A Prolog Specification of Giant Number Arithmetic

TL;DR

This paper introduces hereditarily binary numbers, a tree-based number representation that enables efficient arithmetic operations limited by structural complexity, with a complete declarative Prolog specification.

Contribution

It presents a novel recursive tree-based number system with run-length compression, expanding computational possibilities beyond traditional representations.

Findings

Arithmetic operations are efficient and limited by structural complexity.

The approach enables computations impossible with traditional number representations.

Complete Prolog algorithms are provided for the new number system.

Abstract

The tree based representation described in this paper, hereditarily binary numbers, applies recursively a run-length compression mechanism that enables computations limited by the structural complexity of their operands rather than by their bitsizes. While within constant factors from their traditional counterparts for their worst case behavior, our arithmetic operations open the doors for interesting numerical computations, impossible with traditional number representations. We provide a complete specification of our algorithms in the form of a purely declarative Prolog program.