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Non-negative integer basis algorithms for linear equations with integer coefficients

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Abstract

A new class of non-negative integer basis algorithms for linear equations with integer coefficients is developed. Computer experiments with one of the new algorithms and comparisons with other non-negative integer basis algorithms are reported. When the total run times for collections of common simple examples typical of automated deduction applications are computed, the new algorithm has been found to be significantly faster than previous algorithms.

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References

  1. M. Stickel, ‘A complete unification algorithm for associative-commutative functions’. J. Assoc. Comput. Mach. 28 (1981), 423–434.

    Google Scholar 

  2. M. Livesey and J. Siekmann, ‘Unification of A+C-terms (bags) and A+C+I-terms (sets).’ Institut für Informatik I, Universität Karlsruhe, West Germany, Interner Bericht Nr. 3/76, 1976.

    Google Scholar 

  3. F. Fages, ‘Associative-commutative unification.’ Lect. Notes in Comp. Sci. 170, 7th Int. Conf. on Aut. Ded., Springer-Verlag, Berlin, 1984, 194–208.

    Google Scholar 

  4. A. Fortenbacher, “An algebraic approach to unification under associativity and commutativity.’ Proc. 1st Int. Conf. on Rew. Tech. and Appl., preprint, June 1985.

  5. W. W. Bledsoe and D. W. Loveland, eds., ‘Automated Theorem Proving: After 25 Years.’ Contemporary Math. 29, American Mathematical Society, Providence, 1984.

    Google Scholar 

  6. R. E. Shostak, ed., ‘7th International Conference On Automated Deduction.’ Lect. Notes In Comp. Sci. 170, Springer-Verlag, Berlin, 1984.

  7. P. Gordan, ‘Ueber die Auflösung linearer Gleichungen mit reellen Coefficienten.’ Mathematische Annalen 1873, VI Band, 1 Heft, 23–28.

    Google Scholar 

  8. G. Huet, ‘An algorithm to generate the basis of solutions to homogeneous linear Diophantine equations.’ Inf. Proc. Let. 7, 3 (1987) 144–147.

    Google Scholar 

  9. M. Hack, “Decision problems for Petri nets and vector addition systems.’ Computational Structures Group Memo 95, Project MAC, M.I.T., Cambridge, MA, 1974.

    Google Scholar 

  10. L. Dickson, ‘Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors.’ Amer. J. Math. XXXV (1913), 413–422.

    Google Scholar 

  11. D. König, ‘Sur les correspondances multivoques des ensembles.’ Fund. Math. VII (1926), 114–134.

    Google Scholar 

  12. D. E. Knuth, The Art of Computer Programming, Vol. 1, Fundamental Algorithms, Addison-Wesley Pub. Co., Reading, 1969.

    Google Scholar 

  13. G. Butler and D. Lankford, ‘Dickson's lemma, Hilbert's basis theorem, and applications to completion in commutative Noetherian rings.’ Dept. Math. and Stat., Louisiana Tech University, Ruston, LA 71272, techn rpt., June 1984.

    Google Scholar 

  14. M. Kac and S. Ulam, Mathematics and Logic, Retrospect and Prospects, The New Americal Library, Inc., New York, 1969.

    Google Scholar 

  15. R. Ayoub, An Introduction To The Analytic Theory Of Numbers, Math. Surveys, No. 10, American Mathematical Society, Providence, 1963.

    Google Scholar 

  16. G. Huet, ‘An algorithm to generate the basis of solutions to homogeneous linear diophantine equations,’ IRIA Laboria, Rapport de Recherche No. 174, Janvier 1978.

  17. A. Fortenbacher, ‘Algebraische Unifikation,’ Diplomarbeit, Universität Karlsruhe, 1983.

  18. T. Guckenbiehl and A. Herold, ‘Solving linear diophantine equations,’ technical report, Universität Kaiserslautern, January 1986.

  19. R. Burnes and D. Lankford, “High resolution IBM PC AT timer and Turbo Pascal interface,” Burnes Electronics, P.O. Box 906, Ruston, LA 71273, May 1987.

    Google Scholar 

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Lankford, D. Non-negative integer basis algorithms for linear equations with integer coefficients. J Autom Reasoning 5, 25–35 (1989). https://doi.org/10.1007/BF00245019

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  • DOI: https://doi.org/10.1007/BF00245019

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