Minimal Sofic Shift

• Minimal sofic shift is defined in two senses: one via canonical deterministic presentations using follower-equivalence, and the other via dynamical minimality in group actions.
• The deterministic approach leverages follower separation and state reduction techniques, offering polynomial-time computation for irreducible or synchronizing presentations.
• In the group-shift framework, minimal sofic shifts are constructed by inducing actions from finitely generated groups to more complex, non-finitely-generated groups, revealing sharp complexity boundaries.

Searching arXiv for the specified papers and closely related work to ground the article. “Minimal sofic shift” is used in two technically distinct senses. In the theory of one-dimensional sofic shifts presented by deterministic edge-labeled graphs, it refers to minimality of a deterministic presentation: no other deterministic presentation of the same shift has strictly fewer vertices, or equivalently there is no pair of distinct follower-equivalent states. In the theory of subshifts over countable groups, it refers to dynamical minimality: a subshift $X \subseteq A^G$ has no nonempty proper closed $G$-invariant subsets. Current work develops both meanings: the first through the computational theory of deterministic presentations, and the second through existence results for group actions, including a non-finitely-generated group admitting an infinite minimal sofic shift (Cai et al., 2021, Salo, 9 Jul 2025).

1. Two notions of minimality

For a finite alphabet $\Sigma$ and a one-dimensional shift space $X \subseteq \Sigma^{\mathbb{Z}}$, a labeled graph

$G = (Q_G,E_G,i_G,t_G,L_G)$

with labels in $\Sigma$ is a presentation of $X$ if

$$X = X_G := \{\,L_G(x) : x \in E_G^{\mathbb{Z}} \text{ is a bi-infinite path}\,\}.$$

Such a presentation is essential if every vertex lies on some bi-infinite path, and presentations are assumed essential in the development summarized in (Cai et al., 2021).

In this presentation-theoretic setting, a deterministic presentation, also called right-resolving, is one in which for each state $q \in Q_G$ and each $a \in \Sigma$ there is at most one outgoing edge from $G$0 labeled $G$1. This yields a partial transition action

$G$2

For deterministic $G$3 and $G$4, the follower set is

$G$5

and

$G$6

Two states $G$7 are follower-equivalent, written $G$8, if $G$9. The quotient $\Sigma$0, called the follower-separation, has one vertex $\Sigma$1 for each $\Sigma$2-class and is again a deterministic presentation of the same shift (Cai et al., 2021).

In the group-shift setting, let $\Sigma$3 be a countable discrete group and $\Sigma$4 a finite alphabet. The full $\Sigma$5-shift is

$\Sigma$6

with the product topology and the natural left $\Sigma$7-action

$\Sigma$8

A subshift $\Sigma$9 is a closed $X \subseteq \Sigma^{\mathbb{Z}}$0-invariant subset; it is an SFT if it is defined by finitely many forbidden finite patterns; it is sofic if it is a factor of some SFT cover; and it is minimal if it has no nonempty proper closed $X \subseteq \Sigma^{\mathbb{Z}}$1-invariant subsets (Salo, 9 Jul 2025).

A common source of confusion is that these two meanings of minimality concern different objects. In the first, minimality concerns redundancy of states in a deterministic presentation. In the second, minimality concerns the orbit-closure structure of the $X \subseteq \Sigma^{\mathbb{Z}}$2-system itself. This suggests that the phrase “minimal sofic shift” is context-dependent and must be interpreted from the ambient theory.

2. Minimal deterministic presentations and follower separation

For deterministic presentations of one-dimensional sofic shifts, minimality is controlled by follower sets. A deterministic presentation $X \subseteq \Sigma^{\mathbb{Z}}$3 of a sofic shift $X \subseteq \Sigma^{\mathbb{Z}}$4 is minimal if no other deterministic presentation of $X \subseteq \Sigma^{\mathbb{Z}}$5 has strictly fewer vertices. Equivalently, $X \subseteq \Sigma^{\mathbb{Z}}$6 is minimal if it has no pair of distinct follower-equivalent states (Cai et al., 2021).

The quotient $X \subseteq \Sigma^{\mathbb{Z}}$7 therefore plays the role of a canonical reduction procedure. By identifying states with the same follower set and inheriting edges in the obvious way, one obtains a deterministic presentation of the same shift. In the summary attached to (Cai et al., 2021), this collapse is described as the classical “Myhill–Nerode” construction for sofic shifts.

The formulation “A sofic shift $X \subseteq \Sigma^{\mathbb{Z}}$8 is minimal (among all presentations) if one of its deterministic presentations has no redundant (follower-equivalent) states” appears explicitly in (Cai et al., 2021). In that presentation-theoretic sense one also speaks of its minimal deterministic presentation $X \subseteq \Sigma^{\mathbb{Z}}$9. The significance of this formulation is algorithmic: it converts a structural question about symbolic dynamics into a state-equivalence question for deterministic automata.

3. Irreducible and synchronizing regimes

Two classes admit a particularly clean theory: irreducible deterministic presentations and synchronizing deterministic presentations. In both settings, collapsing follower-equivalent states yields a unique minimal object and can be carried out in polynomial time (Cai et al., 2021).

A deterministic presentation $G = (Q_G,E_G,i_G,t_G,L_G)$0 is irreducible, equivalently strongly connected, if for every $G = (Q_G,E_G,i_G,t_G,L_G)$1 there is a path from $G = (Q_G,E_G,i_G,t_G,L_G)$2 to $G = (Q_G,E_G,i_G,t_G,L_G)$3. Equivalently $G = (Q_G,E_G,i_G,t_G,L_G)$4 is an irreducible shift. A theorem attributed to Lind–Marcus states that if $G = (Q_G,E_G,i_G,t_G,L_G)$5 is an irreducible deterministic presentation, then $G = (Q_G,E_G,i_G,t_G,L_G)$6 is an equivalence, the quotient $G = (Q_G,E_G,i_G,t_G,L_G)$7 is again irreducible and deterministic, and $G = (Q_G,E_G,i_G,t_G,L_G)$8 is the unique, up to isomorphism, minimal irreducible deterministic presentation of $G = (Q_G,E_G,i_G,t_G,L_G)$9. Moreover, $\Sigma$0 can be computed in

$\Sigma$1

time by Hopcroft’s DFA-state-equivalence algorithm, viewing $\Sigma$2 as a DFA with a sink state (Cai et al., 2021).

A word $\Sigma$3 in a deterministic presentation $\Sigma$4 is synchronizing if $\Sigma$5 is a singleton. The presentation is synchronizing if every state $\Sigma$6 admits some $\Sigma$7 with $\Sigma$8. Jonoska (1996) showed that the class of sofic shifts admitting a synchronizing deterministic presentation is strictly larger than the irreducible class, but still admits a unique minimal SDP. More precisely, if $\Sigma$9 is follower-separated and synchronizing, then

$X$0

meaning that whenever $X$1 and $X$2 both lie in $X$3, then $X$4. As a corollary, every synchronizing deterministic presentation has a unique minimal SDP obtained by collapsing follower-equivalent states, and this quotient can be computed in polynomial time (Cai et al., 2021).

The algorithmic template in both cases is identical: compute follower-equivalence via Hopcroft’s DFA-equivalence in $X$5, form the quotient $X$6 in $X$7, and return it. To decide whether $X$8 admits any deterministic presentation with at most $X$9 states in the irreducible or synchronizing case, it suffices to compute $$X = X_G := \{\,L_G(x) : x \in E_G^{\mathbb{Z}} \text{ is a bi-infinite path}\,\}.$$0 and test whether $$X = X_G := \{\,L_G(x) : x \in E_G^{\mathbb{Z}} \text{ is a bi-infinite path}\,\}.$$1 (Cai et al., 2021).

4. Complexity boundary and size bounds

Outside the irreducible and synchronizing settings, the minimality problem changes sharply in complexity. For arbitrary deterministic presentations, possibly reducible and non-synchronizing, the decision problem

$$X = X_G := \{\,L_G(x) : x \in E_G^{\mathbb{Z}} \text{ is a bi-infinite path}\,\}.$$2

$$X = X_G := \{\,L_G(x) : x \in E_G^{\mathbb{Z}} \text{ is a bi-infinite path}\,\}.$$3

is called Minimality in (Cai et al., 2021), and Theorem 6.12 states that Minimality is PSPACE-complete (Cai et al., 2021).

The PSPACE-hardness proof reduces from DFA-UNION. Given DFAs $$X = X_G := \{\,L_G(x) : x \in E_G^{\mathbb{Z}} \text{ is a bi-infinite path}\,\}.$$4 over $$X = X_G := \{\,L_G(x) : x \in E_G^{\mathbb{Z}} \text{ is a bi-infinite path}\,\}.$$5, one constructs $$X = X_G := \{\,L_G(x) : x \in E_G^{\mathbb{Z}} \text{ is a bi-infinite path}\,\}.$$6 by “grafting” each $$X = X_G := \{\,L_G(x) : x \in E_G^{\mathbb{Z}} \text{ is a bi-infinite path}\,\}.$$7 into $$X = X_G := \{\,L_G(x) : x \in E_G^{\mathbb{Z}} \text{ is a bi-infinite path}\,\}.$$8 so that between special markers $$X = X_G := \{\,L_G(x) : x \in E_G^{\mathbb{Z}} \text{ is a bi-infinite path}\,\}.$$9 and $q \in Q_G$0 the acceptance of each $q \in Q_G$1 is tested in parallel, together with an extra state $q \in Q_G$2 whose role is to cover any word not accepted by any $q \in Q_G$3. One then shows that $q \in Q_G$4 admits a 2-state deterministic presentation if and only if

$q \in Q_G$5

Thus checking minimality for $q \in Q_G$6 solves the PSPACE-complete UNION problem. Membership in PSPACE follows because one can verify a $q \in Q_G$7-state candidate presentation in PSPACE by checking language-equality in PSPACE, and therefore Minimality lies in PSPACE by Savitch’s theorem (Cai et al., 2021).

The same source gives two complementary size bounds that delimit what can be expected even inside deterministic and synchronizing frameworks. First, there exist sofic shifts $q \in Q_G$8 whose minimal deterministic presentation has $q \in Q_G$9 states but whose minimal synchronizing deterministic presentation has $a \in \Sigma$0 states. The construction starts from a $a \in \Sigma$1-entry DFA whose union of $a \in \Sigma$2 initial states requires a $a \in \Sigma$3-state minimal DFA, and embeds it into a sofic shift presentation with $a \in \Sigma$4 extra pre-initial states so that the minimal SDP essentially forces the $a \in \Sigma$5 blow-up. Second, there exist deterministic presentations on $a \in \Sigma$6 states whose shortest synchronizing word has length $a \in \Sigma$7 (Cai et al., 2021).

These results establish a sharp boundary. In the irreducible or synchronizing cases, minimality, equality, subshift, and SFT-test are all in $a \in \Sigma$8 via follower-separation. For general deterministic presentations, all these problems, including Minimality, are PSPACE-complete via reductions from DFA-UNION and DFA-INTERSECTION (Cai et al., 2021).

5. Minimal sofic shifts over countable groups

For group actions, minimality is a dynamical property rather than a presentation-theoretic one. A sofic shift $a \in \Sigma$9 is obtained as a continuous $G$00-equivariant image of an SFT $G$01, and minimality means precisely that $G$02 has no nonempty proper closed $G$03-invariant subsets (Salo, 9 Jul 2025).

A central recent existence theorem states that there exists a countable group

$G$04

which is not finitely generated, together with a finite alphabet $G$05 and an infinite subshift $G$06 that is both sofic and minimal under the $G$07-action. This answers Question 7.18(ii) of Doucha–Melleray–Tsankov in the positive (Salo, 9 Jul 2025).

The group is built explicitly. One fixes free groups of rank $G$08,

$G$09

One forms $G$10, so $G$11. By Nielsen–Schreier, $G$12 contains subgroups of every countable rank, so one chooses $G$13 with $G$14. An action $G$15 is then defined by

$G$16

Finally,

$G$17

Because $G$18, the resulting group is not finitely generated (Salo, 9 Jul 2025).

This result is contrasted in (Salo, 9 Jul 2025) with known obstructions: no non-finitely-generated amenable or locally-finite group admits such a shift. It also interacts with the study of generic Cantor actions: for non-finitely-generated $G$19, any projectively isolated sofic shift must be minimal, and the construction provides the first nontrivial example of such minimal sofic shifts.

6. Construction mechanisms and structural significance

The construction of the minimal sofic shift on $G$20 proceeds by building a sofic $G$21-system for

$G$22

and then inducing it to the larger group $G$23 (Salo, 9 Jul 2025).

The base action is defined on the Cantor space $G$24. Thompson’s group $G$25 acts naturally on $G$26 by finite-prefix replacement, and this action is antidiagonally minimal, faithful and center-free, expansive, and computable. Since $G$27 is 2-generated, one chooses surjections

$G$28

so that together the images generate $G$29. Then $G$30 acts on $G$31 by sending each generator to the corresponding element of $G$32, and this action is antidiagonally minimal, expansive, and computable. Extending trivially over $G$33 yields a $G$34-action on $G$35 that remains expansive and faithful (Salo, 9 Jul 2025).

At this point the construction invokes self-simulation. Because $G$36 is a direct product of nonamenable finitely generated groups, every computable expansive action has an SFT cover. Hence the $G$37-system above admits a sofic shift model $G$38. The next step is free extension, or co-induction, from $G$39 to $G$40. Viewing $G$41 as forbidden-pattern definitions on the cosets of $G$42 in $G$43, one defines the induced subshift

$G$44

by requiring that for each left coset $G$45, the restriction of a configuration $G$46 to $G$47, after translation by $G$48, lies in $G$49. Since $G$50 is sofic, its free extension remains sofic (Salo, 9 Jul 2025).

Minimality of the induced system is obtained by a “minimal $G$51-joinings” argument. The original $G$52 has no nontrivial joining with any of its automorphic conjugates arising from $G$53, and this forces the induced $G$54-action on $G$55 to be minimal. Because $G$56 acts trivially and $G$57 acts by permuting the $G$58-coordinates, no new proper invariant subsets appear, so $G$59 is minimal under the full $G$60-action (Salo, 9 Jul 2025).

The proof architecture is summarized by four key lemmas in (Salo, 9 Jul 2025): antidiagonal minimality implies minimality and expansivity; the prefix-replacement action of $G$61 has the required dynamical and computability properties; direct products of finitely generated nonamenable groups are self-simulable in the sense that every computable expansive action is sofic; and minimal $G$62-joinings imply minimal induction. Together these lemmas show that minimal soficity over non-finitely-generated groups can be produced by combining expansive symbolic models, semidirect-product geometry, and induction.

A plausible implication is that the phrase “minimal sofic shift” now spans two mature but only partially overlapping research programs. One studies minimal deterministic presentations through follower-equivalence, canonical quotients, and computational complexity. The other studies minimality as a dynamical property of group subshifts, including existence and construction problems beyond the finitely generated case. The two programs share the language of soficity but organize minimality around different invariants: state complexity in the first case, invariant closed subsets in the second.