Distortion element in the automorphism group of a full shift

2.1 General notions

We have $\mathbb {N} = \{0,1,2,\ldots \}$ , $\mathbb {Z}_+ = \mathbb {N} \setminus \{0\}$ , and $\mathbb {Z}_{\ell } = \mathbb {Z}/\ell \mathbb {Z}$ is integers modulo $\ell $ . For S a finite set, we denote by $\# S$ the cardinality of S. For $i,j \in \mathbb {N}$ , denote and . If $w \in \{0,1,\ldots , k-1\}^*$ , write $\mathrm {v}_k(w)$ for the value $w$ represents in base $k$ (the leftmost digit having the highest significance by default), that is, $\mathrm {v}_k(w) = \sum _{i=0}^{\lvert w\rvert -1} k^{\lvert w\rvert -1-i} w_{i}$ ; and we write $n_{(k)} \in \{0, \ldots , k-1\}^*$ for the number $n \in \mathbb {N}$ written in base $k$ (with length determined from context or specified in text), that is, $\mathrm {v}_k(n_{(k)}) = n$ .

For $\Sigma $ a finite set, called an alphabet, denote by $\Sigma ^* = \bigcup _{n = 0}^{\infty } \Sigma ^n$ the set of finite words over $\Sigma $ . For $w \in \Sigma ^*$ , denote by ${\mathrm {len}}(w)$ the length of $w$ , that is, the integer $n$ such that $w \in \Sigma ^n$ . For a word $w \in \Sigma ^*$ , denote by $\overline {w}$ the reverse (or ‘mirror image’) of $w$ , that is, if $w = w_0 \cdot w_1 \cdots w_{n-1}$ , then $\overline {w} = w_{n-1} \cdot w_{n-2} \cdots w_0$ . For $w \in \Sigma ^n$ and , define $w|_{J} = w_{j_0} \cdot w_{j_1} \cdots w_{j_k}$ as the restriction of $w$ to J, for $J = \{j_0,\ldots ,j_k\}$ and $j_0 \leq \cdots \leq j_k$ . Given $a \in \Sigma $ and , the cylinder $[a]_j$ denotes the set of words $\{w \in \Sigma ^n \mid w_j = a\}$ . Usually our alphabets are non-trivial, by which we mean $\lvert \Sigma \rvert \geq 2$ .

In Lemma 4.10, we denote ${\mathrm {NC}}^1$ for ‘Nick’s Class’ of complexity of level $1$ , that is, the class of languages $L \subseteq \Sigma ^*$ such that L is decidable by Boolean circuits with a polynomial number of gates, with at most two inputs and depth $O(\log n)$ (see for example [Reference Arora and Barak2]). The reader need not be familiar with this class to follow our argument. The main technical result we need is Barrington’s theorem from [Reference Barrington4] (but this is also proved from scratch in our context).

For $a,b$ elements of a group, the commutator of $a$ and $b$ is $[a, b] = a^{-1}b^{-1}ab$ . The conjugation convention is $a^b = b^{-1} \circ a \circ b$ . If $\pi \in \mathrm {Sym}(A)$ is a permutation, we may regard it as a permutation of $A \times B$ by $\pi ((a, b)) = (\pi (a), b)$ . Our groups always act from the left. If $g_1, \ldots , g_n$ are commuting elements of a group, we write $\prod _{i=1}^n g_i$ for their ordered product $g_n \cdots g_1$ . In groups of bijections on a set (which almost all our groups are), we denote composition by $\circ $ .

Given a finitely generated group G generated by the finite set S, a presentation of $g \in G$ is a word $w = s_n \cdots s_1 \in (S \cup S^{-1})^*$ such that $g = s_n \cdots s_1$ , and we write $w \equiv g$ . The word norm $\lVert g \rVert _S$ of $g \in G$ relative to S is then the length of a shortest presentation of $g$ , that is, $\lVert g \rVert _S = \min \{ n \in \mathbb {N} : \text { there exists } w \in (S \cup S^{-1})^n, w \equiv g \}$ . This word norm is also the distance in the Cayley graph of G between $1_G$ and $g$ . In this context, an element $g \in G$ is said to be distorted if $\lVert g^n \rVert _S = o(n)$ .

For sets $X,Y$ , and Z, we say that a map $f : X \to Y$ lifts into $\tilde {f} : X \times Z \to Y \times Z$ (or that $\tilde {f}$ is the lift of $f$ ) if $\tilde {f}(x,z) = (f(x),z)$ . For $S \subseteq X$ a subset of X, and $f : X \to X$ , we call extended restriction of $f$ to S the map $f\hspace {-0.05em}/\hspace {-0.2em}_S : X \to X$ defined as

$$ \begin{align*} f\hspace{-0.05em}/\hspace{-0.2em}_S(x) = \begin{cases} f(x) & \text{if } x \in S, \\ x & \text{otherwise}, \end{cases} \end{align*} $$

that is, $f\hspace {-0.05em}/\hspace {-0.2em}_S$ is the extension of the restriction $f|_S$ back to the full domain X, by fixing elements outside S.

2.2 Subshifts and cellular automata

Let $\Sigma $ be a finite alphabet. An element $x \in \Sigma ^{\mathbb {Z}}$ is called a configuration. An element $w \in \Sigma ^*$ is called a word or a pattern, and a pattern $w \in \Sigma ^*$ is said to appear in a configuration $x \in \Sigma ^{\mathbb {Z}}$ , denoted $w \sqsubseteq x$ if there exists some $i \in \mathbb {Z}$ such that $x_{i+j} = w_j$ for every .

We endow $\Sigma ^{\mathbb {Z}}$ with the product topology. This topology is generated by the cylinders $[a]_j = \{ x \in \Sigma ^{\mathbb {Z}} : x_j = a \}$ for $a \in \Sigma $ and $j \in \mathbb {Z}$ . The left shift $\sigma : \Sigma ^{\mathbb {Z}} \to \Sigma ^{\mathbb {Z}}$ defined by $\sigma (x)_i = x_{i+1}$ is a $\mathbb {Z}$ action on $\Sigma ^{\mathbb {Z}}$ . Closed and shift-invariant subsets X of $\Sigma ^{\mathbb {Z}}$ are called subshifts. For X a subshift and $n \in \mathbb {N}$ , we denote by $\mathcal {L}_n(X)$ the set of finite words of length $n$ that appear in X, and by $\mathcal {L}(X) = \bigcup _{n \in \mathbb {N}} \mathcal {L}_n(X)$ its language. We say that a subshift X is sofic if $\mathcal {L}(X)$ is a regular language. If X and Y are subshifts, a continuous and shift-equivariant map $f : X \to Y$ is called a morphism. It is an endomorphism if $X = Y$ and an automorphism if, in addition, it is bijective (in which case $f^{-1}$ is also an endomorphism). Endomorphisms are sometimes called cellular automata, and automorphisms reversible cellular automata. For $f : X \to Y$ a morphism between two subshifts, its radius (as a cellular automaton) is the minimal $r$ such that $f(x)_i$ is a function of . The biradius of an automorphism is the maximum of the radii of $f$ and $f^{-1}$ .

2.3 Turing machines

In this article, we use Turing machines as a specific kind of action on subshifts. We note that despite the terminology, it is not necessarily helpful to think of them as computational devices. (We will later perform computation in a group of Turing machines, but this computation is not related to the usual type of Turing machine computation, in that the iteration of a single machine is not going to be used to perform computation.)

Let Q be a finite set called the state set, and $\Gamma $ be a finite set called the tape alphabet. In the model of [Reference Kari, Ollinger, Ochmański and Tyszkiewicz32] (this model is equivalent to the usual definition of Turing machines, but handles reversibility better), a Turing machine is a triple $\mathcal {M} = (\Gamma , Q, \Delta )$ , where $\Delta \subseteq (Q \times \{+1,-1\} \times Q) \cup (Q \times \Gamma \times Q \times \Gamma )$ is the transition table. A transition $(q,\delta ,q') \in Q \times \{+1,-1\} \times Q$ is called a move transition, and a transition $(q,a,q',b) \in Q \times \Gamma \times Q \times \Gamma $ is called a matching transition.

In the rest of this paper, we focus on the action of Turing machines on two families of objects: bi-infinite tapes and finite cyclic tapes.

2.3.1 Bi-infinite tapes

In the alphabet $\Gamma \cup (Q \times \Gamma )$ , elements of $H = Q \times \Gamma $ are called heads. Denote by

$$ \begin{align*} \mathcal{X} = \big\{x \in {(\Gamma \cup (Q \times \Gamma))}^{\mathbb{Z}} \mid \text{ for all } i,j \in \mathbb{Z}: i \neq j\big. &\\ &\big.\implies x_i \in \Gamma \vee x_j \in \Gamma \big\} \end{align*} $$

the set of bi-infinite tapes with at most one head somewhere. We can associate to $\mathcal {M}$ its so-called moving-head model [Reference Kůrka35], that is, the binary relation $\rightarrow _{\mathcal {M}}$ on $\mathcal {X}$ defined by $x \rightarrow _{\mathcal {M}} x$ if $x \in \mathcal {X}$ contains no head (that is, $x \in \Gamma ^{\mathbb {Z}}$ ); and if $x \in \mathcal {X}$ contains a head at position, say $i_0 \in \mathbb {Z}$ with $x_{i_0} = (q,a)$ for some $q \in Q$ and $a \in \Gamma $ , then $x \rightarrow _{\mathcal {M}} x'$ if there exists $t \in \Delta $ such that:

$$ \begin{align*} \text{if } t = (q,a,q',b) \in \Delta: & \qquad x^{\prime}_i = \begin{cases} (q',b) & \text{if } i=i_0, \\ x_j & \text{otherwise}; \end{cases} \\ \text{if } t = (q,\delta,q') \in \Delta: & \qquad x^{\prime}_i = \begin{cases} a & \text{if } i=i_0, \\ (q',x_{i}) & \text{if } i=i_0+\delta,\\ x_i & \text{otherwise}. \end{cases} \end{align*} $$

The binary relation $\rightarrow _{\mathcal {M}}$ on $\mathcal {X}$ (denoted $\rightarrow $ for short if the context is clear) is the reachability relation. We write $\rightarrow _{\mathcal {M}}^k$ its $k$ th power, and $\rightarrow _{\mathcal {M}}^*$ its transitive closure. We say that $\mathcal {M}$ reaches the configuration $x'$ from $x$ in $k \in \mathbb {N}$ steps if $x \rightarrow _{\mathcal {M}}^k x'$ . A transition $x \rightarrow _{\mathcal {M}}^* x'$ is called a move.

The machine $\mathcal {M}$ is deterministic if $\rightarrow _{\mathcal {M}}$ defines a partial function, complete deterministic if it defines a total function (which is then continuous and, obviously, shift-commuting), and complete reversible (or reversible for short) if it defines a bijection (which is then a homeomorphism). When $\mathcal {M}$ is complete deterministic (which all our machines are), when using the relation $\rightarrow _{\mathcal {M}}$ as a function, we write it as $T_{\mathcal {M}} : \mathcal {X} \to \mathcal {X}$ , which is an endomorphism of the subshift $\mathcal {X}$ . Similarly, when the machine $\mathcal {M}$ is reversible, it is an automorphism of $\mathcal {X}$ .

2.3.2 Finite cyclic tapes

The set of cyclic configurations of length $\ell \in \mathbb {Z}_+$ is the set

$$ \begin{align*} \mathcal{C}_{\ell} = \big\{ x \in {(\Gamma \cup (Q \times \Gamma))}^{\mathbb{Z}/\ell\mathbb{Z}}\mid \text{for all } i,j \in \mathbb{Z}/\ell\mathbb{Z}, i \neq j\big. &\\ &\big. \implies x_i \in \Gamma \vee x_j \in \Gamma \big\} \end{align*} $$

of finite configurations containing at most one head. We always assume $\ell \geq 2$ in what follows (the case $\ell = 1$ makes sense, but requires notational modifications and is the least interesting case anyway).

The machine $\mathcal {M}$ defines a binary relation $\rightarrow _{\mathcal {M}}$ on $\mathcal {C}_{\ell }$ by considering these finite tapes as cyclic, that is, we define $x \rightarrow _{\mathcal {M}} x$ if $x \in \mathcal {C}_{\ell }$ contains no head (that is, $x \in \Gamma ^{\mathbb {Z}/\ell \mathbb {Z}}$ ); and if $x \in \mathcal {C}_{\ell }$ contains a head at position, say $i_0 \in \mathbb {Z}/\ell \mathbb {Z}$ with $x_{i_0} = (q,a)$ for some $q \in Q$ and $a \in \Gamma $ , then $x \rightarrow _{\mathcal {M}} x'$ if there exists $t \in \Delta $ such that:

$$ \begin{align*} \text{if } t = (q,a,q',b) \in \Delta : & \qquad x^{\prime}_i = \begin{cases} (q',b) & \text{if } i=i_0, \\ x_i & \text{otherwise}; \end{cases} \\ \text{if } t = (q,\delta,q') \in \Delta : & \qquad x^{\prime}_i = \begin{cases} a & \text{if } i=i_0, \\ (q', x_i) & \text{if } i=i_0+\delta \bmod \ell, \\ x_i & \text{otherwise}. \end{cases} \end{align*} $$

As above, the relation $\rightarrow _{\mathcal {M}}$ is called the reachability relation. If $\mathcal {M}$ is complete deterministic, the function $\rightarrow _{\mathcal {M}}$ will be denoted by $T_{\ell ,\mathcal {M}} : \mathcal {C}_{\ell } \to \mathcal {C}_{\ell }$ . Note that it is an endomorphism of the shift action of $\mathbb {Z}$ (or $\mathbb {Z}_{\ell }$ ) which translates the cyclic tape around.

Finally, for any machine $\mathcal {M} = (Q,\Gamma ,\Delta )$ , denote by $m: \mathbb {N} \to \mathbb {N}$ its movement function, that is, $m(n)$ is the maximal number of cells the machine can visit in $n$ steps. More precisely, $m(n)$ is the length $r-l+1$ of the largest interval such that there exists a sequence of $n$ steps of computation $x_0 \rightarrow _{\mathcal {M}} x_1 \rightarrow _{\mathcal {M}} \cdots \rightarrow _{\mathcal {M}} x_n$ (with $x_i \in \mathcal {X} $ ) such that for every position , at least one of the tapes $x_k$ ( $0 \leq k \leq n)$ has its head at position $i$ .

3.1 Action of SMART on cyclic tapes

This section studies the action of SMART on cyclic tapes of length $\ell \geq 2$ . We call initial configurations the following four cyclic configurations:

Proof. By symmetries between ${\blacktriangleright }$ and ${\blacktriangleleft }$ (respectively ${\rhd }$ and ${\lhd }$ ), we prove the result for $C_{\blacktriangleright }$ and $C_{\rhd }$ .

We used moves $M_{\blacktriangleright }(\ell -1),M_{\blacktriangleleft }(\ell -1),M_{\rhd }(\ell -1)$ and $M_{\lhd }(\ell -1)$ in patterns that overlap themselves on their first and last letters in the cyclic tape. This is valid, because the cell containing $s_*$ is only visited at the last (respectively first) step of $M_{\blacktriangleright }$ and $M_{\blacktriangleleft }$ (respectively $M_{\rhd }$ and $M_{\lhd }$ ).

For the last claim, by shift-commutation and bijectivity of the action, it is enough to show that a shifted copy of the initial configuration does not appear before the last step. This is clear from looking at the first columns, which have positive values on all but the first step and the two last steps.

3.2 Encoding SMART configurations

Recall that for the SMART machine, $\Gamma = \{0,1,2\}$ and $Q \simeq \{ {\blacktriangleright },{\blacktriangleleft },{\rhd },{\lhd }\} \times \{1,2\} $ . Lemma 3.4 implies that the set of cyclic SMART configurations of length $\ell $ that contain a head

$$ \begin{align*} \{ x \in (\Gamma \cup (Q \times \Gamma))^{\mathbb{Z}/\ell\mathbb{Z}} \mid \text{ there exists } i \in \mathbb{Z}/\ell\mathbb{Z}, x_i \in Q \times \Gamma \} \end{align*} $$

is conjugate (as a finite dynamical system or as a permutation) to a disjoint union of four (depending on whether the head is in state ${\blacktriangleright }, {\blacktriangleleft }, {\rhd }$ , or ${\lhd }$ ) disjoint systems of counters ranging in (respectively for the position of the head, the second component $\{1,2\}$ of Q, and the tape of alphabet $\Gamma )$ . Each of these counters encodes $2\ell \cdot 3^{\ell }$ different values, which is exactly the length of any SMART cycle by Lemma 3.4.

We pick a natural conjugacy $E_{\ell } : \mathcal {C}_{\ell } \to \mathcal {C}_{\ell }$ , a shift-invariant bijection that encodes a SMART configuration into its orbit position in base $2\ell \cdot 3^{\ell }$ . We refer to the conjugacy $E_{\ell }$ as the encoding map.

More precisely, if $w \in \mathcal {C}_{\ell }$ contains a head, then by Lemma 3.4, there exists some $0 \leq n < 2\ell \cdot 3^{\ell }$ such that $w = (T_{\ell ,\mathcal {S}})^n(C_q)$ for some initial configuration $C_q$ ( $q \in \{{\blacktriangleright },{\blacktriangleleft },{\rhd },{\lhd }\}$ ), and the map $E_{\ell }$ encodes the tuple $(q,n)$ in $\mathcal {C}_{\ell }$ as

where:

• • $q$ is stored in the first component of the state $\{{\blacktriangleright },{\blacktriangleleft },{\rhd },{\lhd }\}$ ;

• • $b \cdot c \in \{1,2\} \cdot \{0,1,2\}^{\ell }$ encodes $n \bmod 2\cdot 3^{\ell }$ , that is, $c$ is a ternary word satisfying $v_{(3)}(c) = n \bmod 3^{\ell }$ , and $b$ stores $(\lfloor n/3^{\ell } \rfloor \bmod 2)+1$ in the second component of the state;

• • $a$ is the quotient of $n$ by $2 \cdot 3^{\ell }$ , and is encoded in how much the cyclic configuration is shifted;

• • $\varepsilon = +1$ if $q \in \{{\blacktriangleright },{\rhd }\}$ (respectively $\varepsilon = -1$ if $q \in \{{\blacktriangleleft },{\lhd }\}$ ) shifts to the right (respectively left) if $q \in \{{\blacktriangleright },{\rhd }\}$ (respectively $q \in \{{\blacktriangleleft },{\lhd }\}$ ),

and if $w \in \mathcal {C}_{\ell }$ contains no head (that is, $w \in \Gamma ^{\mathbb {Z}/\ell \mathbb {Z}})$ , then we set $E_{\ell }(w) = w$ .

In other words, given a configuration $w = (T_{\ell ,\mathcal {S}})^n(C_q)$ for some initial configuration $C_q$ ( $q \in \{{\blacktriangleright },{\blacktriangleleft },{\rhd },{\lhd }\}$ ) and $0 \leq n < 2\ell \cdot 3^{\ell }$ , the map $E_{\ell }$ encodes the tuple $(q,n)$ as plainly (and humanly readable) as possible.

In the next two sections (§§3.3 and 3.4), we detail how this bijection can be computed inductively, that is, we define piecewise-defined bijective maps $F_{\mathrm {init}}$ , $F_{k \to k+1}$ (for $0 \leq k \leq \ell -2$ ), and $F_{\ell ,\mathrm {final}}$ acting on $\mathcal {C}_{\ell }$ such that

$$ \begin{align*} E_{\ell} = F_{\ell,\mathrm{final}} \circ \bigg( \prod_{k=0}^{\ell-2} F_{k \to k+1} \bigg) \circ F_{\mathrm{init}}. \end{align*} $$

The precise definition of these maps is not strictly necessary to understand the rest of this paper (it is only used in Lemmas 4.18 and 4.20, where we need the piecewise defined functions to satisfy the requirements described in §4.2.3). On a first reading, we recommend the reader simply remembers the idea of encoding configurations into a counter, and goes directly to §4 about the finitary distortion of the SMART machine.

3.3 Analysis of SMART configurations

We now explain how, given a cyclic SMART configuration $w$ of length $\ell $ , we determine which orbit it belongs in and its position in this orbit, that is, the number of steps required to obtain it from its corresponding initial configuration $C_{\blacktriangleright },C_{\blacktriangleleft },C_{\rhd }$ or $C_{\lhd }$ .

We say that a cyclic configuration $w \in \mathcal {C}_{\ell }$ is performing the $j\text {th}$ step of computation of $M_{\blacktriangleright }(k)$ (respectively $M_{\blacktriangleleft }(k), M_{\rhd }(k), M_{\lhd }(k)$ ), for $0 \leq k \leq \ell -1$ and $0 \leq j \leq f(k)$ , if it contains the $j\text {th}$ pattern of the sequence of transitions $M_{\blacktriangleright }(k)$ (respectively $M_{\blacktriangleleft }(k), M_{\rhd }(k), M_{\lhd }(k)$ ) of Proposition 3.2. At this point, it may not be clear that this is unique, but this will follow from our argument.

If a configuration is performing some step of computation from one of the moves $M_{\blacktriangleright }(k)$ , $M_{\blacktriangleleft }(k)$ , $M_{\rhd }(k)$ , or $M_{\lhd }(k)$ , we refer to this move as its computation of level k.

3.3.1 Initialization

We call the following patterns special patterns of level k (for $1 \leq k \leq \ell -1$ ):

and the following are special patterns of level $\ell $ :

The latter appear exactly in the shifts of the configurations $\mathcal {S}^{-1}(C_q)$ , for $C_q$ the initial configurations ( $q \in \{{\blacktriangleright },{\blacktriangleleft },{\rhd },{\lhd }\}$ ).

By the proof of Proposition 3.2, we see that if a cyclic configuration contains a special pattern $s({\blacktriangleright }_2,k)$ , $s({\blacktriangleright }_1,k)$ , $s({\blacktriangleleft }_2,k)$ , or $s({\blacktriangleleft }_1,k)$ (respectively $s({\rhd }_2,k)$ , $s({\rhd }_1,k)$ , $s({\lhd }_2,k)$ , or $s({\lhd }_1,k)$ ), then it performs the last two steps of $M_{\blacktriangleright }(k)$ or $M_{\blacktriangleleft }(k)$ respectively (respectively the first two steps of $M_{\rhd }(k)$ or $M_{\lhd }(k)$ ).

Claim 3.5. Given a cyclic configuration $w$ of length $\ell $ containing a head, exactly one of the following holds:

• • $w$ is the shift of an initial configuration;

• • $w$ performs some step of computation of level $0$ from either $M_{\blacktriangleright }(0)$ , $M_{\blacktriangleleft }(0)$ , $M_{\rhd }(0)$ , or $M_{\lhd }(0)$ ;

• • $w$ contains a special pattern of level $1 \leq k \leq \ell $ .

Proof. The patterns of $M_{\blacktriangleright }(0)$ , $M_{\blacktriangleleft }(0)$ , $M_{\rhd }(0)$ , and $M_{\lhd }(0)$ (eight in total), along with the special patterns of every level, disjointly cover all the non-initial configurations with a head (the level is determined by the distance to the nearest non-zero symbol in an appropriate direction).

3.3.2 Inductive analysis $k \to k+1$ (for $k+1 \leq \ell -1$ )

Let $k \leq \ell -2$ be an integer and $w$ be a non-initial cyclic configuration. If $w$ performs some computation of level $k$ (for $0 \leq k \leq \ell -2$ ), then it performs some computation of level $k+1$ : indeed, Figure 2 shows that any computation of level $k$ belongs to some computation of level $k+1$ , and that the latter is uniquely determined by considering the value of two cells (circled on the figure) which are left unmodified by the computation of level $k$ .

Figure 2 Bottom-up analysis of SMART configurations: $k \to k+1$ . Intermediate steps of moves of level $k+1$ (patterns of length $k+3$ ) corresponding to sub-moves of level $k$ . We darken the part of the pattern of length $k+2$ performing the sub-move of level $k$ . Circled ternary letters are not modified by the sub-move of level $k$ , and hence can be used to perform a case-analysis.

Additionally, if we know that $w$ performs the $j\text {th}$ step of some computation of level $k$ (for $0 \leq j \leq f(k)$ ), then the same case-analysis determines $j'$ ( $0 \leq j' \leq f(k+1)$ ) such that $w$ performs the $j'\text {th}$ step of its computation of level $k+1$ .

Example 3.6. For example, consider the cyclic configuration of length $\ell = 6$ :

First, $w$ performs step $0$ of $M_{\blacktriangleright }(0)$ (by simply considering all the computations of level $0$ ). Indeed, the highlighted pattern inside the configuration below is the $0\text {th}$ pattern of the move $M_{\blacktriangleright }(0)$ :

Then, by looking at Figure 2 (four cases, with three subcases each), we deduce successively that:

1. (1) by considering Figure 2 (case $M_{\blacktriangleright }(0)$ , second subcase),

   

   we deduce that $w$ performs step $2 = 0 + (f(0)+1)$ of $M_{\blacktriangleleft }(1)$ . Additionally, the computation extends to the left, that is, the move $M_{\blacktriangleleft }(1)$ appears in the following highlighted pattern:

   

2. (2) by considering Figure 2 (case $M_{\blacktriangleleft }(1)$ , third subcase),

   

   we deduce that $w$ performs step $4 = 2 + 2$ of $M_{\lhd }(2)$ . Additionally, the computation extends to the right and the move $M_{\lhd }(2)$ appears in the following highlighted pattern:

   

3. (3) by considering Figure 2 (case $M_{\lhd }(2)$ , first subcase),

   

   we deduce that $w$ performs step $58 = 4 + (2f(2)+4)$ of $M_{\lhd }(3)$ . Additionally, the computation extends to the right and the move $M_{\lhd }(3)$ appears in the following highlighted pattern (remember that configurations are cyclic):

   

4. (4) by considering Figure 2 (case $M_{\lhd }(3)$ , second subcase),

   

   we deduce that $w$ performs step $218 = 58 + (2f(3)+2)$ of $M_{\blacktriangleleft }(4)$ . Additionally, the computation extends to the left and the move $M_{\blacktriangleleft }(4)$ appears in the following highlighted pattern:

   

5. (5) by considering Figure 2 (case $M_{\blacktriangleleft }(4)$ , second subcase),

   

   we deduce that $w$ performs step $460 = 218 + (f(4)+1)$ of $M_{\blacktriangleright }(5)$ .

   On the final step, we should extend to the right; since we run out of cells on the tape, this means that we should interpret the circled $2$ -cell as both the first and last cell of the $M_{\blacktriangleright }(5)$ -computation, that is, the move $M_{\blacktriangleright }(5)$ appears in the following highlighted self-overlapping pattern:

   

So we claim that $w$ performs step $460 = 218 + (f(4)+1)$ of $M_{\blacktriangleright }(5)$ . And one can verify by a direct calculation (for example, by computer) that

as we just deduced.

3.3.3 Conclusion

Let $w \in \mathcal {C}_{\ell }$ be a cyclic tape of length $\ell $ containing a head. By the claim above, there are three different cases.

Either $w$ is the shift of an initial configuration, in which case the state of $w$ determines which orbit $w$ belongs to, and counting the shift (and multiplying it by $2 \cdot 3^{\ell }$ ) is enough to know the position of the configuration $w$ in its orbit. A similar reasoning applies for configurations $w$ that contain a special pattern of level $\ell $ .

However, assume $w$ contains some computation of level $0$ . Then $w$ contains computations of every level $k$ for $0 \leq k \leq \ell -1$ by the previous induction. Similarly, if $w$ contains a special pattern of level $k$ for some $0 \leq k \leq \ell -1$ , then $w$ corresponds to either the last two steps of $M_{\blacktriangleright }(k)$ or $M_{\blacktriangleleft }(k)$ , or the first two steps of $M_{\rhd }(k)$ or $M_{\lhd }(k)$ . Then $w$ corresponds to computations of every level $k'$ for $k \leq k' \leq \ell -1$ by the previous induction.

Finally, the structure of each SMART cycle (detailed in the proof of Proposition 3.3) enables to conclude about which orbit the configuration $w$ belongs to, and its position in said orbit.

Example 3.7. Consider once again the cyclic configuration of length $\ell = 6$ ,

By the previous example, $w$ performs step $460$ of $M_{\blacktriangleright }(5)$ on the following (self-overlapping) pattern:

Considering the proof of Proposition 3.3, only the orbits of $C_{\rhd }$ and $C_{\lhd }$ have a cell containing tape-letter $2$ that is left unchanged during a computation of level $\ell -1$ . Additionally, out of these two, only the orbit of $C_{\rhd }$ contains the move $M_{\blacktriangleright }(5)$ .

So $w$ belongs in the orbit of $C_{\rhd }$ . From the proof of Proposition 3.3, we deduce that, modulo $2 \cdot 3^{\ell }$ , the position of $w$ in said orbit is $1+460$ . Finally, $2$ being the second cell of the tape, one shift already happened in the orbit: we conclude that the position of $w$ in the orbit of $C_{\rhd }$ is $2\cdot 3^{\ell } + 461 = 1919$ .

Indeed, one can verify by a direct calculation that

as we just deduced.

3.4 Encoding cyclic configurations into their orbit positions in $\mathcal {C}_{\ell }$

Denote by $\mathcal {S}$ the SMART machine introduced above. Recall the encoding map $E_{\ell } : \mathcal {C}_{\ell } \to \mathcal {C}_{\ell }$ defined in §3.2 as

if $w$ contains a head (in which case, there exists $n$ and $q \in \{{\blacktriangleright },{\blacktriangleleft },{\rhd },{\lhd }\}$ such that $w = T_{\ell ,\mathcal {S}}^n(C_q)$ , and we set $b \cdot c \in \{1,2\} \cdot \{0,1,2\}^{\ell }$ to encode $n \bmod 2 \cdot 3^{\ell }$ , and $a$ is the quotient of $n$ by $2 \cdot 3^{\ell }$ ); and if $w \in \mathcal {C}_{\ell }$ contains no head, then we set $E_{\ell }(w) = w$ .

3.4.1 Inductive encoding

In this section, we use the analysis performed in §3.3 to provide a linear-time algorithm that computes this encoding $E_{\ell } : \mathcal {C}_{\ell } \to \mathcal {C}_{\ell }$ inductively.

Figure 3 describes a piecewise-defined bijection $F_{\mathrm {init}} : \mathcal {C}_{\ell } \to \mathcal {C}_{\ell }$ : each case describes how a pattern of length $2$ (e.g. ) is bijectively replaced by another (in the previous example, by ). In other words, if a cyclic configuration $w$ contains a sub-pattern of length $2$ that matches with one case of Figure 3, then $F_{\mathrm {init}}$ replaces this sub-pattern in $w$ by its image in the figure.

Figure 3 Encoding SMART configurations: $F_{\mathrm {init}}$ . $F_{\mathrm {init}}$ rewrites sub-patterns of length $2$ in cyclic SMART configurations.

Similarly, Figures 4 and 5 together describe a piecewise-defined bijection $F_{k \to k+1} : \mathcal {C}_{\ell } \to \mathcal {C}_{\ell }$ that replaces sub-patterns of length $k+4$ . Intuitively, Figure 4 (defining the first half of $F_{k \to k+1}$ ) encodes moves of level $k+1$ into counters, while Figure 5 (the second half of $F_{k \to k+1}$ ) encodes special patterns of level $k+1$ .

Figure 4 Encoding SMART configurations: $F_{k \to k+1}$ (Part 1: $k \to k+1$ ). First half of rewriting cases of $F_{k \to k+1}$ , which rewrites sub-patterns of length $k+4$ . Letters are unmodified. $F_{k \to k+1}$ extends the word $c \in \{0,1,2\}^{k+1}$ with one additional letter and, considering $c$ as a counter, adds a number of steps in accordance with Lemma 3.2. Note that at $k+1 = \ell -2$ (respectively $k+1 = \ell -1$ ), the -cells overlap on each other (respectively the counter), because we reach the length of the cyclic tape.

Figure 5 Encoding SMART configurations: $F_{k \to k+1}$ (Part 2: special $\to k+1$ ). Second half of rewriting cases of $F_{k \to k+1}$ , which rewrites sub-patterns of length $k+4$ . Letters are unmodified. Encodes special configurations of level $k+1$ by replacing the $k+2$ other letters by a counter of $\{0,1,2\}^{k+2}$ in accordance with Lemma 3.2, and preserves special configurations of level $> k+1$ . Note that at $k+1 = \ell -2$ (respectively $k+1 = \ell -1$ ), the -cells overlap on each other (respectively the counter), because we reach the length of the cyclic tape.

Finally, Figure 6 describes a similar bijection $F_{\ell ,\mathrm {final}} : \mathcal {C}_{\ell } \to \mathcal {C}_{\ell }$ .

We can then prove the following lemma.

Lemma 3.8. Let $w$ be a cyclic configuration of $\mathcal {C}_{\ell }$ . Then,

$$ \begin{align*} E_{\ell}(w) = \bigg[ F_{\ell,\mathrm{final}} \circ \prod_{k=0}^{\ell-2} F_{k \to k+1} \circ F_{\mathrm{init}} \bigg] (w). \end{align*} $$

Proof Sketch of proof

Let $w$ be a configuration and $k$ be some integer $k \leq \ell -1$ . If $w$ is neither the shift of an initial configuration nor a special configuration of level $> k$ , then there exists some $q \in \{{\blacktriangleright },{\blacktriangleleft },{\rhd },{\lhd }\}$ and a unique word $p = p_0 \cdots p_{k+1}$ of length $k+2$ such that $p \sqsubseteq w$ and $p$ computes the $j\text {th}$ step of $M_q(k)$ for some $0 \leq j \leq f(k) = 3^{k+2} - 2$ .

Figure 6 Final encoding step of SMART configurations: $F_{\ell ,\mathrm {final}}$ . Rewrites complete cyclic configurations of length $\ell $ . $F_{\ell ,\mathrm {final}}$ acts according to the proof of Proposition 3.3: it maps encodings of level $\ell -1$ to their final encodings, and ‘corrects’ the position the head and shifts the counter when required (in the encodings of ${M_{\blacktriangleleft }(\ell -1)}$ and $M_{\rhd }(\ell -1)$ , or in initial configurations, whose heads were moved when applying $F_{\mathrm {init}}$ ).

Then by induction on $k$ , one sees that the partial composition

$$ \begin{align*} \prod_{k'=0}^{k-1} F_{k' \to k'+1} \circ F_{\mathrm{init}}, \end{align*} $$

when applied on $w$ , replaces $p$ in $w$ by another pattern $p'$ of the same length defined as follows:

where $c \in \{0,1,2\}^{k+1}$ is a ternary counter such that $\mathrm {v}_3(c) = j+1$ . Notice that $0 \leq j \leq f(k) -2$ , where $f(k) = 3^{k+2}-2$ ; so that $1 \leq j +1 \leq f(k)-1$ fits exactly in the space of non-zero counters. The zero counters are reserved for (shifts of) initial and special configurations.

Finally,

$$ \begin{align*}F_{\ell,\mathrm{final}} \circ \prod_{k=0}^{\ell-2} F_{k \to k+1} \circ F_{\mathrm{init}} \end{align*} $$

is equal to $E_{\ell }$ by considering how $F_{\ell ,\mathrm {final}}$ acts in accordance with the structure of the four disjoint cycles of SMART (detailed in the proof of Proposition 3.3).

Example 3.9. Consider once again the cyclic configuration of length $\ell = 6$ from Example 3.6, and let us use the formulas above to encode it into a counter value. This process will roughly mirror Examples 3.6 and 3.7, except that due to our coding convention, the counter value is one larger than the correct one until the very last step. First, we apply $F_{{\mathrm {init}}}$ , which observes based on the highlighted cells

that the first case of $M_{\blacktriangleright }(0)$ applies, and rewrites this as

Next, we apply $F_{0 \to 1}$ . Based on the highlighted cells, the fourth case of $M_{\blacktriangleright }(0)$ applies and we rewrite this as

Next, we apply $F_{1 \to 2}$ . Based on the highlighted cells, the fifth case of $M_{\blacktriangleleft }(1)$ applies and we rewrite this as

Next, we apply $F_{2 \to 3}$ . Based on the highlighted cells, the first case of $M_{\lhd }(2)$ applies and we rewrite this as

Next, we apply $F_{3 \to 4}$ . Based on the highlighted cells, the third case of $M_{\lhd }(3)$ applies and we rewrite this as

Next, we apply $F_{4 \to 5}$ . Based on the highlighted cells, the fourth case of $M_{\blacktriangleleft }(4)$ applies and we rewrite this as

Finally, we apply $F_{{\mathrm {final}}}$ (in Figure 6). We are in the fourth case, where we do not touch the counter and just change the state to ${\rhd }_1$ :

This gives the expected result: as $v_3(122002) = 461$ , and the head is shifted once to the right in the tape, this configuration has position $1 \cdot (2 \cdot 3^{\ell }) + 461 = 1919$ in the orbit of the initial configuration corresponding to state ${\rhd }_1$ .

4.1 Context and results

4.1.1 Decorated SMART on $\mathcal {C}_{\ell }$

Let $\mathcal {S}^{(\textrm {o})} = (Q^{(\textrm {o})},\Gamma ^{(\textrm {o})},\Delta ^{(\textrm {o})})$ be the SMART machine (see §3). Define the decorated version of the SMART machine as $\mathcal {S}_{\mathrm {dec}} = (Q,\Gamma ,\Delta )$ , with

$$ \begin{align*} Q = &\ Q^{(\textrm{o})} \times D \times G, \\ \Gamma = &\ \Gamma^{(\textrm{o})}, \\ \Delta = & \bigcup_{(d,x) \in D \times G} \{ ( (q,d,x),a,(q',d,x),b ) : (q,a,q',b) \in \Delta^{(\textrm{o})}\} \\ & \cup \bigcup_{(d,x) \in D \times G} \{ ((q,d,x),\delta,(q',d,x)) : (q,\delta,q') \in \Delta^{(\textrm{o})} \} \end{align*} $$

for $D = \{{d_{1}},{d_{2}}\}$ and , that is, the states of SMART now carry a state $q^{(\textrm {o})} \in Q^{(\textrm {o})}$ of the original machine $\mathcal {S}$ , a special symbol $d \in D$ called the duck, and a ghost symbol $x \in G$ . We have $\lvert Q\rvert = 96$ .

Technically, the two SMART machines $\mathcal {S}_{\mathrm {dec}}$ and $\mathcal {S}$ are different: for one, they act on different sets of cyclic tapes (since they have different sets of states). However, they have very similar behaviors, as the decorated machine only carries its decoration unmodified in its state while acting on tapes. To refer to the original set of states of SMART, we will use $Q^{(\textrm {o})}$ , and Q will denote $Q = Q^{(\textrm {o})} \times D \times G$ .

Since the remainder of this article only uses the decorated version of SMART, in what follows, $\mathcal {S}$ will refer to the decorated version of the machine, despite lacking the ‘ ${\mathrm {dec}}$ ’ subscript. This should not cause confusion, as the machines act on different sets.

The point of the ghost is to allow us to condition the application of gates, and to build the permutations we perform in §4.2. The duck $d \in \{{d_{1}},{d_{2}}\}$ will be important during intermediate steps of computation in §4.2, to realize piecewise defined functions.

For $S \subseteq \mathcal {C}_{\ell }$ a subset of finite cyclic tapes, we denote by $S[{d_{1}}]$ and $S[{d_{2}}]$ the subsets of the tapes of S containing a head, and whose ducks respectively are $d = {d_{1}}$ and $d = {d_{2}}$ . For $d \in \{{d_{1}},{d_{2}}\}$ and a function $f : S \to S$ , we abuse notation and denote by $f\hspace {-0.05em}/\hspace {-0.2em}_{d}$ the extended restriction $f\hspace {-0.05em}/\hspace {-0.2em}_{S[d]}$ of $f$ to $S[d]$ :

$$ \begin{align*} f\hspace{-0.05em}/\hspace{-0.2em}_{d}(w) = \begin{cases} f(w) & \text{if } w \in S[d], \\ w & \text{otherwise}. \end{cases} \end{align*} $$

4.1.2 Group of Turing machine instructions on finite cyclic tapes

Recall that $\mathcal {C}_{\ell }$ is the set of finite cyclic tapes of length $\ell \geq 2$ (see §2) with states Q and tape-alphabet $\Gamma $ , containing at most one head (that is, a letter in $Q \times \Gamma )$ .

Let $\mathcal {G}_{\ell }$ be the finitely generated subgroup of $\mathrm {Sym}(\mathcal {C}_{\ell })$ generated by state-dependent moves, and the unary gates permuting heads. Formally, for $g \in \mathrm {Sym}(Q \times \Gamma )$ , define the unary gate $\pi _g \in \mathrm {Sym}(\mathcal {C}_{\ell })$ as

$$ \begin{align*} \pi_g(w)_j = \begin{cases} g(w_j) & \text{if } w_j \in (Q \times \Gamma), \\ w_j & \text{if } w_j \in \Gamma, \end{cases} \end{align*} $$

for a cyclic tape $w \in \mathcal {C}_{\ell }$ . In addition, for $q \in Q$ , define the state-dependent right move $\rho _q \in \mathrm {Sym}(\mathcal {C}_{\ell })$ as

$$ \begin{align*} \rho_q(w)_j = \begin{cases} (q,w_j) & \text{if } w_{j-1 \bmod \ell} \in (\{q\} \times \Gamma), \\ \pi_{\Gamma}(w_j) & \text{if } w_{j-1 \bmod \ell} \notin (\{q\} \times \Gamma) \wedge w_j \in (\{q\} \times \Gamma) \cup \Gamma, \\ w_j & \text{if } w_{j-1 \bmod \ell} \notin (\{q\} \times \Gamma) \wedge w_j \in ((Q \setminus \{q\}) \times \Gamma), \end{cases} \end{align*} $$

for $w \in \mathcal {C}_{\ell }$ and $\pi _{\Gamma } : \Gamma \cup (Q \times \Gamma ) \to \Gamma $ the natural projection.

We then define the group $\mathcal {G}_{\ell } \leq \mathrm {Sym}(\mathcal {C}_{\ell })$ generated by these permutations:

$$ \begin{align*} \mathcal{G}_{\ell} = \langle \{\pi_g : g \in \mathrm{Sym}(Q \times \Gamma) \} \cup \{ \rho_q \mid q \in Q\} \rangle. \end{align*} $$

We can see the group $\mathcal {G}_{\ell }$ as the group generated by the instructions of Turing machines: moving heads based on their states, or permuting their values. To ease notation, we denote $\prod _{q \in Q} \rho _q$ by $\rho $ . This (finite) group is equipped with a metric, that is, the word norm given by the generators $\pi _g$ and $\rho _q$ .

It is easy to see that for any reversible Turing machine $\mathcal {M}$ of states Q and tape-alphabet $\Gamma $ , $T_{\ell ,\mathcal {M}}$ is an element of $\mathcal {G}_{\ell }$ . Indeed, a step of computation is the composition of a head permutation $\alpha $ of $Q \times \Gamma $ , followed with state-dependent moves $\beta _{+1}$ and $\beta _{-1}$ :

$$ \begin{align*} \alpha(q,a) & = \begin{cases} (q',b) & \text{if } (q,a,q',b) \in \Delta, \\ (q',a) & \text{if } (q,\pm 1, q') \in \Delta, \end{cases} \\ \beta_{+1} & = \prod_{q' \mid \text{ there exists } q, (q,+1,q') \in \Delta} \rho_{q'}, \\ \beta_{-1} & = \prod_{q' \mid \text{ there exists } q, (q,-1,q') \in \Delta} {\rho_{q'}}^{-1}. \end{align*} $$

Finally, we denote by $\delta (\ell ,n)$ the word norm of $(T_{\ell ,\mathcal {M}})^n$ in $\mathcal {G}_{\ell }$ . In this section, we focus on proving that $\delta (\ell ,n)$ is polynomial in $\ell $ for powers of the decorated SMART machine (even for powers exponential in $\ell $ ).

Remark 4.1. It can be shown that $\mathcal {G}_{\ell }$ is, for large enough $\ell $ , $\lvert Q\rvert $ , and $\lvert \Gamma \rvert $ (in particular for all versions of the SMART machine we consider and for $\ell \geq 2$ ), of bounded index in the automorphism group of $\mathcal {C}_{\ell }$ under the shift action of $\mathbb {Z}_{\ell }$ . This is not particularly useful, however, as what we need it for is to provide a group where the SMART machine corresponds to an element of small word norm (far smaller than the radius of the group).