Representation of One-One Degrees by n-Cylindrical Decision Problems

While investigating the one-one equivalence between various general combinatorial decision problems, CLEAVE [l] initated the concept of system functions. System functions are defined on natural numbers and their values are finite sets of natural numbers. They have many properties in common with those arising from GodeI numbering various combinatorial systems such as Turing machines, Markov algorithms, register machines and semi-Thue systems. Thus the decision problems defined for these combinatorial systems can also be defined for these system functions. Furthermore, if a property holds for a particular decision problem for all system functions, then it also holds for this decision problem for all these combinatorial systems. I n his study of the one-one equivalence between general combinatorial decision problems for system functions, CLEAVE [l] considered only a finite number of decision problems. We extended this study [5, 61 to include an infinite number of decision problems. This was accomplished firstly, by defining a generalized class of formulas in terms of a first-order language so that each formula in this class corresponds to a decision problem for system functions, and secondly, by analyzing these formulas to determine whether the corresponding decision problems for various kinds of system functions are cylinders. As stated by CLEAVE [l], we take this approach for the following reason: If P , and P , are two general combinatorial decision problems which are many-one equivalent and if each instance of PI and P , are cylinders, then they are one-one equivalent. This is the best possible equivalence one can obtain. While attempting to give counterexamples for those decision problems which could not be proved to be cylinders, we defined the concept of an n-cylinder [7]. It is a generalization of Yoma'8 [9] concept of a semi-cylinder. Its definition is as follows: D e f i n i t i o n. A set P is an n-cylinder if and only if there is a recursive function f such that for all and The function f is called the n-cylinder function for P.