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An Investigation of the Concept of Influence in Boolean Functions on Random Dependent and Independent Variables

Salari Nasab, Mojtaba | 2024

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 58199 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Alishahi, Kasra
  7. Abstract:
  8. Boolean functions are multi-variable functions where the values of each variable and the value of the function itself come from a two-valued set—usually {-1, 1} or {0, 1}. These functions are fundamental in theoretical computer science and play a key role in fields such as combinatorics, random graph theory, statistical physics, and social choice theory. By considering each variable of the function as a random variable, Boolean functions can be viewed as random variables, thereby presenting a probabilistic structure. An important question in this context is: "What effect does each variable have on the final value of the function?" When the variables are independent of each other, a satisfactory answer to this question has been provided by defining the concept of "influence" for each variable. Based on this definition, important theorems such as the Margulis-Russo theorem, discrete Poincaré inequality, KKL theorem, and BKKKL theorem have been proven. Using these theorems, it has been shown that a sharp threshold phenomenon occurs in a large class of Boolean functions. Fortunately, when the variables have a positive association, there is an acceptable definition of influence for which similar theorems hold as in the independent case. However, in the case of a negative association between the variables, even defining the concept of influence is highly challenging. In this thesis, we first introduce Boolean functions and discuss the basic concepts related to them. Then, we present the important theorems in two cases: the independence of variables and the positive association between variables. Finally, we discuss the concept of influence in cases where there is a negative association between the variables and propose a new definition that generalizes the traditional definition of influence
  9. Keywords:
  10. Boolean Functions ; Indentation ; Positive Association ; Negative Association ; Sharp Threshold Phenomenon

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