Ooh, that subject line is so deliciously jargony. You are probably wondering what the hell I mean. Suppose you have an automaton consisting of an n-bit register, where each bit's next state is a boolean function of the states of all the other bits. We wish to find a rule (that is, an update function for each bit) and an initial state that maximizes the number of states that the automaton traverses before it repeats a state. The obvious maximum of a 2^n-state trajectory is easily achieved by creating a binary counter. Each bit will be on next time in two cases: if it is now on and some bit to its right is off; or if it is now off and all the bits to its right are on. (This is just the standard carry rule, "flattened" to a boolean function of the present bit values.) But note that this logic is required to take explicit notice that certain bits are off. Suppose we restrict the logic, and permit only monotonic boolean functions? We now expect that the 2^n maximum cannot be achieved, in general. We can still implement an n-bit counter with monotonic update rules, but it takes 2n actual bits. So a lower bound of the maximum achievable orbit of such an automaton is 2^[n/2]. Can anyone tighten these lower and upper bounds of 2^[n/2] and 2^n? -A