Quantum-inspired identification of complex cellular automata

Abstract

Elementary cellular automata (ECA) present iconic examples of complex systems. Though described only by one-dimensional strings of binary cells evolving according to nearest-neighbour update rules, certain ECA rules manifest complex dynamics capable of universal computation. Yet, the classification of precisely which rules exhibit complex behaviour remains somewhat an open debate. Here, we approach this question using tools from quantum stochastic modelling, where quantum statistical memory—the memory required to model a stochastic process using a class of quantum machines—can be used to quantify the structure of a stochastic process. By viewing ECA rules as transformations of stochastic patterns, we ask: Does an ECA generate structure as quantified by the quantum statistical memory, and can this be used to identify complex cellular automata? We illustrate how the growth of this measure over time correctly distinguishes simple ECA from complex counterparts. Moreover, it provides a spectrum on which we can rank the complexity of ECA, by the rate at which they generate structure.

Appendices

Appendix A: Sub-tree reconstruction algorithm

Here, inference of the classical statistical complexity $C_\mu$ is achieved through the sub-tree reconstruction algorithm [19]. It works by explicitly building an $\varepsilon$-machine of a stochastic process, from which $C_\mu$ may readily be deduced. The steps are detailed below.

1. Constructing a tree structure. The sub-tree construction begins by drawing a blank node to signify the start of the process with outputs $y \in \mathcal {A}$. A moving window of size 2L is chosen to parse through the process. Starting from the blank node, 2L successive nodes are created with a directed link for every y in each moving window $\{y_{0:2L}\}$. For any sequence starting from $y_0$ within $\{y_{0:2L}\}$ whose path can be traced with existing directed links and nodes, no new links and nodes are added. New nodes with directed links are added only when the $\{y_{0:2L}\}$ does not have an existing path. This is illustrated in Fig. 8

For example, suppose $y_{0:6} = 000000$, giving rise to six nodes that branch outwards in serial from the initial blank node. If $y_{1:7} = 000001$, the first five nodes gain no new branches, while the sixth node gains a new branch connecting to a new node with a directed link. Each different element of $|\mathcal {A}|^{2L}$ has its individual set of directed links and nodes, allowing a maximum of $|\mathcal {A}|^{2L}$ branches that originate from the blank node.

Fig. 8

The sub-tree reconstruction algorithm, here illustrated for $L=3$

2. Assigning probabilities. The probability for each branch from the first node to occur can be determined by the ratio of the number of occurrences the associated strings to the total number of strings. Correspondingly, this allows each link to be denoted with an output y with its respective transition probability p.

3. Sub-tree comparison. Next, starting from the initial node, the tree structure of L outputs is compared against all other nodes. Working through all reachable L nodes from the initial node, any nodes with identical y|p and branch structure of L size are given the same label. Because of finite data and finite L, a $\chi ^2$ test is used to account for statistical artefacts. The $\chi ^2$ test will merge nodes that have similar-enough tree structures. This step essentially enforces the causal equivalence relation on the nodes.

4. Constructing the $\varepsilon$-machine. It is now possible to analyse each individually labelled node with their single output and transition probability to the next node. An edge-emitting hidden Markov model of the process can then be drawn up. This edge-emitting hidden Markov model represents the (inferred) $\varepsilon$-machine of the process.

5. Computing the statistical complexity. The hidden Markov model associated with the $\varepsilon$-machine has a transition matrix $T_{kj}^y$ giving the probability of the next output being y given we are in causal state $S_j$, and $S_k$ being the causal state of the updated past. The steady state of this (i.e. the eigenvector $\pi$ satisfying $\sum _yT^y\pi =\pi$) gives the steady-state probabilities of the causal states. Taking $P(S_j)=\pi _j$, we then have the Shannon entropy of this distribution gives the statistical complexity:

$$\begin{aligned} C_\mu := H[ P(S_j) ] = -\sum _j P(S_j) \log _2 P(S_j). \end{aligned}$$

(A1)

For parity with the quantum inference protocol, we used $L=6$ when inferring the $C_\mu ^{(t)}$ of the ECA states, and set the tolerance of the $\chi ^2$ test to 0.05.

Appendix B: Convergence with L

We here provide an expanded form of Figs. 5 in 9 regarding the convergence of our measures of complexity with increasing probability window lengths L, now including $L=3:8$, and also showing analogous plots for $C_\mu$ and E alongside $C_q$. We note that the excess entropy E is estimated for finite length windows using $E(L)=LH(X_{0:L-1})-(L-1)H(X_{0:L})$.

Fig. 9

Growth of L-length ($L=3:8)$ estimates of $C_\mu$, $C_q$, and E with time, up to a maximum of $t_{\text {max}} = 10^5$ time steps. Different L correspond to different widths $W=10^3*2^L$