On the Difficulty of Computations

Two practical considerations concerning the use of computing machinery are the amount of information that must be given to the machine for it to perform a given task and the time it takes the machine to perform it. The size of programs and their running time are studied for mathematical models of computing machines. The study of the amount of information (i.e., number of bits) in a computer program needed for it to put out a given finite binary sequence leads to a definition of a random sequence; the random sequences of a given length are those that require the longest programs. The study of the running time of programs for computing infinite sets of natural numbers leads to an arithmetic of computers, which is a distributive lattice