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Special Sessions in Applications of Computer Algebra

ACA 2015: Applications of Computer Algebra pp 149–170Cite as

Algebraic Modelling of Covering Arrays

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 198))

Abstract

We introduce a novel technique to model and compute binary covering arrays, discrete combinatorial structures, based on a tuple-level modelling and using methods arising from linear algebra, commutative algebra and symbolic computation. Concrete instances of covering arrays for given parameters will arise as points in a generated variety of a system of multivariate polynomial equations with Gröbner Bases playing an important role.

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References

  1. Avila-George, H., Torres-Jimenez, J., Hernández, V.: New bounds for ternary covering arrays using a parallel simulated annealing. Math. Probl. Eng. (2012)

    Google Scholar 

  2. Becker, T., Weispfenning, V.: Gröbner bases. In: A Computational Approach to Commutative Algebra. Graduate Studies in Mathematics, vol. 141. Springer, New York (1993)

    Google Scholar 

  3. Borges-Quintana, M., Borges-Trenard, M.A., Fitzpatrick, P., Martínez-Moro, E.: Gröbner bases and combinatorics for binary codes. Appl. Algebra Eng. Commun. Comput. 19(5), 393–411 (2008)

    Article  MATH  Google Scholar 

  4. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3-4), 235–265 (1997). Computational algebra and number theory (London, 1993)

    Google Scholar 

  5. Bracho-Rios, J., Torres-Jimenez, J., Rodriguez-Tello, E.: A new backtracking algorithm for constructing binary covering arrays of variable strength. In: MICAI 2009: Advances in Artificial Intelligence, pp. 397–407. Springer, New York (2009)

    Google Scholar 

  6. Bryce, R.C., Colbourn, C.J.: The density algorithm for pairwise interaction testing. Softw. Test. Verif. Reliab. 17(3), 159–182 (2007)

    Article  Google Scholar 

  7. Buchberger, B.: Bruno Buchberger’s phd thesis 1965: an algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. J. Symb. Comput. 41, 475–511 (2006). doi:10.1016/j.jsc.2005.09.007

  8. Cohen, D.M., Dalal, S.R., Fredman, M.L., Patton, G.C.: The AETG system: an approach to testing based on combinatorial design. IEEE Trans. Softw. Eng. 23(7), 437–444 (1997)

    Article  Google Scholar 

  9. Colbourn, C.: Table for CAN(2,k,2) for k up to 20000. http://www.public.asu.edu/~ccolbou/src/tabby/2-2-ca.html. Accessed 31 Dec 2015

  10. Colbourn, C.J.: Combinatorial aspects of covering arrays. Le Mathematiche LIX(I–II), pp. 125–172 (2004)

    Google Scholar 

  11. Colbourn, C.J.: Covering arrays from cyclotomy. Des. Codes Cryptogr. 55(2–3), 201–219 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Colbourn, C.J., Dinitz, J.H.: Handbook of Combinatorial Designs. CRC Press, Boca Raton (2006)

    Book  MATH  Google Scholar 

  13. Faugere, J.C.: A new efficient algorithm for computing Gröbner bases (f 4). J. Pure Appl. Algebr. 139(1), 61–88 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Faugère, J.C.: A new efficient algorithm for computing grÖbner bases without reduction to zero (f5). In: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation (ISSAC ’02), pp. 75–83. ACM, New York (2002). doi:10.1145/780506.780516

  15. Gonzalez-Hernandez, L., Rangel-Valdez, N., Torres-Jimenez, J.: Construction of mixed covering arrays of strengths 2 through 6 using a tabu search approach. Discret. Math. Algorithm Appl. 4(03), 1250,033 (2012)

    Google Scholar 

  16. Hartman, A., Raskin, L.: Problems and algorithms for covering arrays. Discret. Math. 284(1), 149–156 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hnich, B., Prestwich, S.D., Selensky, E., Smith, B.M.: Constraint models for the covering test problem. Constraints 11(2–3), 199–219 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. IPOG-F: CA(2,10,2). http://math.nist.gov/coveringarrays/ipof/cas/t=2/v=2/ca.2.2^10.txt.zip. Accessed 31 Dec 2015

  19. IPOG-F: CA(2,16,2). http://math.nist.gov/coveringarrays/ipof/cas/t=2/v=2/ca.2.2^16.txt.zip. Accessed 31 Dec 2015

  20. IPOG-F: CA(2,17,2). http://math.nist.gov/coveringarrays/ipof/cas/t=2/v=2/ca.2.2^17.txt.zip. Accessed 31 Dec 2015

  21. IPOG-F: CA(2,9,2). http://math.nist.gov/coveringarrays/ipof/cas/t=2/v=2/ca.2.2^9.txt.zip. Accessed 31 Dec 2015

  22. ITL, N.: Covering array tables. http://math.nist.gov/coveringarrays/. Accessed 31 Dec 2015

  23. Kleitman, D.J., Spencer, J.: Families of k-independent sets. Discret. Math. 6(3), 255–262 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kotsireas, I., Koukouvinos, C., Seberry, J.: Hadamard ideals and hadamard matrices with circulant core. J. Combin. Math. Combin. Comput. 57, 47–63 (2006)

    MathSciNet  MATH  Google Scholar 

  25. Kotsireas, I., Koukouvinos, C., Seberry, J.: Hadamard ideals and Hadamard matrices with two circulant cores. Eur. J. Comb. 27(5), 658–668 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kotsireas, I.S., Kutsia, T., Simos, D.E.: Constructing orthogonal designs in powers of two: Gröbner bases meet equational unification. In: 26th International Conference on Rewriting Techniques and Applications (RTA), June 29 to July 1, 2015, Warsaw, pp. 241–256 (2015)

    Google Scholar 

  27. Koukouvinos, C., Simos, D.E., Zafeirakopoulos, Z.: An algebraic framework for extending orthogonal designs. In: ISSAC ’11: Abstracts of Poster Presentations of the 36th International Symposium on Symbolic and Algebraic Computation, ACM Communications in Computer Algebra, vol. 45, pp. 123–124 (2011)

    Google Scholar 

  28. Koukouvinos, C., Simos, D.E., Zafeirakopoulos, Z.: A Gröbner bases method for complementary sequences. In: Proceedings of Applications of Computer Algebra (ACA), p. 255. Málaga (2013)

    Google Scholar 

  29. Kuhn, R., Kacker, R., Lei, Y., Hunter, J.: Combinatorial software testing. Computer 8, 94–96 (2009)

    Article  Google Scholar 

  30. Kuhn, D.R., Kacker, R.N., Lei, Y.: Introduction to Combinatorial Testing. CRC Press, Boca Raton (2013)

    MATH  Google Scholar 

  31. Lei, Y., Tai, K.C.: In-parameter-order: a test generation strategy for pairwise testing. In: Third IEEE International Proceedings High-Assurance Systems Engineering Symposium, pp. 254–261 (1998)

    Google Scholar 

  32. Lei, Y., Kacker, R., Kuhn, D.R., Okun, V., Lawrence, J.: IPOG: a general strategy for t-way software testing. In: 14th Annual IEEE International Conference and Workshops on the Engineering of Computer-Based Systems (ECBS’07), pp. 549–556 (2007)

    Google Scholar 

  33. Nurmela, K.J.: Upper bounds for covering arrays by tabu search. Discret. Appl. Math. 138(1), 143–152 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Raaphorst, S., Moura, L., Stevens, B.: A construction for strength-3 covering arrays from linear feedback shift register sequences. Des. Codes Cryptogr. 73(3), 949–968 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Seroussi, G., Bshouty, N.H.: Vector sets for exhaustive testing of logic circuits. IEEE Trans. Inf. Theory 34(3), 513–522 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  36. Shorin, V.V., Loidreau, P.: Application of Groebner bases techniques for searching new sequences with good periodic correlation properties. In: Proceedings International Symposium on Information Theory (ISIT), pp. 1196–1200 (2005)

    Google Scholar 

  37. Torres-Jimenez, J., Izquierdo-Marquez, I.: Survey of covering arrays. In: 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), pp. 20–27 (2013)

    Google Scholar 

  38. Torres-Jimenez, J., Rodriguez-Tello, E.: New bounds for binary covering arrays using simulated annealing. Inf. Sci. 185(1), 137–152 (2012)

    Article  Google Scholar 

  39. Torres-Jimenez, J., Izquierdo-Marquez, I., Gonzalez-Gomez, A., Avila-George, H.: A branch & bound algorithm to derive a direct construction for binary covering arrays. In: Advances in Artificial Intelligence and Soft Computing, pp. 158–177. Springer, New York (2015)

    Google Scholar 

  40. Yan, J., Zhang, J.: Backtracking algorithms and search heuristics to generate test suites for combinatorial testing. In: 30th Annual International Computer Software and Applications Conference (COMPSAC’06), vol. 1, pp. 385–394 (2006)

    Google Scholar 

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Acknowledgements

The research presented in the paper has been funded in part by the Austrian Research Promotion Agency (FFG) under Grant 851205 (SPLIT—Security ProtocoL Interaction Testing in practice) and the Austrian COMET K1 Program (FFG).

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Authors and Affiliations

  1. SBA Research, 1040, Vienna, Austria

    Bernhard Garn & Dimitris E. Simos

Authors
  1. Bernhard Garn
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  2. Dimitris E. Simos
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Correspondence to Dimitris E. Simos .

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Editors and Affiliations

  1. Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  2. Institute of Mathematics, University of Valladolid, Valladolid, Spain

    Edgar Martínez-Moro

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Garn, B., Simos, D.E. (2017). Algebraic Modelling of Covering Arrays. In: Kotsireas, I., Martínez-Moro, E. (eds) Applications of Computer Algebra. ACA 2015. Springer Proceedings in Mathematics & Statistics, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-319-56932-1_10

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