The undecidability of the preperfectness of thue systems

Since the word problem for Thue systems was proved to be undecidable by Post and Markov in the 1940's. there has been some interest in developing conditions under which the word problem is decidable. At present it is possible to decide whether a finite Thue system is almost-confluent or Church-Rosser, but testing for preperfectness has been an open problem for years. Here we prove this to be undecidable.