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 $i$ .

3.3.1 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 $M_{\lhd }(k)$ ).

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.

3.4.1 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 ${\rhd }_1$ .

4.1.1 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 $ ).

4.1.3 Main result: finitary distortion of the decorated SMART

Recall that $m: \mathbb {N} \to \mathbb {N}$ is the movement function, that is, $m(n)$ is the maximal number of cells the machine $\mathcal {S}$ can visit in $n$ steps; and that $\delta (\ell ,n)$ is the word norm of $(T_{\ell ,\mathcal {S}})^n$ in $\mathcal {G}_{\ell }$ .

Any finite order T satisfies the latter two items, and any non-trivial state-dependent shift satisfies the first and the third items. Achieving the first two items is already difficult and to our knowledge, these properties have only been explicitly shown (in the reversible case) for the SMART machine and the binary SMART machine [Reference Concha-Vega and Torres-Avilés15]. We expect that the Kari–Ollinger construction in [Reference Kari, Ollinger, Ochmański and Tyszkiewicz32] can be used to produce more examples of machines satisfying these two properties (at least $m(n) = O(n / \log n)$ follows from general principles for all these machines [Reference Guillon, Salo, Dennunzio, Formenti, Manzoni and Porreca28]).

4.1.4 Proof of Lemma 4.2

For the second item, the logarithmic speed of SMART is well known. To sketch a proof, consider the following computation: after less than 18 steps, the head of SMART is in state ${\rhd }_1$ or ${\lhd }_1$ reading a $0$ (ignoring the ghost and the duck). Then SMART is either at the left (for ${\rhd }_1$ ) or right (for ${\lhd }_1$ ) extremity of some word $0^m$ for some $m \geq \log _3(k)+2$ or by [Reference Cassaigne, Ollinger and Torres-Avilés14, Lemma 4], it builds around this position some pattern in the set $C_m$ for $m \geq \log _3(k)+2$ (with the notation of [Reference Cassaigne, Ollinger and Torres-Avilés14, Lemma 4]). Either way, after this point, $k$ steps of computations cannot read more than $\log _3(k)+2$ different cells.

The proof of the third item is a matter of programming powers of the machine efficiently with the generators of $\mathcal {G}_{\ell }$ , which can be considered as the primitive reversible instructions of Turing machines. To achieve this, we encode configurations into their orbit position with the automorphism $E_{\ell }$ (defined in §3.2), perform an addition on these positions (as defined below), and decode back, as summarized in the commuting diagram below:

More precisely, we prove the two following lemmas.

Lemma 4.3. Let $E_{\ell } : \mathcal {C}_{\ell } \to \mathcal {C}_{\ell }$ be the encoding map defined in §3.2. There exists some $\tilde {E}_{\ell } : \mathcal {C}_{\ell } \to \mathcal {C}_{\ell }$ in $\mathcal {G}_{\ell }$ with word norm $O(\ell ^4)$ such that

$$ \begin{align*} w \in \mathcal{C}_{\ell}[{d_{1}}] \implies \tilde{E}_{\ell}(w) = E_{\ell}(w), \end{align*} $$

where $\mathcal {C}_{\ell }[{d_{1}}]$ is the set of cyclic tapes $w \in \mathcal {C}_{\ell }$ having a head with duck $d = {d_{1}}$ .

Note that we say nothing about the action of $\tilde {E}_{\ell }$ on tapes having duck $d = {d_{2}}$ . This restriction comes from our use of the ducking trick to build $\tilde {E}_{\ell }$ , which produces ‘garbage’ (that is, acts with no reasonable interpretation) on heads having duck $d = {d_{2}}$ . See §4.2.3 for more details.

Lemma 4.4. Let $(+n)_{\ell }$ be the bijection of $\mathcal {C}_{\ell }$ that performs the addition of $n \in \mathbb {N}$ in base $2\ell \cdot 3^{\ell }$ on the orbits positions encoded by $E_{\ell }$ . Recall that $(+n\hspace {-0.05em}/\hspace {-0.2em}_{{d_{1}}})_{\ell } : \mathcal {C}_{\ell } \to \mathcal {C}_{\ell }$ is defined as

$$ \begin{align*} (+n\hspace{-0.05em}/\hspace{-0.2em}_{{d_{1}}})_{\ell}(w) = \begin{cases} (+n)_{\ell}(w) & \text{if } w \in \mathcal{C}_{\ell}[{d_{1}}], \\ w & \text{otherwise}. \end{cases} \end{align*} $$

Then $(+n\hspace {-0.05em}/\hspace {-0.2em}_{{d_{1}}})_{\ell }$ belongs to $\mathcal {G}_{\ell }$ with word norm $O(\ell ^3)$ .

We give a more detailed definition of $(+n)_{\ell }$ in §4.3.2. Informally, $(+n)_{\ell }$ adds $n$ to the counter $E_{\ell }(w)$ that encodes the orbit position of the SMART configuration $w$ .

Then, these two lemmas are enough to prove the third item of Lemma 4.2 about the decorated SMART being distorted on cyclic tapes of length $\ell $ (more precisely, $\delta (\ell ,n) = O(\ell ^4)$ ). Indeed, combining the two previous results, we obtain the following lemma.

Lemma 4.5. Denoting again:

$$ \begin{align*} T_{\ell,\mathcal{S}}\hspace{-0.05em}/\hspace{-0.2em}_{{d_{1}}}(w) = \begin{cases} T_{\ell,\mathcal{S}}(w) & \text{if } w \in \mathcal{C}_{\ell}[{d_{1}}], \\ w & \text{otherwise}, \end{cases} \end{align*} $$

then for any $n \in \mathbb {N}$ , $(T_{\ell ,\mathcal {S}}\hspace {-0.05em}/\hspace {-0.2em}_{{d_{1}}})^n$ belongs to $\mathcal {G}_{\ell }$ with word norm $O(\ell ^4)$ .

Proof. Let $(+n\hspace {-0.05em}/\hspace {-0.2em}_{{d_{1}}})_{\ell }$ be given by Lemma 4.4. With Lemma 4.3, one can conjugate $(+n\hspace {-0.05em}/\hspace {-0.2em}_{{d_{1}}})_{\ell }$ with $\tilde {E}_{\ell }$ and obtain a bijection on $\mathcal {C}_{\ell }$ that maps configurations of $\mathcal {C}_{\ell }[{d_{1}}]$ to their $n$ th iterate by $\mathcal {S}$ and is the identity on $\mathcal {C}_{\ell }[{d_{2}}]$ . In other words,

$$ \begin{align*} (T_{\ell,\mathcal{S}}\hspace{-0.05em}/\hspace{-0.2em}_{{d_{1}}})^n = ((+n\hspace{-0.05em}/\hspace{-0.2em}_{{d_{1}}})_{\ell})^{\tilde{E}_{\ell}}. \end{align*} $$

Indeed, the addition of $(+n\hspace {-0.05em}/\hspace {-0.2em}_{{d_{1}}})_{\ell }$ is only performed on heads having duck ${d_{1}}$ , so the garbage generated by $\tilde {E}_{\ell }$ on ducks $d={d_{2}}$ is canceled in the conjugation; additionally, the shift of the tape (which happens when the addition modulo $2 \cdot 3^{\ell }$ overflows) is performed to the right (respectively to the left), exactly like $(T_{\ell ,\mathcal {S}})^{2 \cdot 3^{\ell }}$ acts on configurations $C_{\blacktriangleright }$ and $C_{\rhd }$ (respectively $C_{\blacktriangleleft }$ and $C_{\lhd }$ ).

And this lemma then leads to the following proof.

Proof of Lemma 4.2, third item

Let $d' = {d_{1}} \leftrightarrow {d_{2}} \in \mathrm {Sym}(D)$ be the involution that swaps ducks ${d_{1}}$ and ${d_{2}}$ , and $d = \mathrm {id} \times d' \times \mathrm {id} \times \mathrm {id} \in \mathrm {Sym}(H)$ its lift to $H = Q \times \Gamma $ (where $Q = Q^{(\textrm {o})} \times D \times G$ ). We have

$$ \begin{align*} (T_{\ell,\mathcal{S}})^n = (\pi_d \circ (T_{\ell,\mathcal{S}}\hspace{-0.05em}/\hspace{-0.2em}_{{d_{1}}})^n \circ \pi_d ) \circ (T_{\ell,\mathcal{S}}\hspace{-0.05em}/\hspace{-0.2em}_{{d_{1}}})^n \end{align*} $$

because $(T_{\ell ,\mathcal {S}}\hspace {-0.05em}/\hspace {-0.2em}_{{d_{1}}})^{\pi _d} = T_{\ell ,\mathcal {S}}\hspace {-0.05em}/\hspace {-0.2em}_{{d_{2}}}$ .

4.1.5 Overview

The following subsections deal with the proofs of Lemmas 4.3 and 4.4, which respectively prove that the encoding and the addition can be implemented in $\mathcal {G}_{\ell }$ with respective word norms $O(\ell ^4)$ and $O(\ell ^3)$ .

More precisely, §4.2 contains our main technical results, which define and implement conditional permutations. It also contains an exposition of the ducking trick, a method we use to implement piecewise-defined bijections. Section 4.3 contains the proofs of Lemmas 4.3 and 4.4.

4.2.1 Permutation conditioning

We call conditions the subsets C of $(Q \times \Gamma ) \times \Gamma ^{\ell -1}$ . For $g \in \mathrm {Sym}(Q \times \Gamma )$ , if C (considered as a subset of $\mathcal {C}_{\ell }$ ) is a $\pi _g$ -invariant subset, we can define the bijection $\pi _{g,C} : \mathcal {C}_{\ell } \to \mathcal {C}_{\ell }$ as $\pi _{g,C}(w) = w$ if $w \in \mathcal {C}_{\ell }$ contains no head; and if $w \in \mathcal {C}_{\ell }$ contains a head at position, say, $i_0 \in \mathbb {Z}/\ell \mathbb {Z}$ , we set $\pi _{g,C}(w)$ to

$$ \begin{align*} \pi_{g,C}(w)_i = \begin{cases} w_i & \text{if } i \neq i_0, \\ g(w_i) & \text{if } i=i_0 \text{ and } w_i \cdot w_{i+1} \cdots w_{\ell-1} \cdot w_0 \cdots w_{i-1} \in C, \\ w_i & \text{if } i=i_0 \text{ and } w_i \cdot w_{i+1} \cdots w_{\ell-1} \cdot w_0 \cdots w_{i-1} \notin C. \end{cases} \end{align*} $$

We then call $\pi _{g,C}$ the conditional application of g under condition C.

Recall that the states of Q have a ghost component G. We split Q into two components: $Q = Q' \times G$ (so, with the notation $Q = Q^{(\textrm {o})} \times D \times G$ , we have $Q' = Q^{(\textrm {o})} \times D$ ; but the exact structure of $Q'$ has no importance in this section).

Then, the set $H = Q \times \Gamma $ also splits into $H = H' \times G$ , where $H' = Q' \times \Gamma $ . Note that for any permutation $g \in \mathrm {Sym}(G)$ , the permutation $\mathrm {id}_{H'} \times g$ belongs to $\mathrm {Sym}(H)$ ; and similarly, if $g \in \mathrm {Sym}(H')$ , the permutation $g \times \mathrm {id}_{G}$ belongs to $\mathrm {Sym}(H)$ . Finally, all conditions we define below will be of the form $C = C' \times G$ , for $C' \subseteq H' \times \Gamma ^{\ell -1}$ .

In this section, we prove Lemma 4.7: for gates $g \in \mathrm {Sym}(H')$ and $\pi _{g \times \mathrm {id}}$ -invariant conditions $C = C' \times G$ (for some $C' \subseteq H' \times \Gamma ^{\ell -1}$ ), the conditioned gates $\pi _{g \times \mathrm {id}, C}$ belong to $\mathcal {G}_{\ell }$ . We also provide upper bounds on the word norm of $\pi _{g \times \mathrm {id}, C}$ depending on C. This is essentially Barrington’s theorem [Reference Barrington4].

As a first step, we consider the opposite case: instead of leaving the ghost component of the head intact while permuting, we consider permutations that only permute the ghost component G. As conditions $C = C' \times G$ (for $C' \subseteq H' \times \Gamma ^{\ell -1}$ ) are trivially $\pi _{\mathrm {id} \times g}$ -invariant for any $g \in \mathrm {Sym}(G)$ , the conditioned gates $\pi _{\mathrm {id} \times g, C}$ are always defined and we obtain the following lemma.

Lemma 4.6. For any $g \in \mathrm {Alt}(G)$ and condition $C = C' \times G$ (for some $C' \subseteq H' \times \Gamma ^{\ell -1}$ ), the conditioned gate $\pi _{\mathrm {id} \times g, C}$ belongs to $\mathcal {G}_{\ell }$ . Let $T : \mathcal {P}(H \times \Gamma ^{\ell -1}) \to \mathbb {N}$ be the optimal function such that $\lVert \pi _{\mathrm {id} \times g,C} \rVert \leq T(C)$ for all $g \in \mathrm {Alt}(G)$ . Then T satisfies the following inequalities:

$$ \begin{align*} T([a]_j \times G) & \leq \lvert \min(j, \ell-j)\rvert, \\ T((C' \cap C") \times G) & \leq 2 (T(C' \times G) + T(C" \times G)), \\ T((C' \cup C") \times G) & \leq \begin{cases} T(C' \times G) + T(C" \times G) & \text{if } C' \cap C" = \emptyset, \\ 2 (T(C' \times G) + T(C" \times G) ) + 5 & \text{otherwise}, \end{cases} \\ T((C^{\prime c}) \times G) & \leq T(C' \times G) + 1. \end{align*} $$

Proof. We prove by induction over $C' \subseteq H' \times \Gamma ^{\ell -1}$ that, denoting $C = C' \times G$ , every $\pi _{\mathrm {id} \times g,C}$ (for $g \in \mathrm {Alt}(G)$ ) has word norm that checks the aforementioned inequalities.

Case 1. If $C' = B' \times \Gamma ^{\ell -1}$ for some $B' \subseteq H'$ , then any such $\pi _{\mathrm {id} \times g,B' \times G}$ already appears in the set of generators of $\mathcal {G}_{\ell }$ .

Case 2. If $C' = [a]_j \triangleq H' \times \Gamma ^{j-1} \times \{a\} \times \Gamma ^{\ell -j-1}$ for some and $a \in \Gamma $ , define $B' = (Q' \times \{a\}) \times \Gamma ^{\ell -1}$ . Then one can conjugate $\pi _{\mathrm {id} \times g,B' \times G}$ (which belongs to $\mathcal {G}_{\ell }$ by the first item) with either the right-move $\rho ^{j}$ or the left-move $\rho ^{-(\ell -j)}$ : the resulting permutation applies $g$ on the ghost symbol if and only if $j$ cells away from the head, the content of the tape is $a$ .

Case 3. If $C' = (C_1')^c$ , then $\pi _{\mathrm {id} \times g,C' \times G} = \pi _{\mathrm {id} \times g^{-1},C_1' \times G} \circ \pi _{\mathrm {id} \times g}$ .

Case 4. If $C' = C_1' \cap C_2'$ , we use the ‘commutator trick’: as G has cardinality at least $5$ , $g$ is a commutator by Ore’s theorem [Reference Ore42, Theorem 7], so there exist $g_1,g_2$ such that $g = [g_1,g_2]$ . By the induction hypothesis and a straightforward calculation, we conclude that

$$ \begin{align*} \pi_{\mathrm{id} \times g, (C_1' \cap C_2') \times G} = [\pi_{\mathrm{id} \times g_1,C_1' \times G}, \pi_{\mathrm{id} \times g_2, C_2' \times G}].\end{align*} $$

Case 5. If $C = (C_1' \cup C_2') \times G$ with $C_1' \cap C_2' = \emptyset $ , then

$$ \begin{align*} \pi_{\mathrm{id} \times g,(C_1' \cup C_2') \times G} = \pi_{\mathrm{id} \times g,C_1' \times G} \circ \pi_{\mathrm{id} \times g,C_2' \times G}. \end{align*} $$

Case 6. If $C = (C_1' \cup C_2') \times G$ , then

$$ \begin{align*} \pi_{\mathrm{id} \times g,(C_1' \cup C_2') \times G} = \pi_{\mathrm{id} \times g,(C_1^{\prime c} \cap C_2^{\prime c})^c \times G}. \end{align*} $$

We conclude that $\pi _{\mathrm {id} \times g,C} \in \mathcal {G}_{\ell }$ , and that the provided upper-bounds are correct.

Note that for any $g \in \mathrm {Sym}(H')$ , $g \times \mathrm {id}_{G} \in \mathrm {Sym}(H)$ is an even permutation since $\lvert G\rvert $ is even. Combining this with the previous lemma, we obtain the following result which allows the conditioning of gates depending on conditions of the form $C' \times G$ , and controls their word norm in $\mathcal {G}_{\ell }$ .

We divide the proof of this lemma in three parts.

Proof. Let $g = ((h_1,x_1),(h_2,x_2), (h_3,x_3))$ be a 3-cycle of $\mathrm {Sym}(H' \times G)$ and $C = C' \times G$ be a $\pi _g$ -invariant condition for some $C' \subseteq H' \times \Gamma ^{\ell -1}$ .

Consider the condition $B = B' \times G$ where $B' = C' \cap [h_1]_0$ , and for $y_1,y_2,y_3 \in G$ three distinct elements, let us define the following permutations:

$$ \begin{align*} g^{\prime}_{G} & = (y_1,y_2,y_3) \in \mathrm{Alt}(G), \\ g' & = ((h_1,y_1),(h_1,y_2), (h_1,y_3)) \in \mathrm{Alt}(H' \times G). \end{align*} $$

By Lemma 4.7, $\pi _{\mathrm {id} \times g^{\prime }_{G}, B}$ belongs to $\mathcal {G}_{\ell }$ with word norm $O(T(B)) = O(T(C))$ . However, $\pi _{\mathrm {id} \times g^{\prime }_{G},B} = \pi _{g',C}$ , since $B' = C' \cap [h_1]_0$ . This proves that $\pi _{g,C}$ belongs to $\mathcal {G}_{\ell }$ with word norm $O(T(C))$ , since $g$ and $g'$ (hence $\pi _{g,C}$ and $\pi _{g',C}$ ) are conjugated by the involutions $(h_i,x_i) \leftrightarrow (h_1,y_i)$ of $\mathrm {Sym}(H)$ .

Proof. Let $g' \in \mathrm {Sym}(S \times G)$ , $(h,x) \in H' \times G$ , and $\gamma \in \Gamma ^{\ell -1}$ . Let us denote $g'(h,x) = (h',x')$ . As $g$ is a cycle, there exists $k$ such that $g^k(h) = h'$ . Then $\pi _{g'}((h,x)\cdot \gamma ) = g'(h,x) \cdot \gamma = (g^k(h),x') \cdot \gamma $ , so that

$$ \begin{align*} & \pi_{g'}((h,x) \cdot \gamma) \in C' \times G \\ {\iff}\!\quad & (g^k(h),x') \cdot \gamma \in C' \times G \\ {\iff}\!\quad & \pi_{g}^k((h,x') \cdot \gamma) \in C' \times G \\ {\iff}\!\quad & (h,x') \cdot \gamma \in C' \times G \quad \text{as } C' \times G \text{ is } \pi_g\text{-invariant}\\ {\iff}\!\quad & (h,x) \cdot \gamma \in C' \times G.\\[-39pt] \end{align*} $$

We can now conclude the proof.

We finally state an (optional) rephrasing of Lemma 4.7. Readers with a background in complexity theory will find the following version of the statement useful; it is immediate from the definition of the complexity class ${\mathrm {NC}}^1$ .

4.2.2 Examples

To clarify Lemma 4.7, we consider several examples: the permutation of adjacent letters, lexicographic comparisons, and (cyclic) ternary additions.

Permuting two adjacent tape-letters

Lemma 4.11. Consider the permutation $p \in \mathrm {Sym}(\mathcal {C}_{\ell })$ of two adjacent tape-letters around the head, that is, the shift-equivariant action on patterns of size $2$ :

for any $w = w_0 w_1 \in \Gamma ^2$ and $q \in Q$ .

Then $p$ belongs to $\mathcal {G}_{\ell }$ with constant word norm.

Proof. The permutation $p$ is the composition of finitely many commuting $p_{(a,b)}$ that only permutes adjacent tape-letters when they differ and belong to the set $\{a,b\} \subseteq \Gamma $ , that is, for $a \neq b \in \Gamma $ , the involution $p_{(a,b)} \in \mathrm {Sym}(\mathcal {C}_{\ell })$ is defined as

for any $q \in Q$ . So we only need to prove that every $p_{(a,b)}$ belongs to $\mathcal {G}_{\ell }$ .

Let $a \kern1.3pt{\neq}\kern1.3pt b \kern1.3pt{\in}\kern1.3pt \Gamma $ . Define $g_{\Gamma } \kern1.3pt{\in}\kern1.3pt \mathrm {Sym}(\Gamma )$ as the involution $g_{\Gamma } \kern1.3pt{=}\kern1.3pt a \kern1.3pt{\leftrightarrow}\kern1.3pt b$ . Then ${g' \kern1.3pt{=}\kern1.3pt \mathrm {id}_{Q'} \kern1.3pt{\times}\kern1.3pt g_{\Gamma }}$ is a permutation of $\mathrm {Sym}(H')$ , and $g = g' \times \mathrm {id}_{G}$ is a permutation of $\mathrm {Sym}(H)$ . By Lemma 4.7, both $\pi _{g,[a]_1 \times G}$ and $\pi _{g,[a]_{-1} \times G}$ belong to $\mathcal {G}_{\ell }$ with word norm independent of $\ell $ , and we have $p_{(a,b)} = \Pi $ , where

$$ \begin{align*} \Pi = (\rho^{-1} \circ \pi_{g,[a]_{-1} \times G} \circ \rho) \circ (\pi_{g,[a]_1 \times G}) \circ (\rho^{-1} \circ \pi_{g,[a]_{-1} \times G} \circ \rho) \end{align*} $$

and $\rho $ denotes the right rotation of the head. Indeed, for any $q \in Q$ , we calculate

and nothing of interest (other than the head moving back and forth) happens on other patterns.

Remark 4.12. The previous permutation $p$ can be conditioned on any value of the state $q' \in Q'$ by simply replacing the conditions $[a]_1 \times G$ (respectively $[a]_{-1} \times G$ ) with $([a]_1 \cap [\{q'\} \times \Gamma ]_0) \times G$ (respectively $([a]_{-1} \cap [\{q'\} \times \Gamma ]_0) \times G$ ).

Lexicographic comparisons.

Let us consider the case of conditions that lexicographically compare the content of the tape with arbitrary fixed words (word equalities and comparisons). By reducing the movement of the head, we obtain better upper bounds than what a strict reading of Lemma 4.6 would suggest.

Lemma 4.13. Given a lexicographic equality/comparison ${\sim } \in \{{=},{<},{\leq },{>},{\geq }\}$ and a word $c \in \{0,1,2\}^l$ for some $l \leq \ell $ , consider a condition $C \subseteq H \times \Gamma ^{\ell -1}$ of the form $\sim c$ , that is,

Then $T(C) = O(\lvert c\rvert ^2)$ .

Proof. To prove this result, we use a divide and conquer approach: given $l \leq \ell $ , any integer $l' \leq l$ , and two words $w \in \{0,1,2\}^l$ and $c \in \{0,1,2\}^l$ ,

So if C is the condition $= c$ and $g_1,g_2 \in \mathrm {Alt}(G)$ , then

Taking $l' = \lfloor l/2 \rfloor $ and iterating, we obtain $T(C) = O(l^2) = O(\lvert c\rvert ^2)$ . Similarly, if C is the condition $< c$ , we obtain $T(C) = O(\lvert c\rvert ^2)$ . The other cases follow using elementary Boolean algebra on the operators of $\{{=},{<},{\leq },{>},{\geq }\}$ .

Remark 4.14. The previous lemma can still be used to compare words $w$ and $c$ at non-contiguous positions. Denote by $\bullet $ a wildcard symbol that represents positions which will be ignored when performing the comparison $w \sim c$ , that is, for any two words $w_0 \cdots w_{l-1} \in \{0,1,2\}^l$ and $c_0 \cdots c_{l-1} \in (\{0,1,2\} \cup \{\bullet \})^l$ , if $i_0 < \cdots < i_n$ are the non-wildcard positions in $c$ (that is, $c_i \neq \bullet $ if and only if $i \in \{i_0,\ldots ,i_n\}$ ), we say that $w \sim c$ if the usual lexicographic comparison $w_{i_0} \cdots w_{i_n} \sim c_{i_0} \cdots c_{i_n}$ is true.

Then the previous lemma still holds for comparisons $w \sim c$ for $w \in \{0,1,2\}^l$ and $c \in (\{0,1,2\} \cup \{\bullet \})^l$ .

(Cyclic) ternary addition.

Finally, let us consider ternary additions, that is, adding the number $v_3(k)$ for $k \in \{0,1,2\}^*$ to the $\lvert k\rvert $ letters on the right side of the head by considering them as a counter in base 3.

Lemma 4.15. For $k \in \{0,1,2\}^*$ , $\lvert k\rvert \leq \ell $ , let $\mathrm {add}_{k} \in \mathrm {Sym}(\mathcal {C}_{\ell })$ be defined as the following shift-equivariant action:

where $v_3(a_0' \cdots a_{\lvert k\rvert -1}') = v_3(a_0 \cdots a_{\lvert k\rvert -1}) + v_3(k) \bmod 3^{\lvert k\rvert }$ .

Then $\mathrm {add}_{k}$ belongs to $\mathcal {G}_{\ell }$ with word norm $O(\lvert k\rvert ^3)$ .

Proof. Let $r = \mathrm {id}_Q \times (0,1,2) \in \mathrm {Sym}(Q \times \Gamma )$ denote the rotation of the tape-letter by the 3-cycle $(0,1,2) \in \mathrm {Alt}(\Gamma )$ . To perform additions modulo $3^{\lvert k\rvert }$ , we apply the standard ‘school algorithm’, whose main difficulty consists in performing carries.

The application of a carry only needs to be performed when the addition of the previously added digits has overflowed, that is, when the digits on which the addition has already been performed (from right to left) have a smaller value than the rightmost digits of $k$ . This is exactly what we do.

First, move the head to the right of the counter by applying $\rho ^{\lvert k\rvert -1}$ . Then, apply either $\pi _{r}$ if the last digit of $k$ is $1$ (that is, $k_{\lvert k\rvert -1} = 1$ ), or $\pi _{r^2}$ if $k_{\lvert k\rvert -1} = 2$ , or nothing if $k_{\lvert k\rvert -1} = 0$ .

Then, for , do the following.

1. (1) Move the head to the left with $\rho ^{-1}$ .

2. (2) Apply either $\pi _{r}$ if $k_{\lvert k\rvert -j-1} = 1$ , or $\pi _{r^2}$ if $k_{\lvert k\rvert -j-1} = 2$ , or nothing if $k_{\lvert k\rvert -j-1} = 0$ .

3. (3) Perform the carry: denoting , apply $\pi _{r, < \bullet \:\! k'}$ where $\bullet $ is the wildcard symbol, using the notation of Lemma 4.13 and Remark 4.14.

Then $\mathrm {add}_{k}$ is the composition of $O(\lvert k\rvert )$ permutations of $\mathcal {G}_{\ell }$ , each having word norm $O(\lvert k\rvert ^2)$ (according to Lemma 4.13). This concludes the proof.

Remark 4.16. The addition $\mathrm {add}_{k}$ can be conditioned on any value of the state $q' \in Q'$ by simply intersecting all the conditions for applying $\pi _{r}$ (or $\pi _{r^2}$ ) with $[\{q'\} \times \Gamma ]_0 \times G$ .

4.2.3 The ducking trick

Recall that the set Q splits into $Q = Q' \times G$ , and that $Q'$ itself splits into $Q' = Q^{(\textrm {o})} \times D$ : the states $Q^{(\textrm {o})} = \{{\blacktriangleright },{\blacktriangleleft },{\rhd },{\lhd }\} \times \{1,2\}$ of the original SMART machine, and the duck $D = \{{d_{1}},{d_{2}}\}$ . Recall that for $S \subseteq \mathcal {C}_{\ell }$ , we denote by $S[{d_{1}}]$ (respectively $S[{d_{2}}]$ ) the subset of tapes in S containing a head whose duck is $d = {d_{1}}$ (respectively $d = {d_{2}}$ ).

In this section, we introduce the ducking trick: this method uses this ducking component D to realize piecewise-defined bijections, in particular in the proof of Lemma 4.3 that follows.

Let $f = f^{(\textrm {o})} \times \mathrm {id}_{D \times G} : Q \times \Gamma ^{\ell } \to Q \times \Gamma ^{\ell }$ be a map defined on patterns of length $\ell $ by some $f^{(\textrm {o})} : Q^{(\textrm {o})} \times \Gamma ^{\ell } \to Q^{(\textrm {o})} \times \Gamma ^{\ell }$ . Assume that $f$ is a piecewise-defined bijection, that is, that there exist partitions $\sqcup _{i=1}^p U_i^{(\textrm {o})}$ and $\sqcup _{i=1}^p V_i^{(\textrm {o})}$ of $Q^{(\textrm {o})} \times \Gamma ^{\ell }$ and maps $f_i^{(\textrm {o})} : U_i^{(\textrm {o})} \to V_i^{(\textrm {o})}$ such that $f^{(\textrm {o})} = \bigsqcup f_i^{(\textrm {o})}$ (as the union of graphs of functions).

In what follows, we denote $U_i = U_i^{(\textrm {o})} \times D \times G \subseteq Q \times \Gamma ^{\ell }$ and $V_i = V_i^{(\textrm {o})} \times D \times G \subseteq Q \times \Gamma ^{\ell }$ , and $f_i = f_i^{(\textrm {o})} \times \mathrm {id} : U_i \to V_i$ , so that $f = \bigsqcup _i f_i$ .

Assume the following.

1. (1) There exist maps $g_i \in \mathcal {G}_{\ell }$ with polynomial word norm in $\ell $ such that $g_i$ agrees with $f_i$ on $U_i[{d_{2}}]$ (that is, $g_i|_{U_i[{d_{2}}]} = f_i$ ), and is the identity on $U_i[{d_{1}}]$ (that is, $g_i|_{U_i[{d_{1}}]} = \mathrm {id}$ ).

2. (2) Each $U_i$ has a simple description. Specifically, we should have a Boolean circuit of type ${\mathrm {NC}}^1$ for it.

In our application, the descriptions of the $U_i$ involve only numerical comparisons and direct letter comparisons, and the simplicity is encapsulated in the formulas in §4.2.2. The maps $g_i$ differ from $f_i$ in the following way: instead of having domain $U_i$ , they act on the whole set of patterns of length $\ell $ , and are only required to act on $U_i[{d_{2}}]$ as $f_i$ does (and be the identity on $U_i[{d_{1}}]$ ).

Claim 4.17. (Ducking trick)

Consider $F : Q \times \Gamma ^{\ell } \to Q \times \Gamma ^{\ell }$ defined as

$$ \begin{align*} F(w) = \begin{cases} f(w) & \text{if }w \in Q \times \Gamma^{\ell}[{d_{1}}], \\ f^{-1}(w) & \text{if } w \in Q \times \Gamma^{\ell}[{d_{2}}], \\ w & \text{otherwise}. \end{cases} \end{align*} $$

Under the assumptions above, F belongs to $\mathcal {G}_{\ell }$ with polynomial word norm in $\ell $ .

Proof. Denote by $d' \in \mathrm {Sym}(D)$ the involution $d' = {d_{1}} \leftrightarrow {d_{2}}$ . Then $d = \mathrm {id}_{Q^{(\textrm {o})}} \times d' \times \mathrm {id}_{G} \times \mathrm {id}_{\Gamma } \in \mathrm {Sym}(H)$ is an involution. As the sets $U_i$ are $\pi _d$ -invariant, by Lemma 4.7, each $\pi _{d,U_i}$ belongs to $\mathcal {G}_{\ell }$ with word norm $O(T(U_i))$ (which is polynomial in $\ell $ , because of the ${\mathrm {NC}}^1$ -description).

Now conjugate the permutations $\pi _{d,U_i}$ by their respective $g_i^{-1}$ , to get

$$ \begin{align*} F_i = (\pi_{d,U_i})^{(g_i^{-1})} = g_i \circ \pi_{d,U_i} \circ g_i^{-1}. \end{align*} $$

This is an involution that acts like $f_i$ on $U_i[{d_{1}}]$ (and flips the duck), like $f_i^{-1}$ on $V_i[{d_{2}}]$ (and flips the duck), and is the identity on the rest of $Q \times \Gamma ^{\ell }$ . Indeed, $\pi _{d,U_i}$ exchanges words $w \in U_i[{d_{1}}]$ with $\pi _d(w) \in U_i[{d_{2}}]$ . Conjugating it with $g_i^{-1}$ effectively means that we translate the domain of this permutation by $g_i$ . More precisely, this is the involution

$$ \begin{align*} F_i : g_i(U_i[{d_{1}}]) & \leftrightarrow g_i(U_i[{d_{2}}]), \\ g_i(w) & \leftrightarrow g_i(\pi_d(w)). \end{align*} $$

As $g_i$ acts like $f_i$ on $U_i[{d_{2}}]$ and is the identity on $U_i[{d_{1}}]$ , we have in fact:

$$ \begin{align*} F_i : U[{d_{1}}] & \leftrightarrow V_i[{d_{2}}], \\ w & \leftrightarrow f_i(\pi_d(w)). \end{align*} $$

Additionally, $F_i$ has polynomial word norm because $g_i$ and $\pi _{d,U_i}$ have polynomial word norm.

Finally, each time $F_i$ acts non-trivially, it also flips the duck, so all the $F_i$ have disjoint support. Composing them in any arbitrary order, we obtain that $F = \pi _d \circ \prod _i F_i$ belongs to $\mathcal {G}_{\ell }$ with polynomial word norm in $\ell $ .

4.3.1 Proof of Lemma 4.3

In this section, we use the previous lemmas to implement the inductive encoding of §3.4 in $\mathcal {G}_{\ell }$ . More precisely, recall the statement of Lemma 4.3.

Lemma 4.3.

Let $E_{\ell } : \mathcal {C}_{\ell } \to \mathcal {C}_{\ell }$ be the encoding map defined in §3.2. There exists some $\tilde {E}_{\ell } : \mathcal {C}_{\ell } \to \mathcal {C}_{\ell }$ in $\mathcal {G}_{\ell }$ with word norm $O(\ell ^4)$ such that

$$ \begin{align*} w \in \mathcal{C}_{\ell}[{d_{1}}] \implies \tilde{E}_{\ell}(w) = E_{\ell}(w), \end{align*} $$

where $\mathcal {C}_{\ell }[{d_{1}}]$ is the set of cyclic tapes $w \in \mathcal {C}_{\ell }$ having a head with duck $d = {d_{1}}$ .

This section is dedicated to the proof of this lemma. Informally, we use the inductive decomposition of $E_{\ell }$ into

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

defined in §3.4, and build every $F_{\mathrm {init}}, F_{k \to k+1}$ , and $F_{\ell ,\mathrm {final}}$ in the group $\mathcal {G}_{\ell }$ . Each of these steps being defined as piecewise-defined bijections, we use the ducking trick mentioned in §4.2.3: with the duck $D = \{{d_{1}},{d_{2}}\}$ , we write each bijection as a product of involutions that swap ducks ${d_{1}}$ and ${d_{2}}$ .

Let F be one of $F_{\mathrm {init}}$ , $F_{k \to k+1}$ , or $F_{\ell ,\mathrm {final}}$ . Here, F rewrites patterns as explained in §3.4, and copies all other symbols unchanged. Let be one of the cases of F, that is, one of the rules of pattern rewriting that defines F. (Apart from the proof of Lemma 4.19, there is no need to actually know what U, V, and F precisely are to understand the following statements.)

Define

$$ \begin{align*} \tilde{F}_{U,V} : U[{d_{1}}] & \leftrightarrow V[{d_{2}}] \\ w & \leftrightarrow \pi_d(F(w)) \end{align*} $$

as the involution of $\mathrm {Sym}(\mathcal {C}_{\ell })$ that swaps words $w \in U[{d_{1}}]$ with words $\pi _d (F(w)) \in V[{d_{2}}]$ , where $\pi _d$ is the duck-flipping involution. The involution $\tilde {F}_{U,V}$ applies F forward on tapes with duck ${d_{1}}$ and swaps the duck, and applies F backwards on tapes with duck ${d_{2}}$ and swaps the duck.

The proof relies on the structure of the different cases defined in §3.4.

Example 4.19. On an example, consider to be the following transformation between levels $k$ and $k+1$ (this is the third rewrite in Figure 4):

Then the condition U is the conjunction of the head being in state ${\blacktriangleright }_1$ , on top of the tape-letter $1$ , and the $k+1$ tape-letters on its left being non-zero. By Lemma 4.13, $\pi _{d,U}$ belongs to $\mathcal {G}_{\ell }$ with norm $O(k^2)$ .

To obtain $\tilde {E}_{U,V}$ , we then conjugate $\pi _{d,U}$ with the sequence of inverses of the following permutations, all of which are conditioned on the head having duck ${d_{2}}$ .

1. (1) At first, when $\pi _{d,{U}}$ is conjugated by nothing (that is, the identity), we have

   

   and, again, this permutation is only applied if $c$ is a non-zero counter. (Note that we see here the full support of the permutation.)

2. (2) Let $g' = (1,0) \in \mathrm {Sym}(\Gamma )$ , and $g = \mathrm {id}_{Q^{(\textrm {o})}} \times \{{d_{2}}\} \times \mathrm {id}_{G} \times g' \in \mathrm {Sym}(H)$ . Apply the gate $\pi _g$ , which has word norm $O(1)$ in $\mathcal {G}_{\ell }$ . At the moment, we have the permutation:

   

3. (3) Using the permutation $p$ from Lemma 4.11 (that exchanges two adjacent letters) conditioned on the duck being $d={d_{2}}$ (see Remark 4.12), and moving the head using the generator $\rho $ of $\mathcal {G}_{\ell }$ , we move the counter $c$ one step to its right and the tape-letter $0$ under the head $k+1$ steps to its left. At the moment, we have the permutation:

   

4. (4) For $w \in \{0,1,2\}^{k+2}$ such that $v_3(w) = f(k)+1$ , we apply $\mathrm {add}_{w}$ from Lemma 4.15 (namely, the permutation that adds $f(k)+1$ to the ternary number of length $k+2$ to the right of the head) conditioned on the duck being $d = {d_{2}}$ (see Remark 4.16). It has word norm $O(k^3)$ . Then, apply $\rho ^{-1}$ on ducks $d = {d_{2}}$ . At the moment, we have the permutation:

   

5. (5) Let $g' = {\blacktriangleright }_1^{{d_{2}}} \leftrightarrow {\blacktriangleleft }_1^{{d_{2}}} \in \mathrm {Sym}(Q)$ and $g = g' \times \mathrm {id}_{\Gamma } \in \mathrm {Sym}(H)$ . We apply $\pi _g$ and finally obtain $\tilde {F}_{U,V}$ :