Starting from POST'S [55] different notions of effective reducibility and from the fact proved by FRIEDBERG [23] and MUONIK [49] that there are recursively enumerable (r.e.) sets A , of natural numbers neither of whose decision problems 1.x . x E A , is (Turing-)reducible to the other giving incomparable r.e. degrees of unsolvability, many authors investigated the natural question as to what extent known unsolvability results could be generalized to degree analogues of them. Attention was focused on proving or disproving theorems of the following type: let A be a set of objects (for example natural numbers, machines, algorithms, logical deductive systems) and D,, . . . , D,, functions containing for every element a E A so called decision problems, i.e. pairs (a, D,(a)) with sets D,(a) (as for example halting problems, correctness problems, deducibility problems) ; one can give an effective procedure which associates to every n-tuple of degrees d,, . . . , d,, an object a E A such that D,(a) has degree d , (1 5 i n).One expresses such a fact often also by saying that every n-tuple of degrees can be effectively represented by the decision problems D,, . . ., D,& of elements of A . So we will refer to such a theorem briefly as n-tuple representation theorem for the degrees of the decision problems D, , . . . , D, of elements of A , where the particular notion of degree used has to be specified. I n the case n = 1 we omit "1-tuple". Call a pair ( A , D ) as specified above a general decision problem, i.e. a class of particular decision problems (a, D(a)). As in the case of unsolvability proofs (see for example BOONE [5], MARKOV [43], MATIJASEVIE [46], NOVIKOV [51], POST [54, [56], 1571, TURING [69 I), the main method for proving a representation theorem for the degrees of a general decision problem ( A , D ) is the effective degree preserving reduction t o ( A , D) of another general decision problem (A', D') for which the representation theorem is already known. This means that one gives an effective (normally one-one) mapping f : A' -+ A such that the corresponding decision problems D'(a') and D(f(a')) have the same degree. One then says that (A', D') is degree reducible to ( A , D ) (for the particular notion of degree considered) ; if each of them is degree reducible t o the other one they are said to be degree equivalent. SLUGLETARY [66] linked several effective Turing degree preserving reductions published in the literature for certain specific general decision problems3) together and combined them with own such reductions providing a proof of the Turing equivalence of a great number of general decision problems for algorithmic systems such as partial recursive This work is partly based on and extends the author's Habilitationsschrift [8] presented to the 2, The reader who is not acquainted with notions used in this introduction may look at $ 2 where 3, For exact references see the bibliography op. cit. Fachbereich Mathematik of the University of Miinster i. W. definitions or references to the literature are gi...
