Kruskal's tree theorem

"TREE" redirects here. For the plant form, see Tree. For other uses, see Tree (disambiguation).

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.

A finitary application of the theorem gives the existence of the fast-growing TREE function. TREE(3) is largely accepted to be one of the largest simply defined finite numbers, dwarfing other large numbers such as Graham's number and googolplex.

History

The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-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 TREE.

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 descendant of $v$ if the unique path from the root to $w$ contains $v$, and call $w$ a child of $v$ if additionally the path from $v$ to $w$ contains no other vertex.

Take $(X,{\le }_X)$ to be a partially ordered set. If $T_1$, $T_2$ are rooted trees with vertices labeled in $X$, we say that $T_1$ is inf-embeddable in $T_2$ and write $T_1\le T_2$ if there is an injective map $F$ from the vertices of $T_1$ to the vertices of $T_2$ such that:

• For all vertices $v$ of $T_1$, the label of $v$ is ${\le }_X$ the label of $F(v)$;
• If $w$ is any descendant of $v$ in $T_1$, then $F(w)$ is a descendant of $F(v)$; and
• If $w_1$, $w_2$ are any two distinct children of $v$, then the path from $F(w_1)$ to $F(w_2)$ in $T_2$ contains $F(v)$ (equivalently, $F(w_1)$ and $F(w_2)$ lie in different subtrees.)

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 $T_1,T_2,\dots$ of rooted trees labeled in $X$, there is some $i<j$ so that $T_i\le T_j$.)

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, thus giving the first example of a predicative result with a provably impredicative proof. This case of the theorem is still provable by Π¹
₁-CA₀, but by adding a "gap condition" to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system. 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).

Weak tree function

Suppose that $P(n)$ is the statement:

There is some $m$ such that if $T_1,\dots ,T_m$ is a finite sequence of unlabeled rooted trees where $T_i$ has $i+n$ vertices, then $T_i\le T_j$ for some $i<j$.

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

Friedman defined the following function, which is a weaker version of the TREE function below. For a positive integer $n$, take $\text{FFF}(n)$ to be the largest $m$ so that we have the following:

There is a sequence $T_1,\dots ,T_m$ of rooted trees, where each $T_i$ has $i+n-1$ vertices, such that $T_i\le T_j$ does not hold for any $i<j\le m$.

Friedman computes the first few terms of this sequence as $\text{FFF}(1)=1$, $\text{FFF}(2)=2$, and $\text{FFF}(3)=5$. He also estimates $\text{FFF}(4)$ to be less than 100, while $\text{FFF}(5)$ suddenly explodes to a very large value. Any proof that $\text{FFF}(5)$ exists in Peano arithmetic requires at least $A(10)$^[c] symbols, but it can be proved to exist in ACA₀ with at most 10,000 symbols.

TREE function

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

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

There is a sequence $T_1,\dots ,T_m$ of rooted trees labelled from a set of $n$ labels, where each $T_i$ has at most $i$ vertices, such that $T_i\le T_j$ does not hold for any $i<j\le m$.

Kruskal's theorem asserts that $\text{TREE}(n)$ is finite for all $n$. The TREE function eventually dominates every provably recursive function of the system ACA₀ + Π¹
₂-BI.

The sequence begins $\text{TREE}(1)=1$, $\text{TREE}(2)=3$; before $\text{TREE}(3)$ suddenly explodes to a value so large that many other "large" combinatorial constants, such as Friedman's $n(4)$ and Graham's number,^[b] are extremely small by comparison. A lower bound for $n(4)$, and, hence, an extremely weak lower bound for $\text{TREE}(3)$, is $A^{A(187196)}(1)$^[c], where $A(x)$ is the single-argument version of Ackermann's function, defined as $A(x)=A(x,x)$.

Friedman showed that $\text{TREE}(3)$ is greater than the halting time of any Turing machine that can be proved to halt in ACA₀ + Π¹
₂-BI with at most $2\uparrow \uparrow 1000$^[d] symbols, where $\uparrow \uparrow$ denotes tetration.

See also

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

Notes

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

^{^ 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 $x_i,\dots ,x_{2i}$ is a subsequence of any later block $x_j,\dots ,x_{2j}$. For example $n(1)=3$, $n(2)=11$, and $n(3)>2{\uparrow }^{7197}158386$.

^{^ c} The superscript indicates iteration. For example, $A^3(1)$ would mean computing $A(A(A(1)))$.

^{^ d} Friedman actually writes this as 2^[1000], which denotes an exponential stack of 2's of height 1000 using his notation.

References

Citations

Bibliography