In 1961 Martin Davis, Hilary Putnam and Julia Robinson [2] proved that every recursively enumerable set W is exponential diophantine, i.e. can be represented in the formHere P is a polynomial with integer coefficients and the variables range over positive integers.In 1970 Ju. V. Matijasevič used this result to establish the unsolvability of Hilbert's tenth problem. Matijasevič proved [11] that the exponential relation y = 2x is diophantine This together with [2] implies that every recursively enumerable set is diophantine, i.e. every r.e. set Wcan be represented in the formFrom this it follows that there does not exist an algorithm to decide solvability of diophantine equations. The nonexistence of such an algorithm follows immediately from the existence of r.e. nonrecursive sets.Now it is well known that the recursively enumerable sets W1, W2, W3, … can be enumerated in such a way that the binary relation x ∈ Wv is also recursively enumerable. Thus Matijasevič's theorem implies the existence of a diophantine equation U such that for all x and v,

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The logicians notations 3V3, V333, etc., will be used here to denote classes of arithmetical predicates in prenex form with polynomial matrix. For example the prefix 3V3V3 represents formulas of the form (11) 3x1 vx, 3x3 vz4 3X,[P(X1, x2, 23, x4, xg) = 01.Here the quantified variables xl, x2, x3, x4, x5 are to be understood to range over natural numbers and P = P(a, xl, xz, x3, x,, x5) is a polynomial with integer coefficients. Such formulas define arithmetical sets.We may also use the notation 3V3V3 more restrictedly, to refer only to the class of formulas of the formwhere the universal quantifiers are all bounded by polynomials A(x,), B(xl, x3), with natural coefficients. Such formulas define recursively enumerable sets.In [5] Ju. V. MATIJASEVI~: undertook to classify prefixes as to the decidability or undecidability of the predicates represented. According to his classification a prefix defining only recursive sets was to be considered solvable. A prefix sufficient for the definition of some (at least one) non-recursive set was to be considered unsolvable. This definition places every prefix into one of two disjoint classes. However, in order to be able to refer to prefixes whose status is as yet undetermined (open problems) we require a third category, undecided.I n this paper we continue the classification project initiated by MATIJASEVI~. Since 1971 new results have been obtained so that the number of undecided prefixes has become smaller.MATIJASEVI~! attempted only the classification of prefixes in which V is bounded. Here we take interest also in the other type and the problem of classifying these. The results are different for the two categories. So it is necessary to give two classification tables. Fortunately however, the notation is not so ambiguous as might first be thought.The unbounded type (11) prefixes include the bounded type (I) as a special case. To prove this we need a theorem of R. M. ROBINSON [l2]. P r o p o s i t i o n 1. (R.M. ROBINSON). If P(x, y ) i s a polynomial in x and y , then Vy

The purpose of the present paper is to give a new, simple proof of the theorem of M. Davis, H. Putnam and J. Robinson [1961], which states that every recursively enumerable relation A(a1, …, an) is exponential diophantine, i.e. can be represented in the formwhere a1 …, an, x1, …, xm range over natural numbers and R and S are functions built up from these variables and natural number constants by the operations of addition, A + B, multiplication, AB, and exponentiation, AB. We refer to the variables a1,…,an as parameters and the variables x1 …, xm as unknowns.Historically, the Davis, Putnam and Robinson theorem was one of the important steps in the eventual solution of Hilbert's tenth problem by the second author [1970], who proved that the exponential relation, a = bc, is diophantine, and hence that the right side of (1) can be replaced by a polynomial equation. But this part will not be reproved here. Readers wishing to read about the proof of that are directed to the papers of Y. Matijasevič [1971a], M. Davis [1973], Y. Matijasevič and J. Robinson [1975] or C. Smoryński [1972]. We concern ourselves here for the most part only with exponential diophantine equations until §5 where we mention a few consequences for the class NP of sets computable in nondeterministic polynomial time.