which resembles · · · 002002002002 · · · in a one- dimensional infinite lattice. If the PU is employed, the system falls into an absorbing state in one unit time even if the system size is infinite. However, if we employ the RSU, that is, the master equation, the probability that this configuration evolves into an absorbing state in finite time is zero. Still, the density of active sites is zero in the infinite-time limit (the average density of active sites at time t in the RSU is e − $^{t}$/ 3). As this example suggests, it is appropriate to study the time dependence of the density of active sites. From the above consideration, we define the third order parameter φ$_{3}$ as configuration C$_{t}$ . For the RSU we define ρ ( C ) as the average density of active sites in the steady state of the master equation (3) if the system evolves from the con- figuration C . Obviously, ρ ( C ) = 0 irrespective of update rules if S ( C ) = 0. Although S ( C ) = 1 implies ρ ( C ) > 0, ρ ( C ) generally depends on the update rule. For example, consider a con- figuration (0202) for the TM ˜ F in Eq. (16) with V = 4. Obviously, ρ ( C ) = 1 / 2 for the PU. In the RSU, there are two possible patterns, (0202) and (1012), up to transla- tion. Since the average waiting time to the next jump of the first pattern is 1/2 while that of the second pattern is 1, the probability that the first (second) pattern is found in the steady state is 1 / 3 (2/3). Thus, ρ ( C ) = 1 / 3 when the RSU is employed. Now, we define the second order parameter φ$_{2}$ as where P$_{0}$ is the initial distribution and ζ satisfies Eq. (9). As we have shown, φ$_{2}$ depends on the update rule, which should be contrasted with φ$_{1}$ . Since ρ ( C ) ≤ S ( C ) for any C , φ$_{2}$ cannot be larger than φ$_{1}$ . Thus, φ$_{1}$ = 0 implies φ$_{2}$ = 0. For finite V , φ$_{1}$ = 0 should imply φ$_{2}$ = 0. Thus, a phase transition point for finite V , if it exists, is the same irrespective of whether φ$_{1}$ or φ$_{2}$ is used as an order parameter. Although transition points for finite V do not depend on which order parameter is used, it is nontrivial to an- swer whether or not the infinite-size limit affects this conclusion. This is because there are configurations such that ρ ( C ) → 0 under the infinite V limit, while S ( C ) = 1 for any V . For example, if C = · · · 1111201111 · · · for the TM (16), ρ ( C ) = 1 /V → 0 while S ( C ) = 1 for any V . In Appendix A, we show that it is always possible to construct such a configuration with E = Z$_{m}$ for any symmetric TM. If such configurations exist for any E in the range Z$_{m}$ ≤ E ≤ Z$_{M}$ , we cannot exclude the possi- bility that φ$_{1}$ and φ$_{2}$ for the same initial condition can give different transition points in the infinite-size limit. Presumably, an ignorant preparation for the initial con- dition as independent and identical Poisson distributions would not generate such complications. This might be an interesting question, but we will not pursue the difference in φ$_{1}$ and φ$_{2}$ any further in this paper. where n ( C, V ; t ) is the number of active sites averaged over ensembles at time t when the system evolves from the initial configuration C . For convenience, we will ex- clusively refer to ρ$_{a}$ ( t ) as the activity density (at time t ). Although we did not mention it explicitly, the infinite-size limit above should be understood as lim sup to guarantee the existence of the limit for any sequence of P$_{0}$ . It may be tempting to claim that if S ( C ) = 0 for any finite V , the limit of n ( C, V ; t ) should be In the previous two sections, the infinite-size limit, when necessary, is preceded by the infinite-time limit. Now we discuss the consequence of changing the order of the two limits. If the infinite-size limit is taken first, we cannot assign a unique value to S ( C ), introduced in Sec. III A, irrespec- tive of the update rule. For example, consider the TM However, it is not true in general. For example, consider again the TM (16) with an initial configuration C$_{0}$ , ρ$_{a}$ ( t ) = lim sup V →∞ ∑ C n ( C, V ; t ) V P$_{0}$ ( C ; ζ, V ) , (29) φ$_{3}$ = lim t $_{→∞}$ρ$_{a}$ ( t ) , (30) lim t →∞ lim V →∞ n ( C, V ; t ) V = 0 . (31) z$_{i}$ = { 2 , if 1 ≤ i ≤ V/ 4 , 0 , otherwise. (32) φ$_{2}$ ≡ ∑ C ρ ( C ) P$_{0}$ ( C ; ζ, V ) , (27) (16) with the following initial configuration z$_{i}$ = { 2 , if i (mod 3) = 0 , 0 , otherwise , (28) C. Order parameter φ 3 5 t = 0 : 0000222222222222222222220000 t = 1 : 0001122222222222222222211000 t = 2 : 0001212222222222222222121000 t = 3 : 0002031222222222222221302000 t = 4 : 0010213122222222222213120100 t = 5 : 0011031312222222222131301100 FIG. 2. Time evolution of the DFES with the TM (16) for the initial configuration (32) with V = 80. The PU is employed. At every time step, the length of the middle string of 2’s (underlined) decreases by 2.