Originator: Max Dauchet
Date: April 1991
Is termination of one linear rule decidable?
Is termination of one linear (left and right) rule decidable? Left linearity
alone is not enough for decidability [Dau89].
A less ambitious, long-standing open problem (mentioned in [DJ90])
is decidability for one (length-increasing) monadic (string,
semi-Thue) rule. Termination is undecidable for non-length-increasing
monadic systems of rules [Car91]. For one monadic rule, confluence
is decidable [Kur90][Wra90]. What about confluence of one
Partial results for string rewrite rules have been obtained in
The history of the problem and the attempts to solve it are told in
Linear bounded automata and rewrite systems: Influence of initial
configurations on decision properties.
In Proceedings of the International Joint Conference on Theory
and Practice of Software Development, volume 1: Colloquium on Trees in
Algebra and Programming (Brighton, U.K.), volume 493 of Lecture Notes in Computer
Science, pages 74–89, Berlin, April 1991.
Simulation of Turing machines by a left-linear rewrite rule.
In Nachum Dershowitz, editor, Rewriting Techniques and
Applications, volume 355 of Lecture Notes in Computer
Science, pages 109–120, Chapel Hill, NC, USA, April 1989.
In Jürgen Giesl, editor, 16th International
Conference on Rewriting Techniques, volume 3467 of Lecture Notes in Computer
Science, Nara, Japan, April 2005.
Nachum Dershowitz and Jean-Pierre Jouannaud.
In J. van Leeuwen, editor, Handbook of Theoretical Computer
Science, volume B: Formal Methods and Semantics, chapter 6, pages 243–320.
North-Holland, Amsterdam, 1990.
Termination of string rewriting rules that have one pair of overlaps.
In Robert Nieuwenhuis, editor, 14th International Conference on
Rewriting Techniques, volume 2706 of Lecture Notes in Computer
Science, pages 410–423, Valencia, Spain, June 2003.
Termination und Konfluenz von Semi-Thue-Systems mit nur einer
PhD thesis, Technische Universitat Clausthal, Clausthal, Germany,
Confluence of one-rule Thue systems.
In Proceedings of the First International Workshop on Word
Equations and Related Topics (Tubingen), volume 572 of Lecture Notes in Computer
Science, pages 237–246, Berlin, 1990.