Minimality and Computability of Languages of G-Shifts

Amir, Djamel Eddine; Hellouin de Menibus, Benjamin

doi:10.4230/LIPIcs.ICALP.2025.139

Minimality and Computability of Languages of G-Shifts

Djamel Eddine Amir

Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91400, Orsay, France Benjamin Hellouin de Menibus

Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91400, Orsay, France

Abstract

Motivated by the notion of strong computable type for sets in computable analysis, we define the notion of strong computable type for $G$ -shifts, where $G$ is a finitely generated group with decidable word problem. A $G$ -shift has strong computable type if one can compute its language from the complement of its language. We obtain a characterization of $G$ -shifts with strong computable type in terms of a notion of minimality with respect to properties with a bounded computational complexity. We provide a self-contained direct proof, and also explain how this characterization can be obtained from an existing similar characterization for sets by Amir and Hoyrup, and discuss its connexions with results by Jeandel on closure spaces. We apply this characterization to several classes of shifts that are minimal with respect to specific properties. This provides a unifying approach that not only generalizes many existing results but also has the potential to yield new findings effortlessly. In contrast to the case of sets, we prove that strong computable type for G-shifts is preserved under products. We conclude by discussing some generalizations and future directions.

Keywords and phrases:

shifts, subshifts, minimal shifts, computable language, computability, strong computable type, descriptive complexity

Category:

Track B: Automata, Logic, Semantics, and Theory of Programming

Copyright and License:

2012 ACM Subject Classification:

Theory of computation

\rightarrow

Computability ; Mathematics of computing

\rightarrow

Discrete mathematics ; Mathematics of computing

\rightarrow

Point-set topology

Acknowledgements:

The authors are grateful to anonymous reviewers, as well as to Pierre Guillon and Guillaume Theyssier for discussions and numerous remarks that helped to significantly improve this article. We particularly thank Pierre Guillon for pointing out links with results from [26].

Funding:

We gratefully acknowledge funding from the ANR Project IZES-ANR-22-CE40-0011 (Inside Zero Entropy Systems) for this work.

DOI:

10.4230/LIPIcs.ICALP.2025.139

Event:

52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Editors:

Keren Censor-Hillel, Fabrizio Grandoni, Joël Ouaknine, and Gabriele Puppis

Series and Publisher:

Leibniz International Proceedings in Informatics, Schloss Dagstuhl – Leibniz-Zentrum für Informatik

1 Introduction

Shifts, or shift spaces, are sets of colourings of an infinite regular grid (also called configurations) submitted to local constraints usually given as forbidden patterns. In 1961, Wang first studied colourings of the two-dimensional square grid with finitely many constraints (shifts of finite type or SFT) to study some fragments of predicate calculus [37]. Undecidability phenomena were proved soon after, the first one being the undecidability of the so-called seeded Domino problem [28]: given a partial colouring, can it be extended to a full configuration that satisfies the constraints? The set of such partial colourings is called the language of the shift, so this result means that there are SFT with uncomputable language. This result was later extended to other grids, in particular Cayley graphs of finitely generated groups [12], which is the general setting for this article.

Consequently, understanding which properties make the language of a shift computable or not remains a fundamental question. A well-known folklore result states that the language of a minimal SFT, that is, a shift that contains no other non-empty subshift, is computable. However, later research showed that the finite-type assumption is not necessary: it suffices to be able to enumerate all patterns that do not appear in the shift (a notion known as effectiveness) for the result to hold (see [27, 26, 18]). Multiple other results point to the same phenomenon, where being able to enumerate the complement of the language is enough to compute the language itself, under various assumptions that can be seen as a form of minimality: the subshift satisfies a property $P$ that none of its subshifts satisfies.

Similar results in other areas

The connection between minimality and decidability also extends to problems in group theory, combinatorics, and computable analysis (see for example [22, 33, 34]). Jeandel attempted to develop a unified theory for groups, shifts and combinatorics [26]; we discuss the relationship with our results in Section 3.4.

In the setting of computable topology and descriptive complexity, a similar phenomenon occurs where some minimality assumption ensures that a description of a set “from the outside” is enough to provide a description “from the inside”. Notions of strong computable type and minimality for sets were defined in [5] as follows.

Definition.

A set has strong computable type if one can compute the set from its semi-computable information (a description of its complement).

A set is minimal for some property if it satisfies this property but no proper subset of it satisfies the property.

A characterization of sets which have strong computable type related to minimality was given in [5].

Theorem ([5]).

A set has strong computable type if and only if it is minimal satisfying some property with bounded computational complexity ( $\Sigma_{2}^{0}$ -computable).

Our approach

In this article, motivated by the above results for sets, we define analogous notions of strong computable type and minimality for $G$ -shifts, where $G$ is a finitely generated group with decidable word problem. To simplify, $G$ -shifts will be called shifts.

Definition.

A shift has strong computable type if one can compute its language from the complement of its language.

A shift is minimal for some property if it satisfies this property but no proper subshift of it satisfies the property.

We characterize shifts with strong computable type using minimality, and show that this is implied by the characterization of strong computable type for sets. This motivates the question of classifying shifts according to their computability.

In addition to creating new connections between symbolic dynamics and computable analysis, positive results in this direction will motivate further efforts to find more general theorems that unify the two fields.

Our results

We prove the following main theorem.

Theorem (Theorem 11).

A shift has strong computable type if and only if it is minimal for some property with bounded computational complexity ( $\Sigma_{2}^{0}$ -computable).

We apply our theorem to several classes of shifts which are minimal for some specific properties. This provides a unifying approach, as it not only implies many existing results but also introduces new findings effortlessly.

Proposition (Section 3.5).

The following classes of shifts have strong computable type.

$\blacksquare$

Minimal shifts,
$\blacksquare$

Entropy-minimal shifts with a left-computable entropy,
$\blacksquare$

$P$ -isolated shifts,
$\blacksquare$

Infinite-minimal shifts (periodic-minimal shifts and quasi-minimal shifts).

More details and precise definitions will be provided in the following sections, along with an appendix for clarifications.

Many other natural questions arise, such as: can we obtain other analogous results for shifts to those on computable type for sets ([1, 3, 5, 4, 6, 7, 2, 38, 16])? Furthermore, can we reverse the process by obtaining results for computable type analogous to those on shifts?

Sections are organized as follows

In Section 2, we give a minimal background to understand the main results. In Section 3, we define the notion of strong computable type for shifts, state and prove our main theorem, explain its relation with strong computable type for sets and closure spaces, apply it to classes of shifts and study some properties of strong computable type for shifts. In Section 4, we discuss generalizations of our results and future directions. In Appendix A, we provide more details about topology and descriptive complexity. In Appendix B, we prove the relation between strong computable type for shifts and sets.

2 Preliminaries

To present our results, which are interdisciplinary in nature, as they relate computability and symbolic dynamics to topology and descriptive complexity, we first provide some basic concepts. However, for a more comprehensive understanding, the reader should refer to the relevant references provided throughout the text.

2.1 Basics on groups

We provide some classical basic concepts about groups, particularly the notions of finitely generated groups and the word problem. These results and their applications to shifts can be found in [8].

Definition 1.

An alphabet is a set. A word of length $n\in\mathbb{N}^{*}$ on an alphabet $A$ is an element in the finite product $\Pi_{1\leq i\leq n}A$ , the empty word is of length $0$ .

We denote by $A^{*}$ the set of all finite words on $A$ .

Now, we have the necessary ingredients to define key concepts for groups.

Definition 2.

Let $G$ be a group and $S\subseteq G$ . For $w\in S^{*}$ , we denote the corresponding group element by $w_{G}$ (the evaluation of $w$ as an element of $G$ , where the product is the group operation of $G$ ).

$S$ is a generating set for $G$ if every element in $G$ can be written as a word in $S^{*}$ . $G$ is finitely generated if $S$ is finite.

To simplify, we assume that generating sets are symmetric, that is, $g\in S\Rightarrow g^{-1}\in S$ .

Definition 3.

Let $G$ be a group that is finitely generated by $S$ with identity element $1_{G}$ .

The word problem is the set $\{w\in S^{*}:w_{G}=1_{G}\}$ .

$G$ has decidable word problem if this set is decidable, i.e. there is an algorithm which given a word $w$ , decides on finite time whether $w_{G}$ equals $1_{G}$ . This does not depend of the chosen generating set.

2.2 G-shifts

As outlined in the introduction, shifts of dimension $2$ can be defined as sets of colorings of planes. Specifically, given an alphabet $A$ (a set of colors), a configuration corresponds to an element of $A^{\mathbb{Z}^{2}}$ , i.e., an assignation of colors $(a_{(i,j)})_{(i,j)\in\mathbb{Z}^{2}}$ to every coordinate in the plane, where each $a_{(i,j)}$ is a color in $A$ . A shift of dimension $2$ is a shift-invariant closed set of configurations in $A^{\mathbb{Z}^{2}}$ .

Similarly, this concept can be extended by indexing the colors with elements of a group $G$ rather than $\mathbb{Z}^{2}$ , i.e. configurations $(a_{g})_{g\in G}\in A^{G}$ , which allows us to define $G$ -shifts (see [15], [8]). A configuration $(a_{g})_{g\in G}$ can be seen as a function $G\rightarrow A$ sending $g$ to $a_{g}$ .

Now, let us give precise definitions for $G$ -shifts.

Let $G$ be a group and let $A$ be a finite alphabet. The set $A^{G}=\{x:G\rightarrow A\}$ is called the full shift, and its elements are called configurations. $A^{G}$ can be endowed with the pro-discrete topology, which is metrizable if $G$ is countable and computably metrizable if $G$ is finitely generated. For the latter case, see Appendix A.1 for more details.

Definition 4 ( $G$ -shifts).

A pattern is an element $p\in A^{F}$ , where $F\subseteq G$ is finite, and it determines the cylinder $[p]=\{x\in A^{G}:x|_{F}=p\}$ . Let $x:G\rightarrow A$ be a configuration, $p$ appears on $x$ if there exists some $g\in G$ such that $gx\in[p]$ , where $gx:G\rightarrow A$ is the function sending $h\in G$ to $x(g^{-1}h)$ .

A $𝐆$ -shift is a subset $X\subseteq A^{G}$ which is topologically closed and shift-invariant i.e. for every $h\in G$ and every $x\in X$ , one has $hx\in X$ . We denote by $\mathcal{S}(A^{G})$ the set of all non-empty shifts in $A^{G}$ .

The language $\mathcal{L}(X)$ of a $G$ -shift $X$ in $A^{G}$ (or more generally a set $X\subset A^{G}$ ) is the set of patterns $p:F\to A$ that appear in $X$ , that is, such that $x|_{F}=p$ for some $x\in X$ . Its complement is denoted $\mathcal{L}^{c}(X)$ .

A $G$ -shift $X$ is of finite type (SFT) if it can be defined by a finite set $F$ of forbidden patterns, in the sense that a configuration $x$ is in $X$ if and only if no pattern from $F$ appears in $x$ .

To simplify, we call $G$ -shifts just shifts, the underlying group will be clearly known. In the literature, shifts may be called subshifts since every shift is a subshift of the full shift.

If $X$ is a shift, then a pattern $p$ appears in some $x\in X$ if and only if $X\cap[p]\neq\emptyset$ (this will be used in the proofs).

Note that the set of cylinders $[p]$ is a clopen sub-basis of the pro-discrete topology on $A^{G}$ .

2.3 Effectiveness of G-shifts

Let $G$ be a finitely generated group with decidable word problem, $S\subseteq G$ be a finite generating set and $A$ be a finite alphabet. To define notions of effectiveness on $G$ -shifts (see [23]), let us define some classical notions in computability theory.

Definition 5.

A subset $I\subseteq\mathbb{N}$ is computably enumerable (c.e.) if there exists an algorithm that runs forever and enumerates all elements of $I$ ; equivalently, there is an algorithm that semi-decides $I$ , that is, given $i\in\mathbb{N}$ , the algorithm will enumerate $i$ after a finite time if $i\in I$ and keeps running forever otherwise.

A set is co-computably enumerable (co-c.e.) if its complement is computably enumerable.

A set is computable (decidable) if it is both c.e. and co-c.e..

Let $p\in A^{F}$ be a pattern, that is, a function $F\rightarrow A$ , for some finite set $F\subseteq G$ . Let $f:S^{*}\rightarrow G$ be the function sending a word $w$ to the corresponding group element $w_{G}$ . When manipulated by an algorithm, the pattern $p$ is represented by a function $p^{\prime}:W\rightarrow A$ , with $W\subset S^{*}$ finite, such that $p\circ f|_{W}=p^{\prime}$ and $W\cap f^{-1}(i)\neq\emptyset$ for all $i\in F$ . The set $\mathcal{C}$ of all such functions is countable as they are functions from a finite set of words to a finite alphabet:

\mathcal{C}=\{p\circ f_{F}:p\in A^{F},F\subseteq G\text{ is finite}\}.

Note that $\mathcal{C}$ can be effectively enumerated (using an algorithm), given $S$ and the fact that $G$ has decidable word problem.

Hence the notions of c.e., co-c.e. and computable sets can be extended from subsets of $\mathbb{N}$ to subsets of $\mathcal{C}$ .

Definition 6 (Effectively closed shifts).

A shift $X\subseteq A^{G}$ is effectively closed if it has a co-computably enumerable language.

Such shifts are sometimes simply called effective.

Fact 7 (Proposition 2.1 in [8]).

Equivalently, a shift is effectively closed if it can be defined by a computably enumerable set of forbidden patterns.

In particular, an SFT is effectively closed because a finite set of forbidden patterns is computably enumerable.

This terminology comes from the fact that a shift is effectively closed if and only if it is topologically effectively closed (see [9] and Appendix A.2.1). Since a shift is always topologically closed, an effectively closed shift corresponds to an effectively topologically closed shift.

2.4 Descriptive complexity

In this section, we define the notion of properties of shifts, their descriptive complexity and minimal elements satisfying them.

Let $G$ be a finitely generated group with decidable word problem and let $A$ be a finite alphabet. The set $\mathcal{S}(A^{G})$ of all non-empty shifts in $A^{G}$ can be endowed with a computable metric structure, see Appendix A.2.

When we say that an algorithm is given an enumeration as input, this formally means that the enumeration is given as an oracle to the algorithm (that is, the algorithm has access to it as a special input of infinite length).

A property is a subset of $\mathcal{S}(A^{G})$ .

Definition 8 (See Section 2 in [17]).

The descriptive complexity of a property $P$ can be defined as follows: a property is $\Pi_{1}^{0}$ if it is effectively closed. Equivalently (see Fact 36), there exists an algorithm such that for every $X\in\mathcal{S}(A^{G})$ , given two enumerations of its language $\mathcal{L}(X)$ and of its complement $\mathcal{L}^{c}(X)$ , the algorithm halts if and only if $X\notin P$ (it semi-decides whether $X\notin P$ ).

A property is $\Sigma_{1}^{0}$ if its complement is $\Sigma_{1}^{0}$ (the algorithm semi-decides whether $X\in P$ ). It is $\Sigma_{2}^{0}$ if it is a uniform union of $\Pi_{1}^{0}$ properties, that is, $P=\bigcup_{i\in\mathbb{N}}P_{i}$ and there is an algorithm that, given $i$ , semi-decides if a shift $X$ is not in $P_{i}$ .

Definition 9.

Let $P\subseteq\mathcal{S}(A^{G})$ be a property and $X\in\mathcal{S}(A^{G})$ be a non-empty shift. We say that $X$ is $P$ -minimal if $X\in P$ and, for every non-empty subshift $Y\subsetneq X$ , $Y\notin P$ .

A subshift $Y$ of a shift $X$ can always be obtained by forbidding an additional set of patterns; that is, $Y=X\setminus\bigcup_{p\in P}\bigcup_{g\in G}g[p]$ for some set $P\subseteq\mathcal{L}(X)$ , where for $p:F\to A$ we define $g[p]=\{gx\in A^{G}:x\in A^{G}\text{ and }x|_{F}=p\}$ .

3 Strong computable type for G-shifts

3.1 Definition

We introduce the notion of strong computable type for $G$ -shifts, which originates from a similar concept in computable analysis (see Section 3.3).

As mentioned in the introduction, an effectively closed shift has computable language if it satisfies additional properties, such as being minimal. Identifying such sufficient conditions is of particular interest. We generalize this question as follows: given information on the co-language of a shift (semi-computable or “negative” information), when can we use it to compute its language? This corresponds to asking under which conditions there is an algorithm that enumerates $\mathcal{L}(X)$ when provided with any oracle $O$ that makes $\mathcal{L}(X)^{c}$ computably enumerable.

This definition is formalised as follows, where an oracle is viewed as a subset of $\mathbb{N}$ .

Definition 10 (Strong computable type for $G$ -shifts).

Let $G$ be a finitely generated group with decidable word problem, $A$ be a finite alphabet and $X\subseteq A^{G}$ be a shift.

$X$ has strong computable type if for every oracle $O\subset\mathbb{N}$ , the fact that $X$ is effectively closed relative to $O$ implies that the language of $X$ is computable relative to $O$ .

This is equivalent, by Selman’s theorem [36], to the fact that $\mathcal{L}(X)$ is enumeration-reducible to $\mathcal{L}^{c}(X)$ .

Note that for effectively closed shifts (in particular SFTs) having computable language is equivalent to having strong computable type.

There exist shifts that have strong computable type but do not have computable language. Indeed, a shift $X$ which has a c.e. language which is not co-c.e. clearly has strong computable type; consider for example the shift on $\{0,1\}^{\mathbb{Z}}$ obtained by forbidding all patterns $01^{k}0$ such that the $k$ -th Turing machine doesn’t halt.

This notion is interesting because it provides a characterization that unifies various arguments, offers new results, and connects the computability of languages to the descriptive complexity of topological properties (see Section 3.2).

3.2 Shifts and minimality

In this section, we prove our main theorem, which characterizes shifts with strong computable type by the fact that they must satisfy a property that makes them minimal. This is motivated by Theorem 4.7. in [5].

Theorem 11 (Main theorem).

Let $G$ be a finitely generated group with decidable word problem and let $A$ be a finite alphabet. Let $X\subseteq A^{G}$ be a non-empty shift. The following are equivalent.

1.

$X$ has strong computable type,
2.

There exists a $\Sigma_{2}^{0}$ property $P$ in $\mathcal{S}(A^{G})$ such that $X$ is $P$ -minimal.

$\blacktriangleright$ Remark 12.

From the proof of the implication $2.\Rightarrow 1.$ below, we can see that Theorem 11 holds when the property $P$ is only $\Sigma_{2}^{0}$ relative to $\mathcal{L}^{c}(X)$ (that is, the algorithm used to prove that $P$ is $\Sigma_{2}^{0}$ receives $\mathcal{L}^{c}(X)$ as an additional oracle, where $X$ is the fixed subshift in the theorem statement).

Proof.

By Selman’s theorem, Condition $1 .$ is equivalent to the fact that an effective procedure produces an enumeration of $\mathcal{L}(X)$ from any enumeration of its complement $\mathcal{L}^{c}(X)$ .

$1.\Rightarrow 2$ . Assume that there is an effective procedure (a machine $M$ ) producing an enumeration of $\mathcal{L}(X)$ from any enumeration of $\mathcal{L}^{c}(X)$ , let us define some $\Pi_{1}^{0}$ (and hence $\Sigma_{2}^{0}$ ) property $P$ , for which $X$ is $P$ -minimal.

Let $\varGamma$ be the set of all pairs $(\sigma,p)$ where $\sigma=\sigma_{0}\ldots\sigma_{k}$ is a finite sequence of patterns such that the machine $M$ outputs the pattern $p$ after reading $\sigma$ ; clearly $\varGamma$ is c.e..

Let $\mathcal{U}=\bigcup_{(\sigma,p)\in\varGamma}\mathcal{U}_{(\sigma,p)}$ with

\mathcal{U}_{(\sigma,p)}=\{X^{\prime}\in\mathcal{S}(A^{G}):\forall i\leq k,X^{% \prime}\cap[\sigma_{i}]=\emptyset\text{ and }X^{\prime}\cap[p]=\emptyset\}.

Note that a c.e. union of $\Sigma_{1}^{0}$ properties is a $\Sigma_{1}^{0}$ property. $\mathcal{U}$ is a $\Sigma_{1}^{0}$ property because $\varGamma$ is c.e. and for every $(\sigma,p)\in\Gamma$ , $\mathcal{U}_{(\sigma,p)}$ is a $\Sigma_{1}^{0}$ property (given an enumeration of $\mathcal{L}^{c}(X^{\prime})$ one can semi-decide whether the $\sigma_{i}$ ’s and $p$ are in $\mathcal{L}^{c}(X^{\prime})$ , equivalently whether the corresponding cylinders do not intersect $X^{\prime}$ ).

Let $P$ be the complement of $\mathcal{U}$ , it is hence a $\Pi_{1}^{0}$ property. Let us prove that $X$ is $P$ -minimal.

$X$ is in $P$ because otherwise the machine $M$ fails on $X$ , i.e. after reading a finite sequence of patterns $\sigma$ that are in $\mathcal{L}^{c}(X)$ the machine outputs a pattern $p$ that is in $\mathcal{L}^{c}(X)$ (which contradicts our assumption).

Now, let $Y\subsetneq X$ be a shift and $p\in\mathcal{L}(X)\setminus\mathcal{L}(Y)$ . We have $Y\notin P$ because the machine eventually outputs $p$ after reading some finite sequence $\sigma$ , which implies that $Y\in\mathcal{U}_{(\sigma,p)}$ .

$2.\Rightarrow 1$ . Assume that $P$ is $\Sigma_{2}^{0}$ and that $X$ is $P$ -minimal. Hence, $P=\bigcup_{i\in I}P_{i}$ with $P_{i}\in\Pi_{1}^{0}$ for all $i$ . It is easy to see that there exists some $i_{0}$ such that $X$ is $P_{i_{0}}$ -minimal.

By Fact 37, we can suppose that $P_{i_{0}}$ is $\Pi_{1}^{0}$ in a stronger sense: there exists an algorithm such that for every $Y\in\mathcal{S}(A^{G})$ , given an enumeration of the complement of its language $\mathcal{L}^{c}(Y)$ , the algorithm halts if and only if $Y\notin P$ ; see Appendix A.2.2 for more details.

Given an enumeration of $\{q:[q]\cap X=\emptyset\}$ (that is, an enumeration of $\mathcal{L}^{c}(X)$ ), we need to enumerate $\mathcal{L}(X)$ . Note that $p\in\mathcal{L}(X)$ iff $X^{\prime}=X\setminus\bigcup_{g\in G}g[p]$ is a proper subshift in $X$ ; as $X$ is $P_{i_{0}}$ -minimal, it is equivalent to $X^{\prime}\notin P_{i_{0}}$ .

$X^{\prime}$ can be defined by the set of forbidden patterns $\mathcal{L}^{c}(X)\cup\{p\}$ . There is an algorithm that, for any shift $Y$ , computes an enumeration of $\mathcal{L}^{c}(Y)$ from any enumeration of a set of forbidden patterns that defines $Y$ ; this can be proved exactly as Fact 7 relative to an oracle. In particular, from an enumeration of $\mathcal{L}^{c}(X)\cup\{p\}$ we can compute, uniformly in $p$ , an enumeration of $\mathcal{L}^{c}(X^{\prime})$ .

Since $P_{i_{0}}$ is $\Pi_{1}^{0}$ , from an enumeration of $\mathcal{L}^{c}(X^{\prime})$ we can semi-decide the condition $X^{\prime}\notin P_{i_{0}}$ . Therefore, whether $p\in\mathcal{L}(X)$ is semi-decidable relative to an enumeration of $\mathcal{L}^{c}(X)$ . In other words, from an enumeration of $\mathcal{L}^{c}(X)$ we can compute an enumeration of $\mathcal{L}(X)$ . $\hfill\blacktriangleleft$

If we remove oracles from Theorem 11, we obtain the following characterization of the decidability of languages of effectively closed shifts.

Corollary 13.

Let $G$ be a finitely generated group with decidable word problem and let $A$ be a finite alphabet. Let $X\subseteq A^{G}$ be a non-empty effectively closed shift. The following are equivalent.

1.

$X$ has computable language,
2.

There exists a $\Sigma_{2}^{0}$ property $P$ in $\mathcal{S}(A^{G})$ such that $X$ is $P$ -minimal.

3.3 Relation with strong computable type for sets

The notion of strong computable type was initially defined for sets (see [5]), using the notion of copies in the Hilbert cube $Q=[0,1]^{\mathbb{N}}$ , and generalized the notion of computable type for sets defined in [25]. For a comprehensive understanding of strong computable type for sets, see Amir’s PhD thesis [1].

A copy of a set $X\subseteq Q$ is the image of $X$ by a homeomorphism $Q\rightarrow Q$ (a continuous bijection with a continuous inverse).

$Q$ is a computable metric space, and hence there exists a computable dense sequence $(q_{i})_{i\in\mathbb{N}}$ in $Q$ . Let $(B_{i})_{i\in\mathbb{\mathbb{N}}}$ be an enumeration of all the balls $B(q_{i},r_{j})$ with center $q_{i}$ and radius $r_{j}\in\mathbb{Q}$ .

A compact set $X$ in $Q$ is semi-computable if the set $\{i\in\mathbb{N}:X\cap B_{i}=\emptyset\}$ is c.e.. It is computable if it is semi-computable and in addition the set $\{i\in\mathbb{N}:X\cap B_{i}\neq\emptyset\}$ is c.e.. The same notions can be defined by replacing $Q$ by any space with a computable metric structure.

Now, let us define strong computable type for sets. A compact set $X$ in $Q$ has strong computable type if for every copy $Y$ of it in $Q$ , and every oracle which makes $Y$ semi-computable, $Y$ is also computable using the oracle.

One might attempt to define strong computable type for $G$ -shifts using copies. However, this notion is very restrictive: for example, since any shift without isolated points is homeomorphic to the Cantor set, any such shift is also homeomorphic to an effectively closed shift whose language is not computable, so it does not have strong computable type.

Another possible definition involves using analogs of homeomorphisms in symbolic dynamics, namely conjugacies. However, since conjugate shifts preserve the decidability of languages, proving the decidability of the language of one shift automatically implies the decidability of the languages of all conjugate shifts. Therefore, we define strong computable type for $G$ -shifts without considering either copies or conjugacies.

A similar result for the computability of sets, relating computable type with minimality was proved (see [5, 1]). We prove in Appendix B that Theorem 11 can be obtained by this characterization of strong computable type for sets, though it is not immediate.

3.4 Relation with maximal elements in quasivarieties

In Theorem 5 of [26], Jeandel proves a similar result for quasivarieties, which is a general framework including subshifts as well as finitely generated groups, first-order theories, etc. We compare and contrast it with our results.

Jeandel’s result is a sort of dual of our Theorem 11 as it applies to maximal elements and yields an enumeration reduction that is the opposite direction from strong computable type; this is only a consequence of the fact that it applies to $\mathcal{L}^{c}(X)$ instead of $X$ . Quasivarieties correspond to (in our context) effectively closed properties that are stable under finite unions. Our result does not requires this last assumption, although note that:

$\blacksquare$

in the proof of the implication $1.\Rightarrow 2.$ in Theorem 11, the property $P$ is stable by union, and therefore the theorem would hold with this additional assumption;
$\blacksquare$

properties used in our examples in Section 3.5 are stable by union, so we believe these results could be proved in Jeandel’s framework (indeed, examples from Sections 3.5.5 and 3.5.6 appeared in [26]).

Additionally, Theorem 5 of [26] is not an equivalence; note that the converse given in Theorem 6 in op.cit. is not comparable to our main theorem, and shows in the case of subshifts that the language of a subshift with strong computable type is enumeration-reducible to the language of a minimal subshift.

We are not sure of whether $\Pi_{1}^{0}$ properties and quasivarieties are equivalent in terms of expressing minimality properties (see Question 3). We believe that, at least, $\Pi_{1}^{0}$ properties provide an easier description of shifts with strong computable type for researchers in the symbolic dynamics community.

3.5 Applications

In this section, we provide examples of $G$ -shifts that have strong computable type based on our characterization. Most of these results are generalizations of existing results to arbitrary oracles, and a main interest of our characterization is to unify the underlying arguments.

Let $G$ be a finitely generated group with decidable word problem, and let $A$ be a finite alphabet.

3.5.1 Minimal shifts

Recall from the introduction that a non-empty shift is minimal if it contains no other subshifts except itself and the empty shift.

The following result has been first stated, to the best of our knowledge, in [18]; it is a generalisation of a folklore result for the finite type case.

Theorem 14 ([18]).

An effectively closed minimal shift has computable language.

We prove a generalization of this theorem (relative to any oracles) based on our characterization.

Proposition 15.

A minimal shift has strong computable type.

Proof.

A minimal shift is $P$ -minimal for the property $P=\mathcal{S}(A^{G})$ (remember that the empty shift is not a member of $\mathcal{S}(A^{G})$ by definition). This property is trivially $\Pi_{1}^{0}$ , using an algorithm that never halts. $\hfill\blacktriangleleft$

3.5.2 Entropy-minimal shifts

In this section we suppose in addition that the group $G$ is amenable (we do not give the definition of being amenable as we don’t explicitly use it).

Definition 16.

Let $h$ be the entropy map $\mathcal{S}(A^{G})\rightarrow\mathbb{R}^{+}$ that to a shift associates its topological entropy (see Chapter 9 in [29] for more details of this notion). A $G$ -shift $X$ is called entropy-minimal when every proper subshift $Y\subset X$ satisfies $h(Y)<h(X)$ .

The following result is stated in a recent unpublished work by Carrasco-Vargas, Herrera Nunez and Sablik that appeared in Chapter 6 in Carrasco-Vargas’s PhD Thesis [14].

Theorem 17 ([14]).

An entropy-minimal SFT whose entropy is a computable real number has computable language.

A real number is left-computable if there exists a computable increasing sequence of rational numbers converging to it.

In Proposition 18, we generalize this result in several ways: from SFTs to effectively closed shifts, from computable entropies to left-computable entropies, and finally from having computable languages to having strong computable type, based on our characterization in Theorem 11.

Proposition 18.

An entropy-minimal shift $X$ whose entropy is a real number that is left-computable relative to $\mathcal{L}^{c}(X)$ has strong computable type.

Proof.

Let $X$ be an entropy-minimal shift with $q=h(X)$ . Take the property $P=h^{-1}([q,+\infty))$ . Relative to $\mathcal{L}^{c}(X)$ , $P$ is $\Pi_{1}^{0}$ because $q$ is left-computable and $h$ is upper semi-computable (combine Proposition 6.17 in [14] with Fact 35). Clearly, $X$ is $P$ -minimal, so by Theorem 11 it has strong computable type. $\hfill\blacktriangleleft$

$\blacktriangleright$ Remark 19.

The previous proof also works when $h(X)$ is not left-computable, under the assumption that $\sup_{Y\subsetneq X}h(Y)<h(X)$ . In this case, pick $q$ to be any left-computable number in the interval $[\sup_{Y\subsetneq X}h(Y),h(X)]$ .

Entropy-minimality alone is not sufficient

We give an example of a shift in $\{0,1\}^{\mathbb{Z}}$ that is entropy-minimal and does not have strong computable type.

Definition 20.

Let $\alpha\leq\beta$ be two real numbers. Define the shift $X_{[\alpha,\beta]}\subset\{0,1\}^{\mathbb{Z}}$ by forbidding the set of finite patterns

\{w\in\{0,1\}^{n}:\#_{1}w<\lfloor\alpha n\rfloor\text{ or }\#_{1}w>\lceil\beta n% \rceil\},

where $\#_{1}w$ is the number of occurrences of the symbol $1$ in $w$ . $X_{\alpha}=X_{[\alpha,\alpha]}$ is known as the Sturmian shift of slope $\alpha$ .

See [32, Chapter 2] for more properties of Sturmian configurations and Sturmian shifts. In particular, the language of a Sturmian shift with irrational slope contains exactly $n+1$ different patterns of length $n$ .

Proposition 21.

The shift $X_{[0,\alpha]}$ is entropy-minimal and, for some right-computable value of $\alpha$ , does not have strong computable type.

Proof.

The shift $X_{[0,\alpha]}$ is strongly irreducible, that is, for two patterns $u,v\in\mathcal{L}(X_{[0,\alpha]})$ , we have $u0^{k}v\in\mathcal{L}(X_{[0,\alpha]})$ for some constant $k=\lceil\frac{2}{\alpha}\rceil$ . Indeed, check that $\#_{1}u0v=\#_{1}u+\#_{1}v\leq\lceil\alpha|u|\rceil+\lceil\alpha|v|\rceil\leq% \alpha(|uv|+2)\leq\lceil\alpha(|uv|+k)\rceil$ , and the same argument applies to any subpattern of $u0^{k}v$ . A strongly irreducible shift is entropy-minimal: see Corollary 4.7 in [20].

If $\alpha$ is a right-computable real number (namely, $-\alpha$ is left-computable), then $X_{[0,\alpha]}$ is an effectively closed shift since the set $\{w\in\{0,1\}^{n}:\#_{1}w>\lceil\alpha n\rceil\}$ is c.e.. In particular, $\mathcal{L}(X_{[0,\alpha]})$ is co-computably enumerable.

By an argument that can be found e.g. in the proof of Theorem 3.6 in [21], for a strongly irreducible shift $Y$ , from the number of patterns of size $n$ in $\mathcal{L}(Y)$ we can compute an approximation of $h(Y)$ with a known error rate. In our case, having a right-approximation of number of patterns of size $n$ in $\mathcal{L}(X_{[0,\alpha]})$ , we obtain that $h(X_{[0,\alpha]})$ is right-computable.

Now assume that $X_{[0,\alpha]}$ has computable language. Compute all patterns of length $n$ in $\mathcal{L}(X_{[0,\alpha]})$ and count the maximum number of symbols $1$ that appear in a pattern: this number is $\lfloor\alpha n\rfloor$ , from which we get an approximation of $\alpha$ up to error $1/n$ . Therefore, $\alpha$ is a computable real number in this case.

It follows that $X_{[0,\alpha]}$ , for $\alpha$ right-computable but not computable, is entropy-minimal, effectively closed and its language is not computable, so it does not have strong computable type. $\hfill\blacktriangleleft$

3.5.3 P-isolated shifts

As in the previous section, studying isolated points is motivated by the unpublished work of Carrasco-Vargas, Herrera Nunez and Sablik that appeared in [14].

In their work they find an alternative proof that a minimal SFT has computable language, using the fact that it is isolated in $\mathcal{S}(A^{G})$ .

Similarly, they prove that an entropy-minimal SFT with computable real entropy $q$ has computable language, using the fact that it is isolated in $h^{-1}([q,+\infty))$ , see Definition 16.

The common property between $\mathcal{S}(A^{G})$ and $h^{-1}([q,+\infty))$ is that they are $\Pi_{1}^{0}$ (see Fact 35 in the Appendix for $\mathcal{S}(A^{G})$ ). In general, isolated points in $\Pi_{1}^{0}$ classes are computable: see Fact 2.14 in [19]. It motivates us to generalize the result as follows.

Proposition 22.

Let $X\in\mathcal{S}(A^{G})$ be a non-empty shift and $P$ be a property in $\mathcal{S}(A^{G})$ which is $\Sigma_{2}^{0}$ relative to $\mathcal{L}^{c}(X)$ . If $X$ is isolated in $P$ , then $X$ has strong computable type.

Proof.

Since $X$ is isolated in $P$ , we can find a cylinder $[p]$ such that $X\cap[p]\neq\emptyset$ and $Y\in P\Rightarrow Y\cap[p]=\emptyset$ . The property $C=\{Y:Y\cap[p]\neq\emptyset\}$ is both $\Sigma_{1}^{0}$ and $\Pi_{1}^{0}$ . Let $P^{\prime}=P\cap C=\{X\}$ . $P^{\prime}$ is $\Sigma_{2}^{0}$ relative to $\mathcal{L}^{c}(X)$ because it is the intersection of a property that is $\Sigma_{2}^{0}$ relative to $\mathcal{L}^{c}(X)$ with a $\Sigma_{1}^{0}$ property. Clearly, $X$ is $P^{\prime}$ -minimal, hence it has strong computable type. $\hfill\blacktriangleleft$

3.5.4 Periodic-minimal shifts

Definition 23.

The orbit $\operatorname{Orb}(x)$ of a configuration $x\in\mathcal{A}^{G}$ is the closure of the set $\{g\cdot x:g\in G\}$ . A (strongly) periodic configuration is a configuration $x$ such that the number of elements $\#\operatorname{Orb}(x)$ is finite.

To a shift $X$ we associate the vector $\operatorname{Per}(X)=(\operatorname{Per}_{i}(X))_{i\in_{\mathbb{N}}}$ , where $\operatorname{Per}_{i}(X)$ is the number of periodic points in $X$ with orbit size $i$ or less. This is a classical conjugacy invariant for shifts, usually presented under the form of a Zeta function; see e.g. [31] for more information on this object.

A shift $X$ is period-minimal if, for any subshift $Y\subsetneq X$ , $\operatorname{Per}_{i}(Y)<\operatorname{Per}_{i}(X)$ for some $i\in\mathbb{N}$ . This corresponds to minimality for the $\Pi_{1}^{0}$ property $P_{X}=\{Y:\forall i,\operatorname{Per}_{i}(Y)\geq\operatorname{Per}_{i}(X)\}$ ; therefore a period-minimal shift has strong computable type.

Proposition 24.

A shift is period-minimal if, and only if, it has dense periodic points. These shifts have strong computable type.

Proof.

If $X$ is period-minimal, then the subshift obtained from $X$ by forbidding any pattern $p\in\mathcal{L}(X)$ has strictly less periodic points. This means that any pattern $p\in\mathcal{L}(X)$ appears in some periodic point of $X$ , that is, $[p]\cap X$ contains a periodic point. Since cylinders are a basis of the topology, periodic points are dense in $X$ . Conversely, if periodic points are dense in $X$ , then any strict subshift $Y\subsetneq X$ has strictly less periodic points. $\hfill\blacktriangleleft$ We obtained hence a stronger version of a classical result:

Theorem 25 ([30]).

An effectively closed shift with dense periodic points has computable language.

3.5.5 Quasi-minimal shifts

Quasi-minimal shifts as introduced by Salo [35] are shifts with finitely many distinct subshifts.Let

Theorem 26 ([35], Theorem 9.).

An effectively closed quasi-minimal shift has computable language.

We strenghten this result by proving the following.

Proposition 27.

Quasi-minimal shifts have strong computable type.

Proof.

For a given quasi-minimal shift $X$ , denote $X_{1}\dots X_{n}$ its distinct subshifts and fix $p_{i}\in\mathcal{L}(X)\backslash\mathcal{L}(X_{i})$ for all $1\leq i\leq n$ . $X$ is then minimal for the $\Pi_{1}^{0}$ property $P_{X}=\{Y:\forall i\leq n,p_{i}\in\mathcal{L}(Y)\}$ . $\hfill\blacktriangleleft$ In op.cit, Salo studies a different notion, initially introduced in [18], which is having finitely many distinct minimal subshifts. We do not believe that such shifts must have strong computable type.

3.5.6 Infinite-minimal shifts

It is well-known [10, Theorem 3.8] that a shift is finite if and only if it contains only strongly periodic points. We consider the property for a shift $X$ to be infinite-minimal, that is, $|X|=\infty$ and all subshifts $Y\subsetneq X$ are finite. Equivalently, such a shift is minimal for the property of containing a non-periodic configuration. This property was called just-infinite in [26]. The property of being infinite is known to be $\Pi_{0}^{1}$ in $\mathbb{Z}^{2}$ but not in $\mathbb{Z}^{d}$ , for $d>2$ (see [13]), so we do a direct proof.

Proposition 28.

An infinite-minimal shift has strong computable type.

Proof.

If there are finitely many such $Y$ ’s, then $X$ is quasi-minimal and we conclude by Theorem 26. Otherwise, $X$ contains infinitely many periodic points. If there is a pattern $p$ that appears in no periodic point, the subshift obtained from $X$ by forbidding $p$ contains all periodic points of $X$ , so it is infinite, which contradicts infinite-minimality. We conclude that periodic points are dense in $X$ , which is enough by Theorem 25. $\hfill\blacktriangleleft$

3.5.7 Full extensions

Jeandel and Vanier proved the following characterization.

Theorem 29 ([27]).

A shift $X$ in $A^{\mathbb{Z}^{d}}$ has computable language if and only if it is the projective subdynamics of a minimal effectively closed shift $Y$ (possibly with larger alphabet and dimension); in other words, $\mathcal{L}(X)=\mathcal{L}(Y)\cap\mathcal{L}(A^{\mathbb{Z}^{d}})$ , which are patterns $p:F\to A\in\mathcal{L}(Y)$ such that $F\subset\mathbb{Z}^{d}$ .

We show that if $X$ is the projective subdynamics of a minimal shift $Y$ then it has strong computable type, which corresponds to one direction of the previous result relative to an oracle. It is possible that the proof of [27] can also be generalised for the other direction.

Assume that $Y\subset B^{\mathbb{Z}^{d^{\prime}}}$ with $A\subset B$ and $d\leq d^{\prime}$ . We remark that in the previous theorem, we have a $\Pi_{1}^{0}$ property $P$ for which $X$ is minimal:

\displaystyle P

\displaystyle=\{Z\in\mathcal{S}(A^{\mathbb{Z}^{d}}):Y\backslash\bigcup_{g\in% \mathbb{Z}^{d^{\prime}}}\bigcup_{w\in\mathcal{L}^{c}(Z)}g[w]\neq\emptyset\}.

Notice that since $Z$ is a subshift of $A^{\mathbb{Z}^{d}}$ , $\mathcal{L}^{c}(Z)$ contains patterns $F\to A$ with $F\subset\mathbb{Z}^{d}$ , which are seen as patterns $F\to B$ with $F\subset\mathbb{Z}^{d^{\prime}}$ .

Let us prove that $X$ is $P$ -minimal. $Y\backslash\bigcup_{g\in\mathbb{Z}^{d^{\prime}}}\bigcup_{w\in\mathcal{L}^{c}(X% )}g[w]=Y\neq\emptyset$ so $X\in P$ . Now let $X^{\prime}$ be a proper subshift in $X$ : this means that there exists some pattern $p\in\mathcal{L}(X)\setminus\mathcal{L}(X^{\prime})$ . By assumption, $p\in\mathcal{L}(Y)$ . Since $Y$ is minimal, $Y\backslash\bigcup_{g\in\mathbb{Z}^{d^{\prime}}}\bigcup_{w\in\mathcal{L}^{c}(X% ^{\prime})}g[w]=\emptyset$ hence $X^{\prime}\notin P$ .

As $Y$ is effectively closed and minimal, it has computable language (see Section 3.5.1), which makes the condition $Y\backslash\bigcup_{g\in\mathbb{Z}^{d^{\prime}}}\bigcup_{w\in\mathcal{L}^{c}(Z% )}g[w]=\emptyset$ semi-decidable given an enumeration of $\mathcal{L}^{c}(Z)$ . Hence $P$ is a $\Pi_{1}^{0}$ property.

3.6 Invariance under transformations

Strong computable type is preserved under products and factors. However, it is not preserved under unions and intersections.

3.6.1 Factors and computable morphisms

Let $A$ and $B$ be two finite alphabets. A factor $F:\mathcal{S}(A^{G})\rightarrow\mathcal{S}(B^{G})$ is a surjective morphism between $G$ -shifts that commutes with the shift action. It sends shifts with computable languages in $\mathcal{S}(A^{G})$ to shifts with computable languages in $\mathcal{S}(B^{G})$ because it is a computable surjective function. Using oracles, it sends shifts with strong computable type to shifts with strong computable type. More generally, an image of a shift with strong computable type by a computable morphism has strong computable type.

3.6.2 Products

In contrast to the case of sets, where products do not always preserve strong computable type (see [6]), strong computable type for shifts is preserved under finite products.

Let $A$ and $B$ be two finite alphabets. Let $X$ be a shift in $\mathcal{S}(A^{G})$ and $Y$ be a shift in $\mathcal{S}(B^{G})$ , the product $X\times Y$ is a shift in $\mathcal{S}(A^{G})\times\mathcal{S}(B^{G})\subseteq\mathcal{S}(A^{G}\times B^{% G})\equiv\mathcal{S}((A\times B)^{G})$ .

Proposition 30.

If two shifts $X$ and $Y$ have strong computable type, then $X\times Y$ has strong computable type.

Proof.

Assume that $X\subset A^{G}$ and $Y\subset B^{G}$ . A pattern $w:F\to A\times B\in\mathcal{L}^{c}(X\times Y)$ can be seen as the product of two patterns $u:F\to A$ and $v:F\to B$ such that either $u\in\mathcal{L}^{c}(X)$ or $v\in\mathcal{L}^{c}(Y)$ . For a pattern $p:F\to A$ , $p\in\mathcal{L}^{c}(X)$ is equivalent to the fact that all the pairs $(p,q)$ with $q:F\to B$ are in $\mathcal{L}^{c}(X\times Y)$ . Indeed, if $p\in\mathcal{L}(X)$ , then only pairs such that $q\in\mathcal{L}^{c}(Y)$ will be enumerated and $\mathcal{L}(Y)\neq\emptyset$ . Therefore, given an enumeration of $\mathcal{L}^{c}(X\times Y)$ one can enumerate the elements of $\mathcal{L}^{c}(X)$ .

By symmetry, one can enumerate the elements of $\mathcal{L}^{c}(Y)$ . As $X$ and $Y$ have strong computable type, $\mathcal{L}(X)$ and $\mathcal{L}(Y)$ are computable. Hence $\mathcal{L}(X\times Y)=\mathcal{L}(X)\times\mathcal{L}(Y)$ is computable and $X\times Y$ has strong computable type. $\hfill\blacktriangleleft$

3.6.3 Unions and intersections

Proposition 31.

There exists two shifts $X, Y$ that have strong computable type such that $X\cup Y$ does not have strong computable type. Similarly, there exists two shifts $X^{\prime}$ and $Y^{\prime}$ that have strong computable type such that $X^{\prime}\cap Y^{\prime}$ does not have strong computable type.

Proof.

Consider $X_{[\alpha,\alpha+1/2]}$ and $X_{[\beta-1/2,\beta]}$ for some real numbers $\alpha<\beta-1/2<\alpha+1/2<\beta$ (see Definition 20). It is clear that $X_{[\alpha,\alpha+1/2]}\cup X_{[\beta-1/2,\beta]}=X_{[\alpha,\beta]}$ . If $\alpha$ is left-computable and $\beta$ is right-computable (namely, $-\beta$ is left-computable), but one of them is not computable, then $X_{[\alpha,\beta]}$ is effectively closed but its language is not computable by the same argument as in Proposition 21, so it does not have strong computable type. However, from an enumeration of $\mathcal{L}^{c}(X_{[\alpha,\alpha+1/2]})$ and by counting the number of symbols $1$ in each, it is not hard to compute an upper approximation of $\alpha$ and a lower approximation of $\alpha-1/2$ , and therefore to compute arbitrarily good approximations of $\alpha$ with known error, from which it is straightforward to compute the language of $X_{[\alpha,\alpha+1/2]}$ . It follows that $X_{[\alpha,\alpha+1/2]}$ has strong computable type. The case of $X_{[\beta-1/2,\beta]}$ is symmetric.

For the intersection, apply a similar argument to the same shifts when $\beta-1/2<\alpha<\beta<\alpha+1/2$ . $\hfill\blacktriangleleft$

The next proposition shows that strong computable type is conserved under disjoint unions. It is not enough that the intersection $X\cap Y$ is simple (in a computational sense) since, in the previous proof, we had $X_{[\alpha,\alpha+1/2]}\cap X_{[\beta-1/2,\beta]}=X_{[\beta-1/2,\alpha+1/2]}$ which has computable language.

Proposition 32.

Let $X$ and $Y$ be two shifts with strong computable type such that $X\cap Y=\emptyset$ . Then $X\cup Y$ have strong computable type.

Proof.

In this proof we assume that the underlying group $G$ is generated by the set of generators $S$ and we denote $B_{S}(n)=\{w_{G}:w\in S^{\leq n}\}$ .

Since $X\cap Y=\emptyset$ , a compactness argument shows that there is a $N\in\mathbb{N}$ such that, if $p\in A^{B_{S}(n)}$ for $n>N$ , $[p]\cap X=\emptyset$ or $[p]\cap Y=\emptyset$ . Indeed, if that was not the case, we would get a sequence of patterns with increasing support from which we could extract by compacity an element of $X\cap Y$ .

Denote by $E_{Y}$ the set of patterns defined as follows. For any pattern $p$ of support $F$ ,

$\blacksquare$

if $F\subset B_{S}(N)$ , $p\in E_{Y}$ if and only if $[p]\cap X=\emptyset$ ;
$\blacksquare$

otherwise, let $n$ be the smallest number such that $F\subset B_{S}(n)$ . Then $p\in E_{Y}$ if and only if, for any pattern $p^{\prime}$ of support $B_{S}(N)$ that extends $p$ (that is, $p^{\prime}|_{F\cap B_{S}(N)}=p|_{B_{S}(N)}$ ), $[p^{\prime}|_{B_{S}(N)}]\cap Y\neq\emptyset$ .

Checking whether $p\in E_{Y}$ requires finitely many checks of the form $[q]\cap X=\emptyset$ or $[q]\cap Y\neq\emptyset$ for a pattern $q$ of support included in $B_{S}(N)$ . There are finitely many such patterns, so $E_{Y}$ is a computable set.

Now assume that we receive as input an enumeration of $\mathcal{L}^{c}(X\cup Y)=\mathcal{L}^{c}(X)\cap\mathcal{L}^{c}(Y)$ . We show that $\mathcal{L}^{c}(X\cup Y)\cup E_{Y}=\mathcal{L}^{c}(X)$ . It is clear that $E_{Y}\subset\mathcal{L}^{c}(X)$ by definition of $N$ and $E_{Y}$ . Furthermore, $\mathcal{L}(Y)\cap\mathcal{L}^{c}(X)\subset E_{Y}$ ; in fact the first point includes all “small” patterns in $\mathcal{L}^{c}(X)$ and the second point includes all “large” patterns in $\mathcal{L}(Y)$ .

We compute an enumeration of $\mathcal{L}^{c}(X\cup Y)\cup E_{Y}=\mathcal{L}^{c}(X)$ . Since $X$ has strong computable type, we can compute a enumeration of $\mathcal{L}(X)$ . Doing similarly for $Y$ , we compute an enumeration of $\mathcal{L}(Y)$ . From this we obtain an enumeration of $\mathcal{L}(X\cup Y)=\mathcal{L}(X)\cup\mathcal{L}(Y)$ , so $X\cup Y$ has strong computable type. $\hfill\blacktriangleleft$

The previous proof relies on the fact that, in $A^{G}$ , two disjoint shifts (or more generally two disjoint closed sets) $X$ and $Y$ are computably separable; $E_{Y}$ plays the role of a computable superset of $Y$ that is disjoint from $X$ .

4 Discussion, generalizations and future directions

Several results in this article can be generalized to recursively presented groups.

The notion of strong computable type for shifts can be extended to pairs of shifts, consisting of a shift and a subshift of it, in the same way as the notion of strong computable type for pairs of sets. Similar results can then be proved, and we will address this in a subsequent article that extends this one.

Our results for strong computable type for shifts can be analogously extended to strong computable type for sets. For example, $P$ -isolated sets, where $P$ is a topological $\Sigma_{2}^{0}$ invariant, have strong computable type.

Question 1.

Let us consider further results on the computable type of sets. Can we obtain analogous results for shifts, and vice versa?

The results here also motivate an independent research direction: the study and characterization of the descriptive complexity of properties of shifts, similar to how such properties have been studied for sets in [5, 7].

Question 2.

Can we characterize $\Sigma_{2}^{0}$ and $\Pi_{1}^{0}$ properties of shifts?

Motivated by the more general approach of [26], we would like to better understand the relationship between Jeandel’s and our results and find a unifying argument that can be developed using computable analysis, as we have done here for shifts.

Question 3.

Is it possible to obtain results based on minimality similar to Theorem 11 for other theories, such as group theory and combinatorics, in a unifying framework based on computable analysis?

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[35] Ville Salo. Decidability and universality of quasiminimal subshifts. Journal of Computer and System Sciences, 89:288–314, 2017. doi:10.1016/j.jcss.2017.05.017.
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Appendix A Topology on $A^{G}$ and its hyperspace

A.1 Topology on $A^{G}$

To define $G$ -shifts we need to induce $A^{G}$ with a topology as follows (see [15]).

Definition 33.

Let $G$ be a group and $A$ be an alphabet. We endow $A^{G}$ with the pro-discrete topology obtained by inducing $A$ with the discrete topology (each point in $A$ is open and closed (clopen)) and then taking the product topology.

If $G=\{g_{i}:i\in\mathbb{N}\}$ is countable, this topology is metrizable using the metric

d(x,y)=\inf\{\frac{1}{n}:n\in\mathbb{N},\text{ and }x(i)=y(i)\text{ for all }i% \leq n\}.

If $G$ is finitely generated and $S$ is a generating set, they induce a word length $|\cdot|_{S}:G\rightarrow\mathbb{N}$ such that if $g\in G$ , $|g|_{S}$ is the length of the shortest word in $S^{*}$ with $g=w_{G}$ . Consequently, a word metric $d_{S}$ on $G$ can be defined as follows: for every $(g,h)\in G^{2}$ , $d_{S}(g,h)=|g^{-1}h|_{S}$ . In this case another metric for the pro-discrete topology on $A^{G}$ is

d(x,y)=\inf\{\frac{1}{n}:n\in\mathbb{N},\text{ and }x(g)=y(g)\text{ for all }|% g|_{S}\leq n\}.

Fact 34 (Section 3.3. in [11]).

Let $A$ be an alphabet and $G$ a finitely generated group with decidable word problem. The space $A^{G}$ can be endowed with a computable metric structure corresponding to the distance $d$ defined above: that is, there exists a dense sequence $(x_{i})_{i\in\mathbb{N}}\subseteq A^{G}$ which is uniformly computable (i.e. there exists an algorithm which given any pair $(i,j)\in\mathbb{N}^{2}$ and any $n\in\mathbb{N}$ computes some $\alpha\in\mathbb{Q}$ such that $|d(x_{i},x_{j})-\alpha|<1/n$ ).

A.2 The hyperspace of $A^{G}$

Let $G$ be a finitely generated group with decidable word problem and let $A$ be a finite alphabet. We denote by $\mathcal{K}(A^{G})$ the hyperspace of $A^{G}$ i.e. the space of all non-empty compact subsets of $A^{G}$ . It can be endowed with the induced Hausdorff metric (see Section 2.3. in [1]) which induces also a computable metric structure (see [24]) from the computable metric structure defined in Fact 34. This metric structure induces a topology which is called Vietoris topology, see Section 2.3. in [1].

A property is a subset of $\mathcal{K}(A^{G})$ .

A.2.1 Descriptive complexity for properties in $\mathcal{K}(A^{G})$ and $\mathcal{S}(A^{G})$

Descriptive complexity for properties in $\mathcal{K}(A^{G})$ can be defined in the same way as for properties in $\mathcal{S}(A^{G})$ in Definition 8. As $\mathcal{S}(A^{G})$ is a property in $\mathcal{K}(A^{G})$ it is not hard to see its descriptive complexity.

Fact 35.

$\mathcal{S}(A^{G})$ is a $\Pi_{1}^{0}$ property in $\mathcal{K}(A^{G})$ .

Proof.

Let $X\in\mathcal{K}(A^{G})$ be a set, given an enumeration of $\mathcal{L}(X)$ and its complement, we can semi-decide whether $X\notin\mathcal{S}(A^{G})$ i.e. the fact that $X$ is not shift-invariant. Indeed, eventually the machine will enumerate some pattern $p$ and some $g\in G$ such that $[p]\cap X\neq\emptyset$ and $g[p]\cap X=\emptyset$ . $\hfill\blacktriangleleft$ In Section 3.5, the properties we define in $\mathcal{S}(A^{G})$ can be defined more generally in $\mathcal{K}(A^{G})$ , for instance the property of being non-empty is $\Pi_{1}^{0}$ in $\mathcal{K}(A^{G})$ .

Now, let us discuss the relation between topologies on $\mathcal{K}(A^{G})$ and $\mathcal{S}(A^{G})$ and the descriptive complexity of properties.

As $\mathcal{K}(A^{G})$ is a computable metric space, let $(k_{i})_{i\in\mathbb{N}}$ be a computable dense sequence in $\mathcal{K}(A^{G})$ . One can enumerate all the balls of the form $B(k_{i},r_{j})$ with center $k_{i}$ and radius $r_{j}\in\mathbb{Q}$ , let $(B_{k})_{k\in\mathbb{\mathbb{N}}}$ be such an enumeration. $\mathcal{S}(A^{G})$ can then be endowed with the induced topology as a subspace of $\mathcal{K}(A^{G})$ .

A property is effectively open if there is some c.e. $I\subseteq\mathbb{N}$ , such that $P=\bigcup_{i\in I}B_{i}$ .

A property is effectively closed if it is the complement of an effectively open property.

Here is a reformulation of descriptive complexity using effective topology.

Fact 36.

A property in $\mathcal{K}(A^{G})$ is $\Sigma_{1}^{0}$ iff it is effectively open in $\mathcal{K}(A^{G})$ . It is $\Pi_{1}^{0}$ if it is effectively closed in $\mathcal{K}(A^{G})$ . The same is true for properties in $\mathcal{S}(A^{G})$ .

A.2.2 Descriptive complexity and upper Vietoris topology

We explained in Fact 36 that being $\Pi_{1}^{0}$ in $\mathcal{K}(A^{G})$ is equivalent to being effectively closed in the Vietoris topology induced by the Hausdorff metric. The same is true for properties in $\mathcal{S}(A^{G})$ .

There is a stronger notion that corresponds to $P$ being effectively closed in the upper Vietoris topology. This is equivalent to the existence of an algorithm that, for every $X\in\mathcal{K}(A^{G})$ , given an enumeration of $\mathcal{L}^{c}(X)$ , halts if and only if $X\notin P$ . The difference with Vietoris topology is that the algorithm is given as input only the enumeration of the complement. The same is true for properties in $\mathcal{S}(A^{G})$ .

The reason why we could assume this stronger notion in the proof of Theorem 11 is the following fact, see Proposition 4.4.4. in Amir’s PhD thesis [1].

Fact 37.

If $P$ is a $\Sigma_{2}^{0}$ property in the Vietoris topology, then there exists some property $P^{\prime}$ which is $\Sigma_{2}^{0}$ in the upper Vietoris topology and such that they induce the same minimal elements, i.e. $X$ is $P$ -minimal if and only if $X$ is $P^{\prime}$ -minimal.

Appendix B Another proof of Theorem 11

In [5], a characterization for sets with strong computable type was provided. Our Theorem 11 is motivated by this result and, in fact, can be derived from it (though not directly). Let us explain why.

Let $G$ be a finitely generated group with decidable word problem, and let $A$ be a finite alphabet.

First, let us state the result for sets, reformulated without using copies (see Theorem 4.7. in [5]).

Theorem 38 (A characterization of strong computable type for set).

Let $X$ be a compact subset in $A^{G}$ . The following are equivalent.

1.

For every oracle $O\subset\mathbb{N}$ , if $X$ is semi-computable relative to $O$ , then it is computable relative to $O$ .
2.

There exists a $\Sigma_{2}^{0}$ property $P$ in $\mathcal{K}(A^{G})$ such that $X$ is $P$ -minimal.

Fact 39.

Effectively closed $G$ -shifts are exactly the semi-computable compact shift-invariant subsets of $A^{G}$ . (This is not straightforward; it is a theorem, see [9], since for effectively closed shifts we enumerate words, and for semi-computable subsets, we enumerate balls.)

The results from a recent, yet unpublished work by Carrasco-Vargas, Herrera Nunez and Sablik, (see Proposition 6.12. in [14]) imply that $G$ -shifts with computable languages are exactly the computable shift-invariant compact subsets of $A^{G}$ . To show that, one needs to see (semi-)computable compact subsets of $A^{G}$ as computable points in the (upper) Vietoris topology, as explained in Section 2.3. in [1].

We claim that this result can be relativized in the following way.

Proposition 40.

Let $O\subset\mathbb{N}$ be an oracle, a $G$ -shift has computable language relative to $O$ iff it is a computable point relative to $O$ in the Vietoris topology.

Let us see how to obtain to Theorem 11 from Theorem 38.

Suppose we have $1 .$ in Theorem 11 for some shift $X$ . By Proposition 40 and the relative version of Fact 39, we have $1 .$ in Theorem 38, so we find a $\Sigma_{2}^{0}$ property $P$ in $\mathcal{K}(A^{G})$ such that $X$ is $P$ -minimal. Hence, $X$ is minimal for the property $P^{\prime}=P\cap\mathcal{S}(A^{G})$ which is $\Sigma_{2}^{0}$ in $\mathcal{S}(A^{G})$ ( $2 .$ in Theorem 11). For the other direction, remark that a $\Sigma_{2}^{0}$ property in $\mathcal{S}(A^{G})$ is also $\Sigma_{2}^{0}$ in $\mathcal{K}(A^{G})$ , as $\mathcal{S}(A^{G})$ is $\Pi_{1}^{0}$ in $\mathcal{K}(A^{G})$ by Fact 35.

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Minimality and Computability of Languages of G-Shifts

Abstract

Keywords and phrases:

Category:

Copyright and License:

2012 ACM Subject Classification:

Acknowledgements:

Funding:

DOI:

Event:

Editors:

Series and Publisher:

1 Introduction

Similar results in other areas

Definition.

Theorem ([5]).

Our approach

Definition.

Our results

Theorem (Theorem 11).

Proposition (Section 3.5).

Sections are organized as follows

2 Preliminaries

2.1 Basics on groups

Definition 1.

Definition 2.

Definition 3.

2.2 G-shifts

Definition 4 (G-shifts).

2.3 Effectiveness of G-shifts

Definition 5.

Definition 6 (Effectively closed shifts).

Fact 7 (Proposition 2.1 in [8]).

2.4 Descriptive complexity

Definition 8 (See Section 2 in [17]).

Definition 9.

3 Strong computable type for G-shifts

3.1 Definition

Definition 10 (Strong computable type for G-shifts).

3.2 Shifts and minimality

Theorem 11 (Main theorem).

▶ Remark 12.

Proof.

Corollary 13.

3.3 Relation with strong computable type for sets

3.4 Relation with maximal elements in quasivarieties

3.5 Applications

3.5.1 Minimal shifts

Theorem 14 ([18]).

Proposition 15.

Proof.

3.5.2 Entropy-minimal shifts

Definition 16.

Theorem 17 ([14]).

Proposition 18.

Proof.

▶ Remark 19.

Entropy-minimality alone is not sufficient

Definition 20.

Proposition 21.

Proof.

3.5.3 P-isolated shifts

Proposition 22.

Proof.

3.5.4 Periodic-minimal shifts

Definition 23.

Proposition 24.

Proof.

Theorem 25 ([30]).

3.5.5 Quasi-minimal shifts

Theorem 26 ([35], Theorem 9.).

Proposition 27.

Proof.

3.5.6 Infinite-minimal shifts

Proposition 28.

Proof.

3.5.7 Full extensions

Theorem 29 ([27]).

3.6 Invariance under transformations

3.6.1 Factors and computable morphisms

Definition 4 ( $G$ -shifts).

Definition 10 (Strong computable type for $G$ -shifts).

$\blacktriangleright$ Remark 12.

$\blacktriangleright$ Remark 19.

Appendix A Topology on $A^{G}$ and its hyperspace

A.1 Topology on $A^{G}$

A.2 The hyperspace of $A^{G}$

A.2.1 Descriptive complexity for properties in $\mathcal{K}(A^{G})$ and $\mathcal{S}(A^{G})$