Boolean Differential Calculus

Abstract

The Boolean Differential Calculus extends the Boolean Algebra by evaluating the change behavior of logic functions. Vectorial derivative operations explore the value change between pairs of function values reached by the change of an arbitrary number variables. Different function values in these pairs are indicated by the vectorial derivative, unchanged values 1 by the vectorial minimum, and at least one value 1 by the vectorial maximum. A special case of the vectorial derivative operations are the single derivative operations where only a single variable changes its value. Repeated execution of the same type of a single derivative operation with regard to different variables results in k-fold derivative operations. These operations evaluate the function values of certain subspaces of the given function. Derivative operation can be efficiently calculated a have many applications in determining properties of logic functions, in circuit design, test, and others. The differential of a given Boolean variable is also a Boolean variable that is equal to 1 when the given variable changes its value. Logic equations that commonly use Boolean variables and the associated differentials can be used to describe the edges of a graph. Similarly, to the differential of Boolean variables, differential operations of logic function are defined. Differential operations summarize associated derivative operations for several directions of change.