A Variation on the Zero-One Law

Given a decision problem P and a probability distribution over binary strings, for each n, draw independently an instance xn of P of length n. What is the probability that there is a polynomial time algorithm that solves all instances xn of P ? The answer is: zero or one. Keywords: polynomial time solvability, zero-one law At several meetings, J. Hartmanis asked: If it turns out that NP-hard problems are not solvable in polynomial time, will it mean that there is a "hard" sparse sequence of instances, i.e., a sequence which is hard for every polynomial-time algorithm? A natural next question is: how frequent are such "hard" sequences? If we pick a sequence "at random", what is the chance that this randomly chosen sequence is hard? In principle, it could happen that almost all sequences are hard; it could happen that almost all sequences are easy; in principle, it may seem that a third alternative is also possible: that, say, half of all sequences (or any other portion different from ..