TY - JOUR

T1 - ON DETERMINISTIC FINITE STATE MACHINES in RANDOM ENVIRONMENTS

AU - Ratsaby, Joel

N1 - Publisher Copyright:
Copyright © 2018 Cambridge University Press.

PY - 2019/10/1

Y1 - 2019/10/1

N2 - The general problem under investigation is to understand how the complexity of a system which has been adapted to its random environment affects the level of randomness of its output (which is a function of its random input). In this paper, we consider a specific instance of this problem in which a deterministic finite-state decision system operates in a random environment that is modeled by a binary Markov chain. The system interacts with it by trying to match states of inactivity (represented by 0). Matching means that the system selects the (t + 1)th bit from the Markov chain whenever it predicts at time t that the environment will take a 0 value. The actual value at time t + 1 may be 0 or 1 thus the selected sequence of bits (which forms the system's output) may have both binary values. To try to predict well, the system's decision function is inferred based on a sample of the random environment. We are interested in assessing how non-random the output sequence may be. To do that, we apply the adapted system on a second random sample of the environment and derive an upper bound on the deviation between the average number of 1 bit in the output sequence and the probability of a 1. The bound shows that the complexity of the system has a direct effect on this deviation and hence on how non-random the output sequence may be. The bound takes the form of O(√(2k/n)) where 2k is the complexity of the system and n is the length of the second sample.

AB - The general problem under investigation is to understand how the complexity of a system which has been adapted to its random environment affects the level of randomness of its output (which is a function of its random input). In this paper, we consider a specific instance of this problem in which a deterministic finite-state decision system operates in a random environment that is modeled by a binary Markov chain. The system interacts with it by trying to match states of inactivity (represented by 0). Matching means that the system selects the (t + 1)th bit from the Markov chain whenever it predicts at time t that the environment will take a 0 value. The actual value at time t + 1 may be 0 or 1 thus the selected sequence of bits (which forms the system's output) may have both binary values. To try to predict well, the system's decision function is inferred based on a sample of the random environment. We are interested in assessing how non-random the output sequence may be. To do that, we apply the adapted system on a second random sample of the environment and derive an upper bound on the deviation between the average number of 1 bit in the output sequence and the probability of a 1. The bound shows that the complexity of the system has a direct effect on this deviation and hence on how non-random the output sequence may be. The bound takes the form of O(√(2k/n)) where 2k is the complexity of the system and n is the length of the second sample.

KW - Markov chain

KW - frequency instability

KW - prediction of random binary sequence

KW - subsequence selection

UR - http://www.scopus.com/inward/record.url?scp=85058000369&partnerID=8YFLogxK

U2 - 10.1017/S0269964818000451

DO - 10.1017/S0269964818000451

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AN - SCOPUS:85058000369

SN - 0269-9648

VL - 33

SP - 528

EP - 563

JO - Probability in the Engineering and Informational Sciences

JF - Probability in the Engineering and Informational Sciences

IS - 4

ER -