Slow-growing hierarchy explained

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,). 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:^[1]

g _{0(n)} = 0 g _{\alpha+1} (n) = g _{\alpha(n)} + 1 g _{\alpha(n)} = g _{\alpha[n]} (n) for limit ordinal α.

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.

Example

g _{1(n)} = 1 g _{m(n)} = m g _{\omega(n)} = n g _{\omega+m} (n) = n + m g _{\omega} (n) = mn g _{\omega ^{m}} (n) = n ^{m} g _{\omega ^{\omega}} (n) = n ^{n} g _{\varepsilon _{0}} (n) = n\uparrow\uparrow n

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.^{[2] [3] [4]}

However, Girard proved that the slow-growing hierarchy eventually catches up with the fast-growing one. Specifically, that there exists an ordinal α such that for all integers n

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.^[5] However, for the assignment of fundamental sequences found in the first match up occurs at the level ε₀.^[6] For Buchholz style tree ordinals it could be shown that the first match up even occurs at

\omega ^{2} .

Extensions of the result proved 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.^[7]

The slow-growing hierarchy depends extremely sensitively on the choice of the underlying fundamental sequences.^{[8] [9]}

References

• 1129778. Gallier. Jean H.. Jean Gallier . 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 . 1991. 3. 199–260. 10.1016/0168-0072(91)90022-E. See especially "A Glimpse at Hierarchies of Fast and Slow Growing Functions", pp. 59–64 of linked version.

Notes

1. J. Gallier, What's so special about Kruskal's theorem and the ordinal Γ₀? A survey of some results in proof theory (2012, p.63). Accessed 8 May 2023.
2. Girard . Jean-Yves . Jean-Yves Girard . Π¹₂-logic. I. Dilators . 10.1016/0003-4843(81)90016-4 . 656793 . 1981 . Annals of Mathematical Logic . 0003-4843 . 21 . 2 . 75–219 . free .
3. Book: Cichon. Proof Theory. 1992. Cambridge University Press. 173–193. P. Aczel . H. Simmons . S. Wainer . Termination Proofs and Complexity Characterisations.
4. Cichon . E. A. . Wainer . S. S. . The slow-growing and the Grzegorczyk hierarchies . 10.2307/2273557 . 704094 . 1983 . The Journal of Symbolic Logic . 0022-4812 . 48 . 2 . 399–408 . 2273557 . 1390729 .
5. 2274873 . 608–614 . Wainer . S. S. . Slow Growing Versus Fast Growing . 54 . 2 . The Journal of Symbolic Logic . 1989 . 10.2307/2274873. 19848720 .
6. 10.1016/S0168-0072(97)00033-X . Sometimes slow growing is fast growing . 1997 . Weiermann . A . Annals of Pure and Applied Logic . 90 . 1–3 . 91–99. free .
7. Weiermann . A. . 1995 . Investigations on slow versus fast growing: How to majorize slow growing functions nontrivially by fast growing ones. Archive for Mathematical Logic. 34 . 5 . 313–330 . 10.1007/BF01387511. 34180265 .
8. 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.
9. 10.1002/1521-3870(200108)47:3<397::AID-MALQ397>3.0.CO;2-Y. Γ₀ May be Minimal Subrecursively Inaccessible. Mathematical Logic Quarterly . 47. 3. 397–408. 2001. Weiermann. Andreas.