The Boolean function network is a systematical approach proposed for the inference of gene regulatory networks and related Boolean functions. This procedure utilizes two steps to integrate the hidden Markov model, likelihood ratio test and Boolean functions for discovering direct pairwise relations between genes from time-course transcriptome data. In this compilation, the authors justify the need for this novel approach and describe the inference procedure. Next, an extended Boolean logic is introduced, denoted by LLT, called "star-finite" or "hyperfinite" logic. LLT is closely related to infinitary logics, which have been investigated extensively. Lastly, generalized Boolean functions are introduced, and an overview with regard to constructions of Golay complementary sequences from generalized Boolean functions is given.