  So our interest must be in those decompression functions which do best overall. We might hope to say: \(f\) is better than \(g\) iff for all \(\sigma , C_f (\sigma) \le C_g (\sigma)\). Unfortunately, no function is best in this sense, since for any given string \(\sigma\) with \(f\)-complexity \(\lvert\sigma\rvert - k\), we can design a function \(g\) as follows: on input 1, output \(\sigma\); on input n\(\delta\) (for any \(n)\), output \(f(\delta)\). (Generalising, we can add arbitrarily long prefixes of length \(m\) onto the inputs to \(g\) and have better-than\(-f\) compression for \(2^m\) sequences.) But we can define a notion of complexity near-superiority of \(f\) to \(g\) iff there is some constant \(k\) such that for any string, \(C_f (\sigma) \le C_g (\sigma) + k. f\) is least as good as \(g\), subject to some constant which is independent of the functions in question. If \(f\) and \(g\) are both complexity near-superior to each other, for the same \(k\), we say they are complexity equivalent.