Abstract
The problem of decidability of the first order theory of one-step rewriting was stated in [CCD93]. One can find the problem on the lists of open problems in rewriting in [DJK93] and [DJK95]. In 1995 Ralf Treinen proved that the theory is undecidable.
In this paper we show that the theorem of Treinen is an easy consequence of a powerful tool first used to establish undecidability results in the theory of logic programming [MP92]. Then we use the tool to give strong refinements of the result of Treinen:
We show that (the ∃*∀* part of) the first order theory of one-step rewriting is undecidable for linear Noetherian rewriting systems.
Then we prove, what we consider to be quite a striking result, that (the ∃*∀*part of) the first order theory of one-step rewriting is undecidable even if the rewriting system is Noetherian and right-ground. Up to our knowledge it is the first known undecidable property of right-ground systems.
Partially supported by Polish KBN grant 3 P301 046 07
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Bibliography
A-C Caron, J-L Coquide, M. Dauchet, Encompassment Properties and Automata with Constraints Proceedings of 5th RTA, 1993, (Springer LNCS vol. 690) pp. 328–342
N. Dershowitz, J-P Jouannaud, J. W. Klop, More Problems in Rewriting, Proceedings of 5th RTA, 1993, (Springer LNCS vol. 690) pp. 468–487
N. Dershowitz, J-P Jouannaud, J. W. Klop, Problems in Rewriting III, Proceedings of RTA 95, (Springer LNCS vol. 914) pp 457–471
M. Dauchet, S. Tison. The theory of ground rewrite systems is decidable, Proceedings of 5th IEEE LICS, pp 242–256, IEEE Press 1990
J. Marcinkowski, L. Pacholski, Undecidability of the Horn Clause Implication Problem, Proceedings of 33th IEEE FOCS, pp 354–362 Los Alamitos 1992
J. Marcinkowski, A Horn Clause that Implies an Undecidable Set of Horn Clauses, Proceedings of the 1993 Annual Conference of the European Association of Computer Science Logic. (Springer, 1994), LNCS 832, pp 223–237.
F. Seynhaeve, M. Tommasi, R. Treinen, Grid Structures and Undecidable Properties, to appear in Proceedings of TAPSOFT 97
R. Treinen, The First-Order Theory of One-Step Rewriting is Undecidable, Proceedings of RTA 1996, (Springer LNCS).
S. Vorobyov, Elementary theory of one step rewriting is undecidable, draft 1995.
S. Vorobyov, The first-order theory of one step rewriting in linear Noetherian systems is undecidable, this volume.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Marcinkowski, J. (1997). Undecidability of the first order theory of one-step right ground rewriting. In: Comon, H. (eds) Rewriting Techniques and Applications. RTA 1997. Lecture Notes in Computer Science, vol 1232. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62950-5_75
Download citation
DOI: https://doi.org/10.1007/3-540-62950-5_75
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-62950-4
Online ISBN: 978-3-540-69051-1
eBook Packages: Springer Book Archive