Regional Controllability of Cellular Automata as a SAT Problem

Elementary cellular automata (ECA – acronym also used to define a single automaton) are discrete dynamical systems for which the state of each cell only takes two values and is determined at each time step by its own state and the state of left and right neighbours. The evolution of the states of all cells occurs in parallel.

We first consider a one-dimensional infinite set of cells. The state of each cell $i\in\mathbb{Z}$ at time $t$ is given by a variable $x_{i}^{t}\in\{0,1\}$. Mathematically, the evolution of the cells is defined in terms of a local function $f\{0,1\}^{3}\rightarrow\{0,1\}:$

For a given time $t$, we denote by $\boldsymbol{x}^{t}=(x^{t}_{i})_{i\in\mathbb{Z}}$ the sequence of all cell states, i.e., a configuration. In the rest of the paper we consider only finite configurations since we will focus on a particular set of cells, or a region, not taking into consideration what happens outside it.

Since there are $2^{3}=8$ different neighbourhood states, there are $2^{8}=256$ different ECA rules. It is usual to associate to each ECA $f$ its decimal code $W$, defined by $W(f)=f(0,0,0)2^{0}+\dots+f(1,1,1)2^{7}$  [22, 23]. This amounts to writing the digits of the transition table $f(0,0,0)$ …$f(1,1,1)$ and converting the binary number to a decimal number. As an example, Rule 150 corresponds to the array $1,0,0,1,0,1,1,0$, which is equal to 150 in base ten (see Table 1).

One can apply two symmetries on the rule, the left-right inversion and the one-zero exchange, which gives 88 classes. The rules with smallest decimal code in a class are called minimal CA and it is usual to consider only these rules for studying ECA (see the left column of Table LABEL:tab:fraction).

In our problem, the definition of a regionally-controllable CA will be set for an arbitrary number of cells, with the goal of determining the asymptotic behaviour. For a given number cells $n$, we will focus on the region represented by the set of $n$ cells indexed from 1 to $n$ and try to control the evolution of this region, without setting in advance the number of steps $T$. We are also interested in looking for the minimal value of $T(\boldsymbol{x},\boldsymbol{y})$ for a given pair of configurations $(\boldsymbol{x},\boldsymbol{y})$ and the minimal value for a region size $n$

where $\{0,1\}^{n}$ denotes the set of finite words (configurations) of length $n$ over $\{0,1\}$,

Given a number of cells $n$ and two finite sub-configurations $\boldsymbol{x}=(x_{1},\dots,x_{n})$ and $\boldsymbol{y}=(y_{1},\dots,y_{n})\in\{0,1\}^{n}$, our objective is to determine if there exists an appropriate control sequence on the boundary cells, namely cell $0$ and cell $n+1$, such that the system will evolve from $\boldsymbol{x}$ to $\boldsymbol{y}$ in a finite number of time steps (see Fig. 1 for an illustration).

Mathematically, this amounts to saying that:

A CA is regionally controllable, if there exists $N\in\mathbb{N}^{+}$ such that for every $n\geq N$ and for every pair of configurations $\boldsymbol{x}=\boldsymbol{x}^{0},\boldsymbol{y}\in\{0,1\}^{n}$, there exists $T>0$ and a control vector $\boldsymbol{u}=(u^{0},\dots,u^{T-1})$ where $u^{t}=(x_{0}^{t},x_{n+1}^{t})$ such that

that is, $\boldsymbol{y}$ is reachable from $\boldsymbol{x}$ in $T$ steps, with

In the case where a rule is not regionally controllable, we are interested in determining the reachability ratio ${\rho}(n)$, that is, the ratio of controllable pairs of initial and final configurations of size $n$:

Let us now examine under which conditions elementary cellular automata can be controlled.

To this end, we introduce the notion of a derivative of a Boolean function $f(x,y)$ [24] as:

where $\oplus$ stands for the sum modulo two (equivalent to an exclusive OR). This definition obeys many standard properties of derivatives, like the chain rule [25]. It is possible also to define higher-order derivatives.

By means of Boolean derivatives in zero one can obtain the Ring Sum Expansion of a function [26]:

where $D_{i}$ is the derivative of $f$ with respect to the bits that have value 1 in the binary representation of $i$.

For instance

Rules that have all derivatives of order greater than one equal to zero (i.e., $D_{3}=D_{5}=D_{6}=D_{7}=0$) are called affine, and linear if $f(0,0,0)=0$. For instance, Rule 150 is a linear rule since it can be written as

while Rule 22, although quite similar to Rule 150 (see Table. 1), is not affine:

The Boolean derivatives in zero and the affinity of all minimal ECA are reported in Table 2 of Ref. [21].

We can now define the property of linearity with respect to the periphery of the neighbourhood, which, for ECA, amounts to saying that there exists a function $g$ such that:

In term of Boolean derivatives, a rule is peripherally-linear if $D_{1}=1$ and $D_{3}=D_{5}=D_{7}=0$ or $D_{4}=1$ and $D_{5}=D_{6}=D_{7}=0$. In the ECA space, the peripherally-linear rules are Rules 15, 30, 45, 60, 90, 105, 106, 150, 154, and 170.

As shown in Refs. [17, 15], peripherally-linear rules are fully controllable.

Let us recall the definitions of $\varepsilon$-chains, chain transitivity and chain mixing.

Let $\varepsilon>0$ and let $\boldsymbol{x},\boldsymbol{y}\in A^{\mathbb{Z}}$ where $A$ is the alphabet. An $\varepsilon$-chain ($\varepsilon$-pseudo-orbit) from $\boldsymbol{x}$ to $\boldsymbol{y}$ is a finite sequence of configurations $\{\boldsymbol{x}^{0},\boldsymbol{x}^{1},\dots,\boldsymbol{x}^{t}\}$ with $\boldsymbol{x}^{0}=\boldsymbol{x}$ and $\boldsymbol{x}^{T}=\boldsymbol{y}$, such that for $T>0$ and $d(F(\boldsymbol{x}^{t}),\boldsymbol{x}^{t+1})<\varepsilon$, for $t\in\{0,\dots,T-1\}$, where the distance $d$ is defined as:

$d(\boldsymbol{x},\boldsymbol{y})=2^{-\mathrm{min}\left\{|k|:x_{k}\neq y_{k}\right\}}$

and $F:A^{\mathbb{Z}}\rightarrow A^{\mathbb{Z}}$ is the global transition function.

A cellular automaton is chain-transitive if for all $\boldsymbol{x},\boldsymbol{y}\in A^{\mathbb{Z}}$ and $\varepsilon>0$ there exists an $\varepsilon$-chain from $\boldsymbol{x}$ to $\boldsymbol{y}$ [27]. Similarly, a cellular automaton is considered to be chain-mixing if, for any two configurations $\boldsymbol{x},\boldsymbol{y}\in A^{\mathbb{Z}}$ and $\varepsilon>0$ there exists $T>0$ such that for all $t\geq T$, there exists an $\varepsilon$-chain of length $t$ from $\boldsymbol{x}$ to $\boldsymbol{y}$ [27].

Recall that regional controllability refers to the ability to steer a dynamical system in a specific region from any initial configuration to any desired configuration in that region within a finite time $T$ using appropriate inputs (control). Meanwhile, chain transitivity means that for any two points in the state space, one can find a sequence of pseudo-orbits (arbitrary small jumps) connecting them. The connection between chain transitivity, chain mixing and regional controllability of cellular automata was investigated and a proof has been recently proposed to establish an equivalence between regional controllability for every $n$ ( where $n$ is the size of the controlled region), the chain-transitivity and chain-mixing properties of one-dimensional cellular automata [20]. It has been shown that a CA is regionally controllable for every $n$ if and only if it is chain-transitive and chain-mixing.

The link between these concepts can be understood through graph theory, by modelling controlled cellular automata as directed graph, where nodes denotes the finite configurations in $A^{n}$ and the arcs represent transitions between them, governed by system dynamics or external inputs. In graph terms, the regional controllability concept corresponds to strong connectivity of the state transition graph meaning that any state can be reached from any other state via a directed path between nodes [11, 10]. It has been shown that when this connectivity property is verified for every $n$, it is equivalent to chain transitivity and chain mixing [20]. For a finite cellular automaton the relationship also remains valid.

This means that the value of $\epsilon$ controls the number of cells $n$ that we have in our system and that applying the control sufficiently far from the centre cell (say at distance $n/2$) guarantees that we obtain a valid $\varepsilon$-chain. Mathematically, let us consider a configuration $\boldsymbol{x}^{t}$ and its image $\boldsymbol{x}^{t+1}=F(\boldsymbol{x}^{t})$. Applying a control on $\boldsymbol{x}^{t+1}$ on cells $-n/2$ and $n/2$ transforms this configuration into a configuration $\boldsymbol{y}$. It is easy to see that we have $d(F(\boldsymbol{x}^{t}),\boldsymbol{y})\leq\frac{1}{2^{n/2}}$, which exactly corresponds to the notion of $\epsilon$-chain for $\epsilon=\frac{1}{2^{n/2}}$. In other words, we need to apply the control at a larger distance as the value of $\epsilon$ gets smaller in the $\epsilon$-chain.

According to the equivalence presented in Ref. [20], the methods used to investigate the regional controllability problem can thus also be used to verify the chain transitivity and chain mixing properties.

Recall that we have $n$ cells in the target region with indices ranging from 1 to $n$, and two external cells with index $0$ and $n+1$ as the border controls. In the CA world, $x_{i}^{t}\in\{0,1\}$ describe the state of cell $i\in\{1,n\}$ at time $t$ and $x_{0}^{t}$ and $x^{t}_{n+1}$ respectively describe the state of left and right controls at time $t$. To go to the SAT universe, we map each binary value $x_{i}^{t}$ to a Boolean value $b_{i}^{t}$, with $t\in\mathbb{N}$ and $i\in\{0,\dots,n\}$ such that $b_{i}^{t}={\mathcal{B}}(x_{i}^{t})$, where ${\mathcal{B}}$ is the function from $\{0,1\}$ to $\{\texttt{False},\texttt{True}\}$ such that ${\mathcal{B}}(0)=\texttt{False}$ and ${\mathcal{B}}(1)=\texttt{True}$.

Recall that we fix the values $\boldsymbol{x}=(x_{1},\dots,x_{n})$ and that we want to reach $\boldsymbol{y}=(y_{1},\dots,y_{n})$ in $T$ time steps. This is equivalent to saying that there exists two sequences $(\boldsymbol{x}_{0}^{t})_{t\in\{0,\dots,T-1\}}$ and $(\boldsymbol{x}_{n+1}^{t})_{t\in\{0,\dots,T-1\}}$ such that:

and $x_{i}^{T}=y_{i}$ for all $i\in\{1,\dots,n\}$, that is, $\boldsymbol{y}$ is reachable from $\boldsymbol{x}$ in $T$ steps.

We now need to transform this mathematical relationship into a Boolean formula. Such a formula is usually expressed as a combination of variables, or atoms, and operators. A clause is a special case of formula where atoms and their negations are combined using only OR operators, for instance $f(a,b,c)=a\vee\neg b\vee c$. A formula is in conjunctive normal form (CNF) if it is expressed as a conjunction of clauses, for instance: $f(a,b,c,d)=(a\vee\neg b)\wedge(b\vee\neg c\vee\neg d)$. SAT solvers take a CNF as an input and try to compute an assignment of the variables that makes the formula true, or, when this is not possible, try to find a proof that there this no such assignment.

To encode the CNF that verifies the regional controllability problem, we proceed in two steps (see Fig. 3).

a) For each $i$ and $t$, we encode the condition $x_{i}^{t+1}=f(x_{i-1}^{t},x_{i}^{t},x_{i+1}^{t})$ as a CNF composed of eight clauses which use the four variables $b_{i}^{t+1}$, $b_{i-1}^{t}$, $b_{i}^{t}$, and $b_{i+1}^{t}$.

Indeed, each neighbourhood state will generate a clause:

As $A\implies B$ is equivalent to $\neg A\vee B$, the conditions becomes

that is,

which translates into

where $C[b,1]=b$ and $C[b,0]=\neg b$.

We thus translate the $n\cdot T$ transitions described in Eq. (2) into $8\cdot n\cdot T$ clauses.

b) In a second step, we encode the initial and final conditions $(b_{i}^{0})$ and $(b_{i}^{T})$ for $i\in\{1,\dots,n\}$ and let the control variables free of any constraint.

By combining the conditions obtained in step a) and step b), we obtain a CNF formula $\Phi$ with $8nT+2n$ clauses and such that $\Phi$ is satisfiable if and only if $\boldsymbol{y}$ is reachable from $\boldsymbol{x}$ in $T$ time steps.

We tested the solving abilities of the SAT solvers with the minisat solver [28]. We used a simple script to generate the formula seen above in the form of a CNF expressed in the popular DIMACS format. For a fixed value of $n$ and $T$, we randomly generated a pair of initial and final configurations $\boldsymbol{x},\boldsymbol{y}\in\{0,1\}^{n}$ and asked the solver whether $\boldsymbol{y}$ was reachable from $\boldsymbol{x}$ with a control sequence of length $T$.

Table 2 presents the results for the 88 minimal ECA for the setting $n=40$, $T=100$ and a sample of 100 random pairs of initial and final configurations. The second column ($\tilde{\rho}$) presents the ratio of cases where a solution was found, that is, a rough estimate of the reachability ratio ${\rho}(n)$. We observe that $\tilde{\rho}$ is either equal to 1 or is rather small (less than 10%). The case where $\tilde{\rho}$ equals one corresponds exactly to the ten peripherally-linear rules 15, 30, 45, 60, 90, 105, 106, 150, 154, and 170. For the other rules, we issue the following conjecture :

The reachability ratio ${\rho}(n)$ tends to zero when the number of cells $n$ tends to infinity for all the ECA rules which are not peripherally-linear.

This conjecture is supported by the data shown on Table 3, where one can clearly see that the estimated reachability ratio $\tilde{\rho}(n)$ rapidly decreases with $n$, and becomes of the order of a percent when $n$ is equal to 50. It is an open question to obtain more precise scaling laws with $n$, and a good estimate of the time needed to reach a configuration from another.

Table 2 also presents the CPU time that is used to compute the presence or the absence of a solution, and the number of restarts, conflicts and decisions that were taken by the SAT solver. These figures are only presented to furnish a rough estimate of the difficulty of analysis that was met for each rule. It should be noted that we did not encounter any case where the solver was unable to provide an answer, either positive or negative and that the answer was given quite rapidly ($\approx$ 20 ms per solution on average).

It is also remarkable that the among the rules for which the number of restarts is important, one finds the famous Rule 110, which is known to be Turing-universal. Of course, there is no direct correlation between the computational complexity of a rule and the difficulty to decide its controllability, but we can safely state that the rules which exhibit the richest panel of behaviours (gliders, collisions generating other gliders, etc.) are among the most difficult ones to analyse by the SAT solvers.

In order to gain insights on the relationships between regional controllability and the difficulty to find a solution or to prove that there is none, we now explore a technique of tree-building in order to find the shortest paths between two configurations.