A Remark on Post Normal Systems

Let <italic>N</italic>(<italic>T</italic>) be the normal system of Post which corresponds to the Thue system, <italic>T</italic>, as in Martin Davis, <italic>Computability and Unsolvability</italic> (McGraw-Hill, New York, 1958), pp. 98-100. It is proved that for any recursively enumerable degree of unsolvability, <italic>D</italic>, there exists a normal system, <italic>N</italic><subscrpt><italic>T</italic>(<italic>D</italic>)</subscrpt>, such that the decision problem for <italic>N</italic><subscrpt><italic>T</italic>(<italic>D</italic>)</subscrpt> is of degree <italic>D</italic>. Define a generalized normal system as a normal system without initial assertion. For such a <italic>GN</italic> the decision problem is to determine for any enunciations <italic>A</italic> and <italic>B</italic> whether or not <italic>A</italic> and <italic>B</italic> are equivalent in <italic>GN</italic>. Thus the generalized system corresponds more naturally to algebraic problems. It is proved that for any recursively enumerable degree of unsolvability, <italic>D</italic>, there exists a generalized normal system, <italic>GN</italic><subscrpt><italic>T</italic>(<italic>D</italic>)</subscrpt>, such that the decision problem for <italic>GN</italic><subscrpt><italic>T</italic>(<italic>D</italic>)</subscrpt>, such that the decision problem for <italic>GN</italic><subscrpt><italic>T</italic>(<italic>D</italic>)</subscrpt> is of degree <italic>D</italic>.