TITLE: Computational complexity of the word problem for semi-Thue systems with certain restrictions
QUESTION [6 upvotes]: The word problem (from wikipedia).

Given a semi-Thue system T: = (Σ,R)
  and two words , can u be transformed
  into v by applying rules from R?

This problem is undecidable, but with a certain restriction, it is decidable.
The Restriction:
All the rules in R are of the form A->B where A and B are string of the same length.
What is the computational complexity of this problem?

REPLY [6 votes]: Joel David Hamkins has done most of the work in his answer, but just to wrap it up, the problem is PSPACE complete. As Joel has explained, the given
semi-Thue problem can be simulated by a space-bounded, nondeterministic
Turing machine, and conversely. So the problem is (nondeterministic)
PSPACE complete. But Savitch's theorem says that this is the same as
PSPACE complete.