Quantum logic automata generalizing the edge of chaos in complex systems

Background Historically, although researchers in the science of complex systems proposed the idea of the edge of chaos and/or self-organized criticality as the essential feature of complex organization, they were not able to generalize this concept. Complex organization is regarded at the edge of chaos between the order phase and the chaos phase and a very rare case. Additionally, in cellular automata, the critical property is class IV, which is also rarely found. Therefore, there can be overestimation for natural selection. More recently, developments in cognitive and brain science have led to the free energy principle based on Bayesian inference, while quantum cognition has been established to explain various cognitive phenomena. Since Bayesian inference results in the perspective of a steady state, it can be described in Boolean logic. Considering that quantum logic consists of multiple Boolean logic in terms of lattice theory, the perspective of the free energy principle is the perspective of order, and the perspective of quantum logic might be the perspective of multiple worlds, which is strongly relevant for the edge of chaos. Problem The next question arises whether the perspective derived from quantum logic can be generalized for the complex behavior consisting of both order and chaos, instead of the edge of chaos or self-organized criticality, to reveal the property of critical behavior such as a power-law distribution. Solution In this study, we define quantum logic automata, which entail quantum logic (orthomodular lattice) in terms of lattice theory and have the features of a dynamical system. Because quantum logic automata are applied to a binary sequence, one can estimate the behavior of those automata with respect to patterns and a time series. Here, we show that most of a group of quantum logic automata display class IV-like behavior, in which oscillatory traveling waves collide with each other, leading to complex behavior; moreover, a time series of binary sequences displays 1/f noise. Therefore, one can see that quantum logic automata generalize and expand the idea of the edge of chaos.