New methods for proving the impossibility to solve problems through reduction of problem spaces



Problem solvers are computational systems which make use of different search algorithms for solving problems. Sometimes, while employing such search algorithms, problem solvers may prove to be inefficient and take too great an effort so as to showing that the problem has no solution. For such cases, in this paper we explain a technique which provides a quick proof that finding a solution is actually impossible. This technique results in reducing the number and simplifying the topology of the states which shape a problem space. Hence, we show and prove efficient new techniques intended to find such reductions which may result to be very useful for many problems.


Problem spaces Automatic representation changes Problem solvers 

Mathematics Subject Classifications (2010)

68T20 68T30 


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  1. 1.
    Bacchus F., Yang Q: Downward refinement and the efficiency of hierarchical problem solving. Artif. Intell. 71(1), 43–100 (1994)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bayer, K.M., Michalowski M., Choueiry B.Y., Knoblock, C.A.: Reformulating constraint satisfaction problems to improve scalability. In: Abstraction, Reformulation and Approximation (SARA 2007), pp. 64–79. Springer-Verlag, Berlin-Heidelberg (2007)Google Scholar
  3. 3.
    Bonet B., Geffner H.: Planning as heuristic search. Artif. Intell. 129(1–2) 5–33 (2001)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Blum A., Furst M.: Fast planning through planning graph analysis. Artif. Intell. 90(1–2), 281–300 (1997)MATHCrossRefGoogle Scholar
  5. 5.
    Choueiry, B.Y., Iwasaki, Y., McIlraith, S.: Towards a practical theory of reformulation for reasoning about physical systems. Artif. Intell. 162(1–2), 145–204 (2005)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Culberson J.C., Schaeffer J.: Pattern databases. Comput. Intell. 14(3), 318–334 (1998)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Davis M., Logemann G., Loveland D.: A machine program for theorem-proving. Commun. ACM 5/7, 394–397 (1962)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Do, M.B., Kambhampati, S.: Planning as constraint satisfaction: solving the planning graph by compiling it into CSP. Artif. Intell. 132, 151–182 (2001)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Felner, A., Adler, A.: Solving the 24 puzzle with instance dependent pattern databases. In: SARA 2005, pp. 248–260 (2005)Google Scholar
  10. 10.
    Fikes R., Nilsson N.: STRIPS: a new approach to the application of theorem proving to problem solving. Artif. Intell. 1, 27–120 (1971)CrossRefGoogle Scholar
  11. 11.
    Fink, E.: Changes of Problem Representation: Theory and Experiments. Springer, Berlin (2003)MATHGoogle Scholar
  12. 12.
    Helmert M.: The fast downward planning system. J. Artif. Intell. Res 26, 191–246 (2006)MATHCrossRefGoogle Scholar
  13. 13.
    Hernando, A., De Ledesma, L., Laita, L.M.: A system simulating representation change phenomena while problem solving. Math. Comput. Simul. 78, 89–106 (2008)MATHCrossRefGoogle Scholar
  14. 14.
    Hernando, A., De Ledesma, L., Laita, L.M.: Showing the non-existence of solutions in systems of linear Diophantine equations. Math. Comput. Simul. 79(11), 3211–3220 (2009)MATHCrossRefGoogle Scholar
  15. 15.
    Hoffmann J., Nebel B.: The FF planning system: fast plan generation through heuristic search. J. Artif. Intell. Res. 14, 253–302 (2001)MATHGoogle Scholar
  16. 16.
    Holte, R.C., Mkadmi, T., Zimmer, R.M., MacDonald, A.J.: Speeding up problem solving by abstraction: a graph oriented approach. Artif. Intell. 85, 321–361 (1996)CrossRefGoogle Scholar
  17. 17.
    Holte R.C., Felner A., Newton J., Meshulam R., Furcy D.: Maximizing over multiple pattern databases speeds up heuristic search. Artif. Intell. 170, 1123–1136 (2006)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kautz, H., Selman B.: Blackbox: unifying SAT-based and graph-based planning. In: Proc. IJCAI-99, pp. 318–325. Stockholm, Sweden (1999)Google Scholar
  19. 19.
    Kautz, H., Selman B., Hoffmann, J.: Satplan: planning as satisfiability. In: Special Booklet for the International Planning Competition in the Working Notes of the 16th Intl. Conf. on Automated Planning and Scheduling (ICAPS-06). Monterey, CA (2006)Google Scholar
  20. 20.
    Knoblock C.A: Generating Abstraction Hierarchies: An Automated Approach to Reducing Search in Planning. Kluwer Academic Publishers, Boston, MA (1993)Google Scholar
  21. 21.
    Korf, R.E., Felner A.: Disjoint pattern database heuristics. Artif. Intell. 134(1–2), 9–22 (2002)MATHCrossRefGoogle Scholar
  22. 22.
    Korf, R.E.: Toward a model of representation changes. Artif. Intell. 14, 41–78 (1980)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Newell, A., Simon, H.A.: Human Problem Solving. Prentice-Hall, Englewood Cliffs (1972)Google Scholar
  24. 24.
    Roberts, M., Howe, A.: Learning from planner performance. Artif. Intell. 173, 536–561 (2009)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Sturtevant, N., Jansen, R.: An analysis of map-based abstraction and refinement. In: SARA 2007. LNAI, vol. 4612, pp. 344–358 (2007)Google Scholar
  26. 26.
    van Beek P., Chen X.: CPlan: a constraint programming approach to planning. In: Proc. AAAI-99, pp. 585–590. Orlando, FL (1999)Google Scholar
  27. 27.
    Vossen, T., Ball, M., Lotem, A., Nau, D.: On the use of integer programming models in AI planning. In: Proc. IJCAI-99, pp. 304–309. Stockholm, Sweden (1999)Google Scholar
  28. 28.
    Zhang L., Malik S.: The quest for efficient Boolean satisfiability solvers. In: Proc. Intl. Conference on Computer Aided Verification (CAV-02), pp. 17–36 (2002)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Departamento de Sistemas Inteligentes Aplicados, Escuela Universitaria de InformáticaUniversidad Politécnica de MadridMadridSpain

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