Asymptotic bounds for average-case complexity of Boolean functions

Abstract

Some results obtained in recent years concerning the average-case complexity of computing Boolean functions by straight line programs with conditional stop are considered. These programs generalize the concept of a Boolean circuit and are a natural model of straight line computations, i.e., computations that do not involve conditional branching or indirect addressing, but allow an early termination under a certain condition. Such computations can be informally described as follows. Computations are executed by a processor with memory consisting of cells. The processor is capable of computing some number of elementary functions, which form the computational basis. At any moment of time, each memory cell is accessible to the processor for both reading and writing data. The processor is controlled by a program that is a sequence of elementary instructions of two types. Every instruction of the first type computes the value of a basis function whose arguments are contained in certain memory cells. The computed result is also placed in a memory cell. An instruction of the second type can terminate the execution of a program. Every such instruction has a single argument, which is contained in a memory cell. If the value of the argument is equal to a certain fixed number, say, unity, then the processor terminates the program execution. If the argument has a different value, then the next instruction of the program is executed. There are special memory cells whose values are declared to be the result of executing the program after its termination. A natural measure of the complexity of such a program is the running time averaged over all possible arguments.