Chapters: Fast-Growing Hierarchy, Grzegorczyk Hierarchy, Veblen Function, Slow-Growing Hierarchy, Hardy Hierarchy. Source: Wikipedia. Pages: 28. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy) is an ordinal-indexed family of rapidly increasing functions f: N N (where N is the set of natural numbers {0, 1, ...}, and ranges up to some large countable ordinal). A primary example is the Wainer hierarchy, or LobWainer hierarchy, which is an extension to all < 0. Such hierarchies provide a natural way to classify computable functions according to rate-of-growth and computational complexity. Let be a large countable ordinal such that a fundamental sequence (a strictly increasing sequence of ordinals whose supremum is a limit ordinal) is assigned to every limit ordinal less than . A fast-growing hierarchy of functions f: N N, for <, is then defined as follows: Here f(n) = f(f(...(f(n))...)) denotes the n iterate of f applied to n, and denotes the n element of the fundamental sequence assigned to the limit ordinal . (An alternative definition takes the number of iterations to be n+1, rather than n, in the second line above.) The initial part of this hierarchy, comprising the functions f with finite index (i.e., < ), is often called the Grzegorczyk hierarchy because of its close relationship to the Grzegorczyk hierarchy; note, however, that the former is here an indexed family of functions fn, whereas the latter is an indexed family of sets of functions . (See Points of Interest below.) Generalizing the above definition even further, a fast iteration hierarchy is obtained by taking f0 to be any increasing function g: N N. For limit ordinals not gre...http: //booksllc.net/?id=24947285