Notation: $L_R = \{w\#y\space|\space R(w,y)\}$?

Question

What is the set $$L_R = \{w\#y\space|\space R(w,y)\}$$

Specifically what kind of conditional is $R(w,y)$?
Also what's the purpose of $\#$?

This comes from page 2 of the Clay paper on P vs NP: http://www.claymath.org/sites/default/files/pvsnp.pdf

Answer 1

This set is a formal way of expressing a "checking set." What this is asking is, "Is $y$ a valid output for $w$, where the space of all valid input/output pairs is the binary relation $R$?"

Let $\Sigma_i$ be the alphabet of the input and $\Sigma_o$ be the alphabet of outputs. The checking relation, $R\subseteq\Sigma_i^* \times \Sigma_o^*$ is the binary relation $R(w,y)$ that returns true if $y$ is a valid solution to $x$, and false otherwise.

(Note here that a relation can be expressed as a subset of the parameter spaces, $R\subseteq\Sigma_i^* \times \Sigma_o^*$, or as a function $R(w,y)$ that returns true or false. The functional form $R(w,y)$ can be viewed as the question, "Is $(w,y)\in R$?")

Finally, the symbol $\#$ in this definition is just a specific symbol not in $\Sigma_i$. Because a Turing machine only accepts a single string, you need some way of parameterizing your single input into two parameters, $w$ and $y$.