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Import of XigXag version 1.0 revision 2010.0721. catseye 11 years ago
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1 <head><title>The Xigxag Automaton</title>
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4 <title>The Xigxag Automaton</title>
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5964 is the sequence of configurations starting with <var>s</var> and closed under successive
6065 applications of the transition relation.
6166 Every execution sequences comprises a countably infinite
62 number of configurations -- that is, runs never stop.
63 They may, however, be ultimately periodic -- that is, they may reach some
67 number of configurations — that is, runs never stop.
68 They may, however, be ultimately periodic — that is, they may reach some
6469 configuration to which they always return (a "fixed point.")
6570 But, as I'll attempt to show later, the number of initial
6671 configurations that lead to this is finite (and in fact, pretty small.)</p>
9499
95100 <h2>Investigation</h2>
96101
97 <p>Sure, it's simple, but I find Xigxag moderately interesting -- interesting enough to
102 <p>Sure, it's simple, but I find Xigxag moderately interesting — interesting enough to
98103 devote this section to proving the following property: <strong>Xigxag has exponential growth
99104 almost everywhere</strong>.</p>
100105
326331 "programs" (initial configurations) give rise to a countably infinite
327332 sequence of different configurations during an execution sequence.</p>
328333
329 <p>The fact that Xigxag execution never "halts" is also not helpful --
334 <p>The fact that Xigxag execution never "halts" is also not helpful —
330335 the same is true for all cellular automata, and this hurdle is usually
331336 overcome by attaching a "termination predicate" to the system. That is,
332337 if some configuration meets some condition (e.g. contains some substring,)
337342 alphabet {<code>&lt;</code>, <code>&gt;</code>}*, does there exist a Xigxag execution sequence
338343 where <var>s</var> occurs as a substring in one of the configurations,
339344 but not in the initial configuration? I suspect that if one could show
340 that this problem is decidable -- or, stronger, if one could give an
341 algorithm for determining what that initial configuration is -- that
345 that this problem is decidable — or, stronger, if one could give an
346 algorithm for determining what that initial configuration is — that
342347 would imply that Xigxag is not Turing-complete.</p>
343348
344349 <p>Another approach one could take is to find a very small Turing