  We consider a gas of point particles moving on the one-dimensional line with
a hard-core inter-particle interaction that prevents particle crossings ---
this is usually referred to as single-file motion. The individual particle
dynamics can be arbitrary and they only interact when they meet. Starting from
initial conditions such that particles are uniformly distributed, we observe
the displacement of a tagged particle at time $t$, with respect to the initial
position of another tagged particle, such that their tags differ by $r$. For
$r=0$, this is the usual well studied problem of the tagged particle motion.
Using a mapping to a non-interacting particle system we compute the exact
probability distribution function for the two-tagged particle displacement, for
general single particle dynamics. As by-products, we compute the large
deviation function, various cumulants and, for the case of Hamiltonian
dynamics, the two-particle velocity auto-correlation function.
