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I am confused if "state" should take an "s" or not in the following sentence:
... at a given time-slot, the knowledge of the state(s) of all the machines in ...
Each machine has one 'state' at each time-slot. This 'state' can change from machine to machine at the same time-slot and from slot to slot for the same machine.
In this context, we can assume that each machine has its state, so it's correct to use states as we're talking about different entities (machines) with different and independent aspects (states), even if they were the same state in all machines.
On the other hand, if we were talking about the general state of the set of machines, then the plural would be wrong, considering we're yet talking about different entities (machines) but they have aspects (state) which are dependent on each other.
The "s" in the example is due not to multiple machines, but multiple states. Reading just this passage I would assume each machine my have more than one state, meaning that you could have multiple states across the machines. So, if you have one state across all machines, you would say
at a given time-slot, the knowledge of the state of all the machines in
Whereas if you have multiple states across all machines, you would say
at a given time-slot, the knowledge of the states of all the machines in
