Enrico Formenti

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.

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.

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.

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 Sabliks classification. Moreover, we extend the directional dynamics studies by considering also factor languages and attractors.

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.

Decidable Properties of 2D Cellular Automata

In this paper we study some decidable properties of two-dimensional cellular automata (2D CA). The notion of closingness is generalized to the 2D case and it is linked to permutivity and openness. The major contributions of this work are two deep constructions which have been fundamental in order to prove our new results and we strongly believe it will be a valuable tool for proving other new ones in the near future.

Subshift attractors of cellular automata

A subshift attractor is a two-sided subshift which is an attractor of a cellular automaton. We prove that each subshift attractor is chain-mixing, contains a configuration which is both F-periodic and σ-periodic and the complement of its language is recursively enumerable. We prove that a subshift of finite type is an attractor of a cellular automaton iff it is mixing. We identify a class of chain-mixing sofic subshifts which are not subshift attractors. We construct a cellular automaton whose maximal attractor is a non-sofic mixing subshift, answering a question raised in Maass (1995 Ergod. Theory Dyn. Syst. 15 663–84). We show that a cellular automaton is surjective on its small quasi-attractor which is the non-empty intersection of all subshift attractors of all Fqσp, where q > 0 and .

On the dynamical behavior of chaotic cellular automata

Abstract The shift (bi-infinite) cellular automaton is a chaotic dynamical system according to all the definitions of deterministic chaos given for discrete time dynamical systems (e.g., those given by Devaney [6] and by Knudsen [10]). The main motivation to this fact is that the temporal evolution of the shift cellular automaton under finite description of the initial state is unpredictable . Even tough rigorously proved according to widely accepted formal definitions of chaos, the chaoticity of the shift cellular automaton remains quite counterintuitive and in some sense unsatisfactory. The space-time patterns generated by a shift cellular automaton do not correspond to those one expects from a chaotic process. In this paper we propose a new definition of strong topological chaos for discrete time dynamical systems which fulfills the informal intuition of chaotic behavior that everyone has in mind. We prove that under this new definition, the bi-infinite shift is no more chaotic. Moreover, we put into relation the new definition of chaos and those given by Devaney and Knudsen. In the second part of this paper we focus our attention on the class of additive cellular automata (those based on additive local rules) and we prove that essential transformations [2] preserve the new definition of chaos given in the first part of this paper and many other aspects of their global qualitative dynamics.

Non-uniform cellular automata: Classes, dynamics, and decidability

The dynamical behavior of non-uniform cellular automata is compared with the one of classical cellular automata. Several differences and similarities are pointed out by a series of examples. Decidability of basic properties like surjectivity and injectivity is also established. The final part studies a strong form of equicontinuity property specially suited for non-uniform cellular automata.

On the Dynamics of PB Systems: A Petri Net View

We study dynamical properties of PB systems, a new computational model of biological processes, and propose a compositional encoding of PB systems into Petri nets. Building on this relation, we show that three properties: boundedness, reachability and cyclicity, which we claim are useful in practice, are all decidable.