Abstract
Type introduction is a useful technique for simplifying the task of proving properties of rewrite systems by restricting the set of terms that have to be considered to the well-typed terms according to any many-sorted type discipline which is compatible with the rewrite system under consideration. A property of rewrite systems for which type introduction is correct is called persistent. Zantema showed that termination is a persistent property of non-collapsing rewrite systems and non-duplicating rewrite systems. We extend his result to the more complicated case of equational rewriting. As a simple application we prove the undecidability of AC-termination for terminating rewrite systems. We also present sufficient conditions for the persistence of acyclicity and non-loopingness, two properties which guarantee the absence of certain kinds of infinite rewrite sequences.
Preview
Unable to display preview. Download preview PDF.
References
N. Dershowitz, Termination of Rewriting, Journal of Symbolic Computation 3 (1987) 69–116.
N. Dershowitz and J.-P. Jouannaud, Rewrite Systems, in: Handbook of Theoretical Computer Science, Vol. B (ed. J. van Leeuwen), North-Holland (1990) 243–320
N. Dershowitz and Z. Manna, Proving Termination with Multiset Orderings, Communications of the ACM 22 (1979) 465–476.
M.C.F. Ferreira, Dummy Elimination in Equational Rewriting, Proc. 7th RTA, New Brunswick, LNCS 1103 (1996) 78–92.
M.C.F. Ferreira and H. Zantema, Dummy Elimination: Making Termination Easier, Proc. 10th FCT, Dresden, LNCS 965 (1995) 243–252.
A. Geser, A. Middeldorp, E. Ohlebusch, and H. Zantema, Relative Undecidability in Term Rewriting, Proc. CSL, Utrecht, LNCS (1996). To appear. Available at http://www.score.is.tsukuba.ac.jp/∼ami/papers/cs196.dvi.
J.-P. Jouannaud and M. Muñoz, Termination of a Set of Rules Modulo a Set of Equations, Proc. 7th CADE, Napa, LNCS 170 (1984) 175–193.
D. Kapur and G. Sivakumar, A Total, Ground Path Ordering for Proving Termination of AC-Rewrite Systems, Proc. 8th RTA, Sitges, LNCS (1997). To appear.
J.W. Klop, Term Rewriting Systems, in: Handbook of Logic in Computer Science, Vol. 2 (eds. S. Abramsky, D. Gabbay and T. Maibaum), Oxford University Press (1992) 1–116.
A. Middeldorp and B. Gramlich, Simple Termination is Difficult, Applicable Algebra in Engineering, Communication and Computing 6 (1995) 115–128.
E. Ohlebusch, A Simple Proof of Sufficient Conditions for the Termination of the Disjoint Union of Term Rewriting Systems, Bulletin of the EATCS 49 (1993) 178–183.
J. van de Pol, Modularity in Many-Sorted Term Rewriting Systems, Master's thesis, report INF/SCR-92-37, Utrecht University (1992).
A. Rubio and R. Nieuwenhuis, A Total AC-Compatible Ordering Based on RPO, Theoretical Computer Science 142 (1995) 209–227.
J. Steinbach, Termination of Rewriting: Extensions, Comparison and Automatic Generation of Simplification Orderings, Ph.D. thesis, Universität Kaiserslautern (1994).
H. Zantema, Termination of Term Rewriting: Interpretation and Type Elimination, Journal of Symbolic Computation 17 (1994) 23–50.
H. Zantema and A. Geser, Non-Looping Rewriting, report UU-CS-1996-03, Utrecht University, Department of Computer Science (1996). Available at ftp://ftp.cs.ruu.nl/pub/RUU/CS/techreps/CS-1996/1996-03.ps.gz.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ohsaki, H., Middeldorp, A. (1997). Type introduction for equational rewriting. In: Adian, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 1997. Lecture Notes in Computer Science, vol 1234. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63045-7_29
Download citation
DOI: https://doi.org/10.1007/3-540-63045-7_29
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63045-6
Online ISBN: 978-3-540-69065-8
eBook Packages: Springer Book Archive