Summary
It 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.
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Some of the results presented here are from the doctoral dissertation of the first author, which was supervised by Prof. H. Brakhage at the University of Kaiserslautern
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Bauer, G., Otto, F. Finite complete rewriting systems and the complexity of the word problem. Acta Informatica 21, 521–540 (1984). https://doi.org/10.1007/BF00271645
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DOI: https://doi.org/10.1007/BF00271645