Martin-ll of Tests Can Help Too

We argue that Martin-LL of tests provide useful techniques in proofs involving Kolmogorov complexity. Our examples cover more areas: (1) logical deenability; we prove an analogue of the 0-1 Law for various logics in terms of structures with high Kolmogorov complexity, (2) probabilistic computation: we display a relation between the Kolmogorov complexity of the random strings on which a probabilistic algorithm errs and the probability error of the algorithm as well as a trade-oo relation between the error probability, the length of the random bits, the number of provers and the length of the provers' answers in one-round multiprover interactive proof systems for NP, and (3) convergence rates of probability asymptotics; we nd a general lower bound on the convergence rate of such probabilities in terms of Kolmogorov complexity and provide a concrete example for the probability that a random graph with n vertices is connected.