Kolmogorov had an outline of an attempt using Kolmogorov Complexity to settle P vs NP. I do not unfortunately recall the crux of the argument. This was formulat

Kolmogorov had an outline of an attempt using Kolmogorov Complexity to settle P vs NP. I do not unfortunately recall the crux of the argument. This was formulated in the 1970s, and has not been followed up as far as I know. On the other hand, given Baker-Gill-Solovay and the Natural Proof barriers, no proof settling the question is expected to be easy.