## Departmental Papers (CS)

#### Title

Under Physics-Motivated Constraints, Generally-Non-Algorithmic Computational Problems Become Algorithmically Solvable

1-1-2014

#### Document Type

Book Chapter

Kreinovich V. (2014) Under Physics-Motivated Constraints, Generally-Non-Algorithmic Computational Problems become Algorithmically Solvable. In: Ceberio M., Kreinovich V. (eds) Constraint Programming and Decision Making. Studies in Computational Intelligence, vol 539. Springer, Cham

https://doi.org/10.1007/978-3-319-04280-0_11

#### Abstract

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 A nwith 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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