Non-Linear Permutation Groups on Physarum Polycephalum

One of the modern directions of unconventional computing is presented by biological computing. The new branch of this direction deals with computing on Physarum polycephalum, the one-cell organism building intelligent networks. In this paper, first, we propose the design method of reversible logic gates by using attractants and repellents as outer stimuli for Physarum polycephalum. This method is based on the theory of permutation groups which allows to define logic gates combinations as multiplication of permutation matrices. Reversible logic gates on Physarum polycephalum are exemplified by the NOT and CNOT gates. Second, we propose the new design method to construct reversible logic gates without repellents. In this way, we should appeal to the so-called non-linear permutation groups. These groups contain non-well-founded objects such as infinite streams and their families. The theory of non-linear permutation groups proposed in this paper for the first time can be used for designing reversible logic gates on any behavioral systems.