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28 10 Vol. 28 No. 10 2011 10 Control Theory Applications Oct. 2011 : 1000−8152(2011)10−1513−11 , , (, 100190) : . . . , , . . : ; ; ; ; : O153.2, O172 : A From Boolean algebra to Boolean calculus CHENG Dai-zhan, ZHAO Yin, XU Xiang-ru (Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China) Abstract: Boolean functions are of special importance, though they are the simplest class of finite-valued functions. It is widely applied to many fields, including information, control and so on. The theoretical foundation of Boolean functions is introduced in this paper, involving some main results from Boolean algebra to Boolean Calculus, applications in information and control and some recent frontiers and developments. The semi-tensor product approach to these fields is introduced emphatically. Key words: Boolean function; Boolean algebra; Boolean derivative; Boolean integral; semi-tensor product of matrices 1 (Introduction) 1847, (De Mor- [1], gan). 1854, : Boole , )19 , . D = {0, 1}, f : Dn → D . n, n 2, , [3] . , ¬. . ¬x = 1 − x, x ∈ D. (1) [4, 5], [6] [7] , , ∧; : , ∨; →; [8]. ↔, (1).