Hyperoperations for Science and Technology

The authors called “Hyperoperations” a set of algorithmic tools studied by scientists since Euler's times (18th century). Modern mathematical sectors of computer science and artificial intelligence (AI) have shown that these tools are in close relationship with iteration and recursivity, as well as with mathematical objects such as the Ackermann's Function and the Grzegorczyk Hierarchy. The properties of the fourth rank operation of this hierarchy (Tetration) are analyzed, in view of the interest recently shown in this operation by the scientific community around the world. The definition of the new level zero operation (Zeration), almost coincident with Ackermann's “Successor” operation, has also been proposed. A new number notation format (RRH, the Rubtsov-Romerio Hyperformat) to be used for representing very large (and very small) numbers, is defined, similar to the scientific “floating point notation”, but using tetration instead of exponentiation. The problem of extending tetration to the field of the real numbers is also analyzed, together with some approximation proposals.