A complete and efficiently computable topological classification of D-dimensional linear cellular automata over Zm

Manzini, Giovanni; Margara, Luciano

doi:10.1016/s0304-3975(99)00031-6

Cited by 54 publications

(37 citation statements)

References 11 publications

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“…LCA over n m have been strongly investigated in the case n = 1 and all the dynamical properties have been characterized in terms of the 1 × 1 matrices (i.e., coefficients) defining the local rule, in any dimension too [23,3].…”

Section: Definition 3 (Linear Cellular Automata) a Linear Cellular Amentioning

confidence: 99%

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On the dynamical behaviour of linear higher-order cellular automata and its decidability

Dennunzio

Formenti

Manzoni

et al. 2019

Information Sciences

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Higher-order cellular automata (HOCA) are a variant of cellular automata (CA) used in many applications (ranging, for instance, from the design of secret sharing schemes to data compression and image processing), and in which the global state of the system at time t depends not only on the state at time t − 1, as in the original model, but also on the states at time t − 2, . . . , t − n, where n is the memory size of the HOCA. We provide decidable characterizations of two important dynamical properties, namely, sensitivity to the initial conditions and equicontinuity, for linear HOCA over the alphabet m . Such characterizations extend the ones shown in [23] for linear CA (LCA) over the alphabet n m in the case n = 1. We also prove that linear HOCA of size memory n over m form a class that is indistinguishable from a specific subclass of LCA over n m . This enables to decide injectivity and surjectivity for linear HOCA of size memory n over m using the decidable characterization provided in [2] and [19] for injectivity and surjectivity of LCA over n m . Finally, we prove an equivalence between LCA over n m and an important class of non-uniform CA, another variant of CA used in many applications.

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Section: Definition 3 (Linear Cellular Automata) a Linear Cellular Amentioning

confidence: 99%

“…[23]). Let ( m , F ) be a LCA where the local rule f : ( m ) 2r+1 → m is defined by 2r + 1 coeffiecients m −r , .…”

mentioning

confidence: 99%

On the dynamical behaviour of linear higher-order cellular automata and its decidability

Dennunzio

Formenti

Manzoni

et al. 2019

Information Sciences

Self Cite

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“…In the following we restrict our attention to linear CA; that is, CA which are based on a linear local rule (they are sometimes called additive CA). Despite their apparent simplicity, linear CA may exhibit many complex behaviors (see for example [6,8,12,11,13,16]). Linear CA have been used for pattern generation, design of error correcting codes and cipher systems, generation of hashing functions, etc.…”

Section: Inverse4(f (X) M) → G(x)mentioning

confidence: 99%

Inversion of circulant matrices over $\mathbf{Z}_m$

Бини

Corso²,

Manzini³

et al. 2000

Math. Comp.

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Abstract. In this paper we consider the problem of inverting an n × n circulant matrix with entries over Zm. We show that the algorithm for inverting circulants, based on the reduction to diagonal form by means of FFT, has some drawbacks when working over Zm. We present three different algorithms which do not use this approach. Our algorithms require different degrees of knowledge of m and n, and their costs range, roughly, from n log n log log n to n log 2 n log log n log m operations over Zm. Moreover, for each algorithm we give the cost in terms of bit operations. We also present an algorithm for the inversion of finitely generated bi-infinite Toeplitz matrices. The problems considered in this paper have applications to the theory of linear cellular automata.

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“…In the following we restrict our attention to linear CA, that is, CA which are based on a linear local rule. Despite of their apparent simplicity, linear CA may exhibit many complex behaviors (see for example [6][7][8][9]). Linear CA have been used for pattern generation, design of error correcting codes and cipher systems, generation of hashing functions, etc.…”

Section: Definitions and Notationmentioning

confidence: 99%

Inversion of circulant matrices over Zm

Бини

Corso²,

Manzini

et al. 1998

Automata, Languages and Programming

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In this paper we consider the problem of inverting an n x n circulant matrix with entries over Zm. We show that the algorithm for inverting circulants, based on the reduction to diagonal form by means of FFT, has some drawbacks when working over Zm. We present three different algorithms which do not use this approach. Our algorithms require different degrees of knowledge of m and n, and their costs range --roughly --from n log n log log n to n log 2 n log log n log m operations over Zm. We also present an algorithm for the inversion of finitely generated bi-infinite Toeplitz matrices. The problems considered in this paper have applications to the theory of linear Cellular Automata.

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A complete and efficiently computable topological classification of D-dimensional linear cellular automata over Zm

Cited by 54 publications

References 11 publications

On the dynamical behaviour of linear higher-order cellular automata and its decidability

On the dynamical behaviour of linear higher-order cellular automata and its decidability

Inversion of circulant matrices over $\mathbf{Z}_m$

Inversion of circulant matrices over Zm

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