Talk:Counting problem (complexity)

Why is y less than or equal to c_r?
y, to my understanding, isn't necessarily a number, so what's the meaning of the inequality? I think it's supposed to be y is a subset of the set defined in the cardinality of c_r..

In my understanding the symbol $$y$$ denotes a non-negative integer that can hence be compared to $$c_R(x)$$, the size (that is number of elements) of the set of solutions $$\{y\mid (x,y)\in R\}$$. I believe one needs to be a bit more careful in phrasing the definition, because a priori there is no bound on the number of solutions since in general $$R$$ is a binary relation between infinite sets (e.g. all strings over a finite (usually two-element) alphabet). Of course one may allow infinitely many solution for certain inputs $$x$$. For such $$x$$ every pair $$(x,n)$$ will belong to $$\#R$$. One way to make the number $$c_R(x)$$ always finite is to just postulate this in the definition of $$R$$. Often this is achieved by making more restrictive assumptions such a that there is a function $$f\colon \mathbb{N}\to\mathbb{N}$$ (of a certain kind, e.g. a polynomial function) such that for every input $$x$$ there are only solutions $$y$$ satisfying $$(x,y)\in R$$ with bounded length $$|y|\leq f(|x|)$$ and hence finitely many.

Notation
I think the problem needs to be stated in English, as with this unspecified notation it's basically unintelligible. 81.2.154.16 (talk) 20:30, 7 July 2022 (UTC)