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Computational Hierarchy of Elementary Cellular Automata

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Computational Hierarchy of Elementary Cellular Automata Computational Hierarchy of Elementary Cellular Automata Barbora Hudcova´1;2 and Toma´sˇ Mikolov2 1Charles University, Prague 2Czech Institute of Informatics, Robotics and Cybernetics, CTU, Prague Abstract by Mazoyer and Rapaport (1999). We present the emulation relation between a pair of CA, and we demonstrate the re- The complexity of cellular automata is traditionally mea- sults on a toy class of elementary CA; namely, we present sured by their computational capacity. However, it is diffi- cult to choose a challenging set of computational tasks suit- their computational hierarchy. The results inspired us to de- able for the parallel nature of such systems. We study the fine a new notion of chaos for discrete dynamical systems ability of automata to emulate one another, and we use this connected to their computational capacity. notion to define such a set of naturally emerging tasks. We present the results for elementary cellular automata, although Introducing Cellular Automata the core ideas can be extended to other computational sys- tems. We compute a graph showing which elementary cel- A cellular automaton can be perceived as a k-dimensional lular automata can be emulated by which and show that cer- grid consisting of identical finite state automata with the tain chaotic automata are the only ones that cannot emulate same local neighborhood. They are all updated syn- any automata non-trivially. Finally, we use the emulation no- chronously in discrete time steps according to a fixed local tion to suggest a novel definition of chaos that we believe is suitable for discrete computational systems. We believe our update rule which is a function of the neighbors’ states. A work can help design parallel computational systems that are formal definition can be found in Kari (2005). Turing-complete and also computationally efficient. Basic Notions Introduction We say that Z is a one-dimensional cellular grid and we call its elements the cells. Let S be a finite set. An S- Discrete systems exhibit a wide variety of dynamical behav- configuration of the grid is a mapping c : Z ! S, we ior ranging from ordered and easily predictable to a very write ci = c(i) for each i. We define the nearest-neighbors complex, disordered one. In this paper, we study cellular relative neighborhood of each cell i 2 Z to be the triple automata (CA) that have intriguing visualizations of their (i − 1; i; i + 1). In this paper, we will study 1-dimensional space-time dynamics. Observing them helps us build intu- CA with nearest neighbors. Each such CA is character- ition for distinguishing the different types of dynamics, as ized by a tuple (S; f) where S is a finite set of states and studied by Wolfram (2002). Informally, complex CA pro- f : S3 ! S is a local transition rule of the CA. The global duce higher-order structures, whereas the space-time dia- rule of the CA (S; f) operating on an infinite grid is a map- grams of chaotic CA are seemingly random. Though CA ping F : SZ ! SZ defined as: dynamics has been studied extensively (Wolfram (1984), arXiv:2108.00415v1 [cs.NE] 1 Aug 2021 Kurka˚ (2009), Wuensche and Lesser (2001), Gutowitz F (c)i = f(ci−1; ci; ci+1): (1990), Zenil (2009)), it is still a very difficult problem to formally define the notions of complexity and chaos in CA. For practical purposes, when observing the CA simula- Traditionally, complexity of CA is studied through their tions, we consider the grid to be of finite size with a periodic computational capacity (Toffoli (1977), Cook (2004)), boundary condition. In such a case, we compute the cells in whereas chaos is studied via topological dynamics (Devaney the relative neighborhood modulo the size of the grid. (1989)). As the two approaches are not connected in any obvious way, it is an interesting open problem of whether Elementary CA Elementary cellular automata (ECA) are chaotic CA can compute non-trivial tasks (Martinez et al. 1D nearest-neighbors CA with states S = f0; 1g. We iden- (2013)). tify each local rule f determining an ECA with the Wolfram In this paper, we study the complexity of CA through number of f defined as: their computational capacity; namely, we study their abil- ity to emulate one another; this notion was first introduced 20f(0; 0; 0)+21f(0; 0; 1)+22f(0; 1; 0)+:::+27f(1; 1; 1): We will refer to each ECA as a “rule k” where k is the corre- CA Emulations sponding Wolfram number of its underlying local rule. The Basic Notions class of ECA is a frequently used toy model for studying Definition 1 (Subautomaton). different CA properties due to its relatively small size; there Let ca1 = (S; f), ca2 = (T; g) be two nearest-neighbor 1D are only 256 of them. CA. We say that ca is a subautomaton of ca if there exists n 1 2 For an ECA operating on a cyclic grid of size with a one-to-one encoding e : S ! T such that global rule F we define the trajectory of a configuration n 2 u 2 f0; 1g to be (u; F (u);F (u);:::). The space-time di- e(f(s1; s2; s3)) = g(e(s1); e(s2); e(s3)) agram of such a simulation is obtained by plotting the con- 3 figurations as horizontal rows of black and white squares for all (s1; s2; s3) 2 S . (corresponding to states 1 and 0) with time progressing This simply means that we can embed the rule table of downwards. ca1 into the rule table of ca2; the definition is equivalent to saying that (S; f) is a subalgebra of (T; g). CA Complexity via Computational Capacity Example 2. For an ECA ca = (f0; 1g; f), we define the Classically, the complexity of a CA is demonstrated by its dual ECA ca0 = (f0; 1g; f 0) as computational capacity. Intuitively, we believe that CA ca- 0 pable of computing non-trivial tasks should be more com- f (b1; b2; b3) = 1 − f(1 − b1; 1 − b2; 1 − b3): plex than those that are not. In the past, many different com- 0 putational problems were considered, such as the majority f simply changes the role of 0 and 1 states. The encoding e(b) = 1−b, b 2 f0; 1g witnesses that ca is a subautomaton computation task (Mitchell et al. (2000)) or the firing squad 0 synchronization (Mazoyer (1986)), a detailed overview was of ca and vice versa. written by Mitchell (1998). Nevertheless, the most classi- 0 10 20 30 40 0 10 20 30 40 cal task is the simulation of a computationally universal sys- 0 0 tem (a Turing machine, a tag system, etc.). Over the years, 10 10 many different CA were designed or showed to be Turing 20 20 complete. However, it seems unnatural to demonstrate the 30 30 40 40 complexity of an inherently parallel system by embedding a 50 50 sequential computational model into it. In our opinion, an ideal set of benchmark tasks helping us determine the com- putational capacity of CA should Figure 1: Space-time diagram of ECA rule 110 on the left and its dual rule 137 on the right. • consist of tasks suitable for the parallel computational en- vironment Each ECA contains at most two subautomata from the • be challenging enough ECA class: itself and its dual ECA. Hence, the subautoma- ton relation would not produce a rich hierarchy. It is, how- • for a given CA and a task T , it should be effectively veri- ever, a key concept for defining the CA emulation. Below, fiable whether the CA can compute T . we restrict the terminology to ECA, but the theory can be For a fixed class C of CA, a natural task is the following: generalized to any 1D CA in a straightforward way. + S1 k Given a CA in C; how many other CA in C can it simulate? We will write 2 = f0; 1g and define 2 = k=1 2 to be the set of all finite nonempty binary sequences. The key problem is finding a suitable definition of CA ”sim- Definition 3 (Unravelling a Boolean function). ulating” one-another. Various approaches have been sug- 3 + + gested — simulations can be interpreted via CA coarse- Let f : 2 ! 2. We define fe : 2 ! 2 as grainings (Israeli and Goldenfeld (2006)), or through em- fe(b1; b2; : : : ; bn) = f(b1; b2; b3) : : : f(bn−2; bn−1; bn) bedding a local rule to larger size cell-blocks (Mazoyer and Rapaport (1999), Ollinger (2001)). Many interesting the- for any binary sequence b1b2 : : : bn, n ≥ 3. oretical results stem from such definitions. For instance, By fek we simply mean the composition Ollinger (2001) defined a CA simulation notion which ad- mits a universal CA — one that is able to simulate all other fek = fe◦ fe◦ ::: ◦ fe: | {z } CA with the same dimensionality; it was designed by Cu- k−times lik II and Albert (1987). In this paper, we consider a notion very similar to the one suggested by Mazoyer and Rapaport Each iteration of feshortens the input size by 2, therefore we (1999). Compared to their definition, ours is stricter and, can notice that when we restrict the domain of fek we get thus, slightly easier to verify. We call it CA emulation and k k k k k introduce it in the subsequent section. fe : 2 × 2 × 2 ! 2 : 23k Hence, fek can be interpreted as a ternary function operating Properties of the Emulation Relation on supercells of k bits and we obtain a CA (2k; fek) whose Preservation of Turing Completeness dynamics is completely governed by the simple local rule f.
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