This talk advocates intrinsic universality as a notion to identify simple cellular automata with complex computational behavior. After an historical introduction and proper definitions of intrinsic universality, which is discussed with respect to Turing and circuit universality, we discuss construction methods for small intrinsically universal cellular automata before discussing techniques for proving non universality.
Deep Dive into Intrinsically Universal Cellular Automata.
This talk advocates intrinsic universality as a notion to identify simple cellular automata with complex computational behavior. After an historical introduction and proper definitions of intrinsic universality, which is discussed with respect to Turing and circuit universality, we discuss construction methods for small intrinsically universal cellular automata before discussing techniques for proving non universality.
Universal machines are, in some way, the simplest type of complex machines with respect to computational aspects: the sum of all possible behaviors. Universality is also a convenient tool in computation as a way to transform data, that can be further manipulated by the machine, into code. Therefore, computation universality is one of the basic ingredients of self-reproducing cellular automata first introduced by von Neumann [15] and in the subsequent works of Codd [5] and others 1 to achieve construction universality. Since then, universality has been studied for itself both in the case of two-dimensional and one-dimensional cellular automata. For a detailed historical study, see the survey [19].
In the 60s and 70s, universality was mainly studied for high-dimensional cellular automata (2D, 3D). In this context, it seems natural, to achieve universality, to take inspiration from real-world computers by simulating components of boolean circuits. Wires, gates, clocks, fan-out, signal crossing, etc are embedded into the configuration space of some local rule. Using these components, under the assumption that the family of elements is powerful enough, one obtains a universal cellular automaton under every reasonable hypothesis: from boolean circuits, one can wire finite state machines and memories to simulate sequential machines like Turing machines; or, one can wire finite state machines encoding the local rule of a cellular automaton and put infinitely many copies of that machine on a regular lattice, using wires to connect and synchronize the grid of automata to simulate the behavior of the encoded cellular automaton. This way, Banks [2,3] was able to construct very small universal 2D cellular automata.
In the 70s and 80s, the study of cellular automata shifted to the one-dimensional space, motivated by the formal study of parallel algorithmics and formal languages recognition. In 1D, boolean circuits are no more a natural tool, but, as a configuration looks like a biinfinite tape, simulation of sequential machines like Turing machines is straightforward and provides the basis for a notion of computational (that is Turing) universality. This approach was developed by Smith III [23]. A major difficulty with Turing universality is the lack of a formal precise and general definition. The problem arises from two sources. First, a good commonly accepted formal definition of universality for Turing machines does not seem to exist. Second, encoding finitely described Turing machines configurations into infinite configurations, giving a reasonable halting condition, and a decoding of the result, is a delicate task. For a discussion on this formalization problem, see the study by Durand and Róka [9] of the universality of Conway’s Game of Life [4].
In 1D, one can also consider simulating the cells of a configuration of a simulated cellular automaton by blocks of cells of a configuration of a simulator cellular automaton, leading to a notion of intrinsic universality. This notion, that coincide with the notion of boolean circuit universality in the case of 2D cellular automata, was first pointed out in the one-dimensional case by Banks [2,3] in the conclusion of its 2D construction, then rediscovered by Albert and Čulik II [1]. An attempt of a formal definition was given in Durand and Róka [9]. Whereas intrinsic universality implies Turing universality, one can prove that the converse is false, see Ollinger [16]. Intrinsic universality is the topic of this talk.
The paper continues as follows. In section 2, proper definitions of cellular automata and two definitions of intrinsic universality are proposed together with the main structural results. In section 3, the construction of small universal cellular automata is discussed. In section 4, the more difficult question of non universality is considered.
The set of configurations S Z d is endowed with the Cantor topology, i.e., the product topology over Z d of the discrete topology on S. This topology is metric, compact, and perfect. Under this topology, continuity corresponds to locality, as clopen sets correspond to sets of all configurations having a finite pattern in a given finite set, i.e., if C ⊆ S Z d is a clopen, there exists
Adding invariance by translation, one can enforce uniformity and characterize cellular automata. The translation, or shift, over S with translation vector p ∈ Z d , is the map σ p : S Z d → S Z d , satisfying, for all c ∈ S Z d and z ∈ Z d , σ p (c)(z + p) = c(z).
This theorem allows us to manipulate cellular automata by their global function, composing them, inverting bijective ones, taking cartesian products, etc being sure that the result is still the global function of a cellular automaton. For a proper study of cellular automata and their properties, one can read the survey of Kari [11].
A cellular automaton A is a subautomaton of a cellular automaton B, denoted as A ⊑ B, if there exists an injective map ϕ : The injective bulk
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