Encoding Arguments

We study encoding arguments using partial prefix-free codes, which allow us to prove probabilistic statements using the basic fact that uniformly chosen elements from a set of size n cannot be encoded with fewer than log_2 n bits on average. This technique, in the end, is a more direct version of the incompressibility method from the study of Kolmogorov complexity. We use our technique to produce short original proofs for several results. We also explore extensions of the technique to non-uniform input distributions. Finally, we offer a new manner of encoding which allows for real-valued codeword lengths, thereby refining every other result obtained through encoding.