Optimal Two-Level Boolean Minimization

Find a minimal sum-of-products expression for a Boolean function. Consider a Boolean algebra with elements False and True. A Boolean function f .y1; y2; : : : ; yn/ of n Boolean input variables specifies, for each combination of input variable values, the function’s value. It is possible to represent the same function with various expressions. For example, the first and last expressions in Fig. 1 correspond to the same function. Assuming access to complemented input variables, straightforward implementations of these expressions would require two AND gates and an OR gate for a ^ b _ .a ^ b/ and only a wire for a. Although the implementation efficiency depends on target technology, in general terser expressions enable greater efficiency. Boolean minimization is the task of deriving the tersest expression for a function. Elegant and optimal algorithms exist for solving the variant of this problem in which the expression is limited to two levels, i.e., a layer of AND gates followed by a single OR gate or a layer of OR gates followed by a single AND gate.