The one-one equivalence of some general combinatorial decision problems

1. Introduction. A decision problem for a combinatorial system shall denote a pair ($, S) where is a specified kind of decision problem (e.g. derivability problem, halting problem, etc.) and 5 is a combina-torial system. Two decision problems (<£i, Si), ($ 2 , £2) are sa ^d t0 be of the same many-one degree (of unsolvability) if there exist effective many-one mappings ƒ and g such that each instance of ($1, Si) is reducible to an instance of ($2, S2) via ƒ and each instance of (02, S 2) is reducible to an instance of (i, Si) via g. A general combinatorial decision problem, i.e., a decision problem for a class of combinatorial systems, shall denote a pair (0, C) where 0 is a specified kind of decision problem and C is a class of combina-torial systems (e.g. Turing machines, semi-Thue systems, etc.). A general combinatorial decision problem (0i, G) is many-one reducible to another general combinatorial problem (0 2 , C 2) if there exists an effective one-one mapping \f/ of the problems p associated with (0i, Ci) into the problems associated with (0 2 , C2) such that p is of the same many-one degree as \p(p). (0i, G) and (0 2 , C 2) are said to be many-one equivalent if each is many-one reducible to the other. The reduction of one general combinatorial decision problem to another has been investigated by numerous authors. In particular, W. E. Singletary [15] has combined results of his own and those of others in such a way as to provide an effective proof of the (r.e.) equivalence of a number of general combinatorial decision problems. This former work has lead W. W. Boone to suggest that a stronger form of equivalence might exist between at least some subset of the problems considered. Our aim is to show that a number of these general problems are many-one equivalent. In addition, we indicate that these are, in a sense, best possible results.