Classification of Random Binary Sequences Using Walsh-Fourier Analysis

The concept of a finite binary random sequence does not seem to be covered in the classical foundations of the theory of probability. Solomonoff, Kolmogorov and Chaitin have tried to include this case by considering the lengths of programs required to generate these sequences: a longer program implying more randomness. However this definition is difficult to apply. This paper presents a straightforward procedure using Walsh functions to determine the pattern in a binary sequence. A quantitative measure of randomness has also been proposed. This has been defined as the number of independent data (via the Walsh transform) required to generate the sequence divided by the length of the sequence. However at present this classification procedure is restricted to sequences of length 2k only. When extended to infinite sequences it yields results agreeing with those by the classical probability theory.