Some extremal problems arising from discrete control processes

Abstract

We consider a simple abstract 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 ofn bits and is a “success” or “failure” depending on whether the string produced belongs to a prespecified setL. In an uninfluenced process each bit is chosen by a fair coin toss, and hence the probability of success is ¦L¦/2n. A player called the controller, is introduced who has the ability to intervene in the process by specifying the value of some of the bits of the string. We answer the following questions for 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/Ω(n) where Ω(n) tends to infinity withn (so the probability of success with no interventions is 0(1)) then withO(√n logΩ(n)) interventions the probability of success is 1−0(1).

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

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Supported in part by NSF Grant DMS8703541 and Air Force Office of Scientific Research Grant AFOSR-0271.

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Lichtenstein, D., Linial, N. & Saks, M. Some extremal problems arising from discrete control processes. Combinatorica 9, 269–287 (1989). https://doi.org/10.1007/BF02125896

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