Imperfect random sources and discrete controlled processes

We consider a simple model for a class of discrete control processes, motivated in part by recent work about the behavior of imperfect random sources in computer algorithms. The process produces a string of characters from {0, 1} of length n and is a “success” or “failure” depending on whether the string produced belongs to a prespecified set L. In an uninfluenced process each character is chosen by a fair coin toss, and hence the probability of success is |L|/2ⁿ. We are interested in the effect on the probability of success in the presence of a player (controller) who can intervene in the process by specifying the value of certain characters in the string. We answer the following questions in both worst and average case: (1) how much can the player increase the probability of success given a fixed number of interventions? (2) in terms of |L| what is the expected number of interventions needed to guarantee success? In particular our results imply that if |L|/2ⁿ = 1/w(n) where w(n) tends to infinity with n (so the probability of success with no interventions is o(1)) then with Ο(√nlogw(n)) interventions the probability of success is 1-o(1).

Our main results and the proof techniques are related to a well-known theorem of Kruskal, Katona, and Harper in extremal set theory.