arXiv subjects

Ville Salo

Publications and source records attributed to Ville Salo.

At least 19 recordsLinked to original sources

arXiv2026

Thinking outside the box is useless NFA = FNFA

We study picture-walking automata, namely finite-state automata that accept higher-dimensional parallelotope-/tensor-shaped pictures and are allowed to move in all directions depending on their current state and the currently read symbol. It is a long-standing open problem whether the nondeterministic such automata become stronger if automata are allowed to exit the picture. In this paper, we resolve the problem in full generality: NFA = FNFA, for pictures of any dimension.

cs.FL↗

arXiv2026

On surjunctive and injunctive subshifts of finite type

A dynamical system is said to be surjunctive if every injective endomorphism of the system is surjective and it is said to be injunctive if every surjective endomorphism is injective. An endomorphism of a dynamical system is called pre-injective if its restriction to every homoclinicity class of the phase space is injective. One says that a dynamical system has the Moore property if every surjective endomorphism of the system is pre-injective and that it has the Myhill property if every pre-injective endomorphism is surjective. We give characterisations of surjunctivity and injunctivity for $\Z$-subshifts of finite type in terms of their irreducible components and their Cantor-Bendixson decomposition. We also prove that a $\Z$-subshift of finite type is surjunctive if and only if it has the Moore property and that every injunctive $\Z$-subshift of finite type is surjunctive. This implies in particular that a $\Z$-subshift of finite type has the Moore property whenever it has the Myhill property.

math.DS↗

arXiv2026

Wreath products in the automorphism group of a full shift

We prove that if a subgroup $H$ of the automorphism group $\Aut(Σ^\Z)$ of a non-trivial full shift acts on points of finite support (= points bi-asymptotic to a fixed point) with a free orbit, then for every finitely-generated abelian group $A$, the abstract group $A \wr H$ also embeds in $\Aut(Σ^\Z)$. The groups admitting an action with such a free orbit include $A \wr \Z$ for $A$ a finite abelian group, and finitely-generated free groups. The class of such groups is also closed under commensurability and direct products. We obtain for example that $\Z \wr \Z$, $\Z_2 \wr (\Z_2 \wr \Z)$ and $\Z \wr (\Z_2 \wr \Z)$ embed in $\Aut(Σ^\Z)$. The group $\Z \wr \Z$ is the first example of a finitely-generated torsion-free subgroup of $\Aut(Σ^\Z)$ with infinite cohomological dimension, answering an implicit question of Kim and Roush and an explicit question of the author. We also explore a simpler variant of the construction that gives embeddings of certain Neumann groups, as well as some near-misses to higher iterated wreath products.

math.GR↗

arXiv2026

Self-simulability of graph products

A group is self-simulable if all its computable actions admit SFT covers, which means roughly that they can be implemented with finitely many tiling constraints. We prove that a graph product of infinite finitely-generated groups is self-simulable if and only if its defining graph has no disconnecting clique consisting of amenable groups. In particular, a right-angled Artin group (a.k.a.\ a graph group) is self-simulable if and only if the defining graph has no disconnecting clique. As an application, we obtain that a graph product of infinite finitely-generated groups splits (algebraically, or in a certain geometric sense) over an amenable subgroup if and only if the graph has a disconnecting clique consisting of amenable groups.

math.GR↗

arXiv2026

Two block gluing constructions

We prove two existence results about block gluing in two-dimensional SFTs. First, a large class of functions between exponential and logarithmic can be realized as block gluing functions of two-dimensional SFTs. Second, there exists an aperiodic linearly block gluing two-dimensional SFT with entropy dimension 1. These solve two open questions of Gangloff and Sablik.

math.DS↗

arXiv2026

A Three-Dimensional SFT with Sparse Columns

We construct a nontrivial three-dimensional subshift of finite type whose projective $\Z$-subdynamics, or $\Z$-trace, is 2-sparse, meaning that there are at most two nonzero symbols in any vertical column. The subshift is deterministic in the direction of the subdynamics, so it is topologically conjugate to the set of spacetime diagrams of a partial cellular automaton. We also present a variant of the subshift that is defined by Wang cubes, and one whose alphabet is binary.

math.DS↗

arXiv2026

Contractible subshifts

We introduce the notion of a contractible subshift. This is a strengthening of the notion of strong irreducibility, where we require that the gluings are given by a block map. We show that a subshift is a retract of a full shift if and only if it is a contractible SFT with a fixed point. For many groups, including virtually polycyclic groups, metabelian Baumslag-Solitar groups and the lamplighter group, contractibility implies dense periodic points. We introduce a ``homotopy theory'' framework for working with this notion, and ``contractibility'' is in fact simply an analog of the usual contractibility in algebraic topology. We also explore the symbolic dynamical analogs of homotopy equivalence and strong contractibility of subshifts (corresponding to the notion of equiconnectedness in topology). Contractibility is implied by the map extension property of Meyerovitch, and among SFTs, it implies the finite extension property of Briceño, McGoff and Pavlov. We include thorough comparisons with these classes. We also encounter some new geometric notions, in particular a periodic variant of Gromov's asymptotic dimension of a group.

math.DS↗

arXiv2026

Topological entropy of Turing complete dynamics

We explore the relationship between Turing completeness and topological entropy of dynamical systems. We first prove that a natural class of Turing machines that we call "branching Turing machines" (which includes most of the known examples of universal Turing machines) has positive topological entropy. Motivated by the recent construction of Turing complete Euler flows, we deduce that any Turing complete dynamics with a continuous encoding that simulates a universal branching machine is chaotic. On the other hand, we show that, unexpectedly, universal Turing machines with zero topological entropy (and even zero speed) can be constructed, unveiling the independence of chaos and universality at the symbolic level.

math.DS↗

arXiv2026

Periodic points and residual finiteness of automorphism groups of subshifts

If totally periodic points are dense in a subshift $X$, its automorphism group is residually finite. We show a weak converse: if periodic points are not dense in a subshift $X$, then the automorphism group of $X \times Y$ is not residually finite for full shifts $Y$ (and sufficiently full-shift-like subshifts). On the other hand, we show that the automorphism group of a block gluing $\Z^2$-subshift is always locally embeddable in finite groups (thus sofic). Hochman recently constructed a strongly irreducible $\Z^2$-subshift with no periodic points. Combining our result with this example gives a strongly irreducible $\Z^2$-subshift whose automorphism group is not residually finite, which solves a question of Coornaert and Ceccherini-Silberstein.

math.DS↗

arXiv2025

A geometric obstruction to self-simulation for groups

We introduce a new quasi-isometry invariant for finitely generated groups and show that every group with this property admits a subshift which is effectively closed by patterns and that cannot be realized as the topological factor of any subshift of finite type. We provide several examples of groups with the property, such as amenable groups, multi-ended groups, generalized Baumslag-Solitar groups, fundamental groups of surfaces, and cocompact Fuchsian groups.

math.GR↗

arXiv2025

On the growth of actions of free products

If $G$ is a finitely generated group and $X$ a $G$-set, the growth of the action of $G$ on $X$ is the function that measures the largest cardinality of a ball of radius $n$ in the Schreier graph $Γ(G,X)$. In this note we consider the following stability problem: if $G,H$ are finitely generated groups admitting a faithful action of growth bounded above by a function $f$, does the free product $G \ast H$ also admit a faithful action of growth bounded above by $f$? We show that the answer is positive under additional assumptions, and negative in general. In the negative direction, our counter-examples are obtained with $G$ either the commutator subgroup of the topological full group of a minimal and expansive homeomorphism of the Cantor space; or $G$ a Houghton group. In both cases, the group $G$ admits a faithful action of linear growth, and we show that $G\ast H$ admits no faithful action of subquadratic growth provided $H$ is non-trivial. In the positive direction, we describe a class of groups that admit actions of linear growth and is closed under free products and exhibit examples within this class, among which the Grigorchuk group.

math.GR↗

arXiv2025

Minimal sofic shift on a group that is not finitely-generated

We prove that there exists a group which is not finitely generated, but admits a minimal sofic shift. This answers a question of Doucha, Melleray and Tsankov. The group is of the form $(F_4 \times F_2) \rtimes F_{\infty}$. The construction itself is based on simulation theory and properties of Thompson's~$V$.

math.DS↗

arXiv2025

Word problems and embedding-obstructions in cellular automata groups on groups

We study groups of reversible cellular automata, or CA groups, on groups. More generally, we consider automorphism groups of subshifts of finite type on groups. It is known that word problems of CA groups on virtually nilpotent groups are in co-NP, and can be co-NP-hard. We show that under the Gap Conjecture of Grigorchuk, their word problems are PSPACE-hard on all other groups. On free and surface groups, we show that they are indeed always in PSPACE. On a group with co-NEXPTIME word problem, CA groups themselves have co-NEXPTIME word problem, and on the lamplighter group (which itself has polynomial-time word problem) we show they can be co-NEXPTIME-hard. We show also nonembeddability results: the group of cellular automata on a non-cyclic free group does not embed in the group of cellular automata on the integers (this solves a question of Barbieri, Carrasco-Vargas and Rivera-Burgos); and the group of cellular automata in dimension $D$ does not embed in a group of cellular automata in dimension $d$ if $D > d$ (this solves a question of Hochman).

math.GR↗

arXiv2025

Graph and wreath products of cellular automata

We prove that the set of subgroups of the automorphism group of a two-sided full shift is closed under countable graph products. We introduce the notion of a group action without $A$-cancellation (for an abelian group $A$), and show that when $A$ is a finite abelian group and $G$ is a group of cellular automata whose action does not have $A$-cancellation, the wreath product $A \wr G$ embeds in the automorphism group of a full shift. We show that all free abelian groups and free groups admit such cellular automata actions. In the one-sided case, we prove variants of these results with reasonable alphabet blow-ups.

math.GR↗

arXiv2025

Descriptive Complexity of Sensitivity of Cellular Automata

We study the computational complexity of determining whether a cellular automaton is sensitive to initial conditions. We show that this problem is $Π^0_2$-complete in dimension 1 and $Σ^0_3$-complete in dimension 2 and higher. This solves a question posed by Sablik and Theyssier.

math.DS↗

arXiv2025

Avoshifts, Unishifts and Nondeterministic Cellular Automata

In this paper, we study avoshifts and unishifts on $\mathbb{Z}^d$. Avoshifts are subshifts where for each convex set $C$, and each vector $v$ such that $C \cup \{\vec v\}$ is also convex, the set of valid extensions of globally valid patterns on $C$ to ones on $C \cup \{v\}$ is determined by a bounded subpattern of $C$. Unishifts are the subshifts where for such $C, \vec v$, every $C$-pattern has the same number of $\vec v$-extensions. Cellwise quasigroup shifts (including group shifts) and TEP subshifts are examples of unishifts, while unishifts and subshifts with topological strong spatial mixing are examples of avoshifts. We prove that every avoshift is the spacetime subshift of a nondeterministic cellular automaton on an avoshift of lower dimension up to a linear transformation and a convex blocking. From this, we deduce that all avoshifts contain periodic points, and that unishifts have dense periodic points and admit equal entropy full shift factors.

math.DS↗

arXiv2025

Structure and computability of preimages in the Game of Life

Conway's Game of Life is a two-dimensional cellular automaton. As a dynamical system, it is well-known to be computationally universal, i.e.\ capable of simulating an arbitrary Turing machine. We show that in a sense taking a single backwards step of the Game of Life is a computationally universal process, by constructing patterns whose preimage computation encodes an arbitrary circuit-satisfaction problem, or, equivalently, any tiling problem. As a corollary, we obtain for example that the set of orphans is coNP-complete, exhibit a $6210 \times 37800$-periodic configuration whose preimage is nonempty but contains no periodic configurations, and prove that the existence of a preimage for a periodic point is undecidable. Our constructions were obtained by a combination of computer searches and manual design.

cs.FL↗

arXiv2025

Symbol Frequencies in Surjective Cellular Automata

We study the behavior of probability measures under iteration of a surjective cellular automaton. We solve the following question in the negative: if the initial measure is ergodic and has full support, do all weak-* limit points of the sequence of measures have full support as well? The initial measure of our solution is not a product measure, and in this case the question remains open. To this end, we present a tool for studying the frequencies of symbols in preimages of surjective cellular automata, and prove some basic results about it. However, we show that by itself it is not enough to solve the stricter question in the positive.

math.DS↗