Kruskal's tree theorem

In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.

History

The theorem was conjectured by Andrew Vázsonyi and proved by JosephKruskal  ( 1960 ); a short proof was given by CrispinNash-Williams  ( 1963 ). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR₀ (a second-order arithmetic theory with a form of arithmetical transfinite recursion).

In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs ${\displaystyle {\text{TREE}}(3)}$. A finitary application of the theorem gives the existence of the fast-growing TREE function.

Statement

The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.

Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.

Take X to be a partially ordered set. If T₁, T₂ are rooted trees with vertices labeled in X, we say that T₁ is inf-embeddable in T₂ and write ${\displaystyle T_{1}\leq T_{2}}$ if there is an injective map F from the vertices of T₁ to the vertices of T₂ such that:

• For all vertices v of T₁, the label of v precedes the label of ${\displaystyle F(v)}$;
• If w is any successor of v in T₁, then ${\displaystyle F(w)}$ is a successor of ${\displaystyle F(v)}$; and
• If w₁, w₂ are any two distinct immediate successors of v, then the path from ${\displaystyle F(w_{1})}$ to ${\displaystyle F(w_{2})}$ in T₂ contains ${\displaystyle F(v)}$.

Kruskal's tree theorem then states:

If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T₁, T₂, … of rooted trees labeled in X, there is some ${\displaystyle i<j}$ so that ${\displaystyle T_{i}\leq T_{j}}$.)

Friedman's work

For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR₀, ^[1] thus giving the first example of a predicative result with a provably impredicative proof. ^[2] This case of the theorem is still provable by Π¹
₁-CA₀, but by adding a "gap condition" ^[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system. ^{[4] [5]} Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π¹
₁-CA₀.

Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal). ^[6]

Weak tree function

Suppose that ${\displaystyle P(n)}$ is the statement:

There is some m such that if T₁, ..., T_m is a finite sequence of unlabeled rooted trees where T_i has ${\displaystyle i+n}$ vertices, then ${\displaystyle T_{i}\leq T_{j}}$ for some ${\displaystyle i<j}$.

All the statements ${\displaystyle P(n)}$ are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that ${\displaystyle P(n)}$ is true, but Peano arithmetic cannot prove the statement "${\displaystyle P(n)}$ is true for all n". ^[7] Moreover, the length of the shortest proof of ${\displaystyle P(n)}$ in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.^{[ citation needed ]} The least m for which ${\displaystyle P(n)}$ holds similarly grows extremely quickly with n.

Define ${\displaystyle {\text{tree}}(n)}$, the weak tree function, as the largest m so that we have the following:

There is a sequence T₁, ..., T_m of unlabeled rooted trees, where each T_i has at most ${\displaystyle i+n}$ vertices, such that ${\displaystyle T_{i}\leq T_{j}}$ does not hold for any ${\displaystyle i<j\leq m}$.

It is known that ${\displaystyle {\text{tree}}(1)=2}$, ${\displaystyle {\text{tree}}(2)=5}$, ${\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960}$ (about 844 trillion), ${\displaystyle {\text{tree}}(4)\gg g_{64}}$ (where ${\displaystyle g_{64}}$ is Graham's number), and ${\displaystyle {\text{TREE}}(3)}$ (where the argument specifies the number of labels; see below) is larger than ${\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}$

To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.

TREE function

A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.

By incorporating labels, Friedman defined a far faster-growing function. ^[8] For a positive integer n, take ${\displaystyle {\text{TREE}}(n)}$ ^[a] to be the largest m so that we have the following:

There is a sequence T₁, ..., T_m of rooted trees labelled from a set of n labels, where each T_i has at most i vertices, such that ${\displaystyle T_{i}\leq T_{j}}$ does not hold for any ${\displaystyle i<j\leq m}$.

The TREE sequence begins ${\displaystyle {\text{TREE}}(1)=1}$, ${\displaystyle {\text{TREE}}(2)=3}$, then suddenly, ${\displaystyle {\text{TREE}}(3)}$ explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's ${\displaystyle n(4)}$, ${\displaystyle n^{n(5)}(5)}$, and Graham's number, ^[b] are extremely small by comparison. A lower bound for ${\displaystyle n(4)}$, and, hence, an extremely weak lower bound for ${\displaystyle {\text{TREE}}(3)}$, is ${\displaystyle A^{A(187196)}(1)}$. ^{[c] [9]} Graham's number, for example, is much smaller than the lower bound ${\displaystyle A^{A(187196)}(1)}$, which is approximately ${\displaystyle g_{3\uparrow ^{187196}3}}$, where ${\displaystyle g_{x}}$ is Graham's function.

See also

• Paris–Harrington theorem
• Kanamori–McAloon theorem
• Robertson–Seymour theorem

Notes

^{^ a} Friedman originally denoted this function by TR[n].

^{^ b} n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters x_i,...,x_2i is a subsequence of any later block x_j,...,x_2j. ^[10] ${\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}$.

^{^ c} A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).

References

Citations

1. Simpson 1985, Theorem 1.8
2. Friedman 2002, p. 60
3. Simpson 1985, Definition 4.1
4. Simpson 1985, Theorem 5.14
5. Marcone 2001, p. 8–9
6. Rathjen & Weiermann 1993.
7. Smith 1985, p. 120
8. Friedman, Harvey (28 March 2006), "273:Sigma01/optimal/size", Ohio State University Department of Maths, retrieved 8 August 2017
9. Friedman, Harvey M. (1 June 2000), "Enormous Integers In Real Life" (PDF), Ohio State University, retrieved 8 August 2017
10. Friedman, Harvey M. (8 October 1998), "Long Finite Sequences" (PDF), Ohio State University Department of Mathematics, pp. 5, 48 (Thm.6.8), retrieved 8 August 2017

Bibliography

• Friedman, Harvey M. (2002), Internal finite tree embeddings. Reflections on the foundations of mathematics (Stanford, CA, 1998), Lect. Notes Log., vol. 15, Urbana, IL: Assoc. Symbol. Logic, pp. 60–91, MR   1943303
• Gallier, Jean H. (1991), "What's so special about Kruskal's theorem and the ordinal Γ₀? A survey of some results in proof theory" (PDF), Ann. Pure Appl. Logic, 53 (3): 199–260, doi: 10.1016/0168-0072(91)90022-E , MR   1129778
• Kruskal, J. B. (May 1960), "Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture" (PDF), Transactions of the American Mathematical Society, 95 (2), American Mathematical Society: 210–225, doi:10.2307/1993287, JSTOR   1993287, MR   0111704
• Marcone, Alberto (2001), "Wqo and bqo theory in subsystems of second order arithmetic" (PDF), Reverse Mathematics, 21: 303–330
• Nash-Williams, C. St.J. A. (1963), "On well-quasi-ordering finite trees", Proc. Camb. Phil. Soc., 59 (4): 833–835, Bibcode:1963PCPS...59..833N, doi:10.1017/S0305004100003844, MR   0153601, S2CID   251095188
• Rathjen, Michael; Weiermann, Andreas (1993), "Proof-theoretic investigations on Kruskal's theorem" (PDF), Annals of Pure and Applied Logic, 60 (1): 49–88, doi:10.1016/0168-0072(93)90192-G, MR   1212407
• Simpson, Stephen G. (1985), "Nonprovability of certain combinatorial properties of finite trees", in Harrington, L. A.; Morley, M.; Scedrov, A.; et al. (eds.), Harvey Friedman's Research on the Foundations of Mathematics, Studies in Logic and the Foundations of Mathematics, North-Holland, pp. 87–117
• Smith, Rick L. (1985), "The consistency strengths of some finite forms of the Higman and Kruskal theorems", in Harrington, L. A.; Morley, M.; Scedrov, A.; et al. (eds.), Harvey Friedman's Research on the Foundations of Mathematics, Studies in Logic and the Foundations of Mathematics, North-Holland, pp. 119–136