Abstract
Restrictions of the problem of finding all maximal unifiable or minimal nonunifiable subsets to those of certain sizes are shown to be NP-hard, and consequently inappropriate in general for reducing thrashing by intelligent backtracking in resolution theorem provers and logic program executions. We also show that existing backtrack methods based on the computation of all maximal unifiable subsets of assignments as a means to avoid thrashing are intractable because the solution length of these subsets can increase exponentially with the input length, and we give a corresponding result for minimal nonunifiability. The results apply not only to standard unification, but to a characterized collection of unification algorithms which includes unification without the occurs check. This now justifies the necessity for approximate or heuristic approaches to reduce thrashing in resolution theorem provers and the execution of logic programs.
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References
Bruynooghe, M. and Pereira, L. M., ‘Deduction Revision by Intelligent Backtracking,’ in: J. Campbell (ed.), Implementations of Prolog (Ellis Horwood, Chichester, 1984) pp. 194–215.
Chen, T. Y., Lassez, J.-L. and Port, G. S., ‘Maximal Unifiable Subsets and Minimal Non-unifiable Subsets’, New Generation Computing 4, 133–152 (1986).
Clocksin, W. F. and Mellish, C. S., Programming in Prolog, Third Edition, Springer, Berlin (1987).
Cox, P. T., ‘Finding Backtrack Points for Intelligent Backtracking’, in: J. Campbell, (ed.), Implementations of Prolog (Ellis Horwood, Chichester, 1984) pp. 216–233.
Cox, P. T., ‘On Determining the Causes of Nonunifiability’, Journal of Logic Programming 4, 33–58 (1987).
Dilger, W. and Janson, A., ‘Intelligent Backtracking in Deduction Systems by Means of Extended Unification Graphs’, Journal of Automated Reasoning 2 (1), 43–62 (1986).
Forster, D. R., GTP: A Graph Based Theorem Prover, M.S. Thesis, University of 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, 426–444 (1984).
Forsythe, K. and Matwin, S., ‘Copying of Dynamic Structures in a Pascal Environment’, Software-Practice and Experience 16 (4), 335–340 (April 1986).
Garey, M. R. and Johnson, D. S., Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco (1979).
Mackworth, A. K., ‘Consistency in Networks of Relations’, Artificial Intelligence 8, 99–118 (1977).
Matwin, S. and Pietrzykowski, T., ‘Intelligent Backtracking in Plan-Based Deduction’, IEEE Transactions on Pattern Analysis and Machine Intelligence 7 (6) 682–692 (1985).
Nilsson, N. J., Principles of Artificial Intelligence, Springer (1982).
Paterson, M. S. and Wegman, M. N., ‘Linear Unification’, Journal of Computer and System Sciences 16, 158–167 (1978).
Sato, T., ‘An Algorithm for Intelligent Backtracking’, Proceedings of the RIMS Symposia on Software Science and Engineering, 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 (1978) pp. 216–226.
Wilf, H. S., ‘Backtrack: An O(I) Expected Time Algorithm for the Graph Coloring Problem’, Information Processing Letters 18, 119–121 (1984).
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A version of this paper appears in Proceedings of the Third International Logic Programming Conference, London, Lecture Notes in Computer Science, Springer 225, 107–121 (1986).
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Wolfram, D.A. Intractable unifiability problems and backtracking. J Autom Reasoning 5, 37–47 (1989). https://doi.org/10.1007/BF00245020
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DOI: https://doi.org/10.1007/BF00245020