No abstract

Introduction.A decision problem for a combinatorial system shall denote a pair ($, S) where is a specified kind of decision problem (e.g. derivability problem, halting problem, etc.) and 5 is a combinatorial system. Two decision problems (<£i, Si), ($ 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 combinatorial 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.

No abstract

Introduction.A decision problem for a combinatorial system shall denote a pair (<ï>, S) where $ is a specified kind of decision problem (e.g. word problem, halting problem, etc.) and S is a combina* torial system. Likewise, a general combinatorial decision problem, i.e. a decision problem for a class of combinatorial systems, shall denote a pair (<ï >> C), where * is a specified kind of decision problem and C is a general class of combinatorial systems (e.g. Turing machines, semi-Thue systems, etc.). Clearly, each general combinatorial decision problem P has a class of decision problems for combinatorial systems associated with it. We shall refer to these problems simply as the problems associated with P.There are many papers in the literature which deal with the reduction of one general combinatorial decision problem to another. These papers fall into two general groups. The first group consists of unsolvability proofs such as . The general format of these proofs is the following: Two general combinatorial decision problems Pi and P 2 are considered, where Pi is known to have an associated problem of each r.e. degree of unsolvability. Then an effective one-one mapping \j/ of the problems p associated with Pi into the problems associated with P 2 and uniformly effective reductions of p to \p(p) and of \p(p) to p are given.Our aim here is to link several of these reductions together in such a way as to provide an effective proof of the equivalence of a number of general combinatorial decision problems. Furthermore, all of our reductions will conform to the second format given above and hence for each pair P»-, Pj of general combinatorial decision problems con-

No abstract