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Add a few more notes. Cat's Eye Technologies 10 years ago
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100100
101101 There are also any number of hills, each with a fixed position and radius.
102 These serve as goals; when the leader hedgehog reaches a goal hill, the
103 troupe's quest is over, and they are free to break formation and nibble on
104 clover or whatever it is that hedgehogs do for fun.
102 These serve as goals; when the leader hedgehog reaches a hill, the troupe's
103 quest is over, and they are free to break formation and nibble on clover or
104 whatever it is that hedgehogs do for fun.
105105
106106 Part IV. Computational Class
107107 ----------------------------
194194 real analysis techniques to discover the evolution of the system. Motion of
195195 the hedgehogs would be genuinely continuous except for the points where their
196196 velocities change due to one hedgehog approaching another too closely, etc.
197
198 Nor have we said much about the topology of the space all of these things are
199 in. Naturally, the space could be infinite, or it could be mapped onto a
200 torus so that the hedgehogs "wrap around" at the edges. (The fact that the
201 space is finite is not a problem w.r.t. Turing-completeness, so long as there
202 are still an infinite number of possible coordinates, i.e. the space of real
203 or rational numbers; integers, though, would be problematic in a space of
204 finite extent.) Or the space could be mapped onto the surface of a sphere,
205 with appropriate modifications.
206
207 We have also not constrained the sets of faery rings and hills to be finite;
208 so we could have an infinite number of them in an infinite space -- but for
209 correspondence with a Turing machine, these should probably be restricted to
210 infinite sets. (Otherwise, if you have, say, an uncountable set of faery
211 rings, specified non-constructively, you could encode all possible
212 computations just in them, and the behaviour of the hedgehogs becomes
213 essentially moot.)
214
215 The initial condition could either consist of the troupe having a leader
216 hedgehog with an initial velocity, or a leader-elect hedgehog with an
217 initially-selected faery ring to which it is attracted. The former is
218 probably more "natural" for some definition of "natural".