on the directional dynamics of additive cellular automata we continue the study of cellular automata ca directional dynamics i e the behavior of the joint action of ca and shift maps this notion has been investigated for general ca in the case of expansive dynamics by boyle and lind and by sablik for sensitivity and equicontinuity in this paper we give a detailed classification for the class of additive ca providing non trivial examples for some classes of sablik s classification moreover we extend the directional dynamics studies by considering also factor languages and attractors

kolmogorov complexity and cellular automata classification we present a new approach to cellular automata ca classification based on algorithmic complexity we construct a parameter which is based only on the transition table of ca and measures the randomness of evolutions is better in a certain sense than any other parameter recursively definable on ca tables we investigate the relations between the classical topological approach and ours our parameter is compared with langton s parameter turns out to be theoretically better and also agrees with some practical evidences reported in literature finally we propose a protocol to approximate and make experiments on ca dynamical behavior

additive cellular automata and algebraic series abstract a cellular automation is an array of regularly interconnected identical cells we study here the special case of automata where each cell depends in additive manner on its neighbours the successive states of a given cell form a sequence whose generating series is proved to be always an algebraic series we also exemplify the realization of a given algebraic series by means of an automation as a by product we obtain a relation between additive cellular automata and certain automatic sequences like the paper folding sequence

additive one dimensional cellular automata are chaotic according to devaney s definition of chaos abstract we study the chaotic behavior of a particular class of dynamical systems cellular automata we specialize the definition of chaos given by devaney for general dynamical systems to the case of cellular automata a dynamical system x f is chaotic according to devaney s definition of chaos if its transition map f is sensitive to the initial conditions topologically transitive and has dense periodic orbits on x our main result is the proof that all the additive one dimensional cellular automata defined on a finite alphabet of prime cardinality are chaotic in the sense of devaney

computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

fourier encoding of closed planar boundaries a circular gaussian autoregressive cgar source is used as a model for closed planar curves a class of suboptimal encoding schemes is considered which separately quantize the fourier coefficients of the boundary application of rate distortion theoretic techniques leads to parametric equations describing the optimal encoding bound interpretation of these equations establishes a sampling criterion and a computationally efficient transform encoding scheme for the suboptimal class several variants of this transform encoding scheme are suggested and compared to the encoding bound

some results about the chaotic behavior of cellular automata we study the behavior of cellular automata ca for short in the cantor besicovitch and weyl topologies we solve an open problem about the existence of transitive ca in the besicovitch topology the proof of this result has some interest of its own since it is obtained by using kolmogorov complexity to our knowledge it is the first result about discrete dynamical systems obtained using kolmogorov complexity we also prove that in the besicovitch topology every ca has either a unique periodic point thus a fixed point or an uncountable set of periodic points this result underlines the fact that ca have a great degree of stability it may be considered a further step towards the understanding of ca periodic behavior moreover we prove that in the besicovitch topology there is a special set of configurations the set of toeplitz configurations that plays a role similar to that of spatially periodic configurations in the cantor topology that is it is dense and has a central role in the study of surjectivity and injectivity finally it is shown that the set of spatially quasi periodic configurations is not dense in the weyl topology

nondeterministic cellular automata abstract using purely topological concepts we characterize the class of nondeterministic mappings between shift spaces that give rise to nondeterministic sliding block codes we use such topological notions to investigate basic properties of surjective and reversible nondeterministic cellular automata

solution of some conjectures about topological properties of linear cellular automata we study two dynamical properties of linear d dimensional cellular automata over zm namely denseness of periodic points and topological mixing for what concerns denseness of periodic points we complete the work initiated in theoret comput sci 174 1997 157 theoret comput sci 233 1 2 2000 147 14th annual symp on theoretical aspects of computer science stacs 97 lncs n 1200 springer berlin 1997 pp 427 438 by proving that a linear cellular automata has dense periodic points over the entire space of configurations if and only if it is surjective as conjectured in cattaneo et al 2000 for non surjective linear ca we give a complete characterization of the subspace where periodic points are dense for what concerns topological mixing we prove that this property is equivalent to transitivity and then easily checkable finally we classify linear cellular automata according to the definition of chaos given by devaney in an introduction to chaotic dynamical systems 2nd ed addison wesley reading ma usa 1989

number conserving cellular automata i decidability we prove that definitions of number conserving cellular automata found in literature are equivalent a necessary and sufficient condition for cellular automata to be number conserving is proved using this condition we give a quasi linear time algorithm to decide number conservation

conservation of some dynamical properties for operations on cellular automata we consider the family of all the cellular automata ca sharing the same local rule but having different memories this family contains also all ca with memory m 0 one sided ca which can act both on a z and on a n we study several set theoretical and topological properties for these classes in particular we investigate whether the properties of a given ca are preserved when considering the ca obtained by changing the memory of the original one shifting operation furthermore we focus our attention on the one sided ca acting on a z starting from the one sided ca acting on a n and having the same local rule lifting operation as a particular consequence of these investigations we prove that the long standing conjecture surjectivity dense periodic orbits dpo can be restated in several different but equivalent ways furthermore we give some results on properties conserved under the iteration of the ca global map

on the undecidability of the limit behavior of cellular automata cellular automata ca are discrete dynamical systems and an abstract model of parallel computation the limit set of a cellular automaton is its maximal topological attractor a well known result due to kari says that all nontrivial properties of limit sets are undecidable in this paper we consider the properties of limit set dynamics i e properties of the dynamics of ca restricted to their limit sets there can be no equivalent of kari s theorem for limit set dynamics anyway we show that there is a large class of undecidable properties of limit set dynamics namely all properties of limit set dynamics which imply stability or the existence of a unique subshift attractor as a consequence we have that it is undecidable whether the cellular automaton map restricted to the limit set is the identity map and whether it is closing injective expansive positively expansive and transitive

the second laws of quantum thermodynamics the second law of thermodynamics places constraints on state transformations it applies to systems composed of many particles however we are seeing that one can formulate laws of thermodynamics when only a small number of particles are interacting with a heat bath is there a second law of thermodynamics in this regime here we find that for processes which are approximately cyclic the second law for microscopic systems takes on a different form compared to the macroscopic scale imposing not just one constraint on state transformations but an entire family of constraints we find a family of free energies which generalize the traditional one and show that they can never increase the ordinary second law relates to one of these with the remainder imposing additional constraints on thermodynamic transitions we find three regimes which determine which family of second laws govern state transitions depending on how cyclic the process is in one regime one can cause an apparent violation of the usual second law through a process of embezzling work from a large system which remains arbitrarily close to its original state these second laws are relevant for small systems and also apply to individual macroscopic systems interacting via long range interactions by making precise the definition of thermal operations the laws of thermodynamics are unified in this framework with the first law defining the class of operations the zeroth law emerging as an equivalence relation between thermal states and the remaining laws being monotonicity of our generalized free energies

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

bulking ii classifications of 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

physics like models of computation abstract reversible cellular automata are computer models that embody discrete analogues of the classical physics notions of space time locality and microscopic reversibility they are offered as a step towards models of computation that are closer to fundamental physics

sand automata as cellular automata in this paper we exhibit a strong relation between the sand automata configuration space and the cellular automata configuration space this relation induces a compact topology for sand automata and a new context in which sand automata are homeomorphic to cellular automata acting on a specific subshift we show that the existing topological results for sand automata including the hedlund like representation theorem still hold in this context we give a characterization of cellular automata which are sand automata and study some dynamical behaviors such as equicontinuity furthermore we deal with simple sand automata we show that the classical definition of nilpotency is not meaningful for sand automata then we introduce the suitable new notion of flattening sand automata finally we prove that this simple dynamical behavior is undecidable

on the conjugacy problem of cellular automata abstract cellular automata are topological dynamical systems we consider the problem of deciding whether two cellular automata are conjugate or not we also consider deciding strong conjugacy that is conjugacy by a map that commutes with the shift maps we show that the following two sets of pairs of one dimensional one sided cellular automata are recursively inseparable i pairs where the first cellular automaton has strictly higher entropy than the second one and ii pairs that are strongly conjugate and both have zero topological entropies this implies that the following decision problems are undecidable given two one dimensional one sided cellular automata f and g are f and g conjugate is f a factor of g is f a subsystem of g all of these are undecidable in both strong and weak variants whether the homomorphism is required to commute with the shift or not respectively we also prove the same results for reversible two dimensional cellular automata

fluctuation driven computing on number conserving cellular automata a number conserving cellular automaton ncca is a cellular automaton in which the states of cells are denoted by integers and the sum of all of the numbers in a configuration is conserved throughout its evolution nccas have been widely used to model physical systems that are ruled by conservation laws of mass or energy imai et al 13 showed that the local transition function of ncca can be effectively translated into the sum of a binary flow function over pairs of neighboring cells in this paper we explore the computability of nccas in which the pairwise number flows are performed at fully asynchronous timings despite the randomness that is associated with asynchronous transitions useful computation still can be accomplished efficiently in the cellular automata through the active exploitation of fluctuations 18 specifically certain numbers may flow randomly fluctuating between forward and backward directions in the cellular space as if they were subject to brownian motion because random fluctuations promise a powerful resource for searching through a computational state space the brownian like flow of the numbers allows for efficient embedding of logic circuits into our novel asynchronous ncca

majorization properties of generalized thermal distributions abstract we examine the majorization properties of general thermal like mixed states depending on a set of parameters sufficient conditions which ensure the increase in mixedness and hence of any associated entropic form when these parameters are varied are identified we then discuss those exhibiting a power law distribution showing that they can be characterized by two distinct mixing parameters one associated with temperature and the other with the non extensivity index q illustrative numerical results are also provided

universality and decidability of number conserving cellular automata number conserving cellular automata ncca are particularly interesting both because of their natural appearance as models of real systems and because of the strong restrictions that number conservation implies here we extend the definition of the property to include cellular automata with any set of states in z and show that they can be always extended to usual ncca with contiguous states we show a way to simulate any one dimensional ca through a one dimensional ncca proving the existence of intrinsically universal ncca finally we give an algorithm to decide given a ca if its states can be labeled with integers to produce a ncca and to find this relabeling if the answer is positive

a hierarchical classification of cellular automata abstract a central issue in the theory of cellular automata is classification a classification is a structure imposed on the space of automata which groups together cellular automata with related properties in this paper a classification based on the action of cellular automata on n step markov measures is presented this classification is hierarchical and parametric an algorithm is described which efficiently constructs all rules in a given class at a given level of the hierarchy the utility of the classification is explored in the study of the invariant statistical properties of next nearest neighbor cellular automata on two states per cell

on conservative and monotone one dimensional cellular automata and their particle representation number conserving or conservative cellular automata ca have been used in several contexts in particular traffic models where it is natural to think about them as systems of interacting particles in this article we consider several issues concerning one dimensional cellular automata which are conservative monotone specially non increasing or that allow a weaker kind of conservative dynamics we introduce a formalism of particle automata and discuss several properties that they may exhibit some of which like anticipation and momentum preservation happen to be intrinsic to the conservative ca they represent for monotone ca we give a characterization and then show that they too are equivalent to the corresponding class of particle automata finally we show how to determine for a given ca and a given integer b whether its states admit a b neighborhood dependent relabeling whose sum is conserved by the ca iteration this can be used to uncover conservative principles and particle like behavior underlying the dynamics of some ca compliments at http www dim uchile cl anmoreir ncca

on ergodic one dimensional cellular automata we show that all onto cellular automata defined on the binary sequence space are invariant with respect to the haar measure and that an extensive class of such maps including many nonlinear ones are strongly mixing with respect to the haar measure

exact results for deterministic cellular automata with additive rules deterministic cellular automata ca with additive rules are studied by exploiting the properties of circulant matrices on finite fields complete state transition diagrams for higher order and multidimensional ca on finite lattices are analyzed conditions on the rules which make them reversible are obtained it is shown that all state transition diagrams of the ca have identical trees rooted on cycles general formulae for cycle lengths and multiplicities are given

a complete and efficiently computable topological classification of d dimensional linear cellular automata over z m we study the dynamical behavior of d dimensional linear cellular automata over zm we provide easy to check necessary and sufficient conditions for a d dimensional linear cellular automata over zm to be sensitive to initial conditions positively expansive strongly transitive and equicontinuous as a consequence of our results we have a complete and efficiently computable topological classification of d dimensional linear cellular automata over zm according to the most important dynamical properties studied in the theory of discrete time dynamical systems

classifying circular cellular automata we introduce a hierarchy of linear cellular automata based on their limiting behavior on spatially periodic configurations we show that it is undecidable to which class in the hierarchy a cellular automaton belongs in particular it is undecidable whether all spatially periodic configurations evolve to a fixed point furthermore there is no computable bound on the period lengths of these configurations our arguments are based on a non standard simulation of turing machines on circular cellular automata

dynamical behavior of additive cellular automata over finite abelian groups abstract we study the dynamical behavior of additive d dimensional d 1 cellular automata where the alphabet is any finite abelian group this class of discrete time dynamical systems is a generalization of the systems extensively studied by many authors among which one may list 38 44 41 among our major contributions there is the proof that topologically transitive additive d dimensional cellular automata over a finite abelian group are ergodic this result represents a solid bridge between the world of measure theory and that of topology and greatly extends previous results obtained in 12 44 for linear ca over z m z i e additive ca in which the alphabet is the cyclic group z m z and the local rules are linear combinations with coefficients in z m z in our scenario the alphabet is any finite abelian group and the global rule is any additive map this class of ca strictly contains the class of linear ca over z m z n i e with the local rule defined by n n matrices with elements in z m z which in turn strictly contains the class of linear ca over z m z in order to further emphasize that finite abelian groups are more expressive than z m z we prove that contrary to what happens in z m z there exist additive ca over suitable finite abelian groups which are roots with arbitrarily large indices of the shift map as a relevant consequence of our results we have that for additive d dimensional ca over a finite abelian group ergodic mixing weak ergodic mixing ergodicity topological mixing weak topological mixing topological total transitivity and topological transitivity are all equivalent properties as a corollary we see that invertible transitive additive ca are isomorphic to bernoulli shifts furthermore we prove that surjectivity implies openness for additive d dimensional ca over a finite abelian group hence we get that topological transitivity is equivalent to the well known devaney notion of chaos when d 1 moreover we provide a first characterization of strong transitivity for additive ca which we suspect to be true also for the general case

recovery of the absorption coefficient from diffused reflected light using a discrete diffusive model an algorithm based on a discrete model for photon migration is proposed for recovering the absorption coefficient in a semi infinite slab a pulsed light source is modeled and only reflected signals are used in the reconstruction the diffusion coefficient and speed of light are assumed constant and known the algorithm is based on the result that the earliest reflected signals obtained from a particular voxel depend either in a linear or in a quadratical fashion on the square of the absorption coefficient in that voxel by solving the discrete model in the forward direction in a region where the absorption coefficient has been calculated already the absorption coefficients in the voxels on the boundary of the region are calculated using interpolation results show that simple absorptive regions can be accurately recovered

max plus analysis on some binary particle systems we are concerned with a special class of binary cellular automata i e the so called particle cellular automata pca in this paper we first propose max plus expressions to pca of four neighbors then by utilizing basic operations of the max plus algebra and appropriate transformations pca4 1 4 2 and 4 3 are solved exactly and their general solutions are found in terms of max plus expressions finally we analyze the asymptotic behaviors of general solutions and prove the fundamental diagrams exactly

generalized sub shifts in elementary cellular automata the strange case of chaotic rule 180 abstract we study the dynamical behavior of elementary cellular automaton 180 this rule gives rise to a global dynamics on the phase space of all one dimensional bi infinite configurations which is devaney topologically chaotic the dense sub dynamical system of configurations in background of 0s is a generalized sub shift i e multiple sub shift whose multiplicity constant depends on the initial configuration this sub dynamical system is deeply stable in the sense that the null configuration is a global attractor but with some components of the chaotic behaviour transitivity and unpredictability the dense sub dynamical system of configurations in background of 1s is a fractal like system with strongly chaotic components expansivity

number conserving cellular automata with a von neumann neighborhood of range one we present necessary and sufficient conditions for a cellular automaton with a von neumann neighborhood of range one to be number conserving the conditions are formulated for any dimension and for any set of states containing zero the use of the geometric structure of the von neumann neighborhood allows for computationally tractable conditions even in higher dimensions

on the relationship between fuzzy and boolean cellular automata fuzzy cellular automata fca are continuous cellular automata where the local rule is defined as the fuzzification of the local rule of a corresponding boolean cellular automaton in disjunctive normal form in this paper we are interested in the relationship between boolean and fuzzy models and for the first time we analytically show the existence of a strong connection between them by focusing on two properties density conservation and additivity we begin by showing that the density conservation property extensively studied in the boolean domain is preserved in the fuzzy domain a boolean ca is density conserving if and only if the corresponding fca is sum preserving a similar result is established for another novel spatial density conservation property second we prove an interesting parallel between the additivity of boolean ca and oscillations of the corresponding fuzzy ca around its fixed point in fact we show that a boolean ca is additive if and only if the behaviour of the corresponding fuzzy ca around its fixed point coincides with the boolean behaviour finally we give a probabilistic interpretation of our fuzzification which establishes an equivalence between convergent fuzzy ca and the mean field approximation on boolean ca an estimation of their asymptotic density these connections between the boolean and the fuzzy models are the first formal proofs of a relationship between them

investigating topological chaos by elementary cellular automata dynamics we apply the two different definitions of chaos given by devaney and by knudsen for general discrete time dynamical systems dtds to the case of elementary cellular automata i e 1 dimensional binary cellular automata with radius 1 a dtds is chaotic according to the devaney s definition of chaos iff it is topologically transitive has dense periodic orbits and it is sensitive to initial conditions a dtds is chaotic according to the knudsen s definition of chaos iff it has a dense orbit and it is sensitive to initial conditions we enucleate an easy to check property left or rightmost permutivity of the local rule associated with a cellular automaton which is a sufficient condition for d chaotic behavior it turns out that this property is also necessary for the class of elementary cellular automata finally we prove that the above mentioned property does not remain a necessary condition for chaoticity in the case of non elementary cellular automata

ergodicity transitivity and regularity for linear cellular automata over z m 1 we study the dynamical behavior of d dimensional linear cellular automata over zm we provide an easy to check necessary and sufficient condition for a d dimensional linear cellular automata over zm to be ergodic and topologically transitive as a byproduct we get that for linear cellular automata ergodicity is equivalent to topological transitivity finally we prove that for 1 dimensional linear cellular automata over zm regularity denseness of periodic orbits is equivalent to surjectivity

global properties of cellular automata cellular automata are discrete mathematical systems that generate diverse often complicated behavior using simple deterministic rules analysis of the local structure of these rules makes possible a description of the global properties of the associated automata a class of cellular automata that generate infinitely many aperiodic temporal sequences is defined as is the set of rules for which inverses exist necessary and sufficient conditions are derived characterizing the classes of nearest neighbor rules for which arbitrary finite initial conditions i evolve to a homogeneous state ii generate at least one constant temporal sequence

quasi periodic configurations and undecidable dynamics for tilings infinite words and turing machines we describe turing machines tilings and infinite words as dynamical systems and analyze some of their dynamical properties it is known that some of these systems do not always have periodic configurations we prove that they always have quasi periodic configurations and we quantify quasi periodicity we then study the decidability of dynamical properties for these systems in analogy to rice s theorem for computable functions we derive a theorem that characterizes dynamical system properties that are undecidable as an illustration of this result we prove that topological entropy is undecidable for turing machines and for tilings

fundamental study directional dynamics for cellular automata a sensitivity to initial condition approach a cellular automaton is a continuous function f defined on a full shift a z which commutes with the shift s often to study the dynamics of f one only considers implicitly s however it is possible to emphasize the spatio temporal structure produced by considering the dynamics of the zxn action induced by s f in this purpose we study the notion of directional dynamics in particular we are interested in directions of equicontinuity and expansivity which generalize the concepts introduced by gilman robert h gilman classes of linear automata ergodic theory dynam systems 7 1 1987 105 118 and p kurka petr kurka languages equicontinuity and attractors in cellular automata ergodic theory dynam systems 17 2 1997 417 433 we study the sets of directions which exhibit this special kind of dynamics showing that they induce a discrete geometry in space time diagrams

on the limit sets of cellular automata the limit sets of cellular automata defined by wolfram play an important role in applications of cellular automata to complex systems a number of results on limit sets are proved considering both finite and infinite configurations of cellular automata the main concern of this paper is with testing membership and essential emptiness of limit sets for linear and two dimensional cellular automata

classification of sofic projective subdynamics of multidimensional shifts of finite type motivated by hochman s notion of subdynamics of a z subshift 8 we define and examine the projective subdynamics of z shifts of finite type sfts where we restrict not only the action but also the phase space we show that any z sofic shift of positive entropy is the projective subdynamics of a z2 z sft and that there is a simple condition characterizing the class of zero entropy z sofic shifts which are not the projective subdynamics of any z2 sft we define notions of stable and unstable subdynamics in analogy with the notions of stable and unstable limit sets in cellular automata theory and discuss how our results fit into this framework one dimensional strictly sofic shifts of positive entropy admit both a stable and an unstable realization whereas a particular class of zero entropy z sofics only allows for an unstable realization finally we prove that the union of z subshifts all of which are realizable in z sfts is again realizable when it contains at least two periodic points that the projective subdynamics of z2 sfts with the uniform filling property ufp are always sofic and we exhibit a class of non sofic z subshifts which are not the subdynamics of any z sft

error exponents for entanglement concentration consider entanglement concentration schemes that convert n identical copies of a pure state into a maximally entangled state of a desired size with success probability being close to one in the asymptotic limit we give the distillable entanglement the number of bell pairs distilled per copy as a function of an error exponent which represents the rate of decrease in failure probability as n tends to infinity the formula fills the gap between the least upper bound of distillable entanglement in probabilistic concentration which is the well known entropy of entanglement and the maximum attained in deterministic concentration the method of types in information theory enables the detailed analysis of the distillable entanglement in terms of the error rate in addition to the probabilistic argument we consider another type of entanglement concentration scheme where the initial state is deterministically transformed into a possibly mixed final state whose fidelity to a maximally entangled state of a desired size converges to one in the asymptotic limit we show that the same formula as in the probabilistic argument is valid for the argument on fidelity by replacing the success probability with the fidelity furthermore we also discuss entanglement yield when optimal success probability or optimal fidelity converges to zero in the asymptotic limit strong converse and give the explicit formulae for those cases

on the bernoulli automorphism of reversible linear cellular automata this investigation studies the ergodic properties of reversible linear cellular automata over z m for m n we show that a reversible linear cellular automaton is either a bernoulli automorphism or non ergodic this gives an affirmative answer to an open problem proposed by pivato 20 for the case of reversible linear cellular automata

on computing the entropy of cellular automata we study the topological entropy of a particular class of dynamical systems cellular automata the topological entropy of a dynamical system x f is a measure of the complexity of the dynamics of f over the space x the problem of computing or even approximating the topological entropy of a given cellular automata is algorithmically undecidable ergodic theory dynamical systems 12 1992 255 in this paper we show how to compute the entropy of two important classes of cellular automata namely linear and positively expansive cellular automata in particular we prove a closed formula for the topological entropy of d dimensional d 1 linear cellular automata over the ring zm m 2 and we provide an algorithm for computing the topological entropy of positively expansive cellular automata

communication complexity in number conserving and monotone cellular automata one third of the elementary cellular automata cas are either number conserving nccas or monotone increasing or decreasing in this paper we prove that for all of them we can find linear or constant communication protocols for the prediction problem in other words we are able to give a succinct description for their dynamics this is not necessarily true for general nccas in fact we also show how to explicitly construct from any ca a new ncca which preserves the original communication complexity

ergodic endomorphisms of compact abelian groups we show that for a surjective endomorphism of a compact abelian group ergodicity is equivalent to a condition which impliesr mixing for allr 1 and we characterize such maps algebraically this is then used in proving the ergodicity of an extensive class of endomorphisms of the binary sequence space as a simple corollary it is found that one dimensional linear cellular automata and the accumulator automata arer mixing for allr 1

convergence and aperiodicity in fuzzy cellular automata revisiting rule 90 abstract in this paper we consider a continuous version of cellular automata fuzzy ca obtained by fuzzification of the disjunctive normal form which describes the corresponding boolean rule we concentrate on fuzzy rule 90 whose boolean version deserves some attention for the complex patterns it generates we show that the behavior of fuzzy rule 90 is very simple in that the system always converges to a fixed point in the case of finite support configurations we also show aperiodicity of every temporal sequences extending and complementing jen s result on aperiodicity of boolean rule 90 we finally show and analyze the remarkable fact that depending on the level of state discreteness used to visualize the dynamics of fuzzy rule 90 the display might show after a transient the well known complex boolean behavior instead of the correct convergence to a fixed point the results of the analysis lead not only to a caveat on the dangers of visualization but also an unexpected explanation of the dynamics of boolean rule 90

an analytic function describing the error performance of optimum quantizers corresp for optimizing or analyzing digital source coding systems a model for the mean square quantizing error as a function of the number of levels is necessary an analytic approximation function is presented which is valid for a wide range of important probability density functions

number conserving cellular automata ii dynamics in this second part we study the dynamics of the number conserving cellular automata we give a classification which focuses on pattern divergence and chaoticity moreover we prove that in the case of number conserving cellular automata surjectivity is equivalent to regularity as a byproduct we obtain a strong characterization of the class of cellular automata with bounded evolutions on finite configurations

fundamental study from sandpiles to sand automata we introduce a new model for simulating natural phenomena we address several issues topology basic set properties like injectivity and surjectivity reversibility and decidability questions about a special kind of conservation law called grain conservation and ultimate periodicity

on the dynamics and recursive properties of multidimensional symbolic systems we study the sub dynamics of multidimensional shifts of finite type and sofic shifts and the action of cellular automata on their limit sets such a subaction is always an effective dynamical system i e it is isomorphic to a subshift over the cantor set the complement of which can be written as the union of a recursive sequence of basic sets our main result is that to varying degrees this recursive theoretic condition is also sufficient we show that the class of expansive subactions of multidimensional sofic shifts is the same as the class of expansive effective systems and that a general effective system can be realized modulo a small extension as the subaction of a shift of finite type or as the action of a cellular automaton on its limit set after removing a dynamically trivial set as applications we characterize in terms of their computational properties the numbers which can occur as the entropy of cellular automata and construct sfts and cas with various interesting properties

an order preserving property of additive invariants for takesue type reversible cellular automata we show that for a large and important class of reversible one dimensional cellular automata the set of additive invariants exhibits an algebraic structure more precisely if f and g are one dimensional reversible cellular automata of the kind considered by takesue 1989 1 we show that the component wise maximum on these automata is such that j f email protected email protected g where j f denotes the set of additive invariants of f and denotes the inclusion relation between real subspaces

rice s theorem for the limit sets of cellular automata abstract rice s theorem is a well known result in the theory of recursive functions a corresponding theorem for cellular automata limit sets is proved all nontrivial properties of limit sets of cellular automata cas are shown undecidable the theorem remains valid even if only one dimensional cas are considered

theory of cellular automata a survey this article surveys some theoretical aspects of cellular automata ca research in particular we discuss classical and new results on reversibility conservation laws limit sets decidability questions universality and topological dynamics of ca the selection of topics is by no means comprehensive and reflects the research interests of the author the main goal is to provide a tutorial of ca theory to researchers in other branches of natural computing to give a compact collection of known results with references to their proofs and to suggest some open problems

on optimum and nearly optimum data quantization for signal detection the application of companding approximation theory to the quantization of data for detection of coherent signals in a noisy environment is considered this application is twofold allowing for greater simplicity in both analysis and design of quantizers for detection systems most computational methods for designing optimum most efficient quantizers for signal detection systems are iterative and are extremely sensitive to initial conditions companding approximation theory is used here to obtain suitable initial conditions for this problem furthermore the companding approximation idea is applied to design suboptimum quantizers which are nearly as efficient as optimum quantizers when the number of levels is large in this design iteration is not needed to derive the parameters of the quantizer and the design procedure is very simple in this paper we explore this approach numerically and demonstrate its effectiveness for designing and analyzing quantizers in detection systems

the nilpotency problem of one dimensional cellular automata the limit set of a celullar automaton consists of all the configurations of the automaton that can appear after arbitrarily long computations it is known that the limit set is never empty it contains at least one homogeneous configuration a ca is called nilpotent if its limit set contains just one configuration the present work proves that it is algorithmically undecidable whether a given one dimensional cellular automaton is nilpotent the proof is based on a generalization of the well known result about the undecidability of the tiling problem of the plane the generalization states that the tiling problem remains undecidable even if one considers only so called nw deterministic tile sets that is tile sets in which the left and upper neighbors of each tile determine the tile uniquely the nilpotency problem is known to be undecidable for d dimensional ca for d geq 2 the result is the basis of the proof of rice s theorem for ca limit sets which states that every nontrivial property of limit set

local structure theory for cellular automata abstract the mean field theory for cellular automata wolfram and schulman and seiden is generalized to the local structure theory the local structure theory is a sequence of finitely parameterized models of the statistical features of a cellular automaton s evolution the n th model in the sequence takes into account correlations in terms of the probability of blocks of n states a class of measures the n block measures is introduced the local structure operator of order n maps n block measures to n block measures in a manner which reflects the cellular automaton map on blocks of states the fixed points of the map on measures approximate the invariant measures of the cellular automaton the ability of the local structure theory to model evolution from uncorrelated initial distributions is studied the theory gives exact results in simple cases in more complex cases monte carlo numerical experiments suggest that an accurate statistical portrait of cellular automaton evolution is obtained the invariant measures of a cellular automaton and the stability of these measures may be obtained from the local structure theory the local structure theory appears to be a powerful method for characterization and classification of cellular automata nearest neighbor cellular automata with two states per cell are studied using this method

cellular automaton rules conserving the number of active sites this paper shows how to determine all of the unidimensional two state cellular automaton rules of a given number of inputs which conserve the number of active sites these rules have to satisfy a necessary and sufficient condition if the active sites are viewed as cells occupied by identical particles these cellular automaton rules represent evolution operators of systems of identical interacting particles whose total number is conserved some of these rules which allow motion in both directions mimic ensembles of one dimensional pseudorandom walkers numerical evidence indicates that the corresponding stochastic processes might be non gaussian

statistical mechanics of surjective cellular automata reversible cellular automata are seen as microscopic physical models and their states of macroscopic equilibrium are described using invariant probability measures we establish a connection between the invariance of gibbs measures and the conservation of additive quantities in surjective cellular automata namely we show that the simplex of shift invariant gibbs measures associated to a hamiltonian is invariant under a surjective cellular automaton if and only if the cellular automaton conserves the hamiltonian a special case is the well known invariance of the uniform bernoulli measure under surjective cellular automata which corresponds to the conservation of the trivial hamiltonian as an application we obtain results indicating the lack of non trivial gibbs or markov invariant measures for sufficiently chaotic cellular automata we discuss the relevance of the randomization property of algebraic cellular automata to the problem of approach to macroscopic equilibrium and pose several open questions as an aside a shift invariant pre image of a gibbs measure under a pre injective factor map between shifts of finite type turns out to be always a gibbs measure we provide a sufficient condition under which the image of a gibbs measure under a pre injective factor map is not a gibbs measure we point out a potential application of pre injective factor maps as a tool in the study of phase transitions in statistical mechanical models

convergence to equilibrium in a class of interacting particle systems evolving in discrete time we conjecture that for a wide class of interacting particle systems evolving in discrete time namely conservative cellular automata with piecewise linear flow diagrams relaxation to the limit set follows the same power law at critical points we further describe the structure of the limit sets of such systems as unions of shifts of finite type relaxation to the equilibrium resembles ballistic annihilation with defects propagating in opposite directions annihilating upon collision

realization problems for nonuniform cellular automata given a finite set of local rules the sequences built from them give a set of nonuniform cellular automata it is known 1 that for any such set of local rules the subset of nonuniform cellular automata built from them which give a surjective global map is in fact a sofic subshift for the injective sequences this set is only automatic accepted by a buchi automaton and it is not necessarily open or closed we prove the first corresponding realization result we provide for any sft a set of local rules such that the surjective injective nonuniform cellular automata over them are precisely the sequences in the sft up to a renaming of symbols in fact we prove that sfts are exactly the subshifts which are the set of injective sequences of nonuniform cellular automata for some set of local rules we also consider surjectivity of subclasses of nonuniform cellular automata and realizability questions for other properties in particular number conservation and chain transitivity

a family of sand automata we study some dynamical properties of a family of two dimensional cellular automata those that arise from an underlying one dimensional sand automaton whose local rule is obtained using a latin square we identify a simple sand automaton whose local rule is algebraic and classify this automaton as having equicontinuity points but not being equicontinuous we also show that it is not surjective we generalise some of these results to a wider class of sand automata

predecessor existence problems for finite discrete dynamical systems we study the predecessor existence problem for finite discrete dynamical systems given a finite discrete dynamical system s and a configuration c the predecessor existence or pre problem is to determine whether there is a configuration c such that s has a transition from c to c in addition to the decision version we also study the following variants the predecessor existence or pre problem counting the number of predecessors the unique predecessor existence or upre problem deciding whether there is a unique predecessor and the ambiguous predecessor existence or apre problem given a configuration c and a predecessor c of c deciding whether there is a different predecessor c of c general techniques are presented for simultaneously characterizing the computational complexity of the pre problem and its three variants our hardness results are based on the concept of simultaneous reductions single transformations that can be used to simultaneously prove the hardness of the different variants of the pre problem for their respective complexity classes our easiness results are based on dynamic programming and they extend the previous results on pre problem for one dimensional cellular automata the hardness results together with the easiness results provide a tight separation between easy and hard instances further the results imply similar bounds for other classes of finite discrete dynamical systems including discrete hopfield and recurrent neural networks concurrent state machines systolic networks and one and two dimensional cellular automata our results extend the earlier results of green sutner and orponen on the complexity of the predecessor existence problem and its variants

computation theoretic aspects of cellular automata abstract cellular automata may be viewed as computational models of complex systems they also can be seen as continuous functions on a particular family of compact metric spaces this paper surveys results largely motivated by an attempt to answer questions posed by the latter approach by means of techniques from the former

max min plus expressions for one dimensional particle cellular automata obtained from a fundamental diagram we study one dimensional neighborhood five conservative cellular automata ca referred to as particle cellular automata five particle ca5 we show that evolution equations for particle ca5s that belong to certain types can be obtained in the form of max min plus expressions from a fundamental diagram the obtained equations are transformed into other max min plus expressions by ultradiscrete cole hopf transformations which enable us to analyze the asymptotic behaviors of general solutions the equations in the lagrange representation which describe particle motion are also presented which can also be obtained from a fundamental diagram finally we discuss the generalization to a one dimensional conservative neighborhood n ca i e particle can

a second step towards complexity theoretic analogs of rice s theorem abstract rice s theorem states that every nontrivial language property of the recursively enumerable sets is undecidable borchert and stephan 1997 initiated the search for complexity theoretic analogs of rice s theorem in particular they proved that every nontrivial counting property of circuits is up hard and that a number of closely related problems are spp hard the present paper studies whether their up hardness result itself can be improved to spp hardness we show that their up hardness result cannot be strengthened to spp hardness unless unlikely complexity class containments hold nonetheless we prove that every p constructibly bi infinite counting property of circuits is spp hard we also raise their general lower bound from unambiguous nondeterminism to constant ambiguity nondeterminism

a simple class of asymptotically optimal quantizers a simple class of quantizers is introduced which are asymptotically optimal as the number of quantization levels increases to infinity with respect to a mean r th power absolute error distortion measure these asymptotically optimal quantizers are very easy to compute their performance is evaluated for several distributions and compares favorably with the performance of the optimal quantizers in all cases for which the latter have been computed in addition their asymptotic robustness is studied under location scale and shape mismatch for several families of distributions

pattern growth in elementary cellular automata a classification of elementary cellular automata ca based on their pattern growth is introduced it is shown that this classification is effective that is there exists an algorithm to determine to which class a given ca belongs this algorithm is based on the properties of the local rule of cas not requiring the observation of their evolution furthermore necessary and sufficient conditions to detect all the elementary cas exhibiting a shift like behavior are given these cas have interesting dynamical properties and chaotic characteristics

cellular automata in fuzzy backgrounds abstract the main purpose of this work is to understand some limitations introduced by the classical definitions of cellular automata ca to this end we have defined a new model of cas fuzzy ca s which allows the observation of interesting chaotic properties of elementary cas to date neither a formal nor a precise definition of chaos in cas exists we believe that the proposed model provides a sharper tool to detect which properties can be associated to a chaotic behavior we also define a measure rule entropy which gives information about the ca s dynamics solely on the basis of the rule table and provides theoretical explanations to some of the empirical observations

chaos and ergodicity are decidable for linear cellular automata over z mz n abstract we prove that important properties describing complex behaviours as ergodicity chaos topological transitivity and topological mixing are decidable for one dimensional linear cellular automata lca over z m z n theorem 6 and corollary 7 a large and important class of cellular automata ca which are able to exhibit the complex behaviours of general ca and are used in applications in particular we provide a decidable characterization of ergodicity which is known to be equivalent to all the above mentioned properties in terms of the characteristic polynomial of the matrix associated with lca we stress that the setting of lca over z m z n with n 1 is more expressive gives rise to much more complex dynamics and is more difficult to deal with than the already investigated case n 1 the proof techniques from 23 25 used when n 1 for obtaining decidable characterizations of dynamical and ergodic properties can no longer be exploited when n 1 for achieving the same goal indeed in order to get the decision algorithm algorithm 1 we need to prove a non trivial result of abstract algebra theorem 5 which is also of interest in its own we also illustrate the impact of our results in real world applications concerning the important and growing domain of cryptosystems which are often based on one dimensional lca over z m z n with n 1 as a matter of facts since cryptosystems have to satisfy the so called confusion and diffusion properties ensured by ergodicity and chaos respectively of the involved lca algorithm 1 turns out to be an important tool for building chaotic ergodic one dimensional linear ca over z m z n and hence for improving the existing methods based on them

local rule distributions language complexity and non uniform cellular automata this paper investigates a variant of cellular automata namely n ca indeed n ca are cellular automata which can have different local rules at each site of their lattice the assignment of local rules to sites of the lattice completely characterizes n ca in this paper sets of assignments sharing some interesting properties are associated with languages of bi infinite words the complexity classes of these languages are investigated providing an initial rough classification of n ca

hierarchy of fuzzy cellular automata abstract we elaborate notion of cellular automata in the directions of synchroneity asynchroneity stationarity nonstationarity and fuzzify the principle features of cellular automata as cell state transition delay of transition and choice of a local transitions function through analyzing the properties in various combinations we divide a set of fuzzy cellular automata into 14 classes to pelucide the whole structure of the classes and precise relations between them we prove the complexity hierarchy

the magic rule spaces of neural like elementary cellular automata abstract a subclass of elementary cellular automata ca equivalent to a particular class of neural networks nn is studied the nn are dependent on four real parameters a b c and the subspace of the rule space which is obtained by varying the parameters and the dynamical characteristics of the rules belonging to this subset are determined finally a few nice and magic regularities in this subset are noted and explained

asymptotic majorization of finite probability distributions this paper studies the majorization of high tensor powers of finitely supported probability distributions taking two probability distributions p and q to the n th and m th tensor power respectively in such a way that the power of q majorizes the power of p we ask how large the ratio m n can become it is shown that the supremum of such ratios is equal to the minimal ratio of the alpha renyi entropies of p and q for alpha in 0 infty consideration of this ratio of tensor powers is motivated to the author by the resource theory of quantum entanglement where the supremum of these ratios corresponds to the asymptotic conversion rate of bipartite pure quantum states under exact deterministic locc transformations

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

cluster growth in particle conserving cellular automata a class of simple two dimensional cellular automata with particle conservation is proposed for easy simulations of interacting particle systems the automata are defined by the exchange of states of neighboring cells depending on the configurations around the cells by attributing an energy to a configuration of cells we can select significant rules from the huge number of possible rules and classify them into several groups based on the analogy with a binary alloy by numerical calculations cluster growth is found in two kinds of phases which reveal gas solid coexistence and liquid droplets normalized scaling functions are obtained and dynamical scaling is examined

formal languages and global cellular automaton behavior abstract a connection is made between the description of one dimensional cellular automata as dynamical systems acting on a compact metric space and as computational systems acting on strings operators are defined which determine a bijective correspondence between subshifts of infinite configurations and languages of finite strings under this correspondence families of languages give rise to families of subshifts regular languages correspond to sofic subshifts families of subshifts corresponding to context free context sensitive and recursively enumerable languages are introduced and their closure properties under forward and backward cellular automaton images studied