Algorithmic complexity as a criterion of unsolvability

Complexity and Unsolvability Properties of Nilpotency

A nonassociative algebra is nilpotent if there is some n such that the product of any n elements, no matter how they are associated, is zero. Several related, but more general, notions are left nilpotency, solvability, local nilpotency, and nillity. First the complexity of several decision problems for these properties is examined. In finite-dimensional algebras over a finite field it is shown ...

Algorithmic Folding Complexity

How do we most quickly fold a paper strip (modeled as a line) to obtain a desired mountain-valley pattern of equidistant creases (viewed as a binary string)? Define the folding complexity of a mountain-valley string as the minimum number of simple folds required to construct it. We first show that the folding complexity of a length-n uniform string (all mountains or all valleys), and hence of a...

Algorithmic “Kolmogorov” Complexity

The information content or complexity of an object can be measured by the length of its shortest description. For instance the string ‘01010101010101010101010101010101’ has the short description “16 repetitions of 01”, while ‘11001000011000011101111011101100’ presumably has no simpler description other than writing down the string itself. More formally, the Algorithmic “Kolmogorov” Complexity (...

Algorithmic Randomness and Complexity

Algorithmic Problem Complexity

People solve different problems and know that some of them are simple, some are complex and some insoluble. The main goal of this work is to develop a mathematical theory of algorithmic complexity for problems. This theory is aimed at determination of computer abilities in solving different problems and estimation of resources that computers need to do this. Here we build the part of this theor...