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Abstract:
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The thesis consists of five chapters. In Chapter 1 definitions and well-known results from the theory of Boolean algebras and Boolean functions are given. In Chapter 2 of the thesis some properties of Boolean functions, which preserve constants under finite Boolean algebras, are presented by using the component representation. Their consequences about the number of Boolean’s functions are also given. The theorems which are the generalization of Scognmaiglio’s theorem and Andreoli’s theorem for Boolean functions with one variable, are proved in Chapter 3. The following new notions are introduced for monotone logical functions: the profile, the level, homogeneous, the corresponding matrix, etc. Some properties of these functions are shown and some consequences about the number of homogeneous monotone logical functions are presented. In Chapter 4 the applications of monotone Boolean functions in solving the problems of search theory (a branch of the theory of information) are presented. It is shown that the general problem of a type is, in fact, the problem of identifications of homogeneous monotone Boolean functions of the given profile by checking the value of that function for combinations of values of variables. Optimal or almost optimal solutions for some profiles are shown. It is also shown that monotonic logical functions are natural instrument for the generalization of these problems. Some open problems are presented in Chapter 5. |