This is a typical example of the gap between disciplines. As a process engineer / automation and control technician you're probably aware that a P&ID does not contain enough information to build a control system and as such is not sufficient to act as a 'control specification'. It should however at least has the information to extract a 'sensor and actor list' from it, containing all electrical properties of all the instruments. It might even show the control loops between sensors and actuators. What it does not show is the 'behavior' of the installation: how it all 'runs', the events that trigger actions. This is what needs to be defined to be able to program the software that contains all the control algorithms. Also it is likely that a plant is to be supervised and there will be some form of Human Machine Interface (HMI), eg. SCADA system. This needs to be specified as well; what must be shown on which screen, how to navigate between the screens, how to handle alarms.
Keep in mind that this control specifications also is the basis for a Factory/Site Acceptance: the point where you and the end customer have to verify if the system operates as desired. So whatever you put in it, you must be able to realize and verify.
It is important to think about the 'operator and control strategy/philosophy' of the plant. This defines the way the plant is to be operated and is determined for instance by; training level of the operators, 24hours/7days or 8hours/5days production, supervised or non-supervised production, local or remote operation, continuous or batch production a.s.o.