Hyperalgorithms & Superhyperalgorithms: A Unified Framework for

Higher-Order Computation

Takaaki Fujita 1,∗

1Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan

Email: takaaki.fujita060@gmail.com

Abstract

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 Superhyper-

algorithm. By leveraging the mathematical frameworks of hyperstructures and superhyperstructures, respec-

tively, 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 proper-

ties, laying the groundwork for a unified theory of higher-order algorithms.

Keywords: Algorithm; Multialgorithm; Hyperalgorithm; Superhyperalgorithm