# Concrete calcules of natural number arithmetic for recursion

1. The most usual approach to calculative (effectively calculable) functions is done by register machines or similar storage-based computers like the Abacus or Turing machines. The concrete calcule NU of decimal arbative arithmetic uses the ARBACUS a very simple computer with programs that are coded by decimal numbers that are called  arbors. There are four commands only:  nullification, succession, deletion and duplication.  It turns out that all of the storage-based computers lead to the same functions that are identified as the primitive recursive functions. Special care must be taken that the programs that represent these functions always come to an end. This is accomplished by the ARBATOR calculator. Computers have halting problems, per definition calculators do not.

2.  Another usual approach to calculative functions is to start with primitive recursive functions. However, one has to find a way to put this into a form that does not rely on a pre-knowledge about functions and higher logic. The concrete calcule LAMBDA of decimal pinitive arithmetic allows for such an  access. It is based on a machine that is completely different from the storage-based machines: the PINITOR does not use storages but rather many microprocessors, one for each appearance of a command in the code of primitive recursive functions. The codes are decimal numbers, called pinons , where only the characters  0 1 2 8 9 appear. There are  four kind of commands only: 0 nullification , 1 succession, 2 straight recursion  and 8 composition. The PINITOR is a calculator  which means  that there is no halting problem.

There will be two pdf.files with extensive descriptions of the two calcules.