A Remark on Post Normal Systems

Abstract

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