Abstract
This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension, and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-K\r{u}rka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension~1, but also dimension~1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension~ 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.
Highlights
Introduction and Formal SettingCellular automata (CA for short) are a well-studied model appearing in different research areas under different points of view
It is widely used as a modeling tool of fundamental physical phenomena [8] or high-level phenomena from other disciplines [1, 33, 12]. It is a rich class of symbolic dynamical systems[20] extensively studied both from the topological [25] and ergodic point of view [38]. It is a computational model very close to Turing machines but with a massively parallel feature: this translates into a specific algorithmic complexity theory [41], into universal computational power and existence of
We study three classes which form a hierarchy by inclusion: freezing CA (i.e. CA which are locally decreasing according to some order on states), bounded-change CA (i.e. CA with a global bound on the number of state changes a cell can make in any orbit), and the general class of point-wise convergent CA
Summary
Cellular automata (CA for short) are a well-studied model appearing in different research areas under different points of view. They were in particular considered as theoretical models of bootstrap percolation and a lot of work was dedicated to the experimental and rigorous analysis of the phase transitions they exhibit (see for instance [21]) The fact that these examples are monotone (with respect to the order extended to configurations) is to be taken into account when studying their computational complexity [15, 3]. In a freezing CA, a cell can change at most a finite number of time during its evolutions, precisely at most n − 1 times for a CA with n states This property is our second main definition. Since the other possible state changes are Q → {b, b+}, Q → e, b+ → b and {b, b+} → e, we deduce that F2 is (2t0 + 2)-change and the second item of the theorem follows
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
