Word Problems and Recursively Enumerable Degrees of Unsolvability

Abstract

In [2] it was shown that for every r.e. (i.e., recursively enumerable) Turing degree of unsolvability D there exists a Thue system with word problem of degree D. In [3] the analogous result was shown for finite presentations of groups. Except for minor slips set aright at the end, this corrective note is concerned only with footnote 8, p. 523 of [2] and footnote 4, p. 50 of [3]. In these footnotes it is claimed that, in effect, somewhat sharper results had been proved, viz., the corresponding results for tt (i.e., unbounded truth-table) degrees instead of Turing degrees. Two points should be made: these footnotes do not bear upon the validity of the body of [2] and [3] as they were not used, indeed not even referred to, elsewhere in these papers; moreover, these stronger results about tt degrees are in fact valid, and for presentations having the particular form discussed. What is seriously amiss is the implied claim that both these tt results could be verified by the interested reader once he had understood [2] and [3]. For the weaker result about Thue systems and tt degrees this still seems possible' to the author; but certainly not for the stronger result about finite presentations of groups and tt degrees no matter how dedicated the reader! (In a letter to the present author, A. A. Fridman has explained that his methods of [6] and [7] do not settle the question about finitely presented groups and truth-tables either.) Fortunately all the required arguments have now been supplied by Donald J. Collins in DJC.2' Remark (1) of footnote 8 of [2] claims that Thue system Zs i.e., Z, of [2], is tt-reducible to the r.e. set S of natural numbers. From this follows the existence of a Thue system with word problem of arbitrarily preassigned r.e. tt degree. Remark (1) is correct and can be verified by a straightforward systematic reworking of [2], much like writing a computer program; this