Slow-growing hierarchy

In computability theory, computational complexity theory and proof theory, the slow-growing hierarchy is an ordinal-indexed family of slowly increasing functions g_α: N → N (where N is the set of natural numbers, {0, 1, ...}). It contrasts with the fast-growing hierarchy.

Definition

Let μ be a large countable ordinal such that a fundamental sequence is assigned to every limit ordinal less than μ. The slow-growing hierarchy of functions g_α: N → N, for α < μ, is then defined as follows:

• $g_{0}(n)=0,\,$
• $g_{\alpha +1}(n)=g_{\alpha }(n)+1,\,$
• $g_{\alpha }(n)=g_{\alpha [n]}(n)\,\!$ if α is a limit ordinal.

Here α[n] denotes the n^th element of the fundamental sequence assigned to the limit ordinal α.

The article on the Fast-growing hierarchy describes a standardized choice for fundamental sequence for all α < ε₀.

Relation to fast-growing hierarchy

The slow-growing hierarchy grows much more slowly than the fast-growing hierarchy. Even g_ε0 is only equivalent to f₃ and g_α only attains the growth of f_ε0 (the first function that Peano arithmetic cannot prove total in the hierarchy) when α is the Bachmann–Howard ordinal. (Girard 1981) and (Cichon and Wainer 1983)

However, Girard (1981) proved that the slow-growing hierarchy eventually catches up with the fast-growing one. Specifically, that there exists an ordinal α and integers a, b such that

g_α(n) < f_α(n) < g_α(n + 1)

where f_α are the functions in the fast-growing hierarchy. He further showed that the first α this holds for is the ordinal of the theory ID_<ω of arbitrary finite iterations of an inductive definition. (Wainer 1989). However for the Cichon assignment of fundamental sequences the first match up occurs at the level ε₀ (Weiermann 1997).

Extensions of the result proved in Wainer 1989 to considerably larger ordinals show that there are very few ordinals below the ordinal of transfinitely iterated $\Pi _{1}^{1}$-comprehension where the slow and fast growing hierarchy match up (Weiermann 1995).

For the novice it is worth mentioning that the slow growing hierarchy depends extremely sensitive on the choice of the underlying fundamental sequences (Weiermann 1997 and Weiermann 1999).

Relation to term rewriting

Cichon provided an interesting connection between the slow growing hierarchy and derivation length for term rewriting (Cichon 1990).

References

• Girard, Jean-Yves (1981), "Π¹₂-logic. I. Dilators", Annals of Mathematical Logic, 21 (2): 75–219, doi:10.1016/0003-4843(81)90016-4, ISSN 0003-4843, MR 0656793
• Cichon, E. A.; Wainer, S. S. (1983), "The slow-growing and the Grzegorczyk hierarchies", The Journal of Symbolic Logic, 48 (2): 399–408, doi:10.2307/2273557, ISSN 0022-4812, MR 0704094
• . JSTOR 2274873. {{cite journal}}: Cite journal requires |journal= (help); Missing or empty |title= (help)
• Gallier, Jean H. (1991), "What's so special about Kruskal's theorem and the ordinal Γ₀? A survey of some results in proof theory", Ann. Pure Appl. Logic, 53 (3): 199–260, doi:10.1016/0168-0072(91)90022-E, MR 1129778 PDF's: part 1 2 3. (In particular part 3, Section 12, pp. 59–64, "A Glimpse at Hierarchies of Fast and Slow Growing Functions".)
• Weiermann, A. (1995). Archives of Mathematical Logic. 34: 313–330. {{cite journal}}: Missing or empty |title= (help)
• . doi:10.1016/S0168-0072(97)00033-X. {{cite journal}}: Cite journal requires |journal= (help); Missing or empty |title= (help)
• Weiermann, A. (1999), "What makes a (pointwise) subrecursive hierarchy slow growing?" Cooper, S. Barry (ed.) et al., Sets and proofs. Invited papers from the Logic colloquium '97, European meeting of the Association for Symbolic Logic, Leeds, UK, July 6–13, 1997. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 258; 403-423.