-d Int. Workshop on Advances in Databases and Information Systems (ADBIS’95), vol. 1, pp. 259–263. Moscow (1995)

  7. Beniaminov, E.M.: Algebraic Methods in Database Theory and Representation Of Knowledge. Russian, Nauchnyj Mir, Moscow (2003)

    Google Scholar 

  8. Cannon, J.J., Holt, D.F.: Automorphism group computation and isomorphism testing in finite groups. J. Symbol. Comput. 35(3), 241–267 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Clarendon, Oxford (1985)

  10. Hahn, T. (ed.): International Tables for Crystallography, Volume A: Space Group Symmetry, A, 5th edn, Berlin (2002)

  11. Halmos, P.R.: Algebraic Logic. New York (1969)

  12. Henkin, L., Monk, J.D., Tarski, A.: Cylindric Algebras. North-Holland Publ (1995)

  13. Heuer, A.: Equivalent schemes in semantic, nested relational, and relational database models. Lect. Notes Comput. Sci. 364, 237–253 (1989)

    Article  Google Scholar 

  14. Hulek, K.: Elementary algebraic geometry. Stud. Math. Lib. 20, AMS (2003)

  15. Harju, T., Karhumaki, J.: The equivalence problem of multitape finite automata. Theor. Comput. Sci. 78, 347–355 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Knyazhansky, M.: Categorical model of knowledge base and its applications to knowledge bases equivalence verification. Ph.D. thesis, Bar Ilan University (2010)

  17. Knyazhansky, M., Plotkin, T.: Automorphic equivalence of multi-models recognition. Armen. J. Math. 1(2), 10–24 (2008)

    MathSciNet  Google Scholar 

  18. Knyazhansky, M., Plotkin., T.: Knowledge bases and automorphic equivalence of multi-models versus linear spaces and graphs. Discuss. Math. Gen. Algebra Appl. 29(2) 203–213 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Krasner, M.: Généralisation abstraite de la théorie de Galois. Colloq. Int. CNRS (Algèbre et théorie des nombres) 24, 163–168 (1949)

    MathSciNet  Google Scholar 

  20. Kuros, A.G.: The Theory of Groups. Chelsea, New York (1960)

    Google Scholar 

  21. Lover, R.: Elementary Logic for Software Development. Springer (2008)

  22. MacLane, S.: Categories for the Working Mathematician, 2nd edn. Springer (1998)

  23. Malcev, A.I.: Algebraic Systems. Springer (1973)

  24. Manin, Yu.I.: A course in mathematical logic for mathematicians, 2nd edn. In: Zilber, B., et al. (eds) Graduate Texts in Mathematics, vol. 53, xviii+384 pp. Springer, New York (2010)

    Google Scholar 

  25. Mendelson, E.: Introduction to Mathematical Logic. Chapman & Hall, London, UK (1997)

    MATH  Google Scholar 

  26. O’Brien, E.A.: Isomorphism testing for p-groups. J. Symbol. Comput. 16(3), 305–320 (1993)

    Article  MathSciNet  Google Scholar 

  27. Plotkin, B.: Universal Algebra, Algebraic Logic and Databases. Kluwer, Boston, MA (1993)

    Google Scholar 

  28. Plotkin, B.: Seven Lectures in Universal Algebraic Geometry, vol. 87. Arxiv math GM/0204245 (2002, preprint)

  29. Plotkin, B.I.: Model theoretical and deductive approach to databases from the point of view of algebraic logic and geometry. In: Advances of Databases and Information Systems, pp. 161–170. Springer (1996)

  30. Plotkin, B.I.: Algebra, categories and databases. In: Handbook of Algebra, vol. 2, pp. 81–148. Elsevier, Springer (1999)

    Google Scholar 

  31. Plotkin, B., Plotkin, T.: Geometrical aspect of databases and knowledge bases. Algebra Univers. 46, 131–161 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Plotkin, B., Plotkin, T.: An algebraic approach to knowledge bases equivalence. Acta Appl. Math. 89, 109–134 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Plotkin, B., Plotkin T.: Categories of elementary sets over algebras and categories of elementary algebraic knowledge. Lect. Notes Comput. Sci. 4800, 555–570 (2008)

    Article  MathSciNet  Google Scholar 

  34. Plotkin, T.: Relational databases equivalence problem. In: Advances of Databases and Information Systems, pp. 391–404. Springer (1996)

  35. Rissanen, J.: On the equivalence of database schemes. In: Proc. ACM. Symp. Princ. Of Database Systems, issue 1, pp. 22–26 (1982)

  36. Roney-Dougal, C.M.: Conjugacy of subgroups of the general linear group. Exp. Math. 13(2), 151–163 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sims, C.C.: Computation with Finitely Presented Groups, p. xiii+604. Cambridge University Press, Cambridge, UK (1994)

    Book  MATH  Google Scholar 

  38. Tarski, A.: Logic, Semantics, Metamathematics, 2nd edn (first edition 1956). Haskett Publ. Company (1983)

  39. Ullman, J.D.: Principles of Database and Knowledge-Base Systems. Computer Science, Rockville, MD (1988)

    Google Scholar 

  40. Volklein, H.: Groups as Galois Groups: an Introduction. Cambridge University Press, Cambridge, UK (1996)

    Book  Google Scholar 

Download references

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Correspondence to Marina Knyazhansky.

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Plotkin, T., Knyazhansky, M. Symmetries of knowledge bases. Ann Math Artif Intell 64, 369–383 (2012). https://doi.org/10.1007/s10472-012-9296-8

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