One-Rule Trace-Rewriting Systems and Confluence

Confluence is an undecidable property even for finite noetherian trace-rewriting systems. Here we investigate this property for the special case of trace-rewriting systems with a single rule. For certain classes of one-rule trace-rewriting systems R we present syntactic characterizations that are necessary and sufficient for R to be confluent. Based on these characterizations confluence is easily decidable for these systems. In fact, each system satisfying one of these characterizations is strongly confluent. This raises the question of whether every confluent onerule trace-rewriting system is strongly confluent. We close this paper with an example that answers this question in the negative.

[1]  Hélène Kirchner,et al.  Completion of a Set of Rules Modulo a Set of Equations , 1986, SIAM J. Comput..

[2]  Volker Diekert,et al.  Combinatorics on Traces , 1990, Lecture Notes in Computer Science.

[3]  Max Dauchet,et al.  Termination of Rewriting is Undecidable in the One-Rule Case , 1988, MFCS.

[4]  Klaus U. Schulz.  Word Equations and Related Topics , 1990, Lecture Notes in Computer Science.

[5]  Robert Cori,et al.  Automates et Commutations Partielles , 1985, RAIRO Theor. Informatics Appl..

[6]  Volker Diekert,et al.  On the Knuth-Bendix Completion for Concurrent Processes , 1987, Theor. Comput. Sci..

[7]  Celia Wrathall.  Confluence of One-Rule Thue Systems , 1990, IWWERT.

[8]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[9]  Václav Koubek,et al.  Mathematical Foundations of Computer Science 1988 , 1988, Lecture Notes in Computer Science.

[10]  Jean-Pierre Jouannaud,et al.  Rewrite Systems , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[11]  Christine Duboc,et al.  On Some Equations in Free Partially Commutative Monoids , 1986, Theor. Comput. Sci..

[12]  Friedrich Otto,et al.  On Deciding Confluence of Finite String-Rewriting Systems Modulo Partial Commutativity , 1989, Theor. Comput. Sci..

[13]  Friedrich Otto,et al.  Overlaps in Free Partially Commutative Monoids , 1991, J. Comput. Syst. Sci..

[14]  Paliath Narendran,et al.  Preperfectness is Undecidable for Thue Systems Containing Only Length-Reducing Rules and a Single Commutation Rule , 1988, Inf. Process. Lett..

[15]  Michal P. Chytil,et al.  Mathematical Foundations of Computer Science 1984 , 1984, Lecture Notes in Computer Science.

[16]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[17]  Gerard Huet,et al.  Conflunt reductions: Abstract properties and applications to term rewriting systems , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[18]  Gérard P. Huet,et al.  Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems , 1980, J. ACM.

[19]  Antoni W. Mazurkiewicz,et al.  Traces, Histories, Graphs: Instances of a Process Monoid , 1984, International Symposium on Mathematical Foundations of Computer Science.

[20]  Winfried Kurth,et al.  Termination und Konfluenz von Semi-Thue-Systemen mit nur einer Regel , 1990 .