Sufficientcompleteness, groundreducibility and their complexity
 Deepak Kapur,
 Paliath Narendran,
 Daniel J. Rosenkrantz,
 Hantao Zhang
 … show all 4 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessSummary
The sufficientcompleteness 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 inductionlessinduction method. The sufficientcompleteness property is known to be undecidable in general. In an earlier paper, it was shown to be decidable for constructorpreserving, complete (canonical) term rewriting systems, even when there are relations among constructor symbols. In this paper, the complexity of the sufficientcompleteness property is analyzed for different classes of term rewriting systems. A number of results about the complexity of the sufficientcompleteness property for complete (canonical) term rewriting systems are proved: (i) The problem is coNPcomplete for term rewriting systems with free constructors (i.e., no relations among constructors are allowed), (ii) the problem remains coNPcomplete 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 leftlinear term rewriting systems with relations among constructors, and (iv) for leftlinear complete constructorpreserving rewriting systems, the problem can be decided in steps exponential innlogn wheren is the size of the rewriting system. No better lowerbound for the complexity of the sufficientcompleteness property for complete (canonical) term rewriting system with nonlinear lefthand sides is known. An algorithm for leftlinear complete constructorpreserving rewriting systems is also discussed. Finally, the sufficientcompleteness property is shown to be undecidable for nonlinear complete term rewriting systems with associative functions. These complexity results also apply to the groundreducibility property (also called inductivereducibility) which is known to be directly related to the sufficientcompleteness property.
 Book, R.: Confluent and other types of Thue systems. J. Assoc. Comput. Mach.29, 171–182 (1982)
 Boyer, R.S., Moore, J.S.: A computational logic. New York: Academic Press 1979
 Comon, H.: Sufficientcompleteness, term rewriting systems, and “antiunification”. Proc. of 8th Intl. Conf. on Automated Deduction (Lect. Notes Comput. Sci., Vol. 230, pp. 128–140). Berlin Heidelberg New York: Springer 1986
 Cook, S.A.: Characterizations of pushdown machines in terms of timebounded computers. J. Assoc. Comput. Mach.18, 4–18 (1971)
 Dershowitz, N.: Computing with rewrite systems. Inf. Control65, 122–157 (1985)
 Garey, M.R., Johnson, D.S.: Computers and intractability. W.H. Freeman 1979
 Goguen, J.: How to prove algebraic inductive hypotheses without induction. Proc. of the 5th Conference on Automated Deduction, Les Arcs, France. (Lect. Notes Comput. Sci., Vol. 87, pp. 356–373) Berlin Heidelberg New York: Springer 1980
 Guttag, J.: The specification and application to programming of abstract data types. Department of Computer Science, Univ. of Toronto, Ph.D. Thesis, CSRG59 (1975)
 Guttag, J., Horning, J.J.: The algebraic specification of abstract data types. Acta Inf.10, 27–52 (1978)
 Huet, G., Hullot, J.M.: Proofs by induction in equational theories with constructors. J. Comput. Syst. Sci.25, 239–266 (1982)
 Huet, G., Oppen, D.C.: Equations and rewrite rules: A survey. In: R. Book (ed.) Formal language theory: Perspectives and open problems, pp. 349–405. New York: Academic Press 1980
 Hunt, H.B., Rosenkrantz, D.J.: The complexity of monadic recursion schemes: Exponential time bounds. J. Comput. Syst. Sci.28, 395–419 (1984)
 Jouannaud, J.P., Kounalis, E.: Automatic proofs by induction in equational theories without constructors. Proc. of the IEEE Symposium on Logic in Computer Science, Cambridge, Mass, pp. 358–386, 1986
 Kapur, D., Musser, D.R.: Proof by consistency. Artif. Intell. J.31, 125–157 (1987)
 Kapur, D., Narendran, P., Zhang, H.: Proof by induction using test sets. Proc. of 8th Intl. Conf. on Automated Deduction, Oxford England (Lect. Notes Comput. Sci., Vol. 230, pp. 99–117) Berlin Heidelberg New York: Springer 1986a
 Kapur, D., Narendran, P., Zhang, H.: On sufficientcompleteness and related properties of term rewriting systems. Acta Inf.24, 395–416 (1987)
 Kapur, D., Narendran, P., Zhang, H.: Complexity of sufficientcompleteness. Proc. of 6th Conf. on Foundations of Software Technology and Theoretical Computer Science, New Delhi, India (Lect. Notes Comput. Sci., Vol. 241, pp. 426–442) Berlin Heidelberg New York: Springer 1986b
 Knuth, D., Bendix, P.: Simple word problems in universal algebras. In: J. Leech (ed.). Computational problems in abstract algebra, pp. 263–297. Oxford: Pergamon Press (1970)
 Kounalis, E.: Completeness in data type specifications. Proc. EUROCAL'85, LNCS 204 (Bob F. Caviness, ed.), pp. 348–362. Berlin Heidelberg New York: Springer 1985
 Lankford, D.S., Ballantyne, A.M.: Decision procedures for simple equational theories with commutativeassociative axioms: Complete sets of commutativeassociative reductions. Technical Report ATP39, Dept. of Math. and Computer Science, Univ. of Texas at Austin, Texas, 1977
 Lankford, D.S.: A simple explanation of inductionless induction. Memo MTP14, Dept. of Mathematics, Louisiana Tech. University, Ruston, Louisiana, 1981
 Lassez, J.L., Marriott, K.: Explicit representation of terms defined by counterexamples. J Automat. Reasoning3, 301–318 (1987)
 Lewis, H.: Complexity results for classes of quantificational formulas. J. Comput. System Sci.21, 317–353 (1980)
 Marriott, K.: A note on the complexity of determining the reasonability and the existence of an explicit representation for an implicit generalization. Draft manuscript. Computer Science Dept. University of Melbourne, Australia, 1988
 Minsky, M.L.: Recursive unsolvability of Post's problem of “Tag” and other topics in theory of Turing machines. Ann. Math.74, 437–455 (1961)
 Musser, D.R.: On proving inductive properties of abstract data types. Proc. 7th Principles of Programming Languages, Las Vegas, Nevada 1980
 Nipkow, T., Weikum, G.: A decidability result about sufficientcompleteness of axiomatically specified abstract data types. Proc. of the 6th GI Conf. on Theoretical Computer Science, (Lect. Notes Comput. Sci., Vol. 145, pp. 257–268) Berlin Heidelberg New York: Springer 1982
 Peterson, G.L., Stickel, M.E.: Complete set of reductions for some equational theoris. J. Assoc. Comput. Mach.28, 233–264 (1981)
 Plaisted, D.: Complete problems in the firstorder predicate calculalus. J. Comput. System Sci.29, 8–35 (1984)
 Plaisted, D.: Semantic confluence tests and completion methods. Inf. Control65, 182–215 (1985)
 Thiel, J.J.: Stop loosing sleep over incomplete data type specifications. Proc. of Eleventh Annual ACM Symposium on Principles of Programming Languages, Salt Lake City, Utah 1984
 Title
 Sufficientcompleteness, groundreducibility and their complexity
 Journal

Acta Informatica
Volume 28, Issue 4 , pp 311350
 Cover Date
 19910401
 DOI
 10.1007/BF01893885
 Print ISSN
 00015903
 Online ISSN
 14320525
 Publisher
 SpringerVerlag
 Additional Links
 Topics

 Computer Systems Organization and Communication Networks
 Software Engineering/Programming and Operating Systems
 Data Structures, Cryptology and Information Theory
 Theory of Computation
 Logics and Meanings of Programs
 Information Systems and Communication Service
 Computational Mathematics and Numerical Analysis
 Industry Sectors
 Authors

 Deepak Kapur ^{(1)}
 Paliath Narendran ^{(1)}
 Daniel J. Rosenkrantz ^{(1)}
 Hantao Zhang ^{(2)}
 Author Affiliations

 1. Department of Computer Science, State University of New York at Albany, 12222, Albany, NY, USA
 2. Department of Computer Science, The University of Iowa, 52242, Iowa City, IA, USA