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Small weakly universal Turing machines
"... Abstract. We give small universal Turing machines with statesymbol pairs of (6, 2), (3,3) and (2,4). These machines are weakly universal, which means that they have an infinitely repeated word to the left of their input and another to the right. They simulate Rule 110 and are currently the smallest ..."
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Abstract. We give small universal Turing machines with statesymbol pairs of (6, 2), (3,3) and (2,4). These machines are weakly universal, which means that they have an infinitely repeated word to the left of their input and another to the right. They simulate Rule 110 and are currently the smallest known weakly universal Turing machines. Despite their small size these machines are efficient polynomial time simulators of Turing machines. 1
RULE 110: UNIVERSALITY AND CATENATIONS
"... Abstract. Cellular automata are a simple model of parallel computation. Many people wonder about the computing power of such a model. Following an idea of S. Wolfram [16], M. Cook [3] has proved than even one of the simplest cellular automata can embed any Turing computation. In this paper, we give ..."
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Abstract. Cellular automata are a simple model of parallel computation. Many people wonder about the computing power of such a model. Following an idea of S. Wolfram [16], M. Cook [3] has proved than even one of the simplest cellular automata can embed any Turing computation. In this paper, we give a new highlevel version of this proof using particles and collisions as introduced in [10]. Introduced in the 40s by J. Von Neumann as a parallel model of computation [13], cellular automata consist of many simple entities (cells) disposed on a regular grid. All cells evolve synchronously by changing their state according to the ones of their neighbours. Despite being completely known at the local level, global behavior of a cellular automaton is often impossible to predict (see J. Kari [6]). This comes from the fact that even “simple” cellular automata can exhibit a wide range of complex behaviors. Among those behaviors, one often refers as emergence the fact that “complexity ” of the whole system seems far greater than complexity of its elements. Elementary cellular automata are an example of subclass of “simple ” cellular automata. They are obtained by considering only a one dimensional grid (i.e., a line of cells), two