Some Undecidable Termination Problems for Semi-Thue Systems

We show that the uniform termination problem is undecidable for length-preserving semi-Thue systems having 10 rules. We then give an explicit uniformly-terminating semi-Thue system \(\mathcal{T}\) having 9 rules which is “universal with respect to termination problems” in some sense.It follows that there exists a fixed rule \({u_0},{v_0}\) such that \(\mathcal{T} \cup \left\{ {\left( {{u_0},{v_0}} \right)} \right\}\) has 10 rules and undecidable termination problem.

[1]  Max Dauchet.  Simulation of Turning Machines by a Left-Linear Rewrite Rule , 1989, RTA.

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

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

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

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

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

[7]  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).

[8]  Donald J. Collins.  Word and conjugacy problems in groups with only a few defining relations , 1969 .

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

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

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

[12]  Friedrich Otto,et al.  Cancellation Rules and Extended Word Problems , 1985, Inf. Process. Lett..

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

[14]  Ronald V. Book,et al.  Homogeneous Thue systems and the Church-Rosser property , 1984, Discret. Math..

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

[16]  Max Dauchet,et al.  Automata and Rational Expressions on Planar Graphs , 1988, MFCS.

[17]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .