How irreversible processes are possible? Susskind says that all laws of mechanics are reversible and any valid mechanic law most be reversible: you can always determine the previous state of any physically valid system. However, the simplest operations in computation, reset as well as the binary AND and OR operators, are irreversible. You may say that "this is only a virtual simulation" and everything is reversible in reality. However, because of the dissipation, almost all processes in the world are irreversible. So, despite irreversibility is completely forbidden, everything seems to be irreversible and entropy only increases. I want to understand how this contradiction is resolved (at least in one case).
I also do not understand why do you represent a system configuration space by a box. You say that the state is a point in that box and its trajectory represents the state evolution. However, reversibility requirement demands that the trajectories do not intersect. This means there must be only one trajectory. So, you should rearrange all the dots of the configuration box into a line (it might be a segmented into unconnected circles). However, that might be another question.
 A: Things become irreversible when you start ignoring certain degrees of freedom. What we call heat and friction is just our wilful ignorance of the trajectories of countless atoms.
But the fact that the underlying equations of motion are time symmetric deals with microscopic phenomena. Sure, the time-reversed process is equally probable, which leads into the postulate that all microstates are equally probable (Ergodic hypothesis), but because of entropy, the system will move from unlikely towards likely macrostates.
However, for finite systems, there's always the Poincaré recurrence theorem
. Still, the recurrence time can be incomprehensibly large.
A: 
However, the simplest operations in computation, reset as well as the binary AND and OR operators, are irreversible.

So? Their implementation in terms of CMOS logic is not irreversible, one can trackback the voltage levels. Sure, we can simulate irreversible systems with computers, but these aren't physically valid.

However, because of the dissipiation, almost all processes in the world are irreversible.

Irreversible in the sense that one cannot determine the previous state of the system from its current state. However,  dissipation involves open systems, so the surroundings also matter. If you know the state of the system and the surroundings, you can indeed determine the previous state.

This means there must be only one trajectory.

We can have multiple trajectories that never intersect, for different initial conditions. Take the example of a pendulum with different initial amplitudes. Each trajectory is its own circle.
Of course, in two (phase space) dimensions for a dissipative or Hamiltonian system, the trajectories are either loops or finite spirals. Once you have at least 3 phase space dimensions, more robust dynamics like quasiperiodic dynamics become possible. For example, we can have intertwining trajectories on the surface of a torus; where the trajectories never end or loop back on themselves.
