The emerging field of Nominal Computation Theory is concerned with the theory of Nominal Sets and its applications to Computer Science. We investigate here the impact of nominal sets on the definition of Cellular Automata and on their computational capabilities, with a special focus on the emergent behavioural properties of this new model and their significance in the context of computation-oriented interpretations of physical phenomena. A preliminary investigation of the relations between Nominal Cellular Automata and Wolfram's Elementary Cellular Automata is also carried out.
In this paper, we look at two ways to implement determinisitic one dimensional cellular automata into hyperbolic cellular automata in three contexts: the pentagrid, the heptagrid and the dodecagrid, these tilings being classically denoted by $\{5,4\}$, $\{7,3\}$ and $\{5,3,4\}$ respectively.
We study languages over infinite alphabets equipped with some structure that can be tested by recognizing automata. We develop a framework for studying such alphabets and the ensuing automata theory, where the key role is played by an automorphism group of the alphabet. In the process, we generalize nominal sets due to Gabbay and Pitts.
Nominal sets provide a foundation for reasoning about names. They are used primarily in syntax with binders, but also, e.g., to model automata over infinite alphabets. In this paper, nominal sets are related to nominal renaming sets, which involve arbitrary substitutions rather than permutations, through a categorical adjunction. In particular, the left adjoint relates the separated product of nominal sets to the Cartesian product of nominal renaming sets. Based on these results, we define the new notion of separated nominal automata. These automata can be exponentially smaller than classical …
We present an Angluin-style algorithm to learn nominal automata, which are acceptors of languages over infinite (structured) alphabets. The abstract approach we take allows us to seamlessly extend known variations of the algorithm to this new setting. In particular we can learn a subclass of nominal non-deterministic automata. An implementation using a recently developed Haskell library for nominal computation is provided for preliminary experiments.
Linear acceleration theorems are known for most computational models. Although such results have been proved for two-dimensional cellular automata working on specific neighborhoods, no general construction was known. We present here a technique of linear acceleration for all two-dimensional languages recognized by cellular automata working on complete neighborhoods.
Automata models for data languages (i.e. languages over infinite alphabets) often feature either global or local freshness operators. We show that Bollig et al.'s session automata, which focus on global freshness, are equivalent to regular nondeterministic nominal automata (RNNA), a natural nominal automaton model with explicit name binding that has appeared implicitly in the semantics of nominal Kleene algebra (NKA), an extension of Kleene algebra with name binding. The expected Kleene theorem for NKA is known to fail in one direction, i.e. there are nominal languages that can be accepted by …
A cellular automaton with $n$ states may be used for construction of reversible second-order cellular automaton with $n^2$ states. Reversible cellular automata with hidden parameters discussed in this paper are generalization of such construction and may have number of states $N=n m$ with arbitrary $m$. Further modification produces reversible cellular automata with reduced number of states $N' < N = n m$.
Complexity has been a recurrent research topic in cellular automata because they represent systems where complex behaviors emerge from simple local interactions. A significant amount of previous research has been conducted proposing instances of complex cellular automata; however, most of the proposed methods are based on a careful search or a meticulous construction of evolution rules. This paper presents the emergence of complex behaviors based on reversible cellular automata. In particular, this paper shows that reversible cellular automata represent an adequate framework to obtain complex …
Logics and automata models for languages over infinite alphabets, such as Freeze LTL and register automata, serve the verification of processes or documents with data. They relate tightly to formalisms over nominal sets, such as nondetermininistic orbit-finite automata (NOFAs), where names play the role of data. Reasoning problems in such formalisms tend to be computationally hard. Name-binding nominal automata models such as regular nondeterministic nominal automata (RNNAs) have been shown to be computationally more tractable. In the present paper, we introduce a linear-time fixpoint logic Ba…
University of Leicester（莱斯特大学）；Gran Sasso Science Institute（格兰萨索科学研究所）
We investigate a learning algorithm in the context of nominal automata, an extension of classical automata to alphabets featuring names. This class of automata captures nominal regular languages; analogously to the classical language theory, nominal automata have been shown to characterise nominal regular expressions with binders. These formalisms are amenable to abstract modelling resource-aware computations. We propose a learning algorithm on nominal regular languages with binders. Our algorithm generalises Angluin's L* algorithm with respect to nominal regular languages with binders. We sho…
We define a class of languages of infinite words over infinite alphabets, and the corresponding automata. The automata used for recognition are a generalisation of deterministic Muller automata to the setting of nominal sets. Remarkably, the obtained languages are determined by their ultimately periodic fragments, as in the classical case. Closure under complement, union and intersection, and decidability of emptiness and equivalence are preserved by the generalisation. This is shown by using finite representations of the (otherwise infinite-state) defined class of automata.
We Propose A Novel Automaton Model which uses Arithmetic Operations as the Evolving Rules, each cell has the states of the Natural Numbers k = (N), a radius of r = 1/2 and operates on an arbitrary input size. The Automaton reads an Arithmetic Expression as an input and outputs another Arithmetic Expression. In Addition, we simulate a variety of One Dimensional Cellular Automata Structures with different Dynamics including Elementary Cellular Automata.
In this paper we present two interesting properties of stochastic cellular automata that can be helpful in analyzing the dynamical behavior of such automata. The first property allows for calculating cell-wise probability distributions over the state set of a stochastic cellular automaton, i.e. images that show the average state of each cell during the evolution of the stochastic cellular automaton. The second property shows that stochastic cellular automata are equivalent to so-called stochastic mixtures of deterministic cellular automata. Based on this property, any stochastic cellular autom…
Cellular automata are discrete dynamical systems and a model of computation. The limit set of a cellular automaton consists of the configurations having an infinite sequence of preimages. It is well known that these always contain a computable point and that any non-trivial property on them is undecidable. We go one step further in this article by giving a full characterization of the sets of Turing degrees of cellular automata: they are the same as the sets of Turing degrees of effectively closed sets containing a computable point.
In this paper, we investigate the halting problem for deterministic cellula automata in the pentagrid. We prove that the problem is decidable when the cellular automaton starts its computation from a finite configuration and when it has at most two states, one of them being a quiescent state.
In this dissertation, we study temporally stochasticity in cellular automata and the behavior of such cellular automata. The work also explores the computational ability of such cellular automaton that illustrates the computability of solving the affinity classification problem. In addition to that, a cellular automaton, defined over Cayley tree, is shown as the classical searching problem solver. The proposed temporally stochastic cellular automata deals with two elementary cellular automata rules, say $f$ and $g$. The $f$ is the default rule, however, $g$ is temporally applied to the overall…
This work introduces a new problem, named as, affinity classification problem which is a generalization of the density classification problem. To solve this problem, we introduce temporally stochastic cellular automata where two rules are stochastically applied in each step on all cells of the automata. Our model is defined on 2-dimensional grid having affection capability. We show that this model can be used in several applications like modeling self-healing systems.
We study two-dimensional rotation-symmetric number-conserving cellular automata working on the von Neumann neighborhood (RNCA). It is known that such automata with 4 states or less are trivial, so we investigate the possible rules with 5 states. We give a full characterization of these automata and show that they cannot be strongly Turing universal. However, we give example of constructions that allow to embed some boolean circuit elements in a 5-states RNCA.
We propose the characterization of binary cellular automata using a set of behavioral metrics that are applied to the minimal Boolean form of a cellular automaton's transition function. These behavioral metrics are formulated to satisfy heuristic criteria derived from elementary cellular automata. Behaviors characterized through these metrics are growth, decrease, chaoticity, and stability. From these metrics, two measures of global behavior are calculated: 1) a static measure that considers all possible input patterns and counts the occurrence of the proposed metrics in the truth table of the…
Gauge-invariance is a fundamental concept in Physics -- known to provide mathematical justification for the fundamental forces. In this paper, we provide discrete counterparts to the main gauge theoretical concepts directly in terms of Cellular Automata. More precisely, the notions of gauge-invariance and gauge-equivalence in Cellular Automata are formalized. A step-by-step gauging procedure to enforce this symmetry upon a given Cellular Automaton is developed, and three examples of gauge-invariant Cellular Automata are examined.
In this paper, we present the simulation of a simple, yet significantly powerful, sequential model by cellular automata. The simulated model is called oblivious multi-head one-way finite automata and is characterized by having its heads moving only forward, on a trajectory that only depends on the length of the input. While the original finite automaton works in linear time, its corresponding cellular automaton performs the same task in real time, that is, exactly the length of the input. Although not truly a speed-up, the simulation may be interesting and reminds us of the open question about…
Nominal set plays a central role in a group-theoretic extension of finite automata to those over an infinite set of data values. Moerman et al. proposed an active learning algorithm for nominal word automata with the equality symmetry. In this paper, we introduce deterministic bottom-up nominal tree automata (DBNTA), which operate on trees whose nodes are labelled with elements of an orbit finite nominal set. We then prove a Myhill-Nerode theorem for the class of languages recognized by DBNTA and propose an active learning algorithm for DBNTA. The algorithm can deal with any data symmetry that…
The present work proposes and discusses the category of supported sets which provides a uniform foundation for nominal sets of various kinds, such as those for equality symmetry, for the order symmetry, and renaming sets. We show that all these differently flavoured categories of nominal sets are monadic over supported sets. Thus, supported sets provide a canonical finite way to represent nominal sets and the automata therein, e.g. register automata. Name binding in supported sets is modelled by a functor following the idea of de Bruijn indices. This functor lifts to the well-known abstraction…
While for synchronous deterministic cellular automata there is an accepted definition of reversibility, the situation is less clear for asynchronous cellular automata. We first discuss a few possibilities and then investigate what we call phase space invertible asynchronous cellular automata in more detail. We will show that for each Turing machine there is such a cellular automaton simulating it, and that it is decidable whether an asynchronous cellular automaton has this property or not, even in higher dimensions.
We show that a large number of elementary cellular automata are computationally simple. This work is the first systematic classification of elementary cellular automata based on a formal notion of computational complexity. Thanks to the generality of communication complexity, the perspectives of our method include its application to other natural systems such as neural networks and gene regulatory networks.
The dynamical behavior of non-uniform cellular automata is compared with the one of classical cellular automata. Several differences and similarities are pointed out by a series of examples. Decidability of basic properties like surjectivity and injectivity is also established. The final part studies a strong form of equicontinuity property specially suited for non-uniform cellular automata.
This paper is the second part of a series of two papers dealing with bulking: a way to define quasi-order on cellular automata by comparing space-time diagrams up to rescaling. In the present paper, we introduce three notions of simulation between cellular automata and study the quasi-order structures induced by these simulation relations on the whole set of cellular automata. Various aspects of these quasi-orders are considered (induced equivalence relations, maximum elements, induced orders, etc) providing several formal tools allowing to classify cellular automata.
We study the set of strictly periodic points in surjective cellular automata, i.e., the set of those configurations which are temporally periodic for a given automaton but they not spatially periodic. This set turns out to be dense for almost equicontinuous surjective cellular automata while it is empty for the positively expansive ones. In the class of additive cellular automata, the set of strictly periodic points can be either dense or empty. The latter happens if and only if the cellular automaton is topologically transitive.
We prove that there do not exist positively expansive cellular automata defined on the full k-ary tree shift (for k>=2). Moreover, we investigate some topological properties of these automata and their relationships, namely permutivity, surjectivity, preinjectivity, right-closingness and openness.
Infinite words over infinite alphabets serve as models of the temporal development of the allocation and (re-)use of resources over linear time. We approach omega-languages over infinite alphabets in the setting of nominal sets, and study languages of infinite bar strings, i.e. infinite sequences of names that feature binding of fresh names; binding corresponds roughly to reading letters from input words in automata models with registers. We introduce regular nominal nondeterministic Büchi automata (Büchi RNNAs), an automata model for languages of infinite bar strings, repurposing the previous…
In this paper we give three examples of expending patterns defined by hyperbolic cellular automata whose growth seems to be very similar to the growth of colonies of bacteria.
We consider the typical asymptotic behaviour of cellular automata of higher dimension (greater than 2). That is, we take an initial configuration at random according to a Bernoulli (i.i.d) probability measure, iterate some cellular automaton, and consider the (set of) limit probability measure(s) as time tends to infinity. In this paper, we prove that limit measures that can be reached by higher-dimensional cellular automata are completely characterised by computability conditions, as in the one-dimensional case. This implies that cellular automata have the same variety and complexity of typic…
The paper proposes a simple formalism for dealing with deterministic, non-deterministic and stochastic cellular automata in a unifying and composable manner. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the non-deterministic and stochastic settings. We then provide explicit tools to prove or disprove the existence of such a simulation between two stochastic cellular automata, even though the intrinsic simulation relation is shown to be undecidable in dimension two and higher. The key result behind this is the caracterizatio…
We study cellular automata on regular rooted trees. This includes the characterization of sofic tree shifts in terms of unrestricted Rabin automata and the decidability of the surjectivity problem for cellular automata between sofic tree shifts.
We introduce formal languages over infinite alphabets where words may contain binders. We define the notions of nominal language, nominal monoid, and nominal regular expressions. Moreover, we extend history-dependent automata (HD-automata) by adding stack, and study the recognisability of nominal languages.
This talk advocates intrinsic universality as a notion to identify simple cellular automata with complex computational behavior. After an historical introduction and proper definitions of intrinsic universality, which is discussed with respect to Turing and circuit universality, we discuss construction methods for small intrinsically universal cellular automata before discussing techniques for proving non universality.
We study the problem of sequentializing a cellular automaton without introducing any intermediate states, and only performing reversible permutations on the tape. We give a decidable characterization of cellular automata which can be written as a single left-to-right sweep of a bijective rule from left to right over an infinite tape.
A cellular automaton is a parallel synchronous computing model, which consists in a juxtaposition of finite automata whose state evolves according to that of their neighbors. It induces a dynamical system on the set of configurations, i.e. the infinite sequences of cell states. The limit set of the cellular automaton is the set of configurations which can be reached arbitrarily late in the evolution. In this paper, we prove that all properties of limit sets of cellular automata with binary-state cells are undecidable, except surjectivity. This is a refinement of the classical "Rice Theorem" th…
We study cellular automata with respect to a new communication complexity problem: each of two players know half of some finite word, and must be able to tell whether the state of the central cell will follow a given evolution, by communicating as little as possible between each other. We present some links with classical dynamical concepts, especially equicontinuity, expansiveness, entropy and give the asymptotic communication complexity of most elementary cellular automata.
The goal of this paper is to show why the framework of communication complexity seems suitable for the study of cellular automata. Researchers have tackled different algorithmic problems ranging from the complexity of predicting to the decidability of different dynamical properties of cellular automata. But the difference here is that we look for communication protocols arising in the dynamics itself. Our work is guided by the following idea: if we are able to give a protocol describing a cellular automaton, then we can understand its behavior.
We study the generic limit sets of one-dimensional cellular automata, which intuitively capture their asymptotic dynamics while discarding transient phenomena. As our main results, we characterize the automata whose generic limit set is a singleton, and show that this class is $Σ^0_2$-complete. We also prove that given a CA whose generic limit set is guaranteed to be a singleton, the sole configuration it contains cannot be algorithmically determined.
We prove that the (language of the) asymptotic set (and the nonwandering set) of a one-dimensional cellular automaton can be $\SIGMA^1_1$-hard. We do not go into much detail, since the constructions are relatively standard.
We study the class of asynchronous non-uniform cellular automata (ANUCA) over an arbitrary group universe with multiple local transition rules. We introduce the notion of stable injectivity, stable reversibility, stable post-surjectivity and investigate several dynamical properties of such automata. In particular, we establish the equivalence between reversibility, stable reversibility, and stable injectivity for ANUCA. We also prove the invertibility of several classes of injective and stably injective ANUCA. Counter-examples are given to highlight the differences between cellular automata an…
This paper introduces a simple formalism for dealing with deterministic, non- deterministic and stochastic cellular automata in an unified and composable manner. This formalism allows for local probabilistic correlations, a feature which is not present in usual definitions. We show that this feature allows for strictly more behaviors (for instance, number conserving stochastic cellular automata require these local probabilistic correlations). We also show that several problems which are deceptively simple in the usual definitions, become undecidable when we allow for local probabilistic correl…
We study the sofic tree shifts of $A^{Σ^*}$, where $Σ^*$ is a regular rooted tree of finite rank. In particular, we give their characterization in terms of unrestricted Rabin automata. We show that if $X \subset A^{Σ^*}$ is a sofic tree shift, then the configurations in $X$ whose orbit under the shift action is finite are dense in $X$, and, as a consequence of this, we deduce that every injective cellular automata $τ\colon X \to X$ is surjective. Moreover, a characterization of sofic tree shifts in terms of general Rabin automata is given. We present an algorithm for establishing whether two u…
