Primitive recursive functions versus partial recursive functions: comparing the degree of undecidability

Consider a decision problem whose instance is a function. Its degree of undecidability, measured by the corresponding class of the arithmetic (or Kleene-Mostowski) hierarchy hierarchy, may depend on whether the instance is a partial recursive or a primitive recursive function. A similar situation happens for results like Rice Theorem (which is false for primitive recursive functions). Classical Recursion Theory deals mainly with the properties of partial recursive functions. We study several natural decision problems related to primitive recursive functions and characterise their degree of undecidability. As an example, we show that, for primitive recursive functions, the injectivity problem is Π1-complete while the surjectivity problem is Π2-complete. We compare the degree of undecidability (measured by the level in the arithmetic hierarchy) of several primitive recursive decision problems with the corresponding problems of classical Recursion Theory. For instance, the problem “does the codomain of a function have exactly one element?” is Π1-complete for primitive recursive functions and belongs to the class ∆2 \ (Σ1 ∪ Π1) for partial recursive functions. An important decision problem, “does a given primitive recursive function have at least one zero?” is studied in detail; the input and output restrictions that are necessary and sufficient for the decidability this problem – its “frontiers of decidability” – are established. We also study a more general situation in which a primitive recursive function (the instance of the problem) is a part of an arbitrary “acyclic primitive recursive function graph”. This setting may be useful to evaluate the relevance of a given primitive recursive function as a part of a larger primitive recursive structure.