The Representation of Many—One Degrees by the Word Problem for Thue Systems

The object of this paper is to show that, given any total recursive function g, one can effectively construct a Thue system T such that the decision problem for the range of g and the word problem for T are of the same many-one degree. It follows immediately that any many-one degree containing a recursively enumerable (r.e.) set can be represented by the word problem of some Thue system. The properties of Thue systems were investigated by Post ([4]), who was able to show that the general word problem is undecidable. Later Boone ([1]) was able to establish Turing equivalence between the word problem of Thue systems and the word problem for groups. Shepherdson ([5]) gave a simplified construction showing the Turing equivalence of the decision problem for r.e. sets, the decision problems (halting, derivability, and confluence) for Turing machines, and the word problem for Thue systems. As a refinement of a section of Shepherdson's investigation, Overbeek ([3]) established the many-one equivalence of the decision problem for r.e. sets and the decision problems for Turing machines. This paper is a logical extension of that work, carrying many-one equivalence into the general word problem for Thue systems. Singletary ([2]) has shown that these are the best possible results in the sense that not all r.e. one-one degrees can be represented by the word problem for Thue systems. There will be three major steps in the construction of T. (1) Given a total recursive function g, a Turing machine M* is defined such that the confluence problem for M* restricted to a recursive set of configurations is many-one equivalent to the decision problem for the range of g. (2) A Thue system T* is constructed and a recursive set of words P is defined such that the word problem of T* over P and the restricted confluence problem for M* are of the same many-one degree.