We extend a classical result in ordinary recursion theory to higher recursion theory, namely that every recursively enumerable set can be represented in any model A by some Horn theory, where A can be any model of a higher recursion theory, like primitive set recursion, αrecursion, or β -recursion. We also prove that, under suitable conditions, a set defined through a Horn theory in a set A is ...

Recursion is a complex concept that most novice logic programmers have difficulty grasping. Problems associated with recursion are avoided in imperative languages where iteration is provided as an alternative to recursion. Although difficult to learn, recursion is very easy to use once it is understood. In fact, many problems that have straightforward recursive solutions have very sophisticated...

We introduce a variant of Spector’s bar recursion (called “modified bar recursion”) in finite types to give a realizability interpretation of the classical axiom of countable choice allowing for the extraction of witnesses from proofs of Σ1 formulas in classical analysis. As a second application of modified bar recursion we present a bar recursive definition of the fan functional. Moreover, we ...

In this paper a new method for removing recursion from algorithms is demonstrated. The method for removing recursion is based on algebraic manipulations of a mathematical model of the control flow. The method is not intended to solve all possible recursion removal problems, but instead can be seen as one tool in a larger tool box of program transformations. Our method can handle certain types o...

We give a recursion-theoretic characterization of the complexity classes NC for k ≥ 1. In the spirit of implicit computational complexity, it uses no explicit bounds in the recursion and also no separation of variables is needed. It is based on three recursion schemes, one corresponds to time (time iteration), one to space allocation (explicit structural recursion) and one to internal computati...