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We study one-dimensional cellular automata whose rules are chosen at random from among $r$-neighbor rules with a large number $n$ of states. Our main focus is the asymptotic behavior, as $n \to \infty$, of the longest temporal period…
In this thesis we will work under the premises of the Cellular Automata Interpretation of QM, by Gerard 't Hooft, according to whom particles evolve following the rules of Cellular Automata (CA), a mathematical model consisting of discrete…
Let g >= 1 and let Q be a monic, squarefree polynomial of degree 2g + 1 in Z[x]. For an odd prime p not dividing the discriminant of Q, let Z_p(T) denote the zeta function of the hyperelliptic curve of genus g over the finite field F_p…
Continuous cellular automata (CCAs) have evolved from discrete lookup tables to continuous partial differential equation (PDE) formulations in the search for novel forms of complexity. Despite innovations in qualitative behavior, analytical…
We present an intuitive formalism for implementing cellular automata on arbitrary topologies. By that means, we identify a symmetry operation in the class of elementary cellular automata. Moreover, we determine the subset of topologically…
We establish formulae for the asymptotic growth (with respect to the scaling dimension) of the number of operators in effective field theory, or equivalently the number of $S$-matrix elements, in arbitrary spacetime dimensions and with…
We give asymptotic formulae for random matrix averages of derivatives of characteristic polynomials over the groups USp(2N), SO(2N) and O^-(2N). These averages are used to predict the asymptotic formulae for moments of derivatives of…
This article presents a new characterization of controllability and regional controllability of Deterministic Cellular Automata (CA for short). It focuses on analyzing these problems within the framework of control theory, which have been…
We define the class of rapidly left expansive cellular automata, which contains fractional multiplication automata, Wolfram's Rule 30, and many others. The definition has been shaped by a proposition of Jen on aperiodicity of columns in…
This paper studies two kinds of simulation between cellular automata: simulations based on factor and simulations based on sub-automaton. We show that these two kinds of simulation behave in two opposite ways with respect to the complexity…
Abelian cellular automata (CA) are CA which are group endomorphisms of the full group shift when endowing the alphabet with an abelian group structure. A CA randomizes an initial probability measure if its iterated images weak *-converge…
We systematically study the boundaries of one-dimensional, 2-color cellular automata depending on 4 cells, begun from simple initial conditions. We determine the exact growth rates of the boundaries that appear to be reducible. Morphic…
Let $C_n$ be the $n$th Catalan number. We show that the asymptotic density of the set $\{n: C_n \equiv 0 \mod p \}$ is $1$ for all primes $p$, We also show that if $n = p^k -1$ then $C_n \equiv -1 \mod p$. Finally we show that if $n \equiv…
In the coupon collector problem with $n$ items, the collector needs a random number of tries $T_n\simeq n\ln n$ to complete the collection. Also, after $nt$ tries, the collector has secured approximately a fraction…
In this survey we show how to produce asymptotics of determinants of structured matrices using operator theory methods. We describe the asymptotics for finite Toeplitz matrices, finite Toeplitz plus Hankel matrices and generalizations of…
In a probabilistic cellular automaton in which all local transitions have positive probability, the problem of keeping a bit of information indefinitely is nontrivial, even in an infinite automaton. Still, there is a solution in 2…
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,…
While there has been a keen interest in studying computation at the edge of chaos for dynamical systems undergoing a phase transition, this has come under question for cellular automata. We show that for continuously deformed cellular…
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.…
In the fields of computation and neuroscience, much is still unknown about the underlying computations that enable key cognitive functions including learning, memory, abstraction and behavior. This paper proposes a mathematical and…