Boolean differential calculus 4. Differentials and differential operations 4.1 Differential of a Boolean variable 4.2 Total differential operations 4.3 Partial differential operations 4.4 m fold differential operations The Boolean Differential Calculus (BDC) is a very powerful theory that extends the basic concepts of Boolean Algebras significantly. Its applications are based on Boolean spaces B and B n Boolean operations and basic structures such as Boolean Algebras and Boolean Rings Boolean functions Boolean equations Boolean inequalities incompletely specified Boolean functions and Boolean lattices of Boolean functions. These basics sometimes also called switching theory are widely used in many modern information processing applications. The BDC extends the known concepts and allows the consideration of changes of function values. Such changes can be explored for pairs of function values as well as for whole subspaces. The BDC defines a small number of derivative and differential operations. Many existing theorems are very welcome and allow new insights due to possible transformations of problems. The available operations of the BDC have been efficiently implemented in several software packages. The common use of the basic concepts and the BDC opens a very wide field of applications. The roots of the BDC go back to the practical problem of testing digital circuits. The BDC deals with changes of signals which are very important in applications of the analysis and the synthesis of digital circuits. The comprehensive evaluation and utilization of properties of Boolean functions allow for instance to decompose Boolean functions very efficiently; this can be applied not only in circuit design but also in data mining. Other examples for the use of the BDC are the detection of hazards or cryptography. The knowledge of the BDC gives the scientists and engineers an extended insight into Boolean problems leading to new applications e.g. the use of Boolean lattices of Boolean functions 2. Derivative operations of Boolean functions 2.1 Vectorial derivative operations 2.2 Single derivative operations 2.3 m fold derivative operations 2.4 Derivative operations of XBOOLE 6. Solutions of the exercises 6.1 Solutions of chapter 1 6.2 Solutions of chapter 2 6.3 Solutions of chapter 3 6.4 Solutions of chapter 4 6.5 Solutions of chapter 5 Bibliography Authors biographies Index 3. Derivative operations of lattices of Boolean functions 3.1 Boolean lattices of Boolean functions 3.2 Vectorial derivative operations 3.3 Single derivative operations 3.4 m fold derivative operations 1. Basics of Boolean structures 1.1 Lattices and functions 1.2 Boolean algebras 1.3 Boolean rings 1.4 Boolean equations and inequalities 1.5 Lists of ternary vectors (TVL) 5. Applications 5.1 Properties of Boolean functions 5.2 Solution of a Boolean equation with regard to variables 5.3 Computation of graphs 5.4 Analysis of digital circuits 5.5 Synthesis of digital circuits 5.6 Test of digital circuits 5.7 Synthesis by bi decompositions