Abstract
Previous work has suggested that there is a kind of phase transition between deterministic automata exhibiting periodic behavior and those exhibiting chaotic behavior. However, unlike the usual phase transitions of physics, this transition takes place over a range of values of the parameter rather than at a specific value. The present paper asks whether the transition can be made sharp, either by taking the limit of an infinitely large rule table, or by changing the parameter in terms of which the space of automata is explored. We find strong evidence that, for the class of automata we consider, the transition does become sharp in the limit of an infinite number of symbols, the size of the neighborhood being held fixed. Our work also suggests an alternative parameter in terms of which it is likely that the transition will become fairly sharp even if one does not increase the number of symbols. In the course of our analysis, we find that mean field theory, which is our main tool, gives surprisingly good predictions of the statistical properties of the class of automata we consider.
Highlights
Of the four Wolfram classes [1] of deterministic cellular automata-homogeneous, periodic, chaotic, and complex--those exhibiting complex behavior are for many purposes the most interesting, but they seem to be the most rare, when the rule table for the automaton is large [2]
There is not a unique value of L at which the transition occurs, but rather a range of possible transitbn values. In this res~ct the transition differs from the usual phase transitions of physics, whkh occur at definite values of, say, temperature or magnetk field strength
The gap is assodatect with the transition from perbdk to chaotic behavior, and one ckms not expect the predictions of mean field theory to apply to periodfc automata. only in chaotic automata muld one hope to find enough mixing of the symbols for the assumptions of mean flefd theory to be approximately valid. (See, the paper by McIntosh in this volume, whkh relates automaton behavior not to the prediction of mean ffeld theory but rather to the form of the mean field equation [12].)
Summary
Of the four Wolfram classes [1] of deterministic cellular automata-homogeneous, periodic, chaotic, and complex--those exhibiting complex behavior are for many purposes the most interesting, but they seem to be the most rare, when the rule table for the automaton is large [2]. 1 as [he size of the neighborhood (n) or the number of symbols (k) grows In partkular, it Is not clear whether the spread In entropy for each value of X wIII shrink or wIII stay essentially constant. The rule tables were constructed by proceeding through all rotationally inequivalent neighborhood configurations, and assigning to each one the symbol zero with probability 1-~, and a nonzero zymbol with probability 1. (12) and (13) are the main result of this section They give us the mean-field prediction for the distribution of the density of zeros in terms of known quantities. From this distribution one can obtain the distribution of values of the entropy via Eq (4). The gap is assodatect with the transition from perbdk to chaotic behavior, and one ckms not expect the predictions of mean field theory to apply to periodfc automata. only in chaotic automata muld one hope to find enough mixing of the symbols for the assumptions of mean flefd theory to be approximately valid. (See, the paper by McIntosh in this volume, whkh relates automaton behavior not to the prediction of mean ffeld theory but rather to the form of the mean field equation [12].)
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