The Ackermann functions are not optimal, but by how much?

By taking a closer look at the construction of an Ackermann function we see that between any primitive recursive degree and its Ackermann modification there is a dense chain of primitive recursive degrees. Key words: Primitive recursive degree, Ackermann jump 200 AMS Classification: To be done Preamble Along with a bit of history, in these introductory remarks I will try to give an informal idea of the aims of this paper. I may use some terminology which you, the reader, may not be familiar with. Don’t worry. In the later sections, when we get down to the technical details, I will be more precise. In 1928 Ackermann [1] produced a function (on the natural numbers) that is not primitive recursive but is clearly recursive in some sense. We may view the function as 2-placed together with a parameter a. The lower levels are Aa(0, x) = a + x Aa(1, x) = a × x Aa(2, x) = ax and these are continued by A(i′ , 0) = some suitable value Aa(i′ , x′ ) = Aa(i, z) where z = Aa(i′ , x) for...

• Type: Journal
• Year: 2010
• Where: JSYML
• Authors: Harold Simmons
• Keywords: JSYML 2010, Primitive Recursive, Primitive Recursive Degree, Successor Function