  In this paper, we study the degrees of freedom (DoF) region and sum DoF of
the time-selective $K$-user interference channel in the presence of intelligent
reflecting surfaces (IRSs). We consider three types of IRSs, namely 1) active
IRSs, which are able to amplify, attenuate, and add a phase shift to the
received signal, 2) passive IRSs, which are able to attenuate and add a phase
shift to the received signal, and 3) passive lossless IRSs, which are only able
to add a phase shift to the received signal. The performance of ideal active
IRSs serves as an upper bound for the performance of passive and passive
lossless IRSs. We derive inner and outer bounds for the DoF region and lower
and upper bounds for the sum DoF of the $K$-user interference channel in the
presence of an active IRS and prove that the maximum value $K$ for the sum DoF
can be achieved if the number of IRS elements exceeds a certain finite value.
Then, we introduce probabilistic inner and outer bounds for the DoF region and
probabilistic lower and upper bounds for the sum DoF of the $K$-user
interference channel in the presence of a passive IRS and prove that the lower
bound for the sum DoF asymptotically approaches $K$ as the number of IRS
elements grows large. For the DoF analysis of passive lossless IRSs, first, we
approximate the IRS by a model which we refer to as $\varepsilon$-relaxed
passive lossless IRS and introduce a probabilistic lower bound for the
corresponding sum DoF. We prove that this bound asymptotically tends to $K$. In
addition, we define a relaxed type of DoF called $\rho$-limited DoF. We
introduce a lower bound for the $\rho$-limited sum DoF of the passive lossless
IRS-assisted $K$-user interference channel and prove that this lower bound
asymptotically also tends to $K$.
