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Abstract: . We settle in the negative the following Problem 51 (Comon--Dauchet, RTA'93 [3], RTA'95 [4]). Given an arbitrary finite term rewriting system R, is the first-order theory of one-step rewriting (!R ) decidable? Decidability would imply the decidability of the first-order theory of encompassment (that is, being an instance of a subterm) [1], as well as several known decidability results in rewriting. (It is well known that the theory of ! R is in general undecidable. ) ut The positive... (Update)
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...logic does not employ variables. An alternative proof of the undecidability of one step rewriting has been given by Sergei Vorobyov [Vor95]. He defines one particular rewrite system R and shows the undecidability of the full first order theory of one step rewriting by R. He...
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2: More problems in rewriting - Dershowitz, Jouannaud et al. - 1993
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BibTeX entry: (Update)
Sergei Vorobyov. Elementary theory of one-step rewriting is undecidable (note). Draft, 1995. http://citeseer.ist.psu.edu/vorobyov95elementary.html More
@misc{ vorobyov95elementary,
author = "S. Vorobyov",
title = "Elementary theory of one-step rewriting is undecidable",
text = "Sergei Vorobyov. Elementary theory of one-step rewriting is undecidable
(note). Draft, 1995.",
year = "1995",
url = "citeseer.ist.psu.edu/vorobyov95elementary.html" }
Citations (may not include all citations):788 Rewrite systems - Dershowitz, Jouannaud - 1990
41 Encompassment properties and automata with constraints (context) - Caron, Coquid'e et al. - 1993
32 Theory of Formal Systems (context) - Smullyan - 1961
27 More problems in rewriting - Dershowitz, Jouannaud et al. - 1993
20 Concatenation as a basis for arithmetic (context) - Quine - 1946 DBLP
11 Problems in rewriting III - Dershowitz, Jouannaud et al. - 1995 ACM DBLP
Documents on the same site (http://www.mpi-sb.mpg.de/~sv/index-rew.html):
-Theories of One-Step Rewriting in Linear Noetherian.. - Sergei Vorobyov November (1996) (Correct)
The First-Order Theory of One Step Rewriting in Linear.. - Vorobyov (1997) (Correct)
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