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Examples of a class of one-dimensional constraints where it is in general undecidable whether they can be satisfied. The constraints require that concatenating in some order the blocks shown should yield identical upper and lower strings. In cases (a)-(l) the constraints can be satisfied, and the minimal strings which do so are shown. The plots to the right give the successive differences in length between upper and lower strings when each new block is added; that this difference reaches zero reflects the fact that the constraint is satisfied. Cases (m)-(s) show constraints that cannot be satisfied by strings of any finite length. When the constraints involve more than two blocks there seems in general to be no upper limit on how long a string one may need to consider to tell whether the constraints can be satisfied. Pictures (a), (b), (h) and (j) show the longest minimal strings needed for any of the 4096, 16384, 65536 and 262144 constraints involving blocks with totals of 7, 8, 9 and 10 elements. The general problem of satisfying constraints of the kind shown here is known as the Post Correspondence Problem. Finding the systems on this page required constructing—by computer and otherwise—an immense number of proofs of the impossibility of satisfying particular constraints.

From Stephen Wolfram: A New Kind of Science [citation]