Complexity analysis for algorithmically simple strings

Given a reference computer, Kolmogorov complexity is a well defined function on all binary strings. In the standard approach, however, only the asymptotic properties of such functions are considered because they do not depend on the reference computer. We argue that this approach can be more useful if it is refined to include an important practical case of simple binary strings. Kolmogorov complexity calculus may be developed for this case if we restrict the class of available reference computers. The interesting problem is to define a class of computers which is restricted in a {\it natural} way modeling the real-life situation where only a limited class of computers is physically available to us. We give an example of what such a natural restriction might look like mathematically, and show that under such restrictions some error terms, even logarithmic in complexity, can disappear from the standard complexity calculus. Keywords: Kolmogorov complexity; Algorithmic information theory.