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Continuous automaton
About: Continuous automaton is a research topic. Over the lifetime, 947 publications have been published within this topic receiving 17417 citations.
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28 Sep 2011TL;DR: A construction of a weakly universal cellular automaton in the 3D hyperbolic space with two states is shown, based on a new implementation of a railway circuit in the dodecagrid, which is a truly 3D-one.
Abstract: In this paper, we show a construction of a weakly universal cellular automaton in the 3D hyperbolic space with two states. Moreover, based on a new implementation of a railway circuit in the dodecagrid, the construction is a truly 3D-one. This result under the hypothesis of weak universality and in this space cannot be improved.
6 citations
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17 Aug 2010TL;DR: Estimating the number of regular expressions that have e as a partial derivative, a lower bound of the average number of mergings of states in Apos is computed and its asymptotic behaviour is described.
Abstract: The partial derivative automaton (Apd) is usually smaller than other non-deterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton (Apos). By estimating the number of regular expressions that have e as a partial derivative, we compute a lower bound of the average number of mergings of states in Apos and describe its asymptotic behaviour. This depends on the alphabet size, k, and its limit, as k goes to infinity, is 1/2. The lower bound corresponds exactly to consider the Apd automaton for the marked version of the regular expression, i.e. where all its letters are made different. Experimental results suggest that the average number of states of this automaton, and of the Apd automaton for the unmarked regular expression, are very close to each other.
6 citations
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11 Jul 2016TL;DR: A theoretical model of DNA chemical reaction-diffusion network capable of performing a simple cellular automaton based on well-characterized enzymatic bistable switch, which indicates that the model evolves as expected both in space and time from initial conditions.
Abstract: We introduce a theoretical model of DNA chemical reaction-diffusion network capable of performing a simple cellular automaton. The model is based on well-characterized enzymatic bistable switch that was reported to work in vitro. Our main purpose is to propose an autonomous, feasible, and macro DNA system for experimental implementation.
As a demonstration, we choose a maze-solving cellular automaton. The key idea to emulate the automaton by chemical reactions is assuming a space discretized by hydrogel capsules which can be regarded as cells. The capsule is used both to keep the state uniform and control the communication between neighboring capsules.
Simulations under continuous and discrete space are successfully performed. The simulation results indicate that our model evolves as expected both in space and time from initial conditions. Further investigation also suggests that the ability of the model can be extended by changing parameters. Possible applications of this research include pattern formation and a simple computation. By overcoming some experimental difficulties, we expect that our framework can be a good candidate to program and implement a spatio-temporal chemical reaction system.
6 citations
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TL;DR: In this paper, the authors prove that the Thwaites conjecture holds for the evolution function of a cellular automaton on a torus and show that the cycle length of any cycle is at most a factor of 2.
Abstract: In this paper we prove a conjecture of Brian Thwaites concerning the evolution function of a certain cellular automaton on a torus. In the seventies, when more and more people had access to personal computers, John Conway’s game of life became very popular. Since then, the study of this type of game grew up into the theory of cellular automata. In [4] (see also [1, page 311]) Brian Thwaites proposes a conjecture which leads to such a cellular automaton. Thwaites’s conjecture is: Given any finite sequence of rational numbers, take the positive differences of successive members (including differencing the last member with the first); iteration of this operation eventually produces a set of zeros if and only if the size of the set is a power of 2. Our aim in this note is to prove that Thwaites conjecture holds true. Let a0, ..., ad−1 be the given d rational numbers, which we may think as the heights of d poles situated around a circle. These numbers are replaced at the next step by d rational numbers given by the difference in heights of successive poles, and then the process is repeated. Being an iteration of the same operation, it resembles Conway’s life game. For now the field of play is a 1-dimensional torus and since, as we will see only finitely many numbers which depend on the initial configuration are involved, in the long run, we will end up with a cycle. Finding the lengths of these cycles, which depend mostly on d — the size of the torus —, is the interesting problem. In other words, if d is a power of 2, Thwaites conjecture says that the length of any cycle is equal to 1 (the shortest possible). We examine the lengths of the cycles that occur and provide conditions on d which guarantee that these lengths are small or long. A criterion which tests if a given integer is a period for this evolution function is given in section 3. Received : December 20, 1998; Revised : March 22, 1999. AMS Subject Classification: 11B85, 11A07. 312 C.I. COBELI, M. CRÂŞMARU and A. ZAHARESCU 1 – Proof of Thwaites conjecture We begin by making some notations which set the problem in a clearer framework. Let a0, ..., ad−1 be the given rational numbers. For convenience, we unpack this ordered set of numbers by associating to it the infinite sequence (a0, a1, ...), where the components are defined by ak = ak+d for k ≥ 0 . (1) Let us denote by Qd and by Nd the set of all the sequences with rational and natural components respectively, satisfying (1). The evolution function φ :Qd→Qd is defined by φ(a0, a1, ...) = (a ′ 0, a ′ 1, ...), where a′k = |ak − ak+1| for k ≥ 0 . (2) With these notations, Thwaites conjecture says that for any sequence (a0, a1, ...) ∈ Qd, φ(a0, a1, ...) = (0, 0, ...) for all sufficiently large n ∈ N iff d is a power of 2. (Here φ(n) is the repeated composition of n samples of φ.) Let’s note that all the components of φ(a0, a1, ...) are nonnegative if n ≥ 1 and by multiplying all the components of the initial sequence (a0, a1, ...) by the least common multiple of their denominators, we may assume that the domain of our evolution function is Nd. Let M = max {a0, ..., ad−1}. By the definition of φ, it is easy to see that all the components of φ(a0, a1, ...) are integers belonging to [0,M ]. Because there are only finitely many such periodic sequences in Nd, it follows that given any initial configuration (a0, a1, ...), the repeated application of the evolution function will eventually produce a cycle of sequences which keep repeating. The next lemma shows that after sufficiently many steps we always end up with sequences with components having at most 2 distinct values. Lemma 1. Let d be a positive integer, (a0, a1, ...) a sequence of nonnegative integers satisfying (1) and suppose the function φ is defined as above. Then there is a positive integer a such that for sufficiently large n all the components of φ(a0, a1, ...) belong to {0, a}. Proof: The proof is by (inverse) induction. Let us look at a portion of the sequence of numbers we get at some step. We write them on a line as follows: ..., b, 0, ..., 0, } {{ } s zeros m, ...,m, } {{ } u numbers 0, ..., 0, } {{ } t zeros c, ... (3) A CELLULAR AUTOMATON ON A TORUS 313 Here m is the maximum of all our numbers at this step, b and c are nonzero, m > b, m > c, s ≥ 0, t ≥ 0, u ≥ 1 and the part of the sequence that begins and ends with m contains only 0’s or m’s. Then, after at most s + u + t steps, the maximum of the numbers that are produced out by this portion of the sequence will be ≤ max {m− b,m− c} < m. Of course at a given step the sequence of numbers we obtain might contain several subsequences of the form (3), but what happens is that after at most d steps the maximum of the numbers at that step will be strictly less than m. The lemma then follows by induction. By multiplying all the components of the initial configuration by a−1, where a is given by Lemma 1, we may assume that after sufficiently many steps all the components of the sequences we obtain are 0 or 1. Then our operation (taking the positive differences of successive members of the sequence) is nothing else than addition in the group (Z/2Z,+). Now there is a transparent way to generalize the game by replacing Z/2Z by a more general finite monoid and also by playing on a multidimensional field. The operation in this case is to take the sum (or product if the multiplicative notation is used) of the closest neighbors. We only mention here that if we keep the same group Z/2Z, but play on a multidimensional torus, then we eventually obtain a sequence of zeros if and only if the size of one of the dimensions is a power of 2. This can be showed by following the same lines of proof. Returning to our problem, let us observe that by starting with an arbitrary sequence of 0’s and 1’s, by applying repeatedly the evolution function, we obtain the following table which is filled with the beginning of the sequences obtained in the first few iterations. Step 1 2 3 · · · 0. a0 a1 a2 · · · 1. a0 + a1 a1 + a2 a2 + a3 · · · 2. a0 + a2 a1 + a3 a2 + a4 · · · 3. a0 + a1 + a2 + a3 a1 + a2 + a3 + a4 a2 + a3 + a4 + a5 · · · 4. a0 + a4 a1 + a5 a2 + a6 · · · 5. a0 + a1 + a4 + a5 a1 + a2 + a5 + a6 a2 + a3 + a6 + a7 · · · 6. a0 + a2 + a4 + a6 a1 + a3 + a5 + a7 a2 + a4 + a6 + a8 · · · 7. a0 + a1 + · · · + a7 a1 + a2 + · · · + a8 a2 + a3 + · · · + a9 · · · 8. a0 + a8 a1 + a9 a2 + a10 · · · 9. a0 + a1 + a8 + a9 a1 + a2 + a9 + a10 a2 + a3 + a10 + a11 · · · 10. a0 + a2 + a8 + a10 a1 + a3 + a9 + a11 a2 + a4 + a10 + a12 · · · · · · · · · · · · · · · · · · 314 C.I. COBELI, M. CRÂŞMARU and A. ZAHARESCU Now it is easy to see by induction that in the above table if d=2m then the d-th row (and consequently all that follow after it) contains only 0’s. Also, there are sequences of 0’s and 1’s, namely those containing an odd number of 1’s, for which on the (d−1)-th row all the numbers are equal to 1. Thus, if d is a power of 2 and we start with an arbitrary set of 0’s and 1’s, then the process will produce 0’s in d steps and only for particular ak’s in less then d steps. The outcome in the case d 6= 2m can be also deduced easily by induction. Thus, if we start for example with the periodic sequence given by a0 = 1 and ak = 0 for 1 ≤ k ≤ d−1, then 1 will always be the first number on the rows representing the steps of order a power of 2. Therefore, if d is not a power of 2 then there are sequences which will never produce a set of 0’s. We summarize our result in the following theorem which proves the Thwaites conjecture. Theorem 1. Let d be a positive integer and suppose the evolution function φ is defined as above. Then there is a rational number r>0 such that the repeated application of φ to any initial sequence of rational numbers (a0, a1, ...) satisfying (1) will eventually produce a cycle of sequences with the property (1) with all their components in {0, r}. Moreover, the cycle will contain only the sequence (0, 0, ...) independently on the initial sequence if and only if d is a power of 2. 2 – The length of cycles We assume in this section that the evolution function φ is defined on Ud, where Ud, is the set of all the sequences with components in {0, 1}, satisfying (1). This is not restrictive as we saw above and also has the advantage that it makes φ to be additive. Theorem 1 shows that if d is a power of 2 then the length of any cycle is equal to 1. Suppose from now on that d is not a power of 2. Write d = 2k r with k ≥ 0 and r odd, r > 1. Let s be the order of 2 modulo r. Thus 2s−1 ≡ 0 (mod r). Let F = {0, 1} be the field with 2 elements and let I be the ideal of F [X] generated by Xd−1. Map the sequence a = (a0, ..., ad−1) to the coset
6 citations