An Optimal Algorithm for Finding the Spectrum of the Joint Action of Binary Factors

The paper describes an effective algorithm for calculating the spectrum of joint action (interaction) of binary factors in a binary response. The spectrum of joint action allows us to determine the presence/absence of interaction of factors (Boolean variables) in a given response (Boolean function), as well as to determine the strength of this interaction described by the degree of joint action concept. The need to calculate these characteristics of joint action is obvious in itself, and it is also relevant to the theory of sufficient causes, which is the basic concept of causality in the medical and biological sciences. The paper shows that the proposed algorithm, given its implementation, is optimal in terms of time complexity among all algorithms that compute the spectrum of joint action by definition. Some new theoretical results were obtained and utilised for creation of this algorithm. The proposed implementation uses data parsing, representation of Boolean vectors as decimal numbers and application of bitwise operations to them. It is shown that for this algorithm the time complexity is $O(6^{n})$ , where $n$ is the number of binary factors.