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 *A* *n*with *A* *n* ⊇ *A* *n* + 1 and ⋂nAn=∅" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">⋂nAn=∅⋂nAn=∅, there exists an *N* for which *A* *N* ∩ *T* = ∅; the definition of a *set of random elements* with respect to a probability measure *P* is similar, with the condition ⋂nAn=∅" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">⋂nAn=∅⋂nAn=∅replaced by a more general condition limnP(An)=0" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">limnP(An)=0limnP(An)=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.

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