I am trying to (intuitively) understand the two terms "decidability" and "verifiability".
I have done a reasonable amount of searching and going through the various texts I can put my hands on. However, their intuitive understanding seems to escape me, specially for the second one.
Out of the many definitions found, the following one found in this page, clearly explained decidability to me.
A language is called decidable if there exists a method - any method at all - to determine whether a given word belongs to that language or not.
However, I fail to find a parallel definition for verifiability.
In the Theory of Computation book by Sipser, we find,
P = the class of languages for which membership can be decided quickly.
NP = the class of languages for which membership can be verified quickly.
In light of the above, I want to understand verifiability.
Please provide as many examples as you can, at one moment, I try catch the meaning, in the next one, I get confused again.