Exact distribution of deletion sizes for unavoidable strings

We constructively prove the exact distribution of deletion sizes for unavoidable strings, under the reductive decidability method of Zimin and Bean et al. Bounds such as these on the unique initial reductions of unavoidable strings were instrumental in proving the computational intractability of the reduction algorithm. We also provide the necessa y supporting results, including some useful approximations on the deletion sizes of individual strings. This work improves upon previous results that, although suficient to establish the desired exponential lower bound, were far from optimal.