Decision problems for semi-Thue systems with a few rules

For several decision problems about semi-Thue systems, we try to locate the frontier between the decidable and the undecidable from the point of view of the number of rules. We show that the the Termination Problem, the U-Termination Problem, the Accessibility Problem and the Common-Descendant Problem are undecidable for 3 rules semi-Thue systems. As a corollary we obtain the undecidability of the Post-Correspondence Problem for 7 pairs of words.

[1]  Martin D. Davis,et al.  Computability and Unsolvability , 1959, McGraw-Hill Series in Information Processing and Computers.

[2]  Philip K. Hooper.  The undecidability of the Turing machine immortality problem , 1966, Journal of Symbolic Logic.

[3]  D. J. Collins,et al.  Review: Ju. V. Matijasevic, Sue Walker, Simple Examples of Undecidable Associative Calculi , 1968 .

[4]  V. Borisov.  Simple examples of groups with unsolvable word problem , 1969 .

[5]  W. W. Boone,et al.  Embeddings into Semigroups with Only a Few Defining Relations , 1971 .

[6]  Gérard Huet,et al.  On the Uniform Halting Problem for Term Rewriting Systems , 1978 .

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

[8]  Jean-Jacques Pansiot.  A Note on Post's Correspondence Problem , 1981, Inf. Process. Lett..

[9]  Andrzej Ehrenfeucht,et al.  On the (Generalized) Post Correspondence Problem with Lists of Length 2 , 1981, ICALP.

[10]  D. J. Collins.  A simple presentation of a group with unsolvable word problem , 1986 .

[11]  Nachum Dershowitz,et al.  Termination of Rewriting , 1987, J. Symb. Comput..

[12]  Matthias Jantzen,et al.  Confluent String Rewriting , 1988, EATCS Monographs on Theoretical Computer Science.

[13]  Michel Billaud,et al.  Graph Rewriting Systems with Priorities , 1989, WG.

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

[15]  Anne-Cécile Caron,et al.  Linear Bounded Automata and Rewrite Systems: Influence of Initial Configurations on Decision Properties , 1991, TAPSOFT, Vol.1.

[16]  Max Dauchet,et al.  Simulation of Turing Machines by a Regular Rewrite Rule , 1992, Theor. Comput. Sci..

[17]  Yuri V. Matiyasevich.  Word Problem for Thue Systems with a Few Relations , 1993, Term Rewriting.

[18]  Friedrich Otto,et al.  String-Rewriting Systems , 1993, Text and Monographs in Computer Science.

[19]  Bernhard Gramlich,et al.  Simple Termination is Difficult , 1993, RTA.

[20]  B. Gramlich.  Simple Termination Is Diicult , 1993 .

[21]  Gerard Lallement.  The Word Problem for Thue Rewriting Systems , 1993, Term Rewriting.

[22]  Géraud Sénizergues.  Some Undecidable Termination Problems for Semi-Thue Systems , 1995, Theor. Comput. Sci..

[23]  Hans Zantema,et al.  A Complete Characterization of Termination of Op 1q -> 1r Os , 1995, RTA.

[24]  Géraud Sénizergues.  On the Termination Problem for One-Rule Semi-Thue System , 1996, RTA.

[25]  Masashi Katsura,et al.  On Termination of Confluent One-Rule String-Rewriting Systems , 1997, Inf. Process. Lett..