Acta Informatica

, Volume 28, Issue 4, pp 311–350

Sufficient-completeness, ground-reducibility and their complexity

  • Deepak Kapur
  • Paliath Narendran
  • Daniel J. Rosenkrantz
  • Hantao Zhang
Article

DOI: 10.1007/BF01893885

Cite this article as:
Kapur, D., Narendran, P., Rosenkrantz, D.J. et al. Acta Informatica (1991) 28: 311. doi:10.1007/BF01893885

Summary

The sufficient-completeness property of equational algebraic specifications has been found useful in providing guidelines for designing abstract data type specifications as well as in proving inductive properties using the induction-less-induction method. The sufficient-completeness property is known to be undecidable in general. In an earlier paper, it was shown to be decidable for constructor-preserving, complete (canonical) term rewriting systems, even when there are relations among constructor symbols. In this paper, the complexity of the sufficient-completeness property is analyzed for different classes of term rewriting systems. A number of results about the complexity of the sufficient-completeness property for complete (canonical) term rewriting systems are proved: (i) The problem is co-NP-complete for term rewriting systems with free constructors (i.e., no relations among constructors are allowed), (ii) the problem remains co-NP-complete for term rewriting systems with unary and nullary constructors, even when there are relations among constructors, (iii) the problem is provably in “almost” exponential time for left-linear term rewriting systems with relations among constructors, and (iv) for left-linear complete constructor-preserving rewriting systems, the problem can be decided in steps exponential innlogn wheren is the size of the rewriting system. No better lower-bound for the complexity of the sufficient-completeness property for complete (canonical) term rewriting system with nonlinear left-hand sides is known. An algorithm for left-linear complete constructor-preserving rewriting systems is also discussed. Finally, the sufficient-completeness property is shown to be undecidable for non-linear complete term rewriting systems with associative functions. These complexity results also apply to the ground-reducibility property (also called inductive-reducibility) which is known to be directly related to the sufficient-completeness property.

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Deepak Kapur
    • 1
  • Paliath Narendran
    • 1
  • Daniel J. Rosenkrantz
    • 1
  • Hantao Zhang
    • 2
  1. 1.Department of Computer ScienceState University of New York at AlbanyAlbanyUSA
  2. 2.Department of Computer ScienceThe University of IowaIowa CityUSA