Okay, so we agree that it’s improbable (at least for decision problems) to be able to verify an answer faster than finding it. What you care about are cases where verification is easier, as is conjectured for example for NP (where verification is polynomial, but finding an answer is supposed to not be).
For IP, if we only want to verify any real-world property, I actually have a simple example I give into my intro to complexity theory lectures. Imagine that you are color-blind (precisely, a specific red and a specific green look exactly the same to you). If I have two balls, perfectly similar except one is green and the other is red, I can convince you that these balls are of different colors. It is basically the interactive protocol for graph non-isomorphism: you flip a coin, and depending on the result, you exchange the balls without me seeing it. If I can tell whether you exchanged the balls a sufficient number of times, then you should get convinced that I can actually differentiate them.
Of course this is not necessarily applicable to questions like tastes. Moreover, it is a protocol for showing that I can distinguish between the balls; it does not show why.
It does still require some manipulation ability—we have to be able to experimentally intervene (at reasonable expense). That doesn’t open up all possibilities, but it’s at least a very large space. I’ll have to chew on it some more.
Okay, so we agree that it’s improbable (at least for decision problems) to be able to verify an answer faster than finding it. What you care about are cases where verification is easier, as is conjectured for example for NP (where verification is polynomial, but finding an answer is supposed to not be).
For IP, if we only want to verify any real-world property, I actually have a simple example I give into my intro to complexity theory lectures. Imagine that you are color-blind (precisely, a specific red and a specific green look exactly the same to you). If I have two balls, perfectly similar except one is green and the other is red, I can convince you that these balls are of different colors. It is basically the interactive protocol for graph non-isomorphism: you flip a coin, and depending on the result, you exchange the balls without me seeing it. If I can tell whether you exchanged the balls a sufficient number of times, then you should get convinced that I can actually differentiate them.
Of course this is not necessarily applicable to questions like tastes. Moreover, it is a protocol for showing that I can distinguish between the balls; it does not show why.
That is an awesome example, thank you!
It does still require some manipulation ability—we have to be able to experimentally intervene (at reasonable expense). That doesn’t open up all possibilities, but it’s at least a very large space. I’ll have to chew on it some more.