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Radial View of Continuous Cellular Automata ∗
Paola Flocchini
School of Information Technology and Engineering University of Ottawa, 800 King Edward Ottawa, Ontario, K1N 6N5, Canada firstname.lastname@example.com
Vladimir Cezar
School of Information Technology and Engineering University of Ottawa, 800 King Edward Ottawa, Ontario, K1N 6N5, Canada email@example.com
Abstract. Continuous cellular automata (or coupled map lattices) are cellular automata where the state of the cells are real values in [0, 1] and the local transition rule is a real function. The classical observation medium for cellular automata, whether Boolean or continuous, is the space-time diagram, where successive rows correspond to successive configurations in time. In this paper we introduce a different way to visualize the evolution of continuous cellular automata called Radial Representation and we employ it to observe a particular class of continuous cellular automata called fuzzy cellular automata (FCA), where the local rule is the "fuzzification" of the disjunctive normal form that describes the local rule of the corresponding Boolean cellular automata. Our new visualization method reveals interesting dynamics that are not easily observable with the space-time diagram. In particular, it allows us to detect the quick emergence of spatial correlations among cells and to observe that all circular FCA from random initial configurations appear to converge towards an asymptotic periodic behavior. We propose an empirical classification of FCA based on the length of the observed periodic behavior: interestingly, all the minimum periods that we observe are of lengths one, two, four, or n (where n is the size of a configuration). Keywords: Continuous Cellular Automata, Fuzzy Cellular Automata, Visualization of dynamics, Evolution, Asymptotic Behavior.
Work partially supported by NSERC.
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
Boolean cellular automata have been introduced by Von Neumann as models of self-organizing and reproducing behaviors [22]. The study of their properties and their behavior has interested various disciplines, as diverse as ecology, biology, and theoretical computer science (e.g., [2, 10, 14, 25]). A one dimensional Boolean cellular automaton (CA) is constituted by a collection of cells arranged in an array. Each cell has a state in {0, 1} which changes at successive discrete steps by the iteration of a local transition function that depends on the states of the "neighbouring" cells. The global evolution of a CA is defined by the synchronous update of all states according to the local function applied to each cell. A configuration of the automaton is a description of all cell values at a given time. Cellular automata can be bi-infinite or finite with defined boundary conditions. A typical boundary condition consists of "wrapping around" a finite array (circular CA) or fixing a certain initial size for the array and assuming all the cells beyond the boundaries to be quiescent (zero background). Continuous cellular automata (or Coupled Map Lattices) are cellular automata where the states of the cells are real values in [0, 1], and the local transition rule is a real function. They have have been introduced by Kaneko as simple models with the features of spatiotemporal chaos, and have now applications in many different areas like fluid dynamics, biology, chemistry, etc. (for the definition of couple map lattices and their applications see [12, 13, 23]). A particular type of continuous cellular automata are Fuzzy cellular automata (FCA) where the local transition rule is obtained by "fuzzifying" the disjunctive normal form that describes the local function of the corresponding Boolean CA 1 . In other words, a FCA can be seen as a continuous generalization of a Boolean CA, where the behavior at the extreme of the interval [0, 1] is exactly the same as the behavior of the corresponding Boolean CA; vice-versa, a boolean CA can be considered a discretized version of the corresponding FCA. Fuzzy cellular automata have been introduced in [3, 4] as a framework for understanding complex behaviors. They have been employed for studying the impact that perturbations (e.g. noisy sources, computation errors, mutations, etc.) can have on the evolution of Boolean CA (e.g., see [9]), they have also been studied in relation to pattern recognition (e.g., see [15, 16]) and as a model to generate interesting images mimicking nature (e.g. [5, 20]). Moreover, dynamical properties of Fuzzy CA rules have been analytically studied (e.g., in [8, 18, 19]) to understand the relationship with Boolean cellular automata and the impact of discretization. The analytical study of these systems is generally complex; as a consequence visualization plays an important role in the understanding of their dynamics. Space-time diagrams have been traditionally used to display the dynamics of one dimensional CA: the top-most row corresponding to the initial configuration at time t = 0 and rows corresponding to configurations at successive time steps (see, for example the space-time diagram of two FCA in Figure 1). When visualizing the space-time diagram of continuous CAs, and in particular of FCAs, the interval [0,1] must be discretized, since only a finite number of states can obviously be represented. The interval is divided in k ranges and each is assigned a different colour. It has been shown in [8] that this discretization process could mislead the observer by showing a totally incorrect dynamics. This is the case, for example, of FCA rule 90 where depending on 1 the parity of k, totally different behaviors are displayed. In fact, if the fixed point ( 2 ) happens to be the extreme of a range, the diagram shows an alternation between the two different values around the fixed point; on the other hand, if the fixed point belongs to a range, the diagram shows a quick convergence to
not to be confused with a variant of cellular automata, also called Fuzzy cellular automata, where the fuzziness refers to the fuzzy choice of a deterministic local rule (e.g., see [1])
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
(a) FCA 18 (Space-Time)
(b) FCA 184 (Space-Time)
Figure 1. The first 300 iteration of two Fuzzy Cellular Automata displayed with the space-time diagram.
an homogeneous configuration. There is another aspect of space-time diagrams that could mislead the observer: colours can show nice patterns, which allow the observer to differentiate between an extremely simple dynamics and a more complex one: they however do not give insights into the way a converging rule behaves towards its attractor. In this paper we propose a new way of visualizing continuous CAs (Radial Representation) and we show that by observing their dynamics using this representation we can detect properties of their behavior that were hidden in the space-time diagram. In particular, we observe circular elementary FCA; the radial representation allows us to discover very quickly spatial correlations between neighbouring cells, which suggest the main features of the asymptotic behavior of the rule. Based on this new visualization point of view, we classify their observable behavior. We find that all FCA appear to converge towards asymptotic periodic behaviors: the only lengths of the periods that we have observed are one, two, four, and n. The paper is organized as follows: in Section 2 we give definitions and introduce some terminology, Section 3 briefly outlines some of the existing classifications of cellular automata. In Section 4 we introduce a new visualization method; in Section 5 we propose a classification based on the lengths of the periods displayed by the various rules.
Notations and Preliminaries
A one dimensional bi-infinite Boolean cellular automaton is a collection of cells arranged in a linear bi-infinite lattice. Cells have a Boolean values and they synchronously update their values according to a local rule applied to their neighborhood. A configuration X = (. . . , x−2 , x−1 , x0 , x1 , x2 , . . .) is a description of all cell values at a given time. In the case of elementary cellular automata the neighborhood of a cell consists of the cell itself and its left and right neighbours, thus the local rule has the form: g : {0, 1}3 → {0, 1}. The global dynamics of an elementary one dimensional cellular automata is then defined by: B : {0, 1}Z → {0, 1}Z s.t. ∀X ∈ {0, 1}Z , ∀i ∈ Z, B(X)i = g(xi−1 , xi , xi+1 )
A cellular automata is circular if the lattice is finite and the last cell is considered to be neighbour of
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
the first. In this case the global function has the following form, where the size of the lattice is n and the operations on the indices are modulo n: B : {0, 1}n → {0, 1}n s.t. ∀X ∈ {0, 1}n , ∀i ∈ {0, . . . , n − 1}, B(X)i = g(xi−1 , xi , xi+1 )
In the following we concentrate only on elementary, circular cellular automata. The local rule g : {0, 1}3 → {0, 1} of an elementary Boolean CA can be also given in tabular form by listing the 8 binary tuples corresponding to the 8 possible local configurations a cell can detect in its direct neighborhood, and mapping each tuple to a binary value qi (0 ≤ i ≤ 7): (000, 001, 010, 011, 100, 101, 110, 111) → (q0 , · · · , q7 ). The binary representation (q0 , · · · , q7 ) is often converted into the decimal representation i=1:8 2i−1 qi , and this value is used as the name of the rule. The local rule can also be canonically expressed as a disjunctive normal form: g(v1 , v2 , v3 ) =
i|qi =1 j=1:3 0 1 where dij is the j-th digit, from left to right, of the binary expression of i, and vj (resp. vj ) stands for ¬vj (resp. vj ). For example, the canonical expression of rule 18 = 2 + 16, which is expressed in tabular form as: (000, 001, 010, 011, 100, 101, 110, 111) → (0, 1, 0, 0, 1, 0, 0, 0), is: g(v1 , v2 , v3 ) = (¬v1 ∧ ¬v2 ∧ v3 ) ∨ (v1 ∧ ¬v2 ∧ ¬v3 ). In continuous cellular automaton (also known as Coupled Map Lattices [12]) the states assumed by the cells are real values in [0, 1] and the local rule is any real function. In the case of one-dimensional, circular elementary continuous CA the local rule has the form f : [0, 1]3 → [0, 1], and the corresponding global function is as follows (where the operations on the indices are modulo n):
vj ij
F : [0, 1]n → [0, 1]n
s.t. ∀X ∈ [0, 1]n , ∀i ∈ {0, . . . , n − 1}, F (X)i = f (xi−1 , xi , xi+1 )
A Fuzzy cellular automaton is a particular continuous cellular automata where the local rule is obtained by "fuzzification" of the local rule of a classical Boolean CA. The "fuzzification" consists of a fuzzy extension of the boolean operators AND, OR, and NOT. Depending on which fuzzy operators are used different types of Fuzzy cellular automata can be defined. Among the various possible choices, we consider the following: (a ∨ b) is replaced by (a + b); (a ∧ b) by (ab), and (¬a) by (1 − a). The resulting local rule becomes the following real function that generalizes the original function on {0, 1}3 , where l(a, 0) = 1 − a and l(a, 1) = a: f : [0, 1]3 → [0, 1] s.t. , f (v1 , v2 , v3 ) =
l(vj , di,j )
The usual fuzzification of the expression (a ∨ b) would be max{1, (a + b)} so as to ensure that the result is not larger than 1. Note however, that taking (a + b) for the CA fuzzification does not lead to values greater than 1 since the maximum possible sum is 1 and occurs for rule 255 which contains the sum of all the expressions (000, 001, 010, 011, 100, 101, 110, 111) → (1, 1, 1, 1, 1, 1, 1, 1); any other CA rule is a partial sum and must be bounded by 1.
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
Example 1: Fuzzification. Consider, for example, rule 18 whose canonical Boolean expression is: g(x, y, z) = (¬x ∧ ¬y ∧ z) ∨ (x ∧ ¬y ∧ ¬z). The fuzzification process for the corresponding FCA yields: f (x, y, z) = [(1 − x) · (1 − y) · z] + [x · (1 − y) · (1 − z)] = (1 − y) · (x + z − 2yz). A configuration of a fuzzy cellular automata at a given time consists of the values assumed by the cells at that time: we shall indicate with X 0 = (x0 , x0 , . . . , x0 ) an initial configuration, and with X t 0 1 n−1 a configuration at time t. Definition 2.1. A fixed point P ∈ [0, 1]N for a FCA with global rule F : [0, 1]n → [0, 1]n is a configuration P such that F (P ) = P . Definition 2.2. A rule has a (right) shift behavior if there is a time T such that ∀t > T , F (X t+1 ) = F (x0 , x1 , . . . , xn−1 ) = (x1 , . . . , xn−1 , x0 ). Definition 2.3. A rule is Temporally Periodic with period p if ∀ > 0, ∃T such that ∀t > T : F (X t ) = F (X t+p ) Definition 2.4. A rule is Spatially Periodic with period p if ∀ , ∃T such that ∀t > T , ∀i: xt = xt i i+p Notice that a rule with shifting behavior is temporally periodic with period n; moreover, a spatially periodic rule with period p with shifting behavior is also temporally periodic with the same period p. Definition 2.5. A rule is Asymptotically Periodic in time (or Asymptotically Temporally Periodic) with period p if ∀ > 0 ∃T such that ∀t > T and ∀i |xt − xt+p | < i i Definition 2.6. A rule is Asymptotically Periodic in space (or Asymptotically Spatially Periodic) with period q if ∀ ∃T such that ∀t > T and ∀i |xt − xt | < i i+q A rule which is asymptotically periodic in space with period 1 will be called asymptotically homogeneous. A rule which is asymptotically periodic in time with period 1 will be said to be convergent to a fixed point. n In the following, when a configuration at time t is spatially periodic with smallest period m , we shall t = (α)m , where α = a , . . . , a n is the sequence of values corresponding to the smallest indicate it as X 1 m period. For example, a configuration X = (a, b, c, d, a, b, c, d) of a circular CA of size n = 8 is spatially periodic of minimum period four, and can be indicated as X = (a, b, c, d)2 .
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
Classifications of CA behaviors
The classification of cellular automata has always been considered a fundamental problem. The first attempt to classify cellular automata has been done by Wolfram in [24] where cellular automata are grouped according to the observed behavior of their space-time diagram as follows: - Class 1: automata that evolve to a unique, homogeneous state, after a finite transient, - Class 2: automata whose evolution leads to a set of separated simple stable or periodic structures (space-time patterns), - Class 3: automata whose evolution leads to aperiodic ("chaotic") space-time patterns, - Class 4: automata that evolve to complex patterns with propagative localized structures, sometimes long-lived. Although not formally precise, this classification captures important distinctions among cellular automata. Several other criteria for grouping CA have followed: some based on observable behaviors, some on intrinsic properties of CA rules (e.g., see [6, 7, 11, 21]). Since class membership is undecidable, the observation of the evolution of a CA starting from (possibly all) initial configurations becomes crucial to understand its dynamics. Fuzzy cellular automata have been observed using the classical space-time diagram, where colours represent intervals of real values. Fuzzy cellular automata in boolean backgrounds have been grouped according to their observed behavior in [3, 4] and, in particular, according to the level of "spread" of fuzziness. Essentially the possible observed behaviors are three: 1) Boolean values are destroyed by Fuzzy values; 2) Fuzzy values are destroyed by Boolean values; 3) Fuzzy and Boolean values co-exist forming various patterns in the space-time diagram. Circular Fuzzy cellular automata have never been classified; from the observation of their space-time diagram it is clear that some evolve to a fixed-point, some present a shifting behavior, while some display "nice" patterns (for some examples of patterns see Figures 1). Interestingly, the behavior of Fuzzy cellular automata does not follow the behavior of their boolean counter-part: in fact, rules belonging to the same Wolfram-class can have, in some cases, an observable dynamics very different in the Fuzzy version (see Figure 2). In [8] the reasons for such differences are analytically explained for the case of rule 90. Because of the complex patterns generated by some Fuzzy cellular automata in the space-time diagram, some rules were thought to be more complex than others, possibly having chaotic asymptotic dynamics. The only analytical study of Fuzzy cellular automata has been done by Mingarelli, who conducted a comprehensive analysis of FCA in quiescent backgrounds showing that none of them has a chaotic dynamics ([17, 18, 19]). No studies have been done for the circular case.
Radial Representations
In the following we propose a new way to visualize circular continuous cellular automata. Let k be the number of cells. Consider a unitary circle with center C divided in k equal sectors corresponding to radii r0 , . . . rk−1 . Each cell of the CA corresponds to a radius ri , its value xi corresponds to distance xi from C in ri and is represented by a dot in the corresponding position. For example, cells with value 1 will have a dot in correspondence of the circumference, cells with value 0 will have a dot in the center, and any value between 0 and 1 will have its representational dot plotted inside the circle. A
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
(a) Rule 90 Boolean
(b) Rule 90 Fuzzy
(c) Rule 18 Boolean
(d) Rule 18 Fuzzy
Figure 2. Two rules that display similar behavior in the Boolean setting and very different in the Fuzzy setting.
configuration of the CA then corresponds to a plotting on the circle of the various distances. When we connect the dot on radius ri to the ones on the two neighbouring radii ri±1 (the operations on indices are modulo n) we say that the Radial representation is open, it is closed otherwise. Finally, we also consider a linear version of the radial representation by arranging the values xi on a line (see Figure 3 for an example of the three variants).
Figure 3. Radial representations (open, close, and linear) of a random initial configuration.
As opposed to the space-time diagram, with a radial representation we cannot observe the "life" of a CA in a 2-dimensional picture. In fact, the evolution of a CA corresponds to the time sequence of the plottings of the configurations. In other words, a radial representation provides a dynamic diagram where the observer can see the evolution through time in frames where each frame corresponds to a
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
global configuration in an instant. Example 3: Radial Representations of FCA. Consider FCA rule R110 . An example of closed radial representation is given in Figure 4, where a sequence of configurations of Rule R110 are displayed. In this case the rule reaches quickly to an homogeneous configuration and already after 20 iterations it is clear that all the dots corresponding to the FCA's states are placed on a circle.
(a) Initially
(b) After 3 iterations
(c) After 20 iterations
Figure 4. Some configurations in the evolution of rules R110 with the closed radial representation
Another, perhaps more interesting, example of radial representations is shown in Figure 5 where FCA rule R18 is displayed with both the open and the closed radial representations at different times in the evolution. It is interesting to observe a very quick formation of two curves during the evolution revealing a spatial correlation between the cells that becomes observable with the space-time diagram only after a very large number of iterations.
(a) Initially
(b) After 5 iterations
(c) After 51 iterations
(d) After 370 iterations
Figure 5. Some configurations in the evolution of rules R18 with the open and closed radial representations
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
Experimental Classification of FCA using Radial Representations
In this Section we group elementary circular Fuzzy cellular automata according to their behavior when observed with the radial representations starting from random initial configuraitons. We observed all 256 elementary CAs; the results below refer to the 90 that are not equivalent under simple transformations (conjugation, reflection, and combination of conjugation and reflection). We executed each FCA starting from 50 different random initial configuration for lattices of size 45,90,180. In each initial configuration X 0 = x0 , x0 , . . . , x0 , the states x0 are randomly selected in the interval [0,1]. 0 1 n−1 i The observation of FCA on a computer obviously involves a discretization of the real values, and the values that we can observe are only approximations of the actual ones. Moreover, the calculations are all performed up to a certain decimal digit thus involving rounding and precision errors. In particular, we work with java doubles and truncate our observations at 16 decimal digits. Note that, due to limitations of precision, in any simulation rules that are asymptotically periodic (in time or in space) with some period p will appear to be truly periodic with period p after some initial time. Period 1 (Quiescent) Period 1 (Homogeneous) Period 1 (Heterogeneous) Period 1 for n even (Heterogeneous) Period 2 for all n Period 2 for n even Period 4 for n multiple of 4 Period n (Shifts) R0 , R8 , R32 , R40 , R72 , R104 , R128 , R136 R160 , R168 , R24 , R36 , R152 , R164 , R44 , R56 R74 , R200 R6 , R9 , R22 , R25 , R26 , R30 , R33 , R35 , R37 , R38 , R41 , R45 , R54 , R57 R60 , R61 , R62 , R73 , R90 , R105 , R106 , R110 , R122 , R126 R134 , R150 , R154 R172 . R4 , R12 , R13 , R76 , R77 , R132 , R140 , R204 , R232 R28 R108 , R156 R78 , R94 R1 , R5 , R19 , R23 , R27 , R50 , R51 R178 R18 , R29 , R58 , R146 , R184 R46 R2 , R10 , R15 , R34 , R42 , R130 , R138 , R162 ,R170 R3 , R7 , R11 , R14 , R43 , R142
Table 1. Observed dynamics of Circular Elementary Cellular Automata
In our observations, all rules eventually display periodic behaviors with periods smaller than n, which leads us to speculate that they all have an asymtotically periodic behavior. Interestingly, the lengths of the periods that we have observed are only: p = 1, p = 2, p = 4 and p = n. In the rest of the paper we group the elementary FCAs on the basis of their observed period. As mentioned before, an asymptotic behavior will necessarily be observed as an exact behavior after a certain time due to discretization, while the reverse is not necessarily true. We however conjecture that when we observe
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
periodicity in a FCA, its asymptotic behavior is indeed periodic. In the following, whenever we say that a rule "appears to converge" or "seems to be asymtotically periodic" our statements are only based on empirical observations.
Periods of Length 1: Fixed Points
Homogeneous Configurations
In the majority of Fuzzy cellular automata rules the observed configurations become homogeneous in time, suggesting convergence to an homogeneous fixed point. In all cases, it can be easily shown that the homogeneous configuration reached by each rule is indeed its homogeneous fixed point, i.e., the solution of F (P ) = P when starting from an homogeneous configuration X 0 = (x, x, . . . , x). Fixed points for all rules starting from homogeneous configurations can be easily calculated (a table containing all of them can be found in [17]). Among the rules that reach a quiescent configuration we observe that for rules R0 , R8 , R32 , R40 , R72 , R104 , R128 , R136 , R160 , R168 , R200 this is evident after a very small small number of iterations (less then 100 when n = 180), while for rules R24 , R36 , R152 , R164 it takes considerable more time for the rule to stabilize; finally, for rules R44 R56 and R74 the quiescent state is reached extremely slowly. In our experiments, for several rules the dots quickly arrange in a circle; i.e., the values reach an homogeneous configuration different from zero (see an example in Figure 4). This is the case of Rules R6 , R9 , R22 , R25 , R26 , R30 , R33 , R35 , R37 , R38 , R41 , R45 , R54 , R57 , R60 , R61 , R62 , R73 , R90 , R105 , R106 , R110 , R122 , R126 R134 , R150 , R154 R172 . 5.1.2. Non-homogeneous Configurations
In the case of rule R204 (f (x, y, z) = y), we have that any inital configuration is a fixed point (∀X ∈ [0, 1]n , F (X) = X). Other rules that appear to have an asymtotic convergence to a non-homogeneous fixed point after a few iterations are: R4 , R12 , R13 , R76 , R77 , R132 , R140 , R232 , R28 R108 , R156 .
(a) FCA 4 (Radial)
(b) FCA 10 (Radial)
Figure 6.
(a) a fixed point for rule R4 observed after 364 steps (b) a shifting (i.e. rotating) configuration for rule R10 , observed after 364 steps.
Finally, Rules R94 and R78 merit special attention and are separately described below. Both rules appear to converge to a fixed point, which is homogeneous when the size is odd, while it consists of a periodic configuration, when the size is even. During the evolution, the radial representations show
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
very quickly a spatial correlation between cells through the formation of two co-existing curves with consecutive values belonging to different curves, eventually stabilizing in two concentric circles (see, for Example, two moments in the evolution of Rule R78 in Figure 8).
Period 1 (Fixed Points) f94 (x, y, z) n even n odd f78 (x, y, z) n even n odd x + y − xy + z − 2xz − yz + xyz (a, b) 2 with 2a + 2b − ab = 2 √ (2 − 2)n y + z − xz − yz (a, b) 2 with a + b = 1 ( 1 )n 2
n n n
The Case of Rule R94 . Our simulations show that rule R94 reaches fixed points of the type (a, b) 2 , with 2a + 2b − ab = 2 and a = b, suggesting that it is asymptotically periodic both in time and space. The following Property shows that an exact configuration of this type (which is possibly never reached by the rule) is indeed a fixed point.
Figure 7. Rule R94 after 5000 iteration with 90 cells.
Property 5.1. Consider rule R94 on a configuration of even size n. Let X t = (a, b) 2 . Configuration X t is a fixed point if and only if 2a + 2b − ab = 2. Proof: Rule R94 has the following analytical form: f (x, y, z) = x + y − xy + z − 2xz − yz + xyz. Let a, b, a be three consecutive values in configuration X t ; the only way to obtain a fixed point dynamic with configuration X t is when the local function f satisfies the condition f (a, b, a) = b (and f (b, a, b) = a). We have that f (a, b, a) = a + b − ab + a − 2a2 − ab + a2 b = 2a + b − 2ab − 2a2 + a2 b, thus f (a, b, a) = b when either a = 0 or 2a + 2b − ab = 2. When 2a + 2b − ab = 2, we also have that f (b, a, b) = b and n thus this condition indeed guarantees that (a, b) 2 is a fixed point. When a = 0 it must be f (b, 0, b) = 0, which is verified only when b ∈ {0, 1}. It is easy to see that the homogeneous configuration (0)n is n repelling, the remaining spatially 2-periodic fixed point (0, 1) 2 satisfies also condition 2a + 2b − ab = 2.
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
On the other hand, if the size of the CA is odd, the two curves merge into a single circle suggesting √ asymptotic convergence to an homogeneous fixed point (2 − 2)n , which is the only attracting homogeneous fixed point of rule R94 . The Case of Rule R78 . During the evolution the behavior of the CA is shifting (i.e., the two curves n rotate); however, the CA eventually reaches a spatially periodic fixed point of the form (a, b) 2 , with (a + b) = 1 and a = b, suggesting an asymptotically periodic behavior both in time and space. We can n easily verify that (a, b) 2 is indeed a fixed point for rule R78 under the condition (a + b) = 1.
(a) Rule 78 during the evolution
(b) Rule 78 at the convergence point
Figure 8. Rule R78 with 90 cells observed with Radial representations (a) after 2500 iterations (b) after 14,000 iterations.
Property 5.2. Consider rule R78 on a configuration of even size n. Let X t = (a, b) 2 with a = b. Configuration X t is a fixed point if and only if a + b = 1. Proof: The proof is analogous to the one of property 5.1. Rule R78 has the following analytical form: f (x, y, z) = y + z − xz − yz. Let a, b, a be three consecutive values in configuration X t . We have that f (a, b, a) = b for b + a − a2 − ab = b, which implies that either a + b = 1 or a = 0. In the first case we also have f (b, a, b) = a, in the second case, we must have that b ∈ {0, 1}. Configuration (0)n is repelling, the n remaining spatially 2-periodic fixed point (0, 1) 2 satisfies condition a + b = 1 as well. As for the case of rule R94 , rule R78 reaches an homogeneous configuration when starting our simula1 tions from configurations of odd size. The reached configuration ( 2 )n is the only attracting homogeneous fixed point of the rule. Thus, also in this case, the observed behavior suggests an asymptotic convergence to an homogeneous fixed point.
Periods of Length Two
Some rules show a periodic behavior of length two for any value of n (the size of the initial configuration), while some other rules display this periodicity only when n is even while they reach an homogeneous configuration when n is odd. This is the case of rules where the configuration become spatially periodic in time and the temporal periodicity is actually given by a shifting behavior.
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
Periodicity for all sizes of the initial configuration.
The simplest rule with obvious periodic behavior is the complement rule R51 (f (x, y, z) = 1 − y). Other rules that quickly show a periodic behavior of length two are R1 , R5 , R19 , R23 , R50 , R178 . Rule R27 also become periodic of length two, but it takes a longer time. For this rule, each configuration in the period is homogeneous and is the complement of the previous (i.e., X t = (a)n , X t+1 = (1 − a)n , . . .). This happens regardless of the parity of the initial configuration. It is indeed very simple to see that if X t = (a)n , the next configuration is its complement. In fact, it comes form the observation that, for rule R27 , whose analytical expression is f (x, y, z) = 1 − y − xz + yz we have that f (a, a, a) = (1 − a). 5.2.2. Period two for configurations of even size only
The observed periodic behavior of the rules described below (except for rule R29 ) is actually a shifting behavior occurring on spatially periodic configurations with period two, which causes the behavior to be spatially and temporally 2-periodic. Due to this reason, this dynamics is visible only if the configuration is of even size. Interestingly, if the size is odd, all the rules reach an homogeneous fixed point. From a pictorial point of view, when the size of the configuration is even, after a few time steps they all display two co-existing curves with consecutive values belonging to different curves (see, for an example, Figure 10), and they eventually form two concentric rotating circles. The case of rule R29 is similar in the sense that it also shows spatially periodic configurations of period two, when n is even, and a temporal behavior of period two; in this case, however, the shift-like behavior is slightly more complicated since a value xt becomes the complement of the neighbour xt−1 i i+1 at the previous time step.
Spatially and Temporally periodic f18 n even n odd f58 n even n odd x − xy + z − 2xz − yz + 2xyz (a, b) with 2a + 2b − 2ab = 1
n 2
Spatially and Temporally periodic f146 n even n odd f184 n even n odd x − xy + z − 2xz − yz + 3xyz (a, b) 2 with 2a + 2b − 3ab = 1 ( 1 )n 3 x − xy + yz (a, b) 2 , ∀a, b (a)n , ∀a
n n
(1 −
n 2
2 2 )
x − xy + z − xz (a, b) with a + b = 1 ( 1 )n 2
Shift of Complement f29 n even n odd 1 − z + yz − xy (a, b) 2 ∀a, b (a)n , ∀a
The Case of Rules R18 , R146 , R58 . When the configuration size of is even, these fuzzy cellular automata form two co-existing rotating curves that eventually form two concentric circles with consecutive values belonging to different curves. In other words, the observed configurations become of the form n n (a, b) 2 with a = b (alternating with (b, a) 2 ), thus suggesting an asymptotic periodic behavior both in
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
space and time (see, for example, three different moments in the evolution of Rule R58 ). We can show n that, for these rules, an exact configuration X t = (a, b) 2 has indeed a shifting behavior (and thus a periodic behavior of length two) for a certain relationship between a and b, which depends on the rule.
Figure 9. Three moments in the evolution of Rule R58 with the Radial and the Bar representations.
For example, in the case of rule R18 the observed configurations become of the form (a, b) 2 with n a + b − ab = 1 ; indeed the following property shows that for rule R18 , X t = (a, b) 2 has a shifting 2 n behavior (and thus a periodic behavior of length two) only when a + b − ab = 1 or X t = (0, 1) 2 , 2 analogous properties can be shown for the other rules of this class. Property 5.3. Consider rule R18 on a configuration of even size n. Let X t = (a, b) 2 with a = b. n Configuration X t+1 = (b, a) 2 if and only if a + b − ab = 1 or a = 0, b = 1. 2 Proof: Rule R18 has the following analytical form: f (x, y, z) = x − xy + z − 2xz − yz + 2xyz. Let a, b, a be three consecutive values in configuration X t ; in order for the global dynamics to be shifting, the local rule f must verify the condition f (a, b, a) = a (and f (b, a, b) = b). We have that f (a, b, a) = 2a − 2ab − 2a2 + 2a2 b, which means that it must be either a + b − ab = 1 or a = 0. Condition 2 a + b − ab = 1 verifies also f (b, a, b) = b. If a = 0, solving f (b, 0, b) we have that b must be either 2 0 or 1. Configuration (0)n is a repelling homogeneous fixed point, so the only two spatially 2-periodic n n shifting configurations are (a, b) 2 with a + b − ab = 1 , and (0, 1) 2 . 2 Although configuration (0, 1) 2 is also a fixed point for rule R18 , we never observe it in our experiments n from random initial configurations, which always appear to converge to (a, b) 2 for some a and b such that a + b − ab = 1 . 2 n Analogously to the case of rule R18 , it can be shown that a shifting configuration of the form (a, b) 2 must verify condition 2a + 2b − 3ab = 1 for rule R146 , and condition a + b = 1 for rule R58 . When the configuration size is odd, the radial diagram of these rules still show two co-existing curves with consecutive values belonging to different curves in all places except for one. The two curves eventually merge into one and the rule reaches an homogeneous configuration. Thus, our experiments starting from random configurations suggest asymptotic behavior that becomes homogeneous in space √ and converges to a fixed point in time: rule R18 appears to asymptotically converge to (1 − 22 )n , rule 1 R146 to ( 3 )n , and rule R58 to ( 1 )n . It can be easily shown that these are the only attracting homogeneous 2 fixed points for these rules.
n n
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
The Case of Rule R184 . Rule R184 has the following analytical form: f (x, y, z) = x − xy + yz. This rule is special in many ways. First of all, when starting from homogeneous configurations, any configuration is a fixed point (this is due to the fact that f (a, a, a) = a, ∀a). Furthermore, any 2-periodic configuration gives rise to a periodic behavior (this is due to the fact that f (a, b, a) = a, ∀a, b). As before, when the size of the CA is even, the radial diagram shows 2 co-existing curves with consecutive values belonging to different curves thus reaching a spatially and temporally 2-periodic n configuration (a, b) 2 (i.e., to two rotating concentric circles, as shown in Figure 10). However, the values of a and b are not linked to each other. From the experiments it appears that the CA can be n attracted, depending on the initial configuration to various shifting configuration of the type (a, b) 2 .
Figure 10. Rule R184 after 5500 iteration with a configuration of 90 cells.
When the size of the FCA is odd the rule reaches, in our simulations, an homogeneous configuration (a)n , suggesting an asymptotic behavior that converges to a fixed point in time and becomes homogeneous in space. The value of a is variable and appears to depend on the initial configuration. The Case of Rule R29 . Also this rule has a unique behavior. When the size of the FCA is even, the n observed rule reaches a spatially 2-periodic configuration of the type (a, b) 2 and then alternates with the n shift of its complement (1 − b, 1 − a) 2 still giving rise to a period of length 2. Rule R29 has the following analytical form: f (x, y, z) = 1 − xy − z + yz, which means that f (a, b, a) = 1 − a for any values of a n and b. As a consequence (a, b) 2 is clearly a spatial and temporal period-2 configuration. When the size of theF CA is odd the rule reaches a periodic behavior of the type X t = (a)n , X t+1 = (1 − a)n , . . .. The value of a varies and depends on the initial configuration.
Periods of Length 4: Rule R46
There is only one rule in this class: rule R46 . During the evolution, we observe the formation of four co-existing rotating curves that eventually form four concentric circles (see Figure 11). Similarly to the cases of the rules of the previous subsection, the periodic behavior is due to the shift in time of spatially n 4-periodic configurations of the type (a, b, 1 − a, 1 − b) 4 , thus the requirement of configurations with size multiple of 4. Spatially periodic Shift f46 (x, y, z) n multiple of four n not multiple of four y − xy + z − yz (a, b, c, d) with c = 1 − a, d = 1 − b ( 1 )n 2
n 4
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
Figure 11. Some configurations in the evolution of rules R46 with the Radial representation.
In the following we show that, when the size of the configuration is multiple of four, the four-periodic n configuration (a, b, c, d) 4 is a shifting configuration (and thus resulting in a periodic behavior of length four) only when c = 1 − a and d = 1 − b). Property 5.4. Consider rule R46 on a configuration of size n = 4m. Let X t = (a, b, c, d)m with a = b, then X t+4 = X t iff c = 1 − a and d = 1 − b. Proof: The analytical form of rule R46 is f (x, y, z) = y − xy + z − yz. In order for the configuration to be of the form (a, b, c, d)m , we must have that: f (a, b, c) = c, f (b, c, d) = d and f (c, d, a) = a. Solving the equations, we have that b − ab + c − bc = c for b = 0 or a + c = 1; c − bc + d − cd = d for c = 0 or b + d − 1; d − cd + a − da = a for d = 0 or c + a = 1. Thus, the only spatially 4-periodic shifting configuration (with a = b) is X t = (a, b, 1 − a, 1 − b)m . When the configuration size is even but not a multiple of four or when it is odd, we observe that the CA reaches (very slowly) its only homogeneous fixed point (1/2)n , suggesting an asymptotic convergent behavior (both in time and in space).
Periods of Length n: Shifts and Double Alternating Shifts
Rules R170 (f (x, y, z) = z) is the perfect shift and the radial representation shows a perfect rotation of the initial configuration. A shifting behavior after a short transient is displayed also by rules R2 , R10 , R34 , R42 , R130 , R138 , R162 (Stars). An example of shifting configuration for rule R10 is shown in Figure 6, where the configuration rotates as indicated by the arrows. Rule R15 (f (x, y, z) = (1 − x)) is the complement of a perfect shift: since every configuration is the complement of the shift of the previous, we observe what we call an Double alternating Shift or a shifting periodic behavior: every other configuration is shifted by two positions2 . The radial representation clearly shows both the periodic and the shifting nature of the rule. Similar but more complex behavior have rules R3 , R7 , R11 , R14 , R43 , R142 . In these cases, depending on the initial configuration, and on its size, the dynamics could differ. Consider, for example, the case of rule R43 . For large values of n the
We have seen another double alternating shift, Rule R29 , in subsection 5.2.2.
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
rule appears to converge always to a periodic behavior of period n from a random initial configuration; however, for smaller values of n, under certain circumstances, it could actually become totally Boolean n stabilizing in a Boolean shifting spatially periodic configuration (1100) 4 , thus resulting in a temporal periodic behavior of period four. The detailed analysis of this class of rules is left for subsequent research.
Concluding Remarks
In this paper we have described an experimental classification of circular fuzzy cellular automata based on a new visualization method. The different visualization method has allowed us to observe dynamics that were not easily detectable with the classical space-time diagram. In particular, it has allowed us to see the quick emergence of spatial correlation between cells that was not evident from the space time diagram (compare, for example, snapshots of configurations during the evolution of some rules observed with the classical space-time diagram, and with the Radial representations in Figure 12). We have observed that all circular elementary FCA from random initial configurations of size n display at some point a periodic dynamics with period p ≤ n, and we have grouped them on the basis of the length of their periods. Interestingly, the only observed periods lengths are one, two, four, and n. In the most interesting cases we have also analytically verified that the configurations reached by the FCA in our experiments are indeed periodic points. The empirical observations lead us to conjecture that all circular elementary rules have an asymptotic periodic behavior of the observed length. Several problems are now under investigation. First of all we are now conducting a detail study of each rule to analytically show that the periodic behavior that we observe are indeed asymptotic periodic behaviors and not simply due to discretization. We are also investigating the reasons for FCA to have these particular period lengths. The classifications of infinite configurations and configurations in zero backgrounds are also under investigation. Finally, radial representation has been employed in this paper to observe FCA only; it would be interesting to see if it can be useful in the observation of other types of continuous cellular automata
[1] Adamatzky, A. I.: Hierarchy of Fuzzy Cellular Automata, Fuzzy Sets and Systems, (62), 1994, 167–174. [2] Boccara, N., Cheong, K.: Automata Network Epidemic Models, Cellular Automata and Cooperative Systems (N. Boccara, E. Goles, S. Martinez, P. Picco, Eds.), 396, Kluwer Academic Publishers, 1993. [3] Cattaneo, G., Flocchini, P., Mauri, G., Quaranta-Vogliotti, C., Santoro, N.: Cellular Automata in Fuzzy Backgrounds, Physica D, 105, 1997, 105–120. [4] Cattaneo, G., Flocchini, P., Mauri, G., Santoro, N.: Fuzzy cellular automata and their chaotic behavior, Proc. International Symposium on Nonlinear Theory and its Applications, 4, IEICE, 1993. [5] Coxe, A., Reiter, C.: Fuzzy hexagonal automata and snowflakes, Computers and Graphics, 27, 2003, 447– 454. [6] Culik II, K., Hurd, L. P., Yu, S.: On the limit sets of cellular automatas, SIAM Journal on Computing, 18, 1989, 831–842. [7] Fat` s, N.: Experimental study of elementary cellular automata dynamics using the density parameter, Dise crete Models for Complex Systems, DMCS'03, in Discrete Mathematics and Theoretical Computer Science Proceedings AB , DMTCS 2003, 2003.
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
[8] Flocchini, P., Geurts, F., Mingarelli, A., Santoro, N.: Convergence and aperiodicity in fuzzy cellular automata: revisiting rule 90, Physica D, 42, 2000, 20–28. [9] Flocchini, P., Santoro, N.: The chaotic evolution of information in the interaction between knowledge and uncertainty, Complex Systems, Mechanism of Adaptation (R. J. Stonier, X. H. Yu, Eds.), IOS Press, 1994. [10] Garzon, M.: Models of Massive Parallelism. Analysis of Cellular Automata and Neural Networks, SpringerVerlag, 1995. [11] Gutowitz, H. A.: A hierachical classification of cellular automata, Physica D, 45, 1990, 136–156. [12] Kaneko, K.: Theory and Application of Coupled Map Lattices, John Wiley & Sons Ltd, 1993. [13] Keller, G., Kunzle, M., Nowiki, T.: Some phase transitions in coupled map lattices, Physica D, 59, 1992, 39–51. [14] Langton, C. G.: Studying artificial life with cellular automata, Evolution, Games, and Learning, North Holland, 1986. [15] Maji, P., Chaudhuri, P. P.: Fuzzy cellular automata for modeling pattern classifier, IEICE Transactions on Information and Systems, 88(4), 2005, 691–702. [16] Maji, P., Chaudhuri, P. P.: RBFFCA: A Hybrid Pattern Classifier Using Radial Basis Function and Fuzzy Cellular Automata, Fundamenta Informaticae, 78(3), 2007, 369–396. [17] Mingarelli, A.: A classification scheme for fuzzy cellular automata with applications to ECA, I, 2007, Manuscript. [18] Mingarelli, A.: The global evolution of general fuzzy automata, Journal of Cellular Automata, 1(2), 2006, 141–164. [19] Mingarelli, A.: A study of fuzzy and many-valued logics in cellular automata, Journal of Cellular Automata, 1(3), 2006, 233–252. [20] Reiter, C. A.: Fuzzy automata and life, Complexity, 7(3), 2002, 19–29. [21] Sutner, K.: Classifying circular cellular automata, Physica D, 45, 1990, 386–395. [22] Von Neumann, J.: Theory of Self-Reproducing Automata, University of Illinois Press, Urbana, 1966. [23] Weimar, J.: Cellular Automata for reaction-diffusion systems, 1997, 23(11), Parallel computing, 1699–1751. [24] Wolfram, S.: Universality and complexity in cellular automata, Physica D, 10, 1984, 1–35. [25] Wolfram, S.: Theory and Applications of Cellular Automata, World Scientific, 1986.
P. Flocchini, V. Cezar / Radial View of Continuous Cellular Automata
(a) Rule 18 Space-Time
(b) Rule 18 Open Radial
(c) Rule 18 Closed Radial
(d) Rule 184 Space-Time
(e) Rule 184 Open Radial
(f) Rule 184 Closed Radial
(g) Rule 46 Space-Time
(h) Rule 46 Open Radial
(i) Rule 46 Closed Radial
Figure 12. Rules R18 , R184 , and R46 visualized with the classical space-time diagram in (a) (d) (g), and a configurations in their evolution visualized with the Radial representations. In all cases, the configuration has 180 cells.
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Jun. 10, 2013, 3:56 AM
The Nikkei (NKY) continued with its wild ways today, surging 4.9% after Japanese Q1 GDP growth was revised upwards, and U.S. jobs data came in just right (not too good, not too bad) and helped boost the dollar vs the yen. Hong Kong +0.2%, China closed, India +0.1%. USD-JPY (FXY) +0.9%.| Jun. 10, 2013, 3:56 AM | 2 Comments
Jun. 9, 2013, 9:58 PMJapan revises Q1 GDP up to 4.1% (annualized) from an initial reading of 3.5% sending the Nikkei (EWJ, DXJ) up 3% in early trading. Also helping Japanese stocks is the weak yen (FXY), down 0.4% against the dollar to ¥97.95. Meanwhile, the BOJ begins a two-day policy meeting. Goldman's advice: Initiate "long positions in Nikkei September futures with a target of 14,500 and a stop on a close below 12,700." For what it's worth. | Jun. 9, 2013, 9:58 PM | 6 Comments
Jun. 7, 2013, 7:14 AM
Long Nikkei, short yen continues to unwind with investors in 2013's most popular trade all hitting the exit at the same time. The yen's up another 1.6%, with dollar/yen at ¥95.47 and erasing more than 2 months worth of gains in the last few sessions. FXY +1.3% premarket. The Nikkei tumbled another 1.8% overnight and has also erased 2 months worth of gains. DXJ -2.3% premarket. Earlier: Japan's government pension fund confirms it's cutting JGB exposure and increasing equity holdings.| Jun. 7, 2013, 7:14 AM | 1 Comment
Jun. 7, 2013, 4:08 AMAsian markets fall as investors position themselves for weak U.S. payroll figures today and for Chinese economic data over the weekend. Japanese shares (NKY) enjoy another roller coaster day but end a mere -0.2%, while the dollar-yen is -0.4% as the Japanese currency continues to strengthen. In China, stocks (FXI, CAF) drop for a seventh day in a row amid fears about economic growth and tightening liquidity, while Hong Kong's (EWH) fall of -1.3% puts it in and around correction territory. India +0.5%. | Jun. 7, 2013, 4:08 AM | 1 Comment
Jun. 7, 2013, 12:35 AMAsked and answered: Midday Thursday (U.S. time), Forbes wondered if "Soros and the gang .... have unwound their bets on Japan." Friday morning (Asia time), the answer via Dow Jones Newswire: Soros is "back placing bets in Japan, shorting the yen and snapping up local stocks." | Jun. 7, 2013, 12:35 AM
Jun. 6, 2013, 1:11 PM
When crowded trades go bad: Investors don't wait until Tokyo to sell Japan - Nikkei futures tumble 3.5% as the yen flies higher, now at ¥96.76 and in two weeks erasing two months worth of gains in dollar/yen. The Nikkei has returned to its early April level as well. The unhedged EWJ falls 0.9%, while this year's hottest ETF, the hedged Japan fund (DXJ), slides 3%, now up only 16.5% YTD.| Jun. 6, 2013, 1:11 PM | 5 Comments
Jun. 5, 2013, 3:40 AM
The Nikkei (EWJ) drops sharply once again and takes other Asian bourses with it, with the fall this time because of disappointment with Shinzo Abe's "third arrow" of economic reform. The measures "aren't enough for Japan to achieve sustained economic growth and overcome deflation," says economist Hisashi Yamada, especially as the labor-market reforms don't go far enough. There's also disappointment that Abe isn't proposing to cut corporate taxes, which are amongst the highest in the OECD. Japan -3.8%, Hong Kong -1.1%, China -0.1%, India flat.| Jun. 5, 2013, 3:40 AM | 1 Comment
Jun. 4, 2013, 4:36 PM
"This is not a good time to be exiting the Nikkei (EWJ, DXJ)," says Jeff Gundlach in a presentation (webcast) titled "What in the world is going on?" Markets don't rally like that to be immediately followed by a "monster collapse," he adds. He'd prefer to buy below 13K (closed last night at 13.5K), but the his bull thesis remains intact and he wouldn't be a seller here.| Jun. 4, 2013, 4:36 PM
Jun. 4, 2013, 12:31 PM
"Current conditions represent a buying opportunity for bonds (TLT, LQD) says Jeff Gundlach, appearing on CNBC. Higher interest rates will prove too damaging to certain sectors and he expects the 10-year Treasury - currently at 2.13% - to head back below 2% into the summer and fall. Nikkei (EWJ, DXJ)? In the mid-12s, it's a buying opportunity. Apple (AAPL)? He was a buyer in the low $400s and $500 is a chip shot, but it may never see $700. Chipotle (CMG)? He never got filled on his short at $380, but would give it a shot again.| Jun. 4, 2013, 12:31 PM | 9 Comments
Jun. 3, 2013, 11:24 AM
The yen (FXY +1.4%) surges higher, with dollar/yen crashing through ¥100 all the way to ¥98.97 - the first time in double digits in about a month. Along with the strengthening yen are declining Nikkei 225 futures (NKY), off 2.8%. EWJ -1.9%, DXJ -3.6%. The proprietor of the DXJ, WisdomTree (WETF -3.1%).| Jun. 3, 2013, 11:24 AM | 1 Comment
May 30, 2013, 10:37 PMInvestors rush to buy the tumbling Japanese stock market, with WisdomTree's (WETF) Japan Hedged Equity Fund (DXJ) raking in another $264M over the last week, according to XTF.com, the 3rd largest inflow of any ETF. The crazy turnaround in Nikkei futures this morning on word Japan's national pension fund is considering boosting its equity allocation carries through to Tokyo, the Nikkei +1.8%. | May 30, 2013, 10:37 PM
May 30, 2013, 7:35 AM
Nikkei futures go nuts, reversing most of a 5% overnight plunge on word Japan's public pension fund is considering shifting to a more pro-equity stance. They're off just 0.4% at the moment. NKY -1% premarket.| May 30, 2013, 7:35 AM | 13 Comments
May 30, 2013, 7:24 AM
The recently-plunging Nikkei may have already solved this issue, but Japan's giant public pension fund (GPIF) is considering a change in strategy that would otherwise force it to sell stocks (EWJ, DXJ) and buy JGBs (JGBS) as equities rally and bond prices fall. No significant change in policy has been made since 2006. Of interest: Last night's 5% dive in stocks failed to put in a bid for JGBs, the yield on the 10-year flat at 0.89%.| May 30, 2013, 7:24 AM | 3 Comments
May 29, 2013, 7:28 AM
Skittishness in the markets has the yen (FXY) resuming gains against the dollar, with dollar/yen sliding 1.2% to ¥101.13. After a couple of frightening plunges in the past few sessions, the Nikkei took the yen's gain rather well, falling just 0.9% to 14,189. EWJ -1.7%, DXJ -1.5% premarket. Action to eye: The JGB market as the 10-year yield carves out a new Y/Y high, up 4 bps to 0.93%| May 29, 2013, 7:28 AM | 2 Comments
May 24, 2013, 2:19 PM
Soaring volatility on the Nikkei isn't stopping investors from scooping up options "with something approaching glee," according to SIG's derivatives team (for the uninitiated, higher vols mean higher options prices). Most of the buying in the Japan ETF space is in the form of calls as dip buyers do what they've become accustomed to doing. Two popular ones, EWJ and DXJ are off nearly 10% since Wed. afternoon.| May 24, 2013, 2:19 PM | 2 Comments
May 24, 2013, 3:08 AM
Japanese shares end +0.6% following a volatile session in which stocks rose and fell by over 3% vs the close yesterday, when they slumped 7.3%. The plunge on Thursday was "just a speed bump" given this year's strong returns, says U.S. fund manager Audrey Kaplan. Improving economic conditions "will support the market going forward." Hong Kong flat, China +0.6%, India +0.3%.| May 24, 2013, 3:08 AM | 2 Comments
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submitted by Chunksmommy
Guess the Song! (S2, Round 8) (self.radiohead)
submitted by weerman44The SECOND Guess The Song Guy
Got around to buying In Rainbows Disk 2 (self.radiohead)
submitted by MrSirShpeeWhere's the truth, what's the use?
submitted by AdagioDurus
So, what's up with radiohead.com? (i.redd.it)
submitted by worrywott
Burn the witch is the best radiphead song (self.radiohead)
submitted by valkyrie249Broken hearts burn the witch
My First Listen of *In Rainbows* or: How Radiohead Made me Cry and Question my Existence (self.radiohead)
submitted by PartTimeSociopathOnVideotapeOnVideotapeOnVideotapeOnVideotapeOnVideotape
Just listened to Yorke's solo stuff (self.radiohead)
submitted by smallaustralianyour alarm bells, they should be ringing
Backdrifters - A free 30 track album by Radiohead fans (ragdollcollection.bandcamp.com)
submitted by olinobnizov
In NYC right now, anyone know if the huge AMSP poster is still around (self.radiohead)
submitted by RamiN64
Pathway To Paris (COP 21) album including Thom Yorke performance out in 3 days! (pathwaytoparis.com)
submitted by loicd
Please explain your love for in In Rainbows to me! (self.radiohead)
submitted by Mytwistedword
Other bands that sound like crazy, early Radiohead (self.radiohead)
submitted by Young_Neil_PostmanI'm a reasonable man, get off my case


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Snapchat - Android Apps on Google Play
Life's more fun when you live in the moment :) Happy Snapping!* * *Please note: Snapchatters can always capture or save your messages, such
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YouTube Music lets you watch and listen to a nearly endless catalog in an app designed for music discovery.Enjoy music for free with ads, or
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N.W.A.: Straight Outta Compton - Music on Google Play
Straight Outta Compton is the debut studio album by the Compton, California hip-hop group N.W.A, released August 8, 1988 on group member Eaz
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Expand your music collection with Amazon Music that includes Prime Music. Prime members enjoy unlimited, ad-free access to over a million so
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The sprawling crime epic that changed open-world games forever.Welcome to Liberty City. Where it all began. Rockstar Games celebrates the 10
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Angry Birds and Transformers collide in this action-packed, 3D shoot 'em up adventure! The EggSpark has transformed the eggs into crazed rob
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Fallout Shelter - Android Apps on Google Play
** Google Play Best of 2015 ** Mobile Game of the Year - 2016 DICE Awards Winner 2015 Golden Joystick Best Handheld/Mobile Game"Might just b
Transformers: Earth Wars - Android Apps on Google Play
THE TRANSFORMERS ARE BACK ON EARTH! In this digital version of the classic battle of Good vs Evil, Autobots vs Decepticons, commanders aroun
Marvel: Avengers Alliance 2 - Android Apps on Google Play
ASSEMBLE, STRATEGIZE, FIGHT! Marvel: Avengers Alliance 2 is the massive sequel to the smash hit Marvel: Avengers Alliance, with a robust mob
The Amazing Spider-Man 2
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Harness Spider-Man's powers with Manhattan as your playground! Set shortly after the events of Columbia Pictures' new film, The Amazing Spid


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Bigfoot - The Unknown Territory
- Directed by Karl Kozak
===Story=== When a group of poachers are found mutilated to death in the deep forests of the Pacific Northwest, a mob of revenge hungry hunters pursue the beast while four high school seniors try to save it. What ensues is a life-or-death adventure for those brave enough to enter the woods.
Editor's Note:
Also known as The Unknown and Project Bigfoot
|Miles O'Keeffe
Recent threads
- Favourite classic inspirational films Jul 29
- The Shawshank Redemption (1994) Jul 29
- Carlito's Way (1993) Jul 29
- [Actor] Denzel Washington Jul 29
- Genre: Thriller Movies
- Director: Karl Kozak
- Producer: Diversa Films
Theatrical Release North America Jan 8, 2004


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Now Available, HP Propel free! How will you use it?
"Consumerization" supported by IT is a hot trend these days. How can you know the difference between those products that make big-sky promises and those that provide real business value? How can you be sure that the new knowledge management you purchased will be something valuable for your end users and will make them want to come to your knowledge portal every day? Your current IT strategy might be to have an IT portal, service catalog and knowledge management on premise. Now, imagine a new CIO wants a shift in strategy including delivering all of these on SaaS. How flexible are you to allow for a seamless transition? You don't have to answer the question, but you should think about the situation. This is exactly what is happening to many IT organizations right now. To meet the needs and flexibility required, we have a unique opportunity for you.
We have developed the new Propel service which is now available in a free version. It delivers a single, consumer-friendly, cloud-based portal with a service catalog, knowledge management and IT news to end users, without requiring that you replace back-end systems or processes. HP Propel free is fully functional and is accessible from mobile devices or from the web. This service was announced in December and is available beginning today.
How to start?
Onboarding to HP Propel free is very easy! All you have to do is register through the webpage. You can register up to one thousand end-users from your organization. If you need to give more users access to HP Propel free, we will provide you with a second instance of Propel free.
HP Propel free is multi-tenant. You do not need to worry if you need to give access to your central IT portal or service catalog for a new LOB unit –simply register the new users for the HP Propel free.
We know that your IT is under more pressure to deliver new and existing services with more speed than ever before. If you can't keep pace, your business may find their way around you. With HP Propel you can support your business and deliver services—with the quality and speed they require. Here are just some of the benefits:
Quick startup of the Propel free service.
No upfront investment and operations.
Accelerated time-to-value in delivering your IT services. All while continuing to use all existing service management products—for example HP or third-party products.
The key features of HP Propel free include:
Single portal for one-stop shopping for all IT services—the latest IT news, shopping from a standard service catalog or searching for the latest knowledge articles curated by HP. The software is accessible from the web or mobile device and is available in English, French and German.
Standard catalog—populated with the most frequently used 100 IT services, fulfilled through email requests to ITs existing back-end fulfillment engines.
Knowledge base—immediate access to 500 knowledge articles from HP IT, these are applicable to any user or IT organization.
IT News—keep end users informed and up-to-date. You can load your own RSS feeds to share your latest IT service information with web and mobile users.
How free is it really?
Are you still wondering how "free" HP Propel free is? For customers registered with a corporate email account, I promise it is 100 percent free of charge and HP is committed to 99 percent availability. We will continue updating Propel free so you can expect even more features available for free in the future. Should you need to contact HP customer support you can reach out just like you do with any other HP product offering contract and license. For the full description of HP's commitments and responsibilities as well as customers' responsibilities, please read the HP Propel data sheet at www.hp.com/go/propel.
If you are looking to offer your business users the engagement and consumption experience that compares to those of Internet-based services you need to start with HP Propel free—NOW!
If you need more features and integrations with services providers than what we offer in this free Service, I encourage you to look at our premium services for HP Propel. I will look at these services in one of my next blog posts. For more information about HP Propel:
Watch our HP Propel video http://www.youtube.com/watch?v=3DyVYR5WQvA
Register for Propel free
Vesna Soraic


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Book Review: Linux In A Nutshell
Now into its fifth edition, O'Reilly's 'Linux In A Nutshell' is a single volume reference manual that covers system admin, editors, shells, booting, regular expressions and more. It provides, in essence, all the key ingredients required to run Linux, whether that be as a user, an administrator or a developer. It's also pretty much independent as regards Linux distribution, thought the material has been tested extensively on Red Hat/Fedora, SuSE and Debian.
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GNOME Software 3.22 Will Support Installation of Snaps, Flatpak Repository Files
The GNOME 3.21.4 desktop environment was released last week, which means that many of the default applications and components were updated with bug fixes and various enhancements.
openSUSE Leap 42.2 Now Merged with SUSE Linux Enterprise 12 Service Pack 2
The development cycle of the openSUSE Leap 42.2 operating system continues, and today we would like to inform our readers about the availability of the third and last Alpha build in the series.
Linux 4.7 and Linux 4.8
Ubuntu MATE 16.04.1 LTS Fixes the Raspberry Pi Partition Resizer, Adds MATE 1.14
As part of the Ubuntu 16.04.1 LTS (Xenial Xerus) announcement, Martin Wimpress informs us about the release of the Ubuntu MATE 16.04.1 LTS operating systems for users of Ubuntu MATE 16.04 LTS. Ubuntu MATE 16.04.1 LTS is not a major release, and if your Ubuntu MATE 16.04 LTS (Xenial Xerus) installation is up to date, you already have the latest software updates and security patches that have been injected in the new installation mediums generated mainly for those who want to reinstall or deploy the OS on new systems.


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Tommy Lasorda Portrait Unveiled in Smithsonian Institution
Many's the time when Dodger fans said that former Dodger manager Tommy
Lasorda should be institutionalized, but having the Smithsonian hang
his portrait in its august halls was not what some people had in mind.
Nevertheless, that's what happened today, on Lasorda's 82nd birthday, with Lasorda on hand for the ceremony.
The institution's National Portrait Gallery unveiled the life-sized painting by Everett Raymond Kinstler to coincide with the Dodger's three-day stand with the Washington Nationals in Washington D.C. The portrait will be up in the New Arrivals gallery through November 15.
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Sunday, April 29, 2007
Week 32 Numbers (First Four Nights); How the Sitcoms Did
Households: Fox: 9.8/15.5; CBS: 7.7/12.4; ABC: 7.4/11.9; NBC: 4.7/7.6; The CW: 2.2/3.5; MNT: 0.6/0.9.
Total Viewers: Fox: 16.51 million; CBS: 12.06 million; ABC: 11.05 million; NBC: 7.21 million; The CW: 3.28 million; MNT: 0.84 million.
Adults 18-49: Fox: 6.3/16.7; ABC: 3.7/10.0; CBS: 3.5/9.5; NBC: 2.7/7.1; The CW: 1.4/3.8; MNT: 0.3/0.9.
Analysis: The 32nd week saw Fox in front again with the usual leader American Idol. Sweeps began on Thursday and it is all new stuff until the end of the season! Through the first four nights, Fox has the lead, followed by CBS and ABC battling for second in HH, viewers and 18-49. NBC again remains a distant 4th place in all categories. As usual, Friday and Saturday numbers are not above, since final numbers are not available yet. Tonight has ABC airing all-new episodes and CBS is also all new, the battle will be with them for viewers with ABC in command for 18-49 for sure. Fox's comedies are also all-new (except 8:30) and NBC's has a movie and another two-hour Dateline. Expect ABC to win.
On Monday Dancing with the Stars continued to take over Mondays for ABC. 18.87 million, exact same number as the previous week. CBS's CSI: Miami was all-new at 10pm and did 17.03 million, up a lot from last week's very low number and NBC's Deal or No Deal was way down against Stars, with 10.98 million. To me that show only does really well when their is weak competition, so it really isn't a must-see show. CBS's sitcoms were all-new tonight, except How I Met Your Mother at 8. 8-9pm sitcoms were hurt the most for CBS, doing 5.94 and 6.88 million, down a bit from last week. Two and a Half Men rose from those numbers and did a below average 12.27 million (down from last week), followed by an all-new King of Queens doing 12.63 million, building from the lead-in and up from last week, too. Fox was below average thanks to Drive doing just 4.6 million (down over 1 million from last week, Fox has now pulled this show starting tomorrow), pulling down 24 a bit to a season low 10.41 million. ABC aired The Bachelor from 9:30-11 and did a decent 9.13 million, slightly up from last week, and NBC returned Heroes with 11.96 million (off a bit from its normal numbers), followed by the premiere of new reality show The Real Wedding Crashers at just 6.74 million. The CW's comedies are still under performing on this night, with Chris at 3.01 million, the best of the bunch. Worth noting is The Game is building on its lead-in now at 9:30 and did the second best of the comedies with 2.55 million.
Tuesday was Fox's night. American Idol did 26.55 million at 8 once again and House did 20.81 million, down from last week. ABC was on the map thanks to Dancing with the Stars: The Results doing 16.56 million at 9 (slightly up from last week), with two all-new George Lopez episodes preceding it with 4.45 and 4.89 million. Boston Legal at 10 was at its normal levels with 9.54 million. CBS was third with new episodes of NCIS, The Unit from 8-10 (14.17 and 9.83 million), and an encore of Cold Case aired at 10 with 7.89 million. NBC was fourth for its repeats of both Law & Order spin-offs doing just 4.53 and 7.64 million from 9-11pm (down a lot from last week's repeats), with another Dateline kicking the night off with 4.9 million. The CW aired a new Gilmore Girls (3.75 million) and the finale of Pussycat Dolls (3.43million).
Wednesday again Fox got the boost from that show called American Idol, doing 26.93 million for two hours, down 2 million from last week's one-hour edition...hmm. Bones returns this week. CBS was second for the night with Jericho doing just 7.56 million (down from last week's low) but repeats of Criminal Minds and a new CSI: NY followed and did decent with 9.75 and 11.4 million. ABC followed with According to Jim & Notes from the Underbelly doing OK to open up the night with 5.28 and 4.78 million, followed by a Lost encore from last week doing just 3.78 million (they should air Jim/George repeats here) and an all-new Lost closed the night off with a first place 11.86 million, beating CBS. NBC was again fourth with a repeat of Thank God You're Here doing just 4.46 millio, Crossing Jordan followed with just 6.37 million for a new episode. A new Medium followed built on the lead-in a bit with 7.16 million. The CW was down a bit from normal with America's Next Top Model doing 3.87 million for a clip show, followed by an encore of Tuesday's Pussycat Dolls doing 2.88 million.
On Thursday we saw ABC and CBS dominate...but not as much as usual. ABC's Grey's Anatomy was all-new at 9pm with 20.08 million, a bit down from usual. A new Ugly Betty opened the night off with 9.62 million, also down from normal, but up from last week. ABC closed the night with the season finale of October Road with 9.0 million (first in its slot for 18-49), much better than Men in Trees. CBS's Survivor: Fiji was below normal as well with 13.83 million, but slightly up from last week. A new CSI and an all-new Shark followed with 20.2 and 14.1 million. Fox was third with Are You Smarter Than a 5th Grader? with 10.13 million...and they closed the night with the first of a two-week sweeps episode of Trading Spouses doing just 4.76 million, but it still beat NBC's comedies. NBC was fourth in viewers with all-new episodes of its comedies and an all-new ER, which just did 9.52 million, slightly up from last week. And CW aired new episodes of its Thursday dramas doing 3.88 and 3.33 million.
Friday (numbers not counted in our ratings), saw CBS win the night with its dramas. Everyone else was very distant. ABC aired a repeat of Grey's Anatomy and an all-new Wife Swap from 8-10 and 20/20 built on that and did was second place at 10pm. NBC did terrible from 8-10 with Identity and Raines; Law & Order at 10 was the highest rated show of the night for them but it was still 3rd in its slot. Fox aired encores of House & Bones and they were fourth. The CW did decent for its WWE coverage, but it is not doing as well as in the winter. Overall a blah night that needs a makeover next fall. Maybe Bingo for ABC will be the answer?
Saturday ratings saw CBS (for viewers) and Fox (for 18-49) do decent, with ABC and NBC did way behind. As for tonight, ABC will win with its normal line-up. As for this week in general, let the first full week of sweeps begin!
Last Sunday Sitcoms (counts for week of April 6-22)
- War at Home (Fox) - Just 2.85 million for the series...err season finale.
- King of the Hill (Fox) - Just 3.34 million in its new time period.
- The Simpsons (Fox) - 6.46 million for an all-new episode and a repeat followed with 6.61 million.
- Family Guy (Fox) - 6.23 and 6.72 million for two repeats.
Monday Sitcoms
- How I Met Your Mother (CBS) - A repeat did just 5.94 million, down from the previous week's repeat of 6.41 million.
- Old Christine (CBS) - 6.88 million for a new episode, built from lead-in, but down from previous week's new episode of 6.99 million.
- Two and a Half Men (CBS) - 12.27 million for a new episode, below average, and down from last week's 13.03 million, which was also below normal.
- King of Queens (CBS) - 12.63 million for a new episode and up from last week.
- CW Monday Comedies: Chris, All of Us, Girfriends, The Game. All-new episode of Chris started it off with 3.01 million, followed by new episodes of All of Us at 2.32 million, Girlfriends did 2.34 million and The Game did 2.55 million, wow the latter built on the lead-in again! It'll be back next season I think.
Tuesday Sitcoms
- George Lopez (ABC) - Two new episodes did 4.45 and 4.89 million, that was against Idol though.
Wednesday Sitcoms
- According to Jim (ABC) - New episode did 5.28 million for down from last week's 6.35 million, but this episode was against Idol.
- Notes from the Underbelly (ABC) - 4.78 million for its second airing on its regular night and time. Down from last week's 6.26 million, but this week it was head to head with Idol. This won't happen a lot.
- My Name is Earl (NBC) - 7.49 million, slightly up from last week and slowly going up.
- The Office (NBC) - 7.56 million and a 3.9 18-49 rating. Both built on from the lead-in.
- 30 Rock (NBC) - Just 4.72 million for the season finale.
- Scrubs (NBC) - Ouch just did 4.65 million, down almost a million from last week. What happened?
Saturday Sitcoms
Airing tonight (Sunday) are new episodes of Fox comedies.
Summing it up. Two and a Half Men, My Name is Earl, The Office, and King of Queens were impressive for the week. Everything else was OK (such as Old Christine, According to Jim) or just plain old terrible (30 Rock, Scrubs).
New on DVD/Blu-ray (June/July)
06/14 - Ballers - The Complete First Season (Blu-ray)
06/14 - The League - Season 7 - The Final Fantasy
06/14 - The League - The Complete Series
06/14 - Maude - The Complete Fifth Season
06/14 - Survivor's Remorse - The Complete Second Season
06/21 - Workaholics - Season Six
06/24 - The Monkees - The Complete Series (Blu-ray)
06/28 - Two Guys and a Girl - The Complete Series
07/08 - The Monkees - The Complete Series (Blu-ray)
07/12 - Grady - The Best of Grady
07/26 - Family Mattters - The Complete Seventh Season
07/26 - Where's Huddles? - The Complete Series (WBShop.com)
More TV DVD Releases / DVD Reviews Archive / SitcomsOnline Digest
Sitcoms Airing Tonight
Friday, July 29
Best Friends Whenever - "Derby Little Secret" (Disney Channel, 8:00PM ET/PT)
Cyd and Shelby watch roller derby against Astrid's wishes; they find out Shelby's mom has been lying.
Girl Meets World - "Girl Meets Ski Lodge Part 2" (Disney Channel, 8:30PM ET/PT)
With the group's ski trip winding down, it becomes clear that it's time for Lucas to choose between Riley and Maya.
Love Thy Neighbor - "Love Thy Privacy" (OWN, 9:00PM ET/PT)
Sam is compelled to reveal the true nature surrounding his lack of maturity.
Sitcom Stars on Talk Shows (Week of July 25)
Friday, July 29
- Kristen Bell (House of Lies) - Watch Kristen on repeats of Jimmy Kimmel Live! at 11:35pm on ABC and on The Ellen DeGeneres Show, so check your local listings.
- Dave Franco (Scrubs/Do Not Disturb) - Dave appears on a repeat of Jimmy Kimmel Live! at 11:35pm on ABC.
- Mila Kunis (That '70s Show/Family Guy) - Mila is a guest on a repeat of The Ellen DeGeneres Show, so check your local listings.
- Christina Applegate (Up All Night/Samantha Who?/Married with Children) - Christina chats on a repeat of Rachael Ray, so check your local listings.
- Candace Cameron Bure (Fuller House/Full House) - Candace surprises a real family who have the same story as the Tanners on a repeat of Rachael Ray, so check your local listings.
- Jake Paul (Bizaardvark) - Jake appears on Home & Family on Hallmark Channel at 10am ET/PT.
- Marisa Tomei (A Different World) - Marisa chats with the ladies of The View on ABC at 11am ET/10am CT-PT.
- Kevin Hart (The Big House/Undeclared) - Kevin will be on ABC's Good Morning America sometime between 7-9am.
- Anthony Anderson (black-ish/Guys with Kids/All About the Andersons) - Anthony guest co-hosts Access Hollywood Live, so check your local listings.
- Judy Reyes (Scrubs) - Judy is a guest on Access Hollywood Live, so check your local listings.
- Jerry O'Connell (We Are Men/Carpoolers/My Secret Identity) - Jerry guest co-hosts LIVE! with Kelly, so check your listings.


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Buy houses in Davenant Road, Oxford OX2 - 4 bedrooms
More options
- Pre-owned / New homes No preference
- Ownership type No preference
- Retirement No preference
- Buyer incentives No preference
£500,000 Guide price
Harefields, Oxford OX2Terraced house for sale - 4 bedrooms
Three storey townhouse in a popular location, well finished throughout, and situated conveniently for access to Oxford City & Oxford Parkway Station
Note: Distances are straight line measurements
- Oxford (2.5 miles)
- Islip (2.9 miles)
Hayward Road, Oxford OX2Detached house for sale - 4 bedrooms
A substantial detached family residence in one of North Oxford's most sought after roads. The property is in the school catchment area for the highly desirable Cherwell School. The four double bedroom home offers enormous scope for extension and ...
Note: Distances are straight line measurements
- Oxford (2.7 miles)
- Islip (2.7 miles)
Hernes Road, Oxford OX2Semi-detached house for sale - 4 bedrooms
A beautifully presented Victorian house with a south facing garden in North Oxford well placed for access to Summertown and Oxford Parkway station. The property offers excellent family accommodation with period features including sash windows and ...
Note: Distances are straight line measurements
- Oxford (2.1 miles)
- Islip (3.1 miles)