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

  • Vladik KreinovichEmail author
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Part of the Studies in Computational Intelligence book series (SCI, volume 539)

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 with A n  ⊇ A n + 1 and \(\bigcap\limits_{n} A_n=\emptyset\), 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 \(\bigcap\limits_{n} A_n=\emptyset\) replaced by a more general condition \(\lim\limits_n P(A_n)=0\).

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.

Keywords

constraints computable problems random elements typical elements 

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Texas at El PasoEl PasoUSA

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