Complexity of Sets Obtained as Values of Propositional Formulas

Interpretation of logical connectives as operations on sets of binary strings is considered; the complexity of a set is defined as the minimum of Kolmogorov complexities of its elements. It is readily seen that the complexity of a set obtained by the application of logical operations does not exceed the complexity of the conjunction of their arguments (up to an additive constant). In this paper, it is shown that the complexity of a set obtained by a formula $\Phi$ is small (bounded by a constant) if $\Phi$ is deducible in the logic of weak excluded middle, and attains the specified upper bound otherwise.