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A Universal Cellular Automaton…The idea is that each cell in the three-color cellular automaton is represented by a block of three cells in the two-color cellular automaton. And by looking at neighbors out to distance five on each side, the two-color cellular automaton can update these blocks at each step in direct correspondence with the rules of the three-color cellular automaton. … For given the universal cellular automaton, it is always in effect possible to program this cellular automaton to emulate any other cellular automaton, and therefore to produce whatever behavior the other cellular automaton could produce.
A Universal Cellular Automaton…[No text on this page] The universal cellular automaton emulating rule 30. A total of 848 steps in the evolution of the universal cellular automaton are shown, corresponding to 16 steps in the evolution of rule 30.
A Universal Cellular Automaton…On each page the underlying rules for the universal cellular automaton are exactly the same. … The basic idea is that a block of 20 cells in the universal cellular automaton is used to represent each single cell in the cellular automaton that is being emulated. … The rules for the universal cellular automaton.
A Universal Cellular Automaton…[No text on this page] Details of how the universal cellular automaton emulates rule 254. Each of the blocks in the universal cellular automaton represents a single cell in rule 254, and encodes both the current color of the cell and the form of the rule used to update it.
A Universal Cellular Automaton…[No text on this page] The universal cellular automaton emulating elementary rule 254. Each cell in rule 254 is represented by a block of 20 cells in the universal cellular automaton.
A Universal Cellular Automaton…[No text on this page] The universal cellular automaton emulating elementary rule 90. The underlying rules for the universal cellular automaton are exactly the same as on the previous page .
A Universal Cellular Automaton…And indeed, later in this chapter , I will show an example of a cellular automaton with an extremely simple underlying rule that can nevertheless in the end be seen to be universal. … A Universal Cellular Automaton As our first specific example of a system that exhibits universality, I discuss in this section a particular universal cellular automaton that has been set up to make its operation as easy to follow as possible. The rules for this cellular automaton itself are always the same.
A Universal Cellular Automaton…To update the color of the cell represented by a particular block, what the universal cellular automaton must then do is to determine which of the 8 cases applies to that cell. … The pictures on the last few pages have shown how the universal cellular automaton can in effect be programmed to emulate any cellular automaton whose rules involve nearest neighbors and two possible colors for each cell. But the universal cellular automaton is in no way restricted to emulating only rules that involve nearest neighbors.
A Universal Cellular Automaton…[No text on this page] The universal cellular automaton emulating one step in the evolution of the rule shown above, which involves next-nearest as well as nearest-neighbor cells. … The picture shows the evolution of five cells according to the rule shown, with each cell now being represented by a block of 70 cells in the universal cellular automaton.
A Universal Cellular Automaton…[No text on this page] An example of how a cellular automaton with three possible colors and nearest-neighbor rules can be emulated by a cellular automaton with only two possible colors but a larger number of neighbors (in this case five on each side).
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