Forkable Strings are Rare

A fundamental combinatorial notion related to the dynamics of the Ouroboros proof-of-stake blockchain protocol is that of a forkable string. The original analysis of the protocol [2] established that the probability that a string of length n is forkable, when drawn from a binomial distribution with parameter (1 − )/2, is exp(−Ω(√n)). In this note we provide an improved estimate of exp(−Ω(n)). Definition (Generalized margin and forkable strings). Let η ∈ {0, 1}∗ denote the empty string. For a string w ∈ {0, 1}∗ we define the generalized margin of w to be the pair (λ(w), μ(w)) ∈ Z × Z given by the following recursive rule: (λ(η), μ(η)) = (0, 0) and, for all strings w ∈ {0, 1}∗, (λ(w1), μ(w1)) = (λ(w) + 1, μ(w) + 1) , and