Characterizing asymptotic randomization in abelian cellular automata B. Hellouin de Menibus, V. Salo, G. Theyssier Ergodic Theory and Dynamical Systems (21p.) doi:10.1017/etds.2018.75 Article complet
Abelian cellular automata (CAs) are CAs which are group endomorphisms of the full group shift when endowing the alphabet with an abelian group structure. A CA randomizes an initial probability measure if its iterated images have weak*-convergence towards the uniform Bernoulli measure (the Haar measure in this setting). We are interested in structural phenomena, i.e., randomization for a wide class of initial measures (under some mixing hypotheses). First, we prove that an abelian CA randomizes in Cesàro mean if and only if it has no soliton, i.e., a non-zero finite configuration whose time evolution remains bounded in space. This characterization generalizes previously known sufficient conditions for abelian CAs with scalar or commuting coefficients. Second, we exhibit examples of strong randomizers, i.e., abelian CAs randomizing in simple convergence; this is the first proof of this behaviour to our knowledge. We show, however, that no CA with commuting coefficients can be strongly randomizing. Finally, we show that some abelian CAs achieve partial randomization without being randomizing: the distribution of short finite words tends to the uniform distribution up to some threshold, but this convergence fails for larger words. Again this phenomenon cannot happen for abelian CAs with commuting coefficients.
Cellular Automata have been used since their introduction as a discrete tool of modelization. In many of the physical processes one may modelize thus (such as bootstrap percolation, forest fire or epidemic propagation models, life without death, etc), each local change is irreversible. The class of freezing Cellular Automata (FCA) captures this feature. In a freezing cellular automaton the states are ordered and the cells can only decrease their state according to this “freezing-order”. We investigate the dynamics of such systems through the questions of simulation and universality in this class: is there a Freezing Cellular Automaton (FCA) that is able to simulate any Freezing Cellular Automata, i.e. an intrinsically universal FCA? We show that the answer to that question is sensitive to both the number of changes cells are allowed to make, and geometric features of the space. In dimension 1, there is no universal FCA. In dimension 2, if either the number of changes is at least 2, or the neighborhood is Moore, then there are universal FCA. On the other hand, there is no universal FCA with one change and Von Neumann neighborhood. We also show that monotonicity of the local rule with respect to the freezing-order (a common feature of bootstrap percolation) is also an obstacle to universality.
We study the complexity of signed majority cellular automata on the planar grid. We show that, depending on their symmetry and uniformity, they can simulate different types of logical circuitry under different modes. We use this to establish new bounds on their overall complexity, concretely: the uniform asymmetric and the non-uniform symmetric rules are Turing universal and have a P-complete prediction problem; the non-uniform asymmetric rule is intrinsically universal; no symmetric rule can be intrinsically universal. We also show that the uniform asymmetric rules exhibit cycles of super-polynomial length, whereas symmetric ones are known to have bounded cycle length.
In this article we study the minimum number κ of additional automata that a Boolean automata network (BAN) associated with a given block-sequential update schedule needs in order to simulate a given BAN with a parallel update schedule. We introduce a graph that we call 𝖭𝖤𝖢𝖢 graph built from the BAN and the update schedule. We show the relation between κ and the chromatic number of the 𝖭𝖤𝖢𝖢 graph. Thanks to this 𝖭𝖤𝖢𝖢 graph, we bound κ in the worst case between n / 2 and 2n/3+2 (n being the size of the BAN simulated) and we conjecture that this number equals n / 2. We support this conjecture with two results: the clique number of a 𝖭𝖤𝖢𝖢 graph is always less than or equal to n / 2 and, for the subclass of bijective BANs, κ is always less than or equal to n/2+1.
μ-Limit sets of cellular automata from a computational complexity perspective L. Boyer, M. Delacourt, V. Poupet, M. Sablik, G. Theyssier Journal of Computer and System Sciences (41p.) doi:10.1016/j.jcss.2015.05.004 Article complet
This paper concerns μ-limit sets of cellular automata: sets of configurations made of words whose probability to appear does not vanish with time, starting from an initial μ-random configuration. More precisely, we investigate the computational complexity of these sets and of related decision problems. Main results: first, μ-limit sets can have a Σ3-hard language, second, they can contain only α-complex configurations, third, any non-trivial property concerning them is at least Π3-hard. We prove complexity upper bounds, study restrictions of these questions to particular classes of CA, and different types of (non-)convergence of the measure of a word during the evolution.
In this paper we study the strict majority bootstrap percolation process on graphs. Vertices may be active or passive. Initially, active vertices are chosen independently with probability p. Each passive vertex becomes active if at least half of its neighbors are active (and thereafter never changes its state). If at the end of the process all vertices become active then we say that the initial set of active vertices percolates on the graph. We address the problem of finding graphs for which percolation is likely to occur for small values of p. Specifically, we study a graph that we call r-wheel: a ring of n vertices augmented with a universal vertex where each vertex in the ring is connected to its r closest neighbors to the left and to its r closest neighbors to the right. We prove that the critical probability is 1/4. In other words, if p>1/4 then for large values of r percolation occurs with probability arbitrarily close to 1 as n goes to infinity. On the other hand, if p<1/4 then the probability of percolation is bounded away from 1.
We prove a negative result on the power of a model of algorithmic self-assembly for which finding general techniques and results has been notoriously difficult. Specifically, we prove that Winfree's abstract Tile Assembly Model is not intrinsically universal when restricted to use noncooperative tile binding. This stands in stark contrast to the recent result that the abstract Tile Assembly Model is indeed intrinsically universal when cooperative binding is used (FOCS 2012). Noncooperative self-assembly, also known as 'temperature 1', is where all tiles bind to each other if they match on at least one side. On the other hand, cooperative self-assembly requires that some tiles bind on at least two sides. Our result shows that the change from non-cooperative to cooperative binding qualitatively improves the range of dynamics and behaviors found in these models of nanoscale self-assembly. The result holds in both two and three dimensions; the latter being quite surprising given that three-dimensional noncooperative tile assembly systems simulate Turing machines. This shows that Turing universal behavior in self-assembly does not imply the ability to simulate all algorithmic self-assembly processes. In addition to the negative result, we exhibit a three-dimensional noncooperative self-assembly tile set capable of simulating any two-dimensional noncooperative self-assembly system. This tile set implies that, in a restricted sense, non-cooperative self-assembly is intrinsically universal for itself.
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 correlations, even in dimension one. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the non-deterministic and stochastic settings. Although the intrinsic simulation relation is shown to become undecidable in dimension two and higher, we provide explicit tools to prove or disprove the existence of such a simulation between any two given stochastic cellular automata. Those tools rely upon a characterization of equality of stochastic global maps, shown to be equivalent to the existence of a stochastic coupling between the random sources. We apply them to prove that there is no universal stochastic cellular automaton. Yet we provide stochastic cellular automata achieving optimal partial universality, as well as a universal non-deterministic cellular automaton.
Asymptotically almost all lambda-terms are strongly normalizing R. David, C. Raffali, K. Grygiel, J. Kozik, G. Theyssier, M. Zaionc Logical Methods in Computer Science (38p.) doi:10.2168/LMCS-9(1:2)2013 Article complet
We present quantitative analysis of various (syntactic and behavioral) properties of random lambda-terms. Our main results are that asymptotically all the terms are strongly normalizing and that any fixed closed term almost never appears in a random term. Surprisingly, in combinatory logic (the translation of the lambda-calculus into combinators), the result is exactly opposite. We show that almost all terms are not strongly normalizing. This is due to the fact that any fixed combinator almost always appears in a random combinator.
We study the Monadic Second Order (MSO) Hierarchy over colorings of the discrete plane, and draw links between classes of formula and classes of subshifts. We give a characterization of existential MSO in terms of projections of tilings, and of universal sentences in terms of combinations of “pattern counting” subshifts. Conversely, we characterize logic fragments corresponding to various classes of subshifts (subshifts of finite type, sofic subshifts, all subshifts). Finally, we show by a separation result how the situation here is different from the case of tiling pictures studied earlier by Giammarresi et al.
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 caracterization of equality of stochastic global maps by the existence of a coupling between the random sources. We then prove that there is a universal non-deterministic cellular automaton, but no universal stochastic cellular automaton. Yet we provide stochastic cellular automata achieving optimal partial universality.
We study intrinsic simulations between cellular automata and introduce a new necessary condition for a CA to simulate another one. Although expressed for general CA, this condition is targeted towards surjective CA and especially linear ones. Following the approach introduced by the first author in an earlier paper, we develop proof techniques to tell whether some linear CA can simulate another linear CA. Besides rigorous proofs, the necessary condition for the simulation to occur can be heuristically checked via simple observations of typical space-time diagrams generated from finite configurations. As an illustration, we give an example of linear reversible CA which cannot simulate the identity and which is 'time-asymmetric', i.e. which can neither simulate its own inverse, nor the mirror of its own inverse.
Clandestine Simulations in Cellular Automata P. Guillon, P.-E. Meunier, G. Theyssier JAC 2010 (18p.) Article complet
This paper studies two kinds of simulation between cellular automata: simulations based on factor and simulations based on sub-automaton. We show that these two kinds of simulation behave in two opposite ways with respect to the complexity of attractors and factor subshifts. On the one hand, the factor simulation preserves the complexity of limits sets or column factors (the simulator CA must have a higher complexity than the simulated CA). On the other hand, we show that any CA is the sub-automaton of some CA with a simple limit set (NL-recognizable) and the sub-automaton of some CA with a simple column factor (finite type). As a corollary, we get intrinsically universal CA with simple limit sets or simple column factors. Hence we are able to 'hide' the simulation power of any CA under simple dynamical indicators.
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.
This paper is the first part of a serie of two papers dealing with bulking: a quasiorder on cellular automata comparing space-time diagrams up to some rescaling. Bulking is a generalization of grouping taking into account universality phenomena, giving rise to a maximal equivalence class. In the present paper, we discuss the proper components of grouping and study the most general extensions. We identify the most general space-time transforms and give an axiomatization of bulking quasiorder. Finally, we study some properties of intrinsically universal cellular automata obtained by comparing grouping to bulking.
This paper studies directional dynamics in cellular automata, a formalism previously introduced by the third author. The central idea is to study the dynamical behaviour of a cellular automaton through the conjoint action of its global rule (temporal action) and the shift map (spacial action): qualitative behaviours inherited from topological dynamics (equicontinuity, sensitivity, expansivity) are thus considered along arbitrary curves in space-time. The main contributions of the paper concern equicontinuous dynamics which can be connected to the notion of consequences of a word. We show that there is a cellular automaton with an equicontinuous dynamics along a parabola, but which is sensitive along any linear direction. We also show that real numbers that occur as the slope of a limit linear direction with equicontinuous dynamics in some cellular automaton are exactly the computably enumerable numbers.
The notions of universality and completeness are central in the theories of computation and computational complexity. However, proving lower bounds and necessary conditions remains hard in most of the cases. In this article, we introduce necessary conditions for a cellular automaton to be "universal", according to a precise notion of simulation, related both to the dynamics of cellular automata and to their computational power. This notion of simulation relies on simple operations of space-time rescaling and it is intrinsic to the model of cellular automata. Intrinsinc universality, the derived notion, is stronger than Turing universality, but more uniform, and easier to define and study. Our approach builds upon the notion of communication complexity, which was primarily designed to study parallel programs, and thus is, as we show in this article, particulary well suited to the study of cellular automata: it allowed to show, by studying natural problems on the dynamics of cellular automata, that several classes of cellular automata, as well as many natural (elementary) examples, could not be intrinsically universal.
The study of factoring relations between subshifts or cellular automata is central in symbolic dynamics. Besides, a notion of intrinsic universality for cellular automata based on an operation of rescaling is receiving more and more attention in the literature. In this paper, we propose to study the factoring relation up to rescalings, and ask for the existence of universal objects for that simulation relation. In classical simulations of a system S by a system T , the simulation takes place on a specific subset of configurations of T depending on S (this is the case for intrinsic universality). Our setting, however, asks for every configurations of T to have a meaningful interpretation in S. Despite this strong requirement, we show that there exists a cellular automaton able to simulate any other in a large class containing arbitrarily complex ones. We also consider the case of subshifts and, using arguments from recursion theory, we give negative results about the existence of universal objects in some classes.
Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on higher dimensional CA and aims at showing that the situation is different and more complex starting from dimension 2. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants, the existence of CA having only non-recursive equicontinuous points and the existence of CA having only countably many equicontinuous points. They all show a difference between dimension 1 and higher dimensions. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case. Finally, we show that the set of sensitive CA is only Pi_2 in dimension 1, but becomes Sigma_3-hard for dimension 3.
We study the Monadic Second Order (MSO) Hierarchy over infinite pictures, that is tilings. We give a characterization of existential MSO in terms of tilings and projections of tilings. Conversely, we characterise logic fragments corresponding to various classes of infinite pictures (subshifts of finite type, sofic subshifts).
Cellular automata (CA) are dynamical systems defined by a finite local rule but they are studied for their global dynamics. They can exhibit a wide range of complex behaviours and a celebrated result is the existence of (intrinsically) universal CA, that is CA able to fully simulate any other CA. In this paper, we show that the asymptotic density of universal cellular automata is 1 in several families of CA defined by local symmetries. We extend results previously established for captive cellular automata in two significant ways. First, our results apply to well-known families of CA (e.g. the family of outer-totalistic CA containing the Game of Life) and, second, we obtain such density results with both increasing number of states and increasing neighbourhood. Moreover, thanks to universality-preserving encodings, we show that the universality problem remains undecidable in some of those families.
In this paper, we study amalgamations of cellular automata (CA), i.e. ways of combining two CA by disjoint union. We show that for several families of CA obtained by simple amalgamation operations (including the well-known families of majority and minority CA), the density of a large class of properties follows a zero-one law. Besides, we establish that intrinsic universality in those families is always non-trivial and undecidable for some of them. Therefore we obtain various syntactical means to produce CA which are almost all intrinsically universal. These results extend properties already obtained for captive cellular automata. We additionally prove that there exists some reversible captive CA which are (intrinsically) universal for reversible CA.
Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on 2D CA and aims at showing that the situation is different and more complex. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants and the existence of CA having only non-recursive equicontinuous points. They all show a difference between the 1D and the 2D case. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case.
We study the notion of limit sets of cellular automata associated with probability measures (mu-limit sets). This notion was introduced by P. Kurka and A. Maass. It is a refinement of the classical notion of omega-limit sets dealing with the typical long term behavior of cellular automata. It focuses on the words whose probability of appearance does not tend to 0 as time tends to infinity (the persistent words). In this paper, we give a characterisation of the persistent language for non sensible cellular automata associated with Bernouilli measures. We also study the computational complexity of these languages. We show that the persistent language can be non-recursive. But our main result is that the set of quasi-nilpotent cellular automata (those with a single configuration in their mu-limit set) is neither recursively enumerable nor co-recursively enumerable.
We address the problem of the density of intrinsically universal cellular automata among cellular automata or a subclass of cellular automata. We show that the density of captive cellular automata possessing a given poperty follow a zero-one law for a large class of properties. As a corollary, we show that they are almost all intrinsically universal. We show however that intrinsic universality is undecidable for captive cellular automata. Finally, we show that almost all cellular automata have no non-trivial sub-automaton.
We introduce a natural class of cellular automata characterised by a property of the local transition law without any assumption on the states set. We investigate some algebraic properties of the class and show that it contains intrinsically universal cellular automata. In addition we show that Rice's theorem for limit sets is no longer true for that class, although infinitely many properties of limit sets are still undecidable.
The model of cellular automata is fascinating because very simple local rules can generate complex global behaviors. The relationship between local and global function is subject of many studies. We tackle this question by using results on communication complexity theory and, as a by-product, we provide (yet another) classification of cellular automata.