Abstract
Intelligent backtracking in logic programs analyses unification failure to avoid thrashing, which is an inefficient behaviour of ordinary backtracking. We show that the computation of all maximal unifiable subsets of constraints, as a means to avoid thrashing, is intractable in the sense that the solution length can be non-polynomially related to the input length. We also give a corresponding result for minimal nonunifiability. Restrictions of the problem of finding all maximal unifiable (minimal nonunifiable) subsets to those of certain sizes, for use with heuristics, are shown to be NP-hard. The results apply not only to standard unification but for unification without the occur-check as in many versions of Prolog. This now justifies the necessity for approximate or heuristic approaches in general.
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8. References
Baxter, L.D. The Complexity of Unification, Ph.D. thesis, Department of Computer Science, University of Waterloo, Ontario, Canada, 1976.
Bruynooghe, M. and Pereira, L.M. Deduction Revision by Intelligent Backtracking, in: J.A. Campbell (ed.), Implementations of Prolog, Ellis Horwood, Chichester, 1984.
Chang, J.-H. and Despain, A.M., Semi-Intelligent Backtracking of Prolog based on a Static Data Dependency Analysis, 1985 Symposium on Logic Programming, IEEE, Boston, 10–21.
Chen, T.Y., Lassez, J-L. and Port, G.S. Maximal Unifiable Subsets and Minimal Non-unifiable Subsets, Technical Report 84/16, May, 1985, Department of Computer Science, The University of Melbourne, Australia.
Clocksin, W.F. and Mellish, C.S. Programming in Prolog, Springer, Berlin, 1984.
Colmerauer, A. Prolog and Infinite Trees, in: K.L. Clark and S-A. Tärnlund (eds.), Logic Programming, Academic, New York, 1982.
Cox, P.T. Deduction Plans: a graphical proof procedure for the first-order predicate calculus, Research Report CS-77-28, Ph.D. thesis, Department of Computer Science, University of Waterloo, Ontario, Canada, 1977.
Cox, P.T. Finding Backtrack Points for Intelligent Backtracking, in: J.A. Campbell (ed.), Implementations of Prolog, Ellis Horwood, Chichester, 1984.
Dilger, W. and Janson, A. Unifikationsgraphen fűr Intelligentes Backtracking in Deduktionssystemen, Proceedings of GWAI-83, Dassel, Federal Republic of Germany, 1983.
Forster, D.R. GTP: A Graph Based Theorem Prover, M.S. thesis, University of Waterloo, Waterloo, Ontario, Canada, 1982.
Forsythe, K. and Matwin, S. Implementation Strategies for Plan-Based Deduction, Proceedings of the Seventh International Conference on Automated Deduction, (R.E. Shostak, ed.), Napa, California, USA, Lecture Notes in Computer Science, Springer, 170 (1984).
Garey, M.R. and Johnson, D.S. Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979.
Kowalski, R.A. Logic for Problem Solving, North-Holland, New York, 1979.
Mackworth, A.K. Consistency in Networks of Relations, Artificial Intelligence 8 (1977), 99–118.
Martelli, A. and Montanari, U. Unification in Linear Time and Space: A Structured Presentation, Internal Report B76-16, Istituto di Elaborazione della Informazione, Pisa, 1976.
Matwin, S. and Pietrzykowski, T. Exponential Improvement of Exhaustive Backtracking: Data Structure and Implementation, Proceedings of the Sixth Conference on Automated Deduction, Lecture Notes in Computer Science, Springer, New York, 138 (1982), 240–259.
Matwin, S. and Pietrzykowski, T. Intelligent Backtracking in Plan-Based Deduction, IEEE Transactions on Pattern Analysis and Machine Intelligence, 7, 6 (1985), 682–692.
Nilsson, N.J. Principles of Artificial Intelligence, Springer, 1982.
Paterson, M.S. and Wegman, M.N. Linear Unification, JCSS 16 158–167 (1978).
Pietrzykowski, T. and Matwin, S. Exponential Improvement of Efficient Backtracking: A Strategy for Plan-Based Deduction, Proceedings of the Sixth Conference on Automated Deduction, Lecture Notes in Computer Science, Springer, New York, 138 (1982), 223–239.
Robinson, J.A. A Machine-Oriented Logic Based on the Resolution Principle, Journal of the ACM, 12, 1 (January, 1965), 23–41.
Sato, T. An Algorithm for Intelligent Backtracking, Proceedings of the RIMS Symposia on Software Science, and Engineering, (S. Goto et alia, eds.), Lecture Notes in Computer Science, 147, Springer, 1983.
Schaefer, T.J. The Complexity of Satisfiability Problems, Conference Record of the Tenth Annual ACM Symposium on Theory of Computing, ACM, New York, 216–226, (1978).
Vasey, P.E. A Logic-in-Logic Interpreter, M.Sc. Thesis, 1980, Imperial College of Science and Technology, University of London.
Wilf, H.S. Backtrack: An O(1) Expected Time Algorithm for the Graph Coloring Problem, Information Processing Letters, 18 (1984), 119–121.
Wolfram, D.A., Maher, M.J. and Lassez, J.-L. A Unified Treatment of Resolution Strategies for Logic Programs, Proceedings of the Second International Logic Programming Conference, Uppsala, Sweden, 1984, 263–276.
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Wolfram, D.A. (1986). Intractable unifiability problems and backtracking. In: Shapiro, E. (eds) Third International Conference on Logic Programming. ICLP 1986. Lecture Notes in Computer Science, vol 225. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16492-8_68
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DOI: https://doi.org/10.1007/3-540-16492-8_68
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