#### Publication Date

3-2012

#### 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_{n} for which each A_{n} is a subset of A_{n+1} and the intersection of all A_{n} is empty, there exists an N for which T has no common elements with A_{N}; 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 A_{n} 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.

*Original file: CS-UTEP-10-46*

tr10-46a.pdf (67 kB)

*Short version: CS-UTEP-10-46a*

tr10-46b.pdf (115 kB)

*Extended version: CS-UTEP-10-46b*

## Comments

Technical Report: UTEP-CS-10-46c

Short version (UTEP-CS-10-46a) published with the title:

Under Physics-Motivated Constraints, Generally-Non-Algorithmic Computational Problems Become Algorithmically SolvableShort version published in

Proceedings of the Fourth International Workshop on Constraint Programming and Decision Making CoProD'11, El Paso, Texas, March 17, 2011; full paper to appear inMathematical Structures and Modeling.