In January 2020, Papadimitriou had thought of the pigeon principle for 30 years. He was therefore surprised when a fun conversation with a frequent collaborator led them to a simple touch of the principle that they had never considered: and if there are fewer pigeons than holes? In this case, any pigeon arrangement must leave empty holes. Again, it seems obvious. But does the inversion of the principle of pigeons have interesting mathematical consequences?
It may seem that this principle of “pigon pigon” is just the original by another name. But this is not the case, and its subtly different character has made it a new fruitful tool to classify calculation problems.
To understand the principle of empty pigge, let us return to the example of the bank card, transposed from a football stadium to a concert hall with 3,000 seats – a smaller number than total four -digit pins. The principle of empty pigon dictates that certain possible pins are not at all represented. If you want to find one of these missing pins, however, there does not seem to be better than just asking each person to pin. Until now, the void principle is like its more famous counterpart.
The difference lies in the difficulty of checking the solutions. Imagine that someone says they have found two specific people in the football stadium who have the same pin. In this case, corresponding to the original dovecote scenario, there is a simple way to check this assertion: simply check with the two people in question. But in the case of the concert hall, imagine that someone says that no one has a spindle of 5926. Here it is impossible to check without asking everyone in the public what is their pin. This makes the principle of empty pigon much more annoying for theorists of complexity.
Two months after Papadimitriou began to think about the principle of the pige hole, he spoke of it in a conversation with a potential graduate student. He remembers it alive, because it turned out to be his last conversation in person with anyone before the COVVI-19 locking. Cut at home during the following months, he fought with the implications of the problem for the theory of complexity. Finally, he and his colleagues published a paper About the research problems guaranteed to have solutions due to the empty pigon principle. They were particularly interested in the problems where the pigeons are abundant-that is to say where they are far more numerous than the pigeons. In accordance with a tradition heavy acronyms In the theory of complexity, they nicknamed this class of APEPP problems, for “an abundant polynomial void-pigonhole” principle.
One of the problems of this class was inspired by a famous 70 years by the computer scientist Claude Shannon. Shannon has proven that most calculation problems should be intrinsically difficult to solve, using an argument that was based on the empty pigon principle (although he does not call him so). However, for decades, computer scientists have tried and failed to prove that specific problems are really difficult. As missing bank card pins, difficult problems should be available, even if we cannot identify them.
Historically, researchers have not thought of the process of researching difficult problems as a research problem which could itself be analyzed mathematically. Papadimitriou's approach, which has gathered this process with other research problems connected to the principle of empty pige hole, had an auto-referential flavor characteristic of very recent work In the theory of complexity – he offered a new way of reasoning about the difficulty of proving computer difficulties.
