What's the current state of one-rule semi-Thue system termination problem? Search produces a lot of references, but it's hard to find out if decidability of this problem has been proven or not. 



What's the current state of one-rule semi-Thue system termination problem?
It's still (personal experience) agonisingly difficult. The advance using automata was done by Hans Zantema and his friends for some classes of one-rule systems. Also there is a long paper by Kobayashi and some other guys where they develop the whole theory about termination for complicated classes of 1-rule systems based on splitting words in certain way (and in which it is made more explicit the old proof of Senizerques about termination of the systems of the form $0^p1^q\to 1^r0^s$).


Also in the paper of Victor Guba (that famous 1998 paper about one-relator semigroups) there sits inside the proof of the main theorem where he associates another presentation for the group generated by prefixes, that we have a rewriting system (not necessarily 1-rule) and the question is whether it will terminate. May be something can be squized from there but I failed.


Well, the situation is very strange with 1-rule systems. For some time it was beleived that non-termination happens if we can spot loops. But there are examples showing that shortest possible loops to appear we have to wait superexponentially long (in input of the 1-rule system). It's all wild.


In general it's very good such brave people still exist who are interested in this difficult problem. Just in case my e-mail: victor.maltcev@gmail.com, please do contact me and we could talk more


P.S.: About word problem for 1-relators, it's a mystery. Yet, good people believe that Dehn function for 1-relator semigroups have to be polynomial, and for guys like $\langle a,b:aUb=b\rangle$ even at most quadratic. We've got some examples of 1-relator semigroups which we (unproved) believe cannot admit finite complete rewriting systems


