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Conservation laws in cellular automata
 2002 Nonlinearity 15 1781 [math.DS/0111014
"... If X is a discrete abelian group and A a finite set, then a cellular automaton (CA) is a continuous map F: AX−→A X that commutes with all Xshifts. If φ: A−→R, then, for any a ∈ AX, we define Σφ(a) = ∑ x∈X φ(ax) (if finite); φ is conserved by F if Σφ is constant under the action of F. We characteri ..."
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Cited by 8 (0 self)
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If X is a discrete abelian group and A a finite set, then a cellular automaton (CA) is a continuous map F: AX−→A X that commutes with all Xshifts. If φ: A−→R, then, for any a ∈ AX, we define Σφ(a) = ∑ x∈X φ(ax) (if finite); φ is conserved by F if Σφ is constant under the action of F. We characterize such conservation laws in several ways, deriving both theoretical consequences and practical tests, and provide a method for constructing all onedimensional CA exhibiting a given conservation law. If A is a finite set (with discrete topology), and X an arbitrary indexing set, then AX (the space of all functions X ↦ → A) is compact and totally disconnected in the Tychonoff topology. If (X, +) is a discrete abelian group1 (eg. X = ZD) with identity O, then X acts on itself by translation; this induces a shift action of X on AX: if a = [ax  x∈X] ∈ AX, and u ∈ X, then σu (a) = [bx  x∈X], where bx = a(x+u). A cellular automaton (CA) is a continuous map F: AX − → AX which commutes with all shifts. The CurtisHedlundLyndon Theorem [3] says that F is a CA if and only if there is some finite B ⊂ X (a “neighbourhood of the identity”) and a local map f: AB−→A so that, for all a ∈ A X and x ∈ X, F(a)x = f
On conservative and monotone onedimensional cellular automata and their particle representation
 Theoretical Computer Science
"... Numberconserving (or conservative) cellular automata 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 onedimensional cellular automata which are conse ..."
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Cited by 7 (3 self)
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Numberconserving (or conservative) cellular automata 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 onedimensional cellular automata which are conservative, monotone (specially “nonincreasing”), 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 bneighborhooddependent relabelling whose sum is conserved by the CA iteration; this can be used to uncover conservative principles and particlelike behavior underlying the dynamics of some CA.
Probabilistic cellular automata with conserved quantities
, 2003
"... We demonstrate that the concept of a conservation law can be naturally extended from deterministic to probabilistic cellular automata (PCA) rules. The local function for conservative PCA must satisfy conditions analogous to conservation conditions for deterministic cellular automata. Conservation co ..."
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Cited by 7 (0 self)
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We demonstrate that the concept of a conservation law can be naturally extended from deterministic to probabilistic cellular automata (PCA) rules. The local function for conservative PCA must satisfy conditions analogous to conservation conditions for deterministic cellular automata. Conservation condition for PCA can also be written in the form of a current conservation law. For deterministic nearestneighbour CA the current can be computed exactly. Local structure approximation can partially predict the equilibrium current for nondeterministic cases. For linear segments of the fundamental diagram it actually produces exact results.
A New Dimension Sensitive Property for Cellular Automata
 PROCEEDINGS OF MFCS 2004
, 2004
"... In this paper we study numberdecreasing cellular automata. They form a superclass of standard numberconserving cellular automata. It is wellknown that the property of being numberconserving is decidable in quasilinear time. In this paper we prove that being numberdecreasing is dimension sensi ..."
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Cited by 6 (3 self)
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In this paper we study numberdecreasing cellular automata. They form a superclass of standard numberconserving cellular automata. It is wellknown that the property of being numberconserving is decidable in quasilinear time. In this paper we prove that being numberdecreasing is dimension sensitive i.e. it is decidable for onedimensional cellular automata and undecidable for dimension 2 or greater. There are only few known examples of dimension sensitive properties for cellular automata and this denotes some rich panel of phenomena in this class.
Onedimensional monotone cellular automata
, 2005
"... We derive a necessary and sufficient condition for a onedimensional cellular automaton to be either monotone nonincreasing or nondecreasing. Generalizing the motion representation we introduced for numberconserving rules, we give a systematic way to construct this representation using the expressi ..."
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Cited by 1 (1 self)
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We derive a necessary and sufficient condition for a onedimensional cellular automaton to be either monotone nonincreasing or nondecreasing. Generalizing the motion representation we introduced for numberconserving rules, we give a systematic way to construct this representation using the expression of the current, which appears in the discrete version of the continuity equation, completed by the discrete analogue of the source term. This new representation is general in the sense that it can be used to represent, in a more visual way, any onedimensional cellular automaton rule. A few illustrative examples are presented. 1
Contents
, 2005
"... 1 Background and physical setup for road traffic 3 1.1 Historic origins of cellular automata 4 1.2 Ingredients of a cellular automaton 5 1.3 Road layout and the physical environment 7 ..."
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1 Background and physical setup for road traffic 3 1.1 Historic origins of cellular automata 4 1.2 Ingredients of a cellular automaton 5 1.3 Road layout and the physical environment 7
Enumeration of numberconserving cellular automata rules with two inputs
, 711
"... We show that there exists a onetoone correspondence between the set of numberconserving cellular automata (CA) with q inputs and the set of balanced sequences with q terms. This allows to enumerate numberconserving CA. We also show that numberconserving rules are becoming increasingly rare as t ..."
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We show that there exists a onetoone correspondence between the set of numberconserving cellular automata (CA) with q inputs and the set of balanced sequences with q terms. This allows to enumerate numberconserving CA. We also show that numberconserving rules are becoming increasingly rare as the number of states increases.
Automata 2009 On the Relationship between Boolean and Fuzzy 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 ..."
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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 we analytically show, for the first time, the existence of a strong connection between them by focusing on two properties: density conservation and additivity. We begin by giving a probabilistic interpretation of our fuzzification which leads to two important results. First, it establishes an equivalence between convergent fuzzy CA and the mean field approximation on Boolean CA, an estimation of their asymptotic density. Second, we show 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. Finally, we prove an interesting parallel between additivity of Boolean CA and oscillation of the corresponding fuzzy CA around its fixed point. In fact, we show that a Boolean CA has a certain form of additivity if and only if the behavior of the corresponding fuzzy CA around its fixed point coincides with the Boolean behavior. These connections between the Boolean and the fuzzy models are the first formal proofs of a relationship between them. Keywords: Fuzzy cellular automata, density conservation, additivity.