Autonomous indefiniteness versus external indefiniteness: Theory of weak topped ∩-structure and its application to elementary local cellular automaton

We propose a theory to formalize the indefinite features of living systems in the framework of weak topped ∩-structure. This theory contains two notions of indefiniteness, one is called external indefiniteness and the other is called autonomous indefiniteness. The former is defined as the outside of fixed points of the closure operator and the latter is defined as the difference between the set of fixed points of the weakened closure operator and a given set which defines the closure. This theory is then applied to elementary local cellular automaton (ELCA) in which the time development of its cell is driven by observing the dynamics of its nearest neighbors at the previous time step, followed by taking the closure (or the weak closure) in the appropriate space. The behavior of ELCA is characterized by its algebraic and statistical properties. In particular, we show that self-organized criticality (SOC)-like behavior appears in ELCA driven by the weakened closure operator.