On the relation between representations and computability

Abstract: One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of representing data there exist incomputable functions, the computability of a specific abstract function is never an absolute property, but depends on the representation used for the function domain. We examine the scope of this dependency and provide mathematical criteria to favour some representations over others. As we shall show, there are strong reasons to suggest that computational enumerability should be an additional axiom for computation models. We analyze the link between the techniques and effects of representation changes and those of oracle machines, showing an important connection between their hierarchies. Finally, these notions enable us to gain a new insight on the Church-Turing thesis: its interpretation as the underlying algebraic structure to which computation is invariant.