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Deep learning techniques have recently demonstrated broad success in predicting complex dynamical systems ranging from turbulence to human speech, motivating broader questions about how neural networks encode and represent dynamical rules.…
Bent Boolean functions are important objects in cryptography and coding theory, and there are several general approaches for constructing such functions. Metaheuristics proved to be a strong choice as they can provide many bent functions,…
Given a property of Boolean functions, what is the minimum number of queries required to determine with high probability if an input function satisfies this property or is "far" from satisfying it? This is a fundamental question in Property…
In this paper we propose a rule-independent description of applications of cellular automata rules for one-dimensional additive cellular automata on cylinders of finite sizes. This description is shown to be a useful tool for for answering…
A digit function is presented which provides the $i$th-digit in base $p$ of any real number $x$. By means of this function, formulated within $\mathcal{B}$-calculus, the local, nonlocal and global dynamical behaviors of cellular automata…
Cellular automata are discrete dynamical systems that consist of patterns of symbols on a grid, which change according to a locally determined transition rule. In this paper, we will consider cellular automata that arise from polynomial…
We consider two relatively natural topologizations of the set of all cellular automata on a fixed alphabet. The first turns out to be rather pathological, in that the countable space becomes neither first-countable nor sequential. Also,…
We consider the algebra of invariants of binary forms of degree 9 with complex coefficients, find the 92 basic invariants, give an explicit system of parameters and show the existence of four more systems of parameters with different sets…
Gauge-invariance is a fundamental concept in Physics -- known to provide mathematical justification for the fundamental forces. In this paper, we provide discrete counterparts to the main gauge theoretical concepts directly in terms of…
We propose a chaotic encryption method based on Cellular Automata(CA), specifically on the family called the "Life-Like" type. Thus, the encryption process lying on the pseudo-random numbers generated (PRNG) by each CA's evolution, which…
Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing binary minimal codes from sets, Boolean functions and vectorial Boolean…
The emergence of complex behaviors in cellular automata is an area that has been widely developed in recent years with the intention to generate and analyze automata that produce space-moving patterns or gliders that interact in a periodic…
Probabilistic cellular automata describe the dynamics of classical spin models, which, for sufficiently small temperature $T$, can serve as classical memory capable of storing information even in the presence of nonzero external magnetic…
In this paper, we mainly study linear one-dimensional and two-dimensional elementary cellular automata that generate symmetrical spatio-temporal patterns. For spatio-temporal patterns of cellular automata from the single site seed, we…
Exactly ergodicity in boundary-driven semi-infinite cellular automata (CA) are investigated. We establish all the ergodic rules in CA with 3, 4, and 5 states. We analytically prove the ergodicity for 12 rules in 3-state CA and 118320 rules…
For many cellular automata, it is possible to express the state of a given cell after $n$ iterations as an explicit function of the initial configuration. We say that for such rules the solution of the initial value problem can be obtained.…
Various non-classical approaches of distributed information processing, such as neural networks, computation with Ising models, reservoir computing, vector symbolic architectures, and others, employ the principle of collective-state…
We present a conceptual framework for extending homomorphic encryption beyond arithmetic or Boolean operations into the domain of intuitionistic logic proofs and, by the Curry-Howard correspondence, into the domain of typed functional…
Any algorithm (in the sense of Gurevich's abstract-state-machine axiomatization of classical algorithms) operating over any arbitrary unordered domain can be simulated by a dynamic cellular automaton, that is, by a pattern-directed cellular…
Canalizing functions have important applications in physics and biology. For example, they represent a mechanism capable of stabilizing chaotic behavior in Boolean network models of discrete dynamical systems. When comparing the class of…