Title:

System functions and their decision problems

The study of System Functions arose as a consequence of the investigation of oneone equivalence between general combinatorial decision problems. For each combinatorial system such as a Semi Thus System, Thus 1 System, Turing Machine, Markov Algorithm, etc., there exists a corresponding system function. Thus, decision problems defined for these combinatorial systems can also be defined for system functions. Here, a Generalized Class of Formulas is defined in terms of a firstorder language such that each formula in this class corresponds to a decision problem for system functions. Consequently, a large number of decision problems are scanned simultaneously in order to investigate oneone equivalence between general combinatorial decision problems. These formulas are analysed to determine whether they are (a) nonsimple; (b) either finite, cofinite, or cylinders. Also, several simple decision problems and nonrecursive and noncylindrical decision problems for system functions are constructed. Furthermore, the concept of semicylinder has been extended to that of ncylinder in order to construct some of these nonrecursive and noncylindrical decision problems. As a consequence of the investigation on degrees of combinatorial decision problems, some independence and dependence relations between decision problems for system functions are obtained. Finally, an application of our concept of ncylinder in Recursive Function Theory is given.
