Funções parciais recursivas e funções parcialmente Turing-computáveis: uma prova de equivalência

Abstract

In the thirties of the last century, several formal versions for the intuitive notion of algorithmic function were offered. Among them, the version of the recursive functions and the version of the Turing-computable functions. Posteriorly, such versions were extended in order to also include the partial algorithmic functions, giving rise, in this way, to the version of the partial recursive functions and to the version of the partially Turing-computable functions. In this context, this research, located into Computability Theory domain and built in the light of theoretical assumptions of Davis (1982), Mendelson (2009), Dias & Weber (2010), Rogers (1987), Soare (1987), Cooper (2004), among others, is intended to rebuild the proof that the given formal versions referred to the intuitive notion of partial algorithmic function, despite being conceptually distinct, they are extensionally equivalents in the sense that they determine the same set of theoretical-numerical functions. As a part of this rebuilding, we shall prove, in na unprecedented way, using quintuples, that every partial recursive function is partially Turing-computable. In the literature, this theorem is proved by means of a set of quadruples. However, defining a lower cardinality set constructed by quintuples, it is possible to prove it in a smaller time interval, which representes a gain from the computational point of view. Besides presenting this alternative proof, posed by the Church-Turing thesis that the set of partial recursive functions includes all the partial algorithmic functions, we shall investigate if this set itself and its infinite subsets are or are not algorithmic. In this survey, we shall demonstrate, in arithmetical terms, with the aid of Rice‟s theorem, that although the set of partial recursive functions is algorithmic, all its subsets which are different from the empty set are not, among which are the set of recursive functions and the set of primitive recursive functions.