We consider blind, deterministic, finite automata equipped with a register which stores an element of a given monoid, and which is modified by right multiplication by monoid elements. We show that, for monoids M drawn from a large class including groups, such an automaton accepts the word problem of a group H if and only if H has a finite index subgroup which embeds in the group of units of M. In the case that M is a group, this answers a question of Elston and Ostheimer.
For any group $G$ and set $A$, a cellular automaton over $G$ and $A$ is a transformation $τ: A^G \to A^G$ defined via a finite neighborhood $S \subseteq G$ (called a memory set of $τ$) and a local function $μ: A^S \to A$. In this paper, we assume that $G$ and $A$ are both finite and study various algebraic properties of the finite monoid $\text{CA}(G,A)$ consisting of all cellular automata over $G$ and $A$. Let $\text{ICA}(G;A)$ be the group of invertible cellular automata over $G$ and $A$. In the first part, using information on the conjugacy classes of subgroups of $G$, we give a detailed de…
During the last decades, classical models in language theory have been extended by control mechanisms defined by monoids. We study which monoids cause the extensions of context-free grammars, finite automata, or finite state transducers to exceed the capacity of the original model. Furthermore, we investigate when, in the extended automata model, the nondeterministic variant differs from the deterministic one in capacity. We show that all these conditions are in fact equivalent and present an algebraic characterization. In particular, the open question of whether every language generated by a …
Valence automata are a generalization of various models of automata with storage. Here, each edge carries, in addition to an input word, an element of a monoid. A computation is considered valid if multiplying the monoid elements on the visited edges yields the identity element. By choosing suitable monoids, a variety of automata models can be obtained as special valence automata. This work is concerned with the accepting power of valence automata. Specifically, we ask for which monoids valence automata can accept only context-free languages or only languages with semilinear Parikh image, resp…
For an arbitrary group $G$ and arbitrary set $A$, we define a monoid structure on the set of all uniformly continuous functions $A^G\to A$ and then we show that it is naturally isomorphic to the monoid of cellular automata $\mathrm{CA}(G, A)$. This gives a new equivalent definition of a cellular automaton over the group $G$ with alphabet set $A$. We use this new interpretation to give a simple proof of the theorem of Curtis-Hedlund.
This work studies which storage mechanisms in automata permit decidability of the emptiness problem. The question is formalized using valence automata, an abstract model of automata in which the storage mechanism is given by a monoid. For each of a variety of storage mechanisms, one can choose a (typically infinite) monoid $M$ such that valence automata over $M$ are equivalent to (one-way) automata with this type of storage. In fact, many important storage mechanisms can be realized by monoids defined by finite graphs, called graph monoids. Examples include pushdown stacks, partially blind cou…
We prove that if $M$ is a monoid and $A$ a finite set with more than one element, then the residual finiteness of $M$ is equivalent to that of the monoid consisting of all cellular automata over $M$ with alphabet $A$.
For a group $G$ and a set $A$, let $\text{End}(A^G)$ be the monoid of all cellular automata over $A^G$, and let $\text{Aut}(A^G)$ be its group of units. By establishing a characterisation of surjunctuve groups in terms of the monoid $\text{End}(A^G)$, we prove that the rank of $\text{End}(A^G)$ (i.e. the smallest cardinality of a generating set) is equal to the rank of $\text{Aut}(A^G)$ plus the relative rank of $\text{Aut}(A^G)$ in $\text{End}(A^G)$, and that the latter is infinite when $G$ has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for…
We prove that the class of finitely presented inverse monoids whose Schützenberger graphs are quasi-isometric to trees has a uniformly solvable word problem, furthermore, the languages of their Schützenberger automata are context-free. On the other hand, we show that there is a finitely presented inverse monoid with hyperbolic Schützenberger graphs and an unsolvable word problem.
For a group $G$ and a finite set $A$, denote by $\text{CA}(G;A)$ the monoid of all cellular automata over $A^G$ and by $\text{ICA}(G;A)$ its group of units. We study the minimal cardinality of a generating set, known as the rank, of $\text{ICA}(G;A)$. In the first part, when $G$ is a finite group, we give upper bounds for the rank in terms of the number of conjugacy classes of subgroups of $G$. The case when $G$ is a finite cyclic group has been studied before, so here we focus on the cases when $G$ is a finite dihedral group or a finite Dedekind group. In the second part, we find a basic lowe…
We use the notion of a partial action of a monoid to introduce a generalization of automata, which we call "a preautomaton". We study properties of preautomata and of languages recognized by preautomata.
Motivated by the Babai conjecture and the Cerny conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with $n$ states in this class, we prove that the reset thresholds are upper-bounded by $2n^2-6n+5$ and can attain the value $\tfrac{n(n-1)}{2}$. In addition, we study diameters of the pair digraphs of permutation automata and construct $n$-state permutation automata with diameter $\tfrac{n^2}{4} + o(n^2)$.
In this paper we investigate the special automata over finite rank free groups and estimate asymptotic characteristics of sets they accept. We show how one can decompose an arbitrary regular subset of a finite rank free group into disjoint union of sets accepted by special automata or special monoids. These automata allow us to compute explicitly generating functions, $λ-$measures and Cesaro measure of thick monoids. Also we improve the asymptotic classification of regular subsets in free groups.
We construct group codes over two letters (i.e., bases of subgroups of a two-generated free group) with special properties. Such group codes can be used for reducing algorithmic problems over large alphabets to algorithmic problems over a two-letter alphabet. Our group codes preserve aperiodicity of inverse finite automata. As an application we show that the following problems are PSpace-complete for two-letter alphabets (this was previously known for large enough finite alphabets): The intersection-emptiness problem for inverse finite automata, the aperiodicity problem for inverse finite auto…
A monoid $M$ is called surjunctive if every injective cellular automata with finite alphabet over $M$ is surjective. We show that all finite monoids, all finitely generated commutative monoids, all cancellative commutative monoids, all residually finite monoids, all finitely generated linear monoids, and all cancellative one-sided amenable monoids are surjunctive. We also prove that every limit of marked surjunctive monoids is itself surjunctive. On the other hand, we show that the bicyclic monoid and, more generally, all monoids containing a submonoid isomorphic to the bicyclic monoid are non…
The index of a subgroup of a group counts the number of cosets of that subgroup. A subgroup of finite index often shares structural properties with the group, and the existence of a subgroup of finite index with some particular property can therefore imply useful structural information for the overgroup. A developed theory of cosets in inverse semigroups exists, originally due to Schein: it is defined only for closed inverse subsemigroups, and the structural correspondences between an inverse semigroup and a closed inverse subsemigroup of finite index are weaker than in the group case. Neverth…
A general result of Epstein and Thurston implies that all link groups are automatic, but the proof provides no explicit automaton. Here we show that the groups of all torus links are groups of fractions of so-called Garside monoids, i.e., roughly speaking, monoids with a good theory of divisibility, which allows us to reprove that those groups are automatic, but, in addition, gives a completely explicit description of the involved automata, thus partially answering a question of D.F.Holt.
We define compact automata and show that every language has a unique minimal compact automaton. We also define recognition of languages by compact left semitopological monoids and construct the analogue of the syntactic monoid in this context. For rational languages this reduces to the usual theory of finite automata and finite monoids.
The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been many attempts to find notions of conjugacy in semigroups that would be useful in special classes of semigroups occurring in various areas of mathematics, such as semigroups of matrices, operator and topological semigroups, free semigroups, transition monoids for automata, semigroups given by presentations with prescribed properties, monoids of graph endomorphisms, etc. In this paper we study four notions of conjugacy for semigroups, their interconnec…
We use language theory to study the rational subset problem for groups and monoids. We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups. In particular, it passes through free products amalgamated over finite subgroups and HNN extensions with finite associated subgroups. We provide a simple proof of a result of Grunschlag showing that the decidability of this problem is a virtual property. We prove further that the problem is decidable for a direct product of a group G with a monoid M if and only if membership is uniformly decid…
We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to …
In the 1980's Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups. In this paper we attempt to apply the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for ``sew…
We show that the automorphism group of a one-dimensional full shift (the group of reversible cellular automata) does not satisfy the Tits alternative. That is, we construct a finitely-generated subgroup which is not virtually solvable yet does not contain a free group on two generators. We give constructions both in the two-sided case (spatially acting group $\Z$) and the one-sided case (spatially acting monoid $\N$, alphabet size at least eight). Lack of Tits alternative follows for several groups of symbolic (dynamical) origin: automorphism groups of two-sided one-dimensional uncountable sof…
We present an exposition of the theory of finite automata augmented with a multiply-only register storing an element of a given monoid or group. Included are a number of new results of a foundational nature. We illustrate our techniques with a group-theoretic interpretation and proof of a key theorem of Chomsky and Schutzenberger from formal language theory.
Generation by lazy and invertible cellular automata over the Klein four-group
Universidad de Guadalajara, Centro Universitario de Ciencias Exactas e Ingenierías（瓜达拉哈拉大学，精确科学与工程学院）；ChronoAI Pte Ltd（Chronai私人有限公司）；The Omega Institute（欧米茄研究所）；National University of Singapore (NUS)（新加坡国立大学）
We study automaton structures, i.e. groups, monoids and semigroups generated by an automaton, which, in this context, means a deterministic finite-state letter-to-letter transducer. Instead of considering only complete automata, we specifically investigate semigroups generated by partial automata. First, we show that the class of semigroups generated by partial automata coincides with the class of semigroups generated by complete automata if and only if the latter class is closed under removing a previously adjoined zero, which is an open problem in (complete) automaton semigroup theory stated…
Let $G$ be a countable monoid and let $A$ be an Artinian group (resp. an Artinian module). Let $Σ\subset A^G$ be a closed subshift which is also a subgroup (resp. a submodule) of $A^G$. Suppose that $Γ$ is a finitely generated monoid consisting of pairwise commuting cellular automata $Σ\to Σ$ that are also homomorphisms of groups (resp. homomorphisms of modules) with monoid binary operation given by composition of maps. We show that the valuation action of $Γ$ on $Σ$ satisfies a natural intrinsic shadowing property. Generalizations are also established for families of endomorphisms of admissib…
For any group $G$ and any set $A$, a cellular automaton (CA) is a transformation of the configuration space $A^G$ defined via a finite memory set and a local function. Let $\text{CA}(G;A)$ be the monoid of all CA over $A^G$. In this paper, we investigate a generalisation of the inverse of a CA from the semigroup-theoretic perspective. An element $τ\in \text{CA}(G;A)$ is von Neumann regular (or simply regular) if there exists $σ\in \text{CA}(G;A)$ such that $τ\circ σ\circ τ= τ$ and $σ\circ τ\circ σ= σ$, where $\circ$ is the composition of functions. Such an element $σ$ is called a generalised i…
For a finite group $G$ and a finite set $A$, we study various algebraic aspects of cellular automata over the configuration space $A^G$. In this situation, the set $\text{CA}(G;A)$ of all cellular automata over $A^G$ is a finite monoid whose basic algebraic properties had remained unknown. First, we investigate the structure of the group of units $\text{ICA}(G;A)$ of $\text{CA}(G;A)$. We obtain a decomposition of $\text{ICA}(G;A)$ into a direct product of wreath products of groups that depends on the numbers $α_{[H]}$ of periodic configurations for conjugacy classes $[H]$ of subgroups of $G$. …
We consider Turing machines as actions over configurations in $Σ^{\mathbb{Z}^d}$ which only change them locally around a marked position that can move and carry a particular state. In this setting we study the monoid of Turing machines and the group of reversible Turing machines. We also study two natural subgroups, namely the group of finite-state automata, which generalizes the topological full groups studied in the theory of orbit-equivalence, and the group of oblivious Turing machines whose movement is independent of tape contents, which generalizes lamplighter groups and has connections t…
In this paper we show how string rewriting methods can be applied to give a new method of computing double cosets. Previous methods for double cosets were enumerative and thus restricted to finite examples. Our rewriting methods do not suffer this restriction and we present some examples of infinite double coset systems which can now easily be solved using our approach. Even when both enumerative and rewriting techniques are present, our rewriting methods will be competitive because they i) do not require the preliminary calculation of cosets; and ii) as with single coset problems, there are m…
