Hyperalgorithms & Superhyperalgorithms: A Unified Framework for Higher-Order Computation

An algorithm is a finite, well-defined computational procedure that transforms inputs into outputs through a structured sequence of steps, guaranteeing termination and correctness. A multialgorithm comprises multiple algorithms augmented with a selection mechanism that dynamically chooses the most appropriate procedure based on input characteristics or contextual conditions. While these concepts have deep roots in computer science and beyond, this paper introduces two novel generalizations: the Hyperalgorithm and the Superhyperalgorithm. By leveraging the mathematical frameworks of hyperstructures and superhyperstructures, respectively, we extend the classical notion of computation to higher-order operations on sets and iterated powersets. We present formal definitions, illustrative examples, and a preliminary analysis of their computational properties, laying the groundwork for a unified theory of higher-order algorithms.