Negative Results of Computable Analysis Disappear If We Restrict Ourselves to Random (Or, More Generally, Typical) Inputs

Authors

Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

3-2012

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 Solvable

Short 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 in Mathematical Structures and Modeling.

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.