Phase Transitions in the Computational Complexity of "Elementary'' Cellular Automata

A study of the computational complexity of determining the "Garden-of-Eden" states (i.e., states without any pre-images) of one-dimensional cellular automata (CA) is reported. The work aims to relate phase transitions in the computational complexity of decision problems, with the type of dynamical behavior exhibited by the CA time evolution. This is motivated by the observation of critical behavior in several computationally hard problems, e.g., the Satisfiability problem (SAT) [1]. The focus is on "legal" CA rules, i.e., those which obey the quiescence and symmetry conditions [2]. A relation exists between the problem of "Garden-of-Eden" states determination and the SAT problem. Several CA rules (e.g., CA rules 4, 22 and 54) are studied in detail to establish the occurrence of phase transition. Finite-size scaling exponents corresponding to the critical behavior are obtained. Based on these exponents, a new quantitative classification of "elementary" cellular automata into 5 classes is proposed.