It is well known that many computational problems are, in general, not algorithmically solvable: e.g., it is not possible to algorithmically decide whether two computable real numbers are equal, and it is not possible to compute the roots of a computable function. We propose to constraint such operations to certain "sets of typical elements" or "sets of random elements".
In our previous papers, we proposed (and analyzed) physics-motivated definitions for these notions. In short, a set T is a set of typical elements if for every definable sequences of sets An for which each An is a subset of An+1 and the intersection of all An is empty, there exists an N for which T has no common elements with AN; the definition of a set of random elements with respect to a probability measure P is similar, with the condition that the intersection of all An is empty replaced by a more general condition that this intersection has probability 0.
In this paper, we show that if we restrict computations to such typical or random elements, then problems which are non-computable in the general case -- like comparing real numbers or finding the roots of a computable function -- become computable.