Complexity theory has long been a challenging field for computer scientists, with many problems remaining unsolved despite decades of research. Recently, researchers have turned to “reverse mathematics” to better understand why some problems are so difficult. This approach involves analyzing the basic assumptions that underlie mathematical proofs and using them as axioms in new ways.
One notable example is the “equality problem,” a game where two players try to determine whether their strings of 0s and 1s are identical with minimal communication. Researchers have long known that this problem has a lower bound, meaning that at least some amount of communication is required to solve it. But what’s surprising is that the proof for this lower bound relies heavily on the “pigeonhole principle,” a simple statement about counting.
Using reverse mathematics, three researchers were able to prove that the equality problem and the pigeonhole principle are equivalent within a specific set of axioms called PV1. This means that the two seemingly unrelated concepts have more in common than initially thought.
The findings not only shed light on the limits of PV1 but also provide new insights into the nature of complexity theory itself. Researchers hope that this work will help them understand why certain problems are so difficult to solve, and perhaps even lead to breakthroughs in areas like cryptography and coding theory.
While there’s still much to be discovered in the realm of reverse mathematics, it’s clear that this approach is gaining traction among researchers. By examining the fundamental assumptions underlying mathematical proofs, they’re unlocking new connections between seemingly disparate concepts – a promising step towards a deeper understanding of the complexities of complexity theory.
Source: https://www.quantamagazine.org/reverse-mathematics-illuminates-why-hard-problems-are-hard-20251201