In computer science theory some textbooks introduce fractional/quotient regular languages. In particular, given L1 and L2 languages on the same alphabet, the right quotient of L1 with L2 is defined as
L1/L2 = {x : xy ∈ L1 for some y ∈ L2}.
The left quotient is defined in a similar way. I was wondering if there are some real-life cases in which quotient languages can be useful. Any ideas?