On computing the Lyapunov exponents of reversible cellular automata

We consider the problem of computing the Lyapunov exponents of reversible cellular automata (CA). We show that the class of reversible CA with right Lyapunov exponent $2$ cannot be separated algorithmically from the class of reversible CA whose right Lyapunov exponents are at most $2-\delta$ for some absolute constant $\delta>0$. Therefore there is no algorithm that, given as an input a description of an arbitrary reversible CA $F$ and a positive rational number $\epsilon>0$, outputs the Lyapunov exponents of $F$ with accuracy $\epsilon$. We also compute the average Lyapunov exponents (with respect to the uniform measure) of the CA that perform multiplication by $p$ in base $pq$ for coprime $p,q>1$.


Introduction
A cellular automaton (CA) is a model of parallel computation consisting of a uniform (in our case one-dimensional) grid of finite state machines, each of which receives input from a finite number of neighbors. All the machines use the same local update rule to update their states simultaneously at discrete time steps. The following question of error propagation arises naturally: If one changes the state at some of the coordinates, then how long does it take for this change to affect the computation at other coordinates that are possibly very far away? Lyapunov exponents provide one tool to study the asymptotic speeds of error propagation in different directions. The concept of Lyapunov exponents originally comes from the theory of differentiable dynamical systems, and the discrete variant of Lyapunov exponents for CA was originally defined in [17].
The Lyapunov exponents of a cellular automaton F are interesting also when one considers F as a topological dynamical system, because they can be used to give an upper bound for the topological entropy of F [18]. It is previously known that the entropy of one-dimensional cellular automata is uncomputable [8] (and furthermore from [5] it follows that there exists a single cellular automaton whose entropy is uncomputable), which gives reason to suspect that also the Lyapunov exponents are uncomputable in general.
The uncomputability of Lyapunov exponents is easy to prove for (not necessarily reversible) cellular automata by using the result from [9] which says that nilpotency of cellular automata with a spreading state is undecidable. We will prove the more specific claim that the Lyapunov exponents are uncomputable even for reversible cellular automata. In the context of proving undecidability results for reversible CA one cannot utilize undecidability of nilpotency for nonreversible CA. An analogous decision problem, the (local) immortality problem, has been used to prove undecidability results for reversible CA [15]. We will use in our proof the undecidability of a variant of the immortality problem, which in turn follows from the undecidability of the tiling problem for 2-way deterministic tile sets. This result has been published previously in the proceedings of UCNC 2019 [13].
In the other direction, there are interesting classes of cellular automata whose Lyapunov exponents have been computed. In [3] a closed formula for the Lyapunov exponents of linear one-dimensional cellular automata is given. We present results concerning the family of multiplication automata that simulate multiplication by p in base pq for coprime p, q > 1. It is easy to find out their Lyapunov exponents and it is more interesting to consider a measure theorical variant, the average Lyapunov exponent. We compute the average Lyapunov exponents (with respect to the uniform measure) for these automata. This computation is originally from the author's Ph.D. thesis [12].

Preliminaries
For sets A and B we denote by B A the collection of all functions from A to B.
A finite set A of letters or symbols is called an alphabet. The set A Z is called a configuration space or a full shift and its elements are called configurations. An element x ∈ A Z is interpreted as a bi-infinite sequence and we denote by x[i] the symbol that occurs in x at position i. A factor of x is any finite sequence where i, j ∈ Z, and we interpret the sequence to be empty if j < i. Any finite sequence w = w [1]w [2] · · · w[n] (also the empty sequence, which is denoted by and we denote by w Z ∈ A Z the configuration in which w occurs at all positions of the form in (i ∈ Z). The set of all words over A is denoted by A * , and the set of non-empty words is The set of words of length n is denoted by A n . For a word w ∈ A * , |w| denotes its length, i.e. |w| = n ⇐⇒ w ∈ A n .
If A is an alphabet and C is a countable set, then A C becomes a compact metrizable topological space when endowed with the product topology of the discrete topology of A (in particular a set S ⊆ A C is compact if and only if it is closed). In our considerations C = Z or C = Z 2 . We define the shift Every cylinder is an open set of X and the collection of all cylinders form a basis for the topology of X.
Occasionally we consider configuration spaces (A 1 × A 2 ) Z and then we may We may use the terminology that x 1 is on the upper layer or on the A 1 -layer, and similarly that x 2 is on the lower layer or on the A 2 -layer. Definition 2.1. Let X ⊆ A Z be a subshift. We say that the map F : X → X is a cellular automaton (or a CA) on X if there exist integers m ≤ a (memory and anticipation) and a local rule f : If we can choose f so that −m = a = r ≥ 0, we say that F is a radius-r CA and if we can choose m = 0 we say that F is a one-sided CA. A one-sided CA with anticipation 1 is called a radius-1 2 CA. We can extend any local rule f : The CA-functions on X are characterized as those continuous maps on X that commute with the shift [6]. We say that a CA F : X → X is reversible if it is bijective. Reversible CA are homeomorphisms on X. The book [14] is a standard reference for subshifts and cellular automata on them.
For a given reversible CA F : X → X and a configuration x ∈ X it is often helpful to consider the space-time diagram of x with respect to F . Formally it is the map θ ∈ A Z 2 defined by θ(i, −j) = F j (x)[i]: the minus sign in this definition signifies that time increases in the negative direction of the vertical coordinate axis. Informally the space-time diagram of x is a picture which depicts elements of the sequence (F i (x)) i∈Z in such a way that F i+1 (x) is drawn below F i (x) for every i.
The definition of Lyapunov exponents is from [17,18]. For a fixed subshift Then for given cellular automaton These have shift-invariant versions Λ ± n (x, F ) = max i∈Z Λ ± n (σ i (x), F ). The quantities are called (when the limits exist) respectively the right and left Lyapunov exponents of x (with respect to F ). A global version of these are the quantities that are called respectively the right and left (maximal) Lyapunov exponents of F . These limits exist by an application of Fekete's subadditive lemma (e.g. Lemma 4.1.7 in [14]). Occasionally we consider cellular automata from the measure theoretical point of view. For a subshift X we denote by Σ(C X ) the sigma-algebra generated by the collection of cylinders C X . It is the smallest collection of subsets of X which contains all the elements of C X and which is closed under complement and countable unions. A measure on X is a countably additive function µ : A measure µ on X is completely determined by its values on cylinders. We say that a cellular automaton F : X → X is measure preserving if µ(F −1 (S)) = µ(S) for all S ∈ Σ(C).
On full shifts A Z we are mostly interested in the uniform measure determined by µ(Cyl(w, i)) = |A| −|w| for w ∈ A + and i ∈ Z. By Theorem 5.4 in [6] any surjective CA F : A Z → A Z preserves this measure. For more on measure theory of cellular automata, see [16].
Measure theoretical variants of Lyapunov exponents are defined as follows. Given a measure µ on X and for n ∈ N, let I + n,µ (F ) = x∈X Λ + n (x, F )dµ and I − n,µ (F ) = x∈X Λ − n (x, F )dµ. Then the quantities are called respectively the right and left average Lyapunov exponents of F (with respect to the measure µ).
We will write W ± s (x), Λ ± n (x), λ + (x), I + n,µ and I + µ when X and F are clear by the context.

Tilings and Undecidability
In this section we recall the well-known connection between cellular automata and tilings on the plane. We use this connection to prove an auxiliary undecidability result for reversible cellular automata.

Definition 3.1.
A Wang tile is formally a function t : {N, E, S, W } → C whose value at I is denoted by t I . Informally, a Wang tile t should be interpreted as a unit square with edges colored by elements of C. The edges are called north, east, south and west in the natural way, and the colors in these edges of t are t N , t E , t S and t W respectively. A tile set is a finite collection of Wang tiles.

Definition 3.2.
A tiling over a tile set T is a function η ∈ T Z 2 which assigns a tile to every integer point of the plane. A tiling η is said to be valid if neighboring tiles always have matching colors in their edges, i.e. for every (i, j) ∈ Z 2 we have η(i, j) N = η(i, j + 1) S and η(i, j) E = η(i + 1, j) W . If there is a valid tiling over T , we say that T admits a valid tiling.
We say that a tile set T is NE-deterministic if for every pair of tiles t, s ∈ T the equalities t N = s N and t E = s E imply t = s, i.e. a tile is determined uniquely by its north and east edge. A SW-deterministic tile set is defined similarly. If T is both NE-deterministic and SW-deterministic, it is said to be 2-way deterministic.
The tiling problem is the problem of determining whether a given tile set T admits a valid tiling.
A 2-way deterministic tile set T can be used to construct a complete tile set. Namely, let C be the set of colors which appear in tiles of T , let X ⊆ C × C be the set of pairs of colors which do not appear in the northeast of any tile and let Y ⊆ C × C be the set of pairs of colors which do not appear in the southwest of any tile. Since T is 2-way deterministic, there is a bijection p : X → Y . Let T ∁ be the set of tiles formed by matching the northeast corners X with the southwest corners Y via the bijection p. Then the tile set A = T ∪ T ∁ is complete.
Every complete 2-way deterministic tile set A determines a local rule f : configurations F j (x) are diagonals of η going from northwest to southeast and the diagonal corresponding to F j+1 (x) is below the diagonal corresponding to F j (x).

Definition 3.5. A cellular automaton
Generalizing the definition in [15], we call the following decision problem the (p, q)-local immortality problem: given a reversible CA F : We now adapt the proof of Theorem 3.6 to get the following result, which we will use in the proof of Theorem 4.1.
Lemma 3.7. The (1, 5)-local immortality problem is undecidable for reversible radius-1 2 CA. Proof. We will reduce the problem of Theorem 3.3 to the (1, 5)-local immortality problem. Let T be a 2-way deterministic tile set and construct a complete tile Figure 1: The tile sets T 1 (first row) and T 2 (second row). These are originally from [15] (up to a reflection with respect to the northwest -southeast diagonal).
set T ∪ T ∁ as indicated above. Then also Figure 1) is a complete tile set. 1 We denote the blank tile of the set T 1 by t b and call the elements of R = A 1 \ (T × {t b }) arrow tiles. As indicated above, the tile set A 1 determines a reversible radius- and e ∈ A Z 2 , i.e. G simulates G 1 in the upper layer. We construct involutive CA J 1 , J 2 and H of memory 0 with local rules j 1 : We will show that T admits a valid tiling if and only if F is (1, 5)-locally immortal with respect to B.
Assume first that T admits a valid tiling η. Then by choosing Assume then that T does not admit any valid tiling and for a contradiction assume that x is a (1, 5)-witness. Let θ be the space i.e. η can be recovered from the upper layer of θ by applying a suitable linear transformation on the space-time diagram. In drawing pictorial representations of θ we want that the heads and tails of all arrows remain properly matched in neighboring coordinates, so we will use tiles with "bent" labelings, see Figure 2. Since T does not admit valid tilings, it follows by a compactness argument that η is an arrow tile. Since θ contains a "bent" version of η, it follows that In Figure 3 we present the space-time diagram θ with arrow markings of tiles from T 1 and T 2 replaced according to the Figure 2.
In Figure 3 we have also marked the sets N and E. Other features of the figure become relevant in the next paragraph.  The minimal distance between a tile in N and an arrow tile in E situated on the same horizontal line in θ is denoted by d 1 > 0. Then, among those arrow tiles in E at horizontal distance d 1 from N , there is a tile with minimal vertical distance d 2 > 0 from N (see Figure 3). Fix p, q ∈ Z so that θ(p, q − 2) is one such tile and in particular (p − d 1 , q − 2), (p, q − 2 − d 2 ) ∈ N . Then θ(p, q − j) contains an arrow for −2 ≤ j ≤ 2, because if there is a j ∈ [−2, 2) such that θ(p, q − j) does not contain an arrow and θ(p, q − j − 1) does, then θ(p, q − j − 1) must contain one of the three arrows on the left half of Figure 2. These three arrows continue to the southwest, so then also θ(p − 1, q − j − 2) contains an arrow. Because θ(p ′ , q ′ ) ∈ B for (p ′ , q ′ ) ∈ N , it follows that (p − 1, q − j − 2) / ∈ N does not contain an arrow for 0 < i ≤ d 1 , −2 ≤ j ≤ 2 by the minimality of d 1 and d 2 . Now consider the A 2 -layer of θ. For the rest of the proof let y = F −q (x). Assume that π 2 (θ(p−i, q)) = π 2 (y[p−i]) is non-zero for some i ≥ 0, (p−i, q) ∈ E, and fix the greatest such i, i.e. π 2 (y[s]) = 0 for s in the set We start by considering the case π 2 (y[p − i]) = 1. Denote From the choice of (p, q) it follows that π 1 (θ(s, q − 1)) = π 1 (G(y)[s]) are not arrow tiles for s ∈ I 1 , and therefore we can compute step by step that and π 2 (θ(p − (i + 1), q − 1)) = 1. By repeating this argument inductively we see that the digit 1 propagates to the lower left in the space-time diagram as indicated by Figure 4 and eventually reaches N , a contradiction. If on the other hand π 2 (θ(p−i, q)) = 2, a similar argument shows that the digit 2 propagates to the upper left in the space-time diagram as indicated by Figure 4 and eventually reaches N , also a contradiction.
As in the previous paragraph we see that π 2 (θ(p, q − 2)) = 1. This occurrence of the digit 1 propagates to the lower left in the space-time diagram as indicated by the right side of Figure 5 and eventually reaches N , a contradiction.

Remark 3.8.
It is possible that the (p, q)-local immortality problem is undecidable for reversible radius-1 2 CA whenever p ∈ N and q ∈ N + . We proved this in the case (p, q) = (1, 5) but for our purposes it is sufficient to prove this just for some p > 0 and q > 0. The important (seemingly paradoxical) part will be that for (1, 5)-locally immortal radius-1 2 CA F the "local immortality" travels to the right in the space-time diagram even though in reality there cannot be any information flow to the right because F is one-sided.

Uncomputability of Lyapunov exponents
In this section we will prove our main result saying that there is no algorithm that can compute the Lyapunov exponents of a given reversible cellular automaton on a full shift to an arbitrary precision.
To achieve greater clarity we first prove this result in a more general class of subshifts. For the statement of the following theorem, we recall for completeness that a sofic shift X ⊆ A Z is a subshift that can be represented as the set of labels of all bi-infinite paths on some labeled directed graph. This precise definition will not be of any particular importance, because the sofic shifts that we construct are of very specific form. We will appeal to the proof of the following theorem during the course of the proof of our main result. Theorem 4.1. For reversible CA F : X → X on sofic shifts such that λ + (F ) ∈ [0, 5 3 ] ∪ {2} it is undecidable whether λ + (F ) ≤ 5 3 or λ + (F ) = 2. Proof. We will reduce the decision problem of Lemma 3.7 to the present problem. Let G : A Z 2 → A Z 2 be a given reversible radius-1 2 cellular automaton and B ⊆ A 2 some given set. Let A 1 = {0, , ←, →, ւ, ց} and define a sofic shift Y ⊆ A Z 1 as the set of those configurations containing a symbol from Q = {←, →, ւ, ց} in at most one position. We will interpret elements of Q as particles going in different directions at different speeds and which bounce between walls denoted by . Let S : Y → Y be the reversible radius-2 CA which does not move occurrences of and which moves ← (resp. →, ւ, ց) to the left at speed 2 (resp. to the right at speed 2, to the left at speed 1, to the right at speed 1) with the additional condition that when an arrow meets a wall, it changes into the arrow with the same speed and opposing direction. More precisely, S is the CA with memory 2 and anticipation 2 determined by the local rule f : and π 1 : X → Y , π 2 : X → A Z 2 be the natural projections π i (x 1 , x 2 ) = x i for x 1 ∈ Y, x 2 ∈ A Z 2 and i ∈ {1, 2}. Let x 1 ∈ Y and x 2 ∈ A Z 2 be arbitrary. We define reversible CA G 2 , F 1 : X → X by G 2 (x 1 , x 2 ) = (x 1 , G 10 (x 2 )), F 1 (x 1 , x 2 ) = (S(x 1 ), x 2 ). Additionally, let F 2 : X → X be the involution which maps (x 1 , x 2 ) as follows: F 2 replaces an occurrence of →0 ∈ A 2 1 in x 1 at a coordinate i ∈ Z by an occurrence of ւ ∈ A 2 1

(and vice versa) if and only if
and otherwise F 2 makes no changes. Finally, define The reversible CA F works as follows. Typically particles from Q move in the upper layer in the intuitive manner indicated by the map S and the lower layer is transformed according to the map G 10 . There are some exceptions to the usual particle movements: If there is a particle → which does not have a wall immediately at the front and x 2 does not satisfy a local immortality condition in the next 10 time steps, then → changes into ւ and at the same time leaves behind a wall segment . Conversely, if there is a particle ւ to the left of the wall and x 2 does not satisfy a local immortality condition, ւ changes into → and removes the wall segment. We will show that λ + (F ) = 2 if G is (1, 5)-locally immortal with respect to B and λ + (F ) ≤ 5 3 otherwise. Intuitively the reason for this is that if x, y ∈ X are two configurations that differ only to the left of the origin, then the difference between F i (x) and F i (y) can propagate to the right at speed 2 only via an arrow → that travels on top of a (1, 5)-witness. Otherwise, a signal that attempts to travel to the right at speed 2 is interrupted at bounded time intervals and forced to return at a slower speed beyond the origin before being able to continue its journey to the right. We will give more details.
Assume first that G is (1, 5)-locally immortal with respect to B. Let x 2 ∈ A Z 2 be a (1, 5)-witness and define On the other hand, F has memory 2 so necessarily λ + (F ) = 2. Assume then that there are no (1, 5)-witnesses for G. Let us denote Since there are no (1, 5)-witnesses, by a compactness argument we may fix some N ∈ N + such that C(2N ) = ∅. We claim that λ + (F ) ≤ 5 3 , so let us assume that 2 ) ∈ X is a sequence of configurations such that Λ + n (x (n) , F ) = s n n where (s n ) n∈N tends to λ + . There exist y (n) = (y First assume that there are arbitrarily large n ∈ N for which x (n) 1 [i] ∈ {0, } for i > −s n n and consider the subsequence of such configurations x (n) (starting with sufficiently large n). Since G is a one-sided CA, it follows that π 2 (F tn (x (n) ))[j] = π 2 (F tn (y (n) ))[j] for j ≥ 0. Therefore the difference between x (n) and y (n) can propagate to the right only via an arrow from Q, so without loss of generality (by swapping x (n) and y (n) if necessary) π 1 (F tn (x (n) ))[j n ] ∈ Q for some 0 ≤ t n ≤ n and j n ≥ i n − 1. Fix some such t n , j n and let w n ∈ Q tn+1 be such that w n (i) is the unique state from Q in the configuration F i (x (n) ) for 0 ≤ i ≤ t n . The word w n has a factorization of the form w n = u(v 1 u 1 · · · v k u k )v (k ∈ N) where v i ∈ {→} + , v ∈ {→} * and u i ∈ (Q \ {→}) + , u ∈ (Q \ {→}) * . By the choice of N it follows that all v i , v have length at most N and by the definition of the CA F it is easy to see that each u i contains at least 2(|v i | − 1) + 1 occurrences of ւ and at least 2(|v i | − 1) + 1 occurrences of ց (after → turns into ւ, it must return to the nearest wall to the left and back and at least once more turn into ւ before turning back into →. If → were to turn into ← instead, it would signify an impassable wall on the right). If we denote by k n the number of occurrences of → in w n , then k n ≤ |w n |/3 + O(1) (this upper bound is achieved by assuming that |v i | = 1 for every i) and s n n ≤ |w n | + 2k n ≤ |w n | + 2 3 After dividing this inequality by n and passing to the limit we find that λ + (F ) ≤ 5 3 . 2 Next assume that there are arbitrarily large n ∈ N for which x (n) 1 [i] ∈ Q for some i > −s n n. The difference between x (n) and y (n) can propagate to the right only after the element from Q in x (n) reaches the coordinate −s n n, so without loss of generality there are 0 < t n,1 < t n,2 ≤ n and i n ≥ 0 such that π 1 (F tn,1 (x (n) ))[−s] ∈ Q for some s ≥ s n n and π 1 (F tn,2 (x (n) ))[i n ] ∈ Q. From this the contradiction follows in the same way as in the previous paragraph.
We are ready to prove the result for CA on full shifts.
be as in the proof of the previous theorem. We will adapt the conveyor belt construction from [4] to define a CA F ′ on a full shift which simulates F and has the same right Lyapunov exponent as F .
Let (c i , e i ) ∈ (Σ × Σ) * ((Q × Σ) ∪ (Σ × Q))(Σ × Σ) * ∪ (Σ × Σ) * be the word that is derived from w i by removing the symbols from ∆. The pair (c i , e i ) can be seen as a conveyor belt by gluing the beginning of c i to the beginning of e i and the end of c i to the end of e i . The map F ′ 1 will shift arrows like the map F 1 , and at the junction points of c i and e i the arrow can turn around to the opposite side of the belt. More precisely, define the permutation ρ : and for a word u ∈ A * 1 let ρ(u) denote the coordinatewise application of ρ.
The map F 1 : X → X extends naturally to configurations of the form y: y can contain infinitely many arrows, but they all point in the same direction and occur in identical contexts. By applying F 1 to y we get a new configuration of the form . From this we extract the pair (c ′ i , e ′ i ), and by adding plusses and minuses to the left and right of the arrow (or in the same coordinates as in (c i , e i ) if there is no occurrence of an arrow) we get a word w ′ i which is of the same form as w i . We define F ′ 1 : Clearly F ′ 1 is shift invariant, continuous and reversible.
We define the involution F ′ 2 : A Z → A Z as follows. For x ∈ A Z and j ∈ {1, 2} F ′ 2 replaces an occurrence of →0 in π 1,j (x) at coordinate i ∈ Z by an occurrence of ւ (and vice versa) if and only if π ∆ (x)[i + 1] = − and and otherwise F 2 makes no changes. F ′ 2 simulates the map F 2 and we check the condition π ∆ (x)[i + 1] = − to ensure that F ′ 2 does not transfer information between neighboring conveyor belts.
Finally, we define The reversible CA F ′ simulates F : X → X simultaneously on two layers and it has the same right Lyapunov exponent as F .
The following corollary is immediate.

Corollary 4.3.
There is no algorithm that, given a reversible CA F : A Z → A Z and a rational number ǫ > 0, returns the Lyapunov exponent λ + (F ) within precision ǫ.

Lyapunov Exponents of Multiplication Automata
In this section we present a class of multiplication automata which perform multiplication by nonnegative numbers in some integer base. After the definitions and preliminary lemmas we compute their average Lyapunov exponents. For this section denote Σ n = {0, 1, . . . , n − 1} for n ∈ N, n > 1. To perform multiplication using a CA we need be able to represent a nonnegative real number as a configuration in Σ Z n . If ξ ≥ 0 is a real number and ξ = ∞ i=−∞ ξ i n i is the unique base-n expansion of ξ such that ξ i = n − 1 for infinitely many i < 0, we define config n (ξ) ∈ Σ Z n by config n (ξ)[i] = ξ −i for all i ∈ Z. In reverse, whenever x ∈ Σ Z n is such that x[i] = 0 for all sufficiently small i, we define For words w = w [1]w [2] · · · w[k] ∈ Σ k n we define analogously Clearly real n (config n (ξ)) = ξ and config n (real n (x)) = x for every ξ ≥ 0 and every x ∈ Σ Z n such that x[i] = 0 for all sufficiently small i and x[i] = n − 1 for infinitely many i > 0.
For integers p, n ≥ 2 where p divides n let g p,n : Σ n × Σ n → Σ n be defined as follows. Let q be such that pq = n. Digits a, b ∈ Σ pq are represented as a = a 1 q + a 0 and b = b 1 q + b 0 , where a 0 , b 0 ∈ Σ q and a 1 , b 1 ∈ Σ p : such representations always exist and they are unique. Then n (a, b) = g p,n (a 1 q + a 0 , b 1 q + An example in the particular case (p, n) = (3, 6) is given in Figure 6. We define the CA Π p,n : , so Π p,n has memory 0 and anticipation 1. The CA Π p,n performs multiplication by p in base n in the sense of the following lemma.
We omit the proof of the lemma, which can be found for example in [10]. The idea of the proof is to notice that the local rule g p,n mimics the usual school multiplication algorithm. Because p divides n, the carry digits cannot propagate arbitrarily far to the left.
For the statement of the following lemmas, which were originally proved in [11], we define a function int : Σ + pq → N by i.e. int(w) is the integer having w as a base-pq representation.
Lemma 5.2. Let w 1 , w 2 ∈ Σ k pq for some k ≥ 2 and let t > 0 be a natural number. Then Proof. Let x i ∈ Σ Z pq (i = 1, 2) be such that x i [−(k − 1), 0] = w i and x i [j] = 0 for j < −(k − 1) and j > 0. From this definition of x i it follows that int(w i ) = real pq (x i ). Denote y i = Π p,pq (x i ). We have Also note that int(g p,pq (w i )) < (pq) k−1 .
For the proof of the first part, assume that int(w 1 ) < q t . Combining this with the observations above yields int(g p,pq (w 1 )) ≤ ⌊int(w 1 )/q⌋ < q t−1 .

Conclusions
We have shown that the Lyapunov exponents of a given reversible cellular automaton on a full shift cannot be computed to arbitrary precision. Ultimately this turned out to follow from the fact that the tiling problem for 2-way deterministic Wang tile sets reduces to the problem of computing the Lyapunov exponents of reversible CA. Note that the result does not restrict the size of the alphabet A of the CA F : A Z → A Z whose Lyapunov exponents are to be determined. Standard encoding methods might be sufficient to solve the following problem. In our constructions we controlled only the right exponent λ + (F ) and let the left exponent λ − (F ) vary freely. Controlling both Lyapunov exponents would be necessary to answer the following. Problem 6.2. Is it decidable whether the equality λ + (F ) + λ − (F ) = 0 holds for a given reversible cellular automaton F : A Z → A Z ?
We mentioned in the introduction that there exists a single CA whose topological entropy is an uncomputable number. We ask whether a similar result holds also for the Lyapunov exponents. Problem 6.3. Does there exist a single cellular automaton F : A Z → A Z such that λ + (F ) is an uncomputable number?
By an application of Fekete's lemma the limit that defines λ + (F ) is actually the infimum of a sequence whose elements are easily computable when F : A Z → A Z is a CA on a full shift. This yields the natural obstruction that λ + (F ) has to be an upper semicomputable number. We are not aware of a cellular automaton on a full shift that has an irrational Lyapunov exponent (see Question 5.7 in [2]), so constructing such a CA (or proving the impossibility of such a construction) should be the first step. This problem has an answer for CA F : X → X on general subshifts X, and furthermore for every real t ≥ 0 there is a subshift X t and a reversible CA F t : X t → X t such that λ + (F t ) = λ − (F t ) = t [7].
In the previous section we computed that the average Lyapunov exponent I − µ of the multiplication automaton Π p,pq is equal to log pq p when p, q > 1 are coprime integers. This in particular shows that average Lyapunov exponents can be irrational numbers. We do not know whether such examples can be found in earlier literature.