For if any sequence is going to satisfy the constraint one can show that there must already be a sequence of limited length that does so—and if necessary one can find this sequence by explicitly looking at all possibilities.

And for systems with sufficiently simple behavior—say repetitive or nested—the pictures on page 744 indicate that one can typically determine the outcome with an amount of effort that is essentially proportional to the length of this digit sequence.

In effect what it means is that any question about the behavior of any other universal system can be encoded as a statement in the axiom system—and if the answer to the question can be established by watching the evolution of the other universal system for any finite number of steps then it must also be able to be established by giving a proof of finite length in the axiom system.

But often the number of theorems increases rapidly with the length of proof—and in most cases an infinite number of theorems can eventually be proved.

In logic, however, all proofs are in effect ultimately of limited length.

For despite its reputation for generality I argued at length in the previous section that the whole field of mathematics that we as

.) • Is there a cuboid in which all edges and all diagonals are of integer length?

Packing deformable objects If one pushes together identical deformable objects in 2D they tend to arrange themselves in a regular hexagonal array—and this configuration is known to minimize total boundary length.

But if one tries to look at growth rates on scales that are not small compared to characteristic lengths associated with curvature then one again sees exponential growth—just as in the case of a uniform tessellation without hexagons.

And when the notion of algorithmic information content as the length of a shortest program (see page 1067 ) emerged in the 1960s it was suggested that this might be an appropriate definition for complexity. Several other definitions used in specific fields in the 1960s and 1970s were also based on sizes of descriptions: examples were optimal orders of models in systems theory, lengths of logic expressions for circuit and program design, and numbers of factors in Krohn–Rhodes decompositions of semigroups. … Sometimes these have been based on observations of humans trying to understand or verify systems, but more often they have just been based for example on simple properties of networks that define the flow of control or data—or in some cases on the length of documentation needed.)