I am reading a paper and there is a picture here:
just beneath the "while" word, you would see "|" symbol. What is it? I have searched http://web.uvic.ca/~salam/math_symbols.html and https://www.wikizero.com/en/List_of_logic_symbols
I cannot find it.
There are several common ways to denote a set of particular elements. The | is a vertical pipe, which can be typed as "\mid" in LaTeX. Another equivalent option is to use a colon.
So, if we wanted to talk about the even natural numbers, any of these would work. $$ \begin{split} \{ n \in \mathbb{N} \mid n \text{ is even} \}\\ \{ n \in \mathbb{N} : n \text{ is even}\}\\ \{ 2n : n \in \mathbb{N} \}\\ \{ n \in \mathbb{N} \mid (\exists m) [n = 2m]\} \end{split} $$ This is called set builder notation. It is not formally defined, but is a part of informal mathematical language that has developed along with the need to specify sets inside mathematical texts.
The other use of $|$ in logic is to represent the Sheffer stroke, but that is not at all what is suggested in the question above. In programming, $|$ is sometimes used for OR or for bitwise OR. Wikipedia has a longer list of uses in mathematics in their article on "vertical bar".