# Breaking of the Trade-Off Principle between Computational Universality and Efficiency by Asynchronous Updating

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Trade-Off Principle in Synchronous ECA

**B**

^{n}with

**B**= {0, 1} and a transition rule f

_{r}:

**B**

^{3}→

**B**, where f

_{r}is synchronously updated to all cells and r represents the rule number mentioned below. The transition rule with synchronous updating is expressed as

_{i}

^{t}

^{+1}= f

_{r}(a

_{i}

_{−1}

^{t}, a

_{i}

^{t}, a

_{i+}

_{1}

^{t}).

_{r}is adapted to all cells in

**B**

^{n}(i.e., global adaption), then we assign the global use by G(f

_{r}):

**B**

^{n}

^{+2}→

**B**

^{n}such that

_{1}

^{t}

^{+1}, a

_{2}

^{t}

^{+1}, …, a

_{n}

^{t}

^{+1}) = G(f

_{r}) (a

_{0}

^{t}, a

_{1}

^{t}, …, a

_{n}

_{+1}

^{t}).

**B**,

_{s}= f

_{r}(x, y, z)

^{7}

_{s}

_{=0}2

^{s}d

_{s}.

_{1}= d

_{4}= 1 and d

_{s}= 0 with s ≠ 1, 4. There are 256 rules in ECA since there are 2 possible outputs for 8 inputs of a triplet.

**B**

^{n}with random boundary conditions, reachable states are determined by a transition rule. For the case of R0, only one state consists of all 0 for any initial states; this implies that (0, 0, …, 0) = G(f

_{0})(a

_{0}, a

_{1}, …, a

_{n}

_{+1}) for any (a

_{0}, a

_{1}, …, a

_{n}

_{+1}) ∈

**B**

^{n}

^{+2}. By contrast, R204, of which d

_{2}= d

_{3}= d

_{6}= d

_{7}= 1 and d

_{0}= d

_{1}= d

_{4}= d

_{5}= 0, can show that (a

_{1}, a

_{2}, …, a

_{n}) = G(f

_{204})(a

_{0}, a

_{1}, …, a

_{n}

_{+1}) for any (a

_{0}, a

_{1}, …, a

_{n}

_{+1}) ∈

**B**

^{n}

^{+2}and that all possible states can be reached if an adequate initial condition is prepared. It is easy to see that R204 shows a locally frozen pattern (class 2). For R90 or R150, all possible states can be reached, although the generated patterns are chaotic (class 3). Thus, the ratio of reachable states for all possible initial conditions can reveal the computational universality. Given 2

^{n}all possible initial states with random boundary conditions, the computational universality of rule r, U(r), is defined by

_{R}(r) = {G(f

^{Tr})(a

_{0}, a

_{1}, …, a

_{n}

_{+1}) ∈

**B**

^{n}| (a

_{1}, …, a

_{n}) ∈

**B**

^{n}, (a

_{0}, a

_{n}

_{+1}) ∈

**R**(

**B**

^{2})}

_{R}(r)

_{N}(r) = U(r)/2

^{n}

**R**(

**B**

^{2}) represents one element set randomly determined from

**B**

^{2}, and superscript T represents T numbers iteration of f

_{r}. If n = 2, then U(0) = #{(0, 0)} = 1, and U(204) = #{(0, 0), (0, 1), (1, 0), (1, 1)} = 4 U

_{N}(r) represents the normalized computational universality. Here, we call elements of a set, S

_{R}(r), reachable states.

_{R}(r), the average time to reach X represented by τ

_{r}(X) is expressed as

**B*** =

**B**

^{n}×

**R**(

**B**

^{2}), T(G(f

^{T}

_{r})(Y) = X) implies time T such that G(f

^{T}

_{r})(Y) = X. Since the time T is computed for any Y ∈

**B***, it can lead to G(f

^{T}

_{r})(Y) ≠ X. At that case, if G(f

^{T}

_{r})(Y) = X is not obtained within 2

^{n}time steps, then T(G(f

^{T}

_{r})(Y) = X) is a constant value, T

_{θ}. For the case of R204 in which any initial condition is not changed by the transition, G(f

_{r})(Y) = Y with T = 1 and then for any X ∈ S

_{R}(r), τ

_{r}(X) = (1 + T

_{θ}(#

**B*** − 1)). The computational efficiency is defined by

_{N}(r) for all rules in ECA. Since E(r) reveals the average time to reachable states, the smaller E(r) is, the more efficient E(r) is. Thus, the minimal point of E(r) for each computational universality reveals the maximal efficiency for each computational universality. This maximal efficiency is why the solid line representing the lower margin of a cloud of (U

_{N}(r), E(r)) shows the relationship between the computational universality and efficiency. The greater the universality is, the less the efficiency is. It is clear that the solid line shows the trade-off between the computational universality and efficiency.

## 3. The Trade-Off Breaking by Asynchronous Updating

_{i}

^{t}

^{+1}= f

_{r}(a

_{i}

_{−1}

^{t}, a

_{i}

^{t}, a

_{i+}

_{1}

^{t}) with 1 − p;

= a

_{i}

^{t}with p.

_{N}(r) for asynchronous ECA with the probability, compared to the trade-off between E(r) and U

_{N}(r) in synchronous ECA. For the sake of comparison, the lower margin of the distribution of (U

_{N}(r), E(r)) obtained for synchronous ECA is expressed as a monotonous increasing step function. The interval [0.0, 1.0] is divided into m subintervals. The kth subinterval, Int

_{k}, is [(k − 1)∗1.0/m, k∗1.0/m]. In each subinterval,

_{SUB-MIN}(k) = min{E(r) | U

_{N}(r) ∈ Int

_{k}}, if there is an element U

_{N}(r) exists;

= max{E(r) | r = 0, …, 255}, otherwise.

_{MIN}(k) is defined by

_{MIN}(k) = min{E

_{SUB-MIN}(s) | k ≤ s ≤ m}

_{N}(r), E(r)) is expressed as Equation (13), and m = 52. In each graph, the horizontal and vertical lines are the same as those in Figure 2. In Figure 4, all pairs of (U

_{N}(r), E(r)) obtained by synchronous updating are hidden by bars above the increasing step function. The pairs of (U

^{A}

_{N}(r), E

^{A}(r)) obtained by asynchronous updating with the probability p are represented by circles below the increasing step function. It is easy to see that asynchronous updating with a wide region of p entails breaking the trade-off.

^{A}

_{N}(r), E

^{A}(r)) breaking the trade-off obtained by synchronous ECA, which is represented by circles below the lower margin of the distribution of (U

_{N}(r), E(r)) with synchronous updating. The breaking degree, D

_{B}(p) for ECA asynchronously updated with the probability, p, is defined by

_{B}(p) = #{E

^{A}(r) | E

^{A}(r) < E

_{MIN}(k), k = 1, …, m}/256,

_{1}= 1 is replaced by d

_{1}= 0. Here, the transition rule approximated for a pair of binary sequences, (a

_{1}

^{t}, a

_{2}

^{t}, …, a

_{n}

^{t}) and (a

_{1}

^{t}

^{+1}, a

_{2}

^{t}

^{+1}, …, a

_{n}

^{t}

^{+1}) is called an apparent rule. For R18, one can see various apparent changes in the transition rule, as shown in Table 1. If p = 0.0, then the apparent rule is the same as the transition rule, R18. The larger p is, the more apparent the change in d

_{s}is. The lowest row shows the case of p = 1.0, which leads to the apparent rule being R204. In 0 < p < 1, time development can be interpreted to be generated by various apparent rules showing classes 1, 2 and 3 in time and space. That is why a cluster-like pattern is generated by mixing with up class 1, 2 and 3 transitions.

^{*}

_{s}with s = 0, 1 …, 7, is adapted to the binary sequence. It results in a pair of binary sequence such as

_{1}

^{t}, a

_{2}

^{t}, …, a

_{n}

^{t}); (a

_{1}

^{t}

^{+1}, a

_{2}

^{t}

^{+1}, …, a

_{n}

^{t}

^{+1}).

_{1}

^{t}, a

_{2}

^{t}, …, a

_{n}

^{t}) is divided into multiple segments,

_{1}

^{t}), (2, a

_{2}

^{t}), …, (m, a

_{m}

^{t})}, {(m+1, a

_{m}

_{+1}

^{t}), (m+2, a

_{m}

_{+2}

^{t}), …, (h, a

_{h}

^{t})}, …, {…, (n, a

_{n}

^{t})},

_{u}

^{t}), (u+1, a

_{u}

_{+1}

^{t}), …, (w, a

_{w}

^{t})}, for any s ∈ {0, …, 7}, if there exists (a

_{k}

_{−1}

^{t}, a

_{k}

^{t}, a

_{k}

_{+1}

^{t}) ∈

**B**

^{3}such that s = 4a

_{k}

_{−1}

^{t}+ 2a

_{k}

^{t}+ a

_{k}

_{+1}

^{t}, k ∈ {u, u + 1, …, w},

_{s}= a

_{k}

^{t}

^{+1}

_{s}= d

^{*}

_{s}.

_{s}with s = 0, 1 …, 7 and that a sequence, (a

_{u}

^{t}, a

_{u}

_{+1}

^{t}, …, a

_{w}

^{t}); (a

_{u}

^{t}

^{+1}, a

_{u}

_{+1}

^{t}

^{+1}, …, a

_{w}

^{t}

^{+1}) can be interpreted as a transition generated by the synchronous updating of a single transition rule. Thus, segmentation (10) implies the approximation of which each segment can be generated by a single transition rule and a whole sequence can be synchronously generated by multiple transition rules.

_{k}

^{t}

^{+1}= a

_{k}

^{t}. In Figure 6, a transition generated by asynchronous updating is divided into three segments. Algorithmically, the segmentation is implemented from left to right. From the first cell, one can determine d

_{1}= a

_{1}

^{t}

^{+1}= 1, and then d

_{2}= a

_{2}

^{t}

^{+1}= 0, d

_{4}= a

_{3}

^{t}

^{+1}= 1. At the fourth cell at t+1, one obtains d

_{1}= a

_{4}

^{t}

^{+1}= 0 and that conflicts with d

_{1}= (a

_{1}

^{t}

^{+1}) = 1. That is why the first segment is terminated by the third cell at t+1, which is expressed as {(0, a

_{0}

^{t}), (1, a

_{1}

^{t}), (2, a

_{2}

^{t}), (3, a

_{3}

^{t})}. For a transition rule, only d

_{1}, d

_{2}and d

_{4}are determined, and d

_{0}, d

_{3}, d

_{5}, d

_{6}and d

_{7}are not determined in that segment, {(0, a

_{0}

^{t}), (1, a

_{1}

^{t}), (2, a

_{2}

^{t}), (3, a

_{3}

^{t})}. The undetermined value d

_{s}for a transition rule is represented by the blue cell in Figure 6. By definition (18), the undetermined d

_{s}is substituted by d

^{*}

_{s}, which is defined by R18 in Figure 6. Thus, for the first segment in Figure 6, one can obtain R18. Similarly, it results in three segments, and the second and third segments are approximated by R16 and R6, respectively.

^{4}cells whose values are randomly set, the segmentation procedure is run. This process results in N

_{1}segments and N

_{2}transition rules. By using N

_{1}segments and N

_{2}transition rules, the approximated time development is emulated. First, at each cell, it is probably determined whether the segment is cut or not, with the probability of N

_{1}/10

^{4}(segmentation process). Second, a transition rule randomly chosen from N

_{2}transition rules is applied to each segment, and the state of cells is updated (update process). Both segmentation and update processes are performed for each time step, which leads to time development, as shown in the right diagram of each pair. Clearly, the synchronous updating of multiple rules can emulate the time development of asynchronous updating of a single rule. In other words, the behavior of asynchronous CA can be estimated by synchronous CA with multiple rules.

^{4}cells whose states are randomly determined, the asynchronous updating of the transition rule with probability p is applied to 10

^{4}cells. For a pair of binary sequences of the initial configuration and results of application of the transition rule, the segmentation process is applied. This process results in a pair of the number of rules and the number of segments. Figure 8 shows the normalized number of segments (N

_{1}/10

^{4}) against p for some of the asynchronous updating of the transition rules, R110, R50, R90 and R18. The data for each transition rule are approximated by a polynomial function: for R110, y = 0.1081x

^{4}− 0.1643x

^{3}− 0.5596x

^{2}+ 0.6087 x + 0.0048, R

^{2}= 0.99594; for R50, y = −2.7514x

^{4}+ 5.9832x

^{3}− 4.492x

^{2}+ 1.2555x + 0.0143, R

^{2}= 0.98422; for R90, y = −1.1571x

^{4}+ 2.6079x

^{3}− 2.2464x

^{2}+ 0.7964 x + 0.0051, R

^{2}= 0.99293; and for R18, y = −2.0516x

^{4}+ 4.4103x

^{3}− 3.3139x

^{2}+ 0.9666x + 0.0016, R

^{2}= 0.98778.

_{2}/256) against p for each transition rule. The data for each transition rule are approximated by a polynomial function: for R110, y = −0.3159x

^{4}+ 0.706x

^{3}− 0.5401x

^{2}+ 0.1619; for R50, y = −2.7529x

^{4}+ 4.7656x

^{3}− 3.7832x

^{2}+ 1.9264x + 0.0054, R

^{2}= 0.98541; for R90, y = −0.6852x

^{4}+ 1.5296x

^{3}− 1.1948x

^{2}+ 0.3817x + 0.0208, R

^{2}= 0.90307; and for R18, y = −2.0516x

^{4}+ 4.4103x

^{3}− 3.3139x

^{2}+ 0.9666x + 0.0016, R

^{2}= 0.98778.

^{4}+ 1.9304x

^{3}− 1.4477x

^{2}+ 0.5047x + 0.0047, R

^{2}= 0.99594, and y = −0.6488x

^{4}+ 1.333x

^{3}− 1.0644x

^{2}+ 0.4345x + 0.0125, R

^{2}= 0.98811, respectively. The normalized number of rules and segments in the approximation might contribute to an increase in the computational efficiency since it can increase the diversity of the configurations. However, it is not necessary that the diversity of configurations is implemented by the diversity of rules. R90 and R150 can compute any configurations if the corresponding initial condition is prepared.

_{B}(p), is very high (R

^{2}= 0.82076), whereas the correlation between #Rules/#Segments and D

_{B}(p) is very low (R

^{2}= 0.56706). This finding suggests that the diversity resulting from a smaller number of transition rules (i.e., class 3 or 4-like behavior) contributes to breaking the trade-off compared with the diversity resulting from a large number of transition rules (i.e., class 1 or 2-like behavior). In other words, although there are both effects of generalists with high #Segments/#Rules and specialists with high #Rules/#Segment in asynchronous updating, only effect from generalists can contribute to break of the trade-off.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The computational universality and computational efficiency. The cardinality of a set of a(t + 1) in a return map represents the computational universality, U(r). The computational efficiency E(r) is defined by the average time to reach reachable states.

**Figure 2.**E(r) plotted against U

_{N}(r) for all rules in ECA. Each circle represents a coordinate (U

_{N}(r), E(r)) for each transition rule r. The solid curve shows the trade-off between the computational universality and computational efficiency. The parameter is set by the following: n = 8, T = 7, and T

_{θ}= 70.

**Figure 3.**Time development of asynchronous ECA with the probability. The horizontal and vertical lines represent space and time, respectively. The dot and blank represent a state of a cell, 1 and 0, respectively. The transition rule of ECA is R22.

**Figure 4.**Breaking the trade-off between the computational universality and efficiency by asynchronous ECA with the probability, p. Pairs, (U

^{A}

_{N}(r), E

^{A}(r)) breaking the trade-off is represented by circles. In each diagram, the horizontal and vertical lines represent the computational universality and efficiency, respectively.

**Figure 5.**Breaking degree of the trade-off between the computational universality and efficiency plotted against the probability by which asynchronous updating is implemented.

**Figure 6.**Schematic diagram of the approximation for a pair of binary sequences generated by the asynchronous updating of a single rule (R18) approximated by the synchronous updating of multiple rules (R18 + R16 + R6). States 1 and 0 in a cell are represented by filled and blank squares, respectively. The symbols “Syn” and “Asyn” represent synchronous and asynchronous updating, respectively. See text for the detailed discussion.

**Figure 7.**Four examples of the approximation for the transition of asynchronous CA by the transition of synchronous multiple CAs. In each diagram pair, the left diagram shows the time development by asynchronous CA, and the right diagram shows that by synchronous multiple CAs. The transition rule number and the probability defined by Equation (6) are shown at the bottom middle of each pair. The dot represents a cell whose state is 1, and the blank represents a cell whose state is 0.

**Figure 8.**Normalized number of segments (N

_{1}/10

^{4}) against the probability p in the approximation of asynchronous updating by the synchronous updating of multiple rules. Each curve represents a single transition rule, R110 (yellow), R50 (gray), R90 (orange) and R18 (blue).

**Figure 9.**Normalized number of transition rules (N

_{2}/256) against the probability p in the approximation of asynchronous updating by the synchronous updating of multiple rules. Each curve represents a single transition rule, R110 (yellow), R50 (gray), R90 (orange) and R18 (blue).

**Figure 10.**Normalized number of segments and normalized number of transition rules against p averaged over all 256 transition rules. These parameters are obtained from the approximation of the asynchronous updating of a single rule approximated by the synchronous updating of multiple rules.

**Figure 11.**#Segments/#Rules plotted against p. The data are approximated by y = −126.41x

^{6}+ 403.31x

^{5}− 506.65x

^{4}+ 314.36x

^{3}− 98.944x

^{2}+ 14.135x + 0.2609, R

^{2}= 0.96549. The inner graph shows the linear regression between D

_{B}(p) and #Segments/#Rules.

**Table 1.**The apparent change in a transition rule originated from R18. If a transition rule R18 is not adapted to cells with some probability, then some d

_{s}s are changed, and the apparent rule number is changed. The columns d

_{s}represent the apparent transition due to the asynchronous updating with the probability. The column AR represents the apparent rule number and their corresponding classes.

d_{0} | d_{1} | d_{2} | d_{3} | d_{4} | d_{5} | d_{6} | d_{7} | AR | Class |
---|---|---|---|---|---|---|---|---|---|

000 | 001 | 010 | 011 | 100 | 101 | 110 | 111 | ||

0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | R18 | 3 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | R0 | 1 |

0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | R90 | 3 |

0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | R146 | 3 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | R128 | 1 |

0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | R76 | 2 |

0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | R204 | 2 |

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## Share and Cite

**MDPI and ACS Style**

Gunji, Y.-P.; Uragami, D.
Breaking of the Trade-Off Principle between Computational Universality and Efficiency by Asynchronous Updating. *Entropy* **2020**, *22*, 1049.
https://doi.org/10.3390/e22091049

**AMA Style**

Gunji Y-P, Uragami D.
Breaking of the Trade-Off Principle between Computational Universality and Efficiency by Asynchronous Updating. *Entropy*. 2020; 22(9):1049.
https://doi.org/10.3390/e22091049

**Chicago/Turabian Style**

Gunji, Yukio-Pegio, and Daisuke Uragami.
2020. "Breaking of the Trade-Off Principle between Computational Universality and Efficiency by Asynchronous Updating" *Entropy* 22, no. 9: 1049.
https://doi.org/10.3390/e22091049