Finite complete rewriting systems and the complexity of the word problem

SummaryIt is well known that the word problem for a finite complete rewriting system is decidable. Here it is shown that in general this result cannot be improved. This is done by proving that each sufficiently rich complexity class can be realized by the word problem for a finite complete rewriting system. Further, there is a gap between the complexity of the word problem for a finite complete rewriting system and the complexity of the least upper bound for the lengths of the chains generated by this rewriting system, and this gap can get arbitrarily large. Thus, the lengths of these chains do not give any information about the complexity of the word problem. Finally, it is shown that the property of allowing a finite complete rewriting system is not an invariant of finite monoid presentations.

[1]  守屋 悦朗,et al.  J.E.Hopcroft, J.D. Ullman 著, "Introduction to Automata Theory, Languages, and Computation", Addison-Wesley, A5変形版, X+418, \6,670, 1979 , 1980 .

[2]  J. C. Shepherdson,et al.  Machine Configuration and Word Problems of Given Degree of Unsolvability , 1965 .

[3]  Colm Ó'Dúnlaing.  Undecidable questions related to Church-Rosser Thue systems , 1983, Theor. Comput. Sci..

[4]  Gérard Huet,et al.  On the Uniform Halting Problem for Term Rewriting Systems , 1978 .

[5]  Bruno Buchberger,et al.  Computer algebra symbolic and algebraic computation , 1982, SIGS.

[6]  Günther Bauer.  Zur Darstellung von Monoiden durch konfluente Regelsysteme , 1981 .

[7]  D. McIlroy.  Algebraic Simplification , 1966, CACM.

[8]  G. Herman.  Strong Computability and Variants of the Uniform Halting Problem , 1971 .

[9]  M. Newman.  On Theories with a Combinatorial Definition of "Equivalence" , 1942 .

[10]  Maurice Nivat,et al.  Congruences parfaites et quasi-parfaites , 1971 .

[11]  G. Huet,et al.  Equations and rewrite rules: a survey , 1980 .

[12]  Friedrich Otto,et al.  Finite Complete Rewriting Systems for the Jantzen Monoid and the Greendlinger Group , 1984, Theor. Comput. Sci..

[13]  A. Grzegorczyk.  Some classes of recursive functions , 1964 .

[14]  Albert R. Meyer,et al.  Word problems requiring exponential time(Preliminary Report) , 1973, STOC.

[15]  Ronald V. Book,et al.  Confluent and Other Types of Thue Systems , 1982, JACM.

[16]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[17]  Matthias Jantzen,et al.  Thue Systems and the Church-Rosser Property , 1984, MFCS.

[18]  Gérard P. Huet,et al.  Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems , 1980, J. ACM.

[19]  Patrick Horster.  Reduktionssysteme, formale Sprachen und Automatentheorie , 1983 .

[20]  Michael Machtey.  On the Density of Honest Subrecursive Classes , 1975, J. Comput. Syst. Sci..

[21]  Paliath Narendran,et al.  A Finite Thue System with Decidable Word Problem and without Equivalent Finite Canonical System , 1985, Theor. Comput. Sci..

[22]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[23]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[24]  Andrzej Mostowski.  Review: A. Markov, Impossibility of Algorithms for Recognizing Some Properties of Associative Systems , 1952 .

[25]  Martin D. Davis,et al.  Computability and Unsolvability , 1959, McGraw-Hill Series in Information Processing and Computers.

[26]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .