I haven't thought about how to go about proving it or finding a counterexample (I probably don't have the right background), but it seems intuitive to me that, given some representation of a Boolean function, say sum of products, the minimal SOP expression is somehow related to the Kolmogorov Complexity of the binary string representation. I've done some searching but can't find anything. If you don't know of a relationship but know of related references, please share.
The relationship is that a short SOP expression implies a low Kolmogorov complexity, but not vice versa. If you have a short SOP expression, then that is a concise way to compress the string, hence Kolmogorov complexity is low. However, there might be strings that can be compressed in another way even though the SOP expression is long, and those strings would have a low Kolmogorov complexity but a large SOP expression.