The many-one equivalence of some general combinatorial decision problems

Abstract

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

The Representation of Many-One Degrees by the Word Problem for Thue Systems

“…(1) W X RW 2 The following procedure reduces the word problem for T to the word problem for T* over P. A careful examination of this reduction will convince the reader that it is, in fact, many-one.…”

“…either W x or W 2 does not contain an R, then it is decidable whether T^Hjr-W£ by Statement(2). and W^ can be effectively determined and contain no occurrences of R.(C) Both W 3 and W± must contain at least one occurrence of an L or the word problem for W3 and W^ is decidable by Statement (3).…”

Many-one degrees associated with problems of tag

“…This function will be used in later rules. We shall now define those rules which are directly related to the instructions of R If 7 f is R k (n), for k e (0, 1, 2, 3,4) and n e (1, Associated with (5) and ( -+x*,0 ifj=2i*+P/2, -> D, 0 otherwise. Associated with this we have the standard tag rule yf -> 2)w*-a<j C *2)3''/*+a«-i.…”

Combinatorial systems defined over one- and two-letter alphabets

“…More recent work has had as its objective the proof that these Turing degree results can be strengthened to many-one degrees. Examples of papers falling into this category are [6], [7], [8], [9], [10], and [12]. The constructions presented in each of the above lead to combinatorial systems in which the alphabets are finite but not uniformly bounded.…”