Funções recursivas primitivas: caracterização e alguns resultados para esta classe de funções

Abstract

The class of primitive recursive functions is not a formal version to the class of algorithmic functions, we study this special class of numerical functions due to the fact of that many of the functions known as algorithmic are primitive recursive. The approach on the class of primitive recursive functions aims to explore this special class of functions and from that, present solutions for the following problems: (1) given the class of primitive recursive derivations, is there an algorithm, that is, a mechanical procedure for recognizing primitive recursive derivations? (2) Is there a universal function for the class of primitive recursive functions? If so, is this function primitive recursive? (3) Are all the algorithmic functions primitive recursive? To provide solutions to these issues, we base on the hypothetical-deductive method and argue based on the works of Davis (1982), Mendelson (2009), Dias e Weber (2010), Rogers (1987), Soare (1987), Cooper (2004), among others. We present the theory of Turing machines which is a formal version to the intuitive notion of algorithm, and after that the famous Church-Turing tesis which identifies the class of algorithmic functions with the class of Turing-computable functions. We display the class of primitive recursive functions and show that it is a subclass of Turing-computable functions. Having explored the class of primitive recursive functions we proved as results that there is a recognizer algorithm to the class of primitive recursive derivations; that there is a universal function to the class of primitive recursive functions which does not belong to this class; and that not every algorithmic function is primitive recursive.