In previous posts about graph minors, we reviewed the definitions of graph minors and topological minors, then looked at a relationship between the two; and explored whether there was a relationship between the computational hardness of a set of forbidden subgraphs and the graph class defined as all graphs that avoid the set of forbidden subgraphs. Both of those questions are fundamentally related to the Graph Minor Theorem of Robertson and Seymour. Today, I’d like to take a step back and look big-picture at the Graph Minor Theorem, and its relationship to computer science, by featuring an answer Timothy Chow gave to a question Ryan Williams asked about the role of the Axiom of Choice in TCS.
Ryan’s question, which he asked in October 2010, was, “What interesting theorems in TCS rely on the Axiom of Choice (or, alternatively the Axiom of Determinacy)?” The Axiom of Choice, of course, is a (sometimes controversial) axiom extending Zermelo-Fraenkel set theory, and at this point is generally accepted as part of the foundations of mathematics. Intuitively, the Axiom of Choice says that anytime you have a mathematical construction in which there is a set of nonempty bins, then it is legal to produce a set that contains exactly one item from each bin. While this might seem “obviously true,” once this intuition is applied to infinitely-many bins, there are nonconstructive and perhaps counter intuitive consequences. So it is natural to wonder, in the world of TCS where the sets of bins we encounter are unlikely to be “pathologically infinite,” whether we really need the Axiom of Choice to prove anything. (I won’t discuss the Axiom of Determinacy, except to say that it is consistent with the Zermelo-Fraenkel axioms, but inconsistent with the Axiom of Choice, and set theorists sometimes put it forth as a “better” axiom, but it has not been widely accepted by mathematicians outside of logic.)
Janne Korhonen answered Ryan’s question by noting that the proof of the Graph Minor Theorem fundamentally uses Kruskal’s Tree Theorem, which in turn uses the Axiom of Choice. A fine answer, and one might think that was the end of the story — but one would be wrong, as Timothy Chow demonstrated in a followup answer which I will now quote from.
What about arithmetical statements whose proof requires something like Koenig’s lemma or Kruskal’s tree theorem? Don’t these require a weak form of the axiom of choice? The answer is that it depends on exactly how you state the result in question. For example, if you state the graph minor theorem in the form, “given any infinite set of unlabeled graphs, there must exist two of them such that one is a minor of the other,” then some amount of choice is needed to march through your infinite set of data, picking out vertices, subgraphs, etc. However, if instead you write down a particular encoding by natural numbers of the minor relation on labeled finite graphs, and phrase the graph minor theorem as a statement about this particular partial order, then the statement becomes arithmetical and doesn’t require AC in the proof.
So the question comes down to, “What do we really mean in theoretical computer science when we talk about a mathematical proof of an infinitary statement?” To quote from Chow’s answer again:
Most people feel that the “combinatorial essence” of the graph minor theorem is already captured by the version that fixes a particular encoding, and that the need to invoke AC to label everything, in the event that you’re presented with the general set-theoretic version of the problem, is sort of an irrelevant artifact of a decision to use set theory rather than arithmetic as one’s logical foundation. If you feel the same way, then the graph minor theorem doesn’t require AC. (See also this post by Ali Enayat to the Foundations of Mathematics mailing list, written in response to a similar question that I once had.)
So, in the final analysis, at least in Chow’s opinion, there appears to be no example of a TCS theorem that requires the Axiom of Choice to prove. His final suggestion to Ryan is that it may be more fruitful to ask whether there are TCS statements that require large cardinal axioms to prove, instead of the Axiom of Choice. Harvey Friedman has apparently demonstrated some artificial examples of finitary graph theory statements that require the consistency of large cardinals to prove. So if artificial examples exist already, perhaps natural examples are around the corner.